GannLSVZO Indicator [Algo Alert]The Volume Zone oscillator breaks up volume activity into positive and negative categories. It is positive when the current closing price is greater than the prior closing price and negative when it's lower than the prior closing price. The resulting curve plots through relative percentage levels that yield a series of buy and sell signals, depending on level and indicator direction.
The Gann Laplace Smoothed Volume Zone Oscillator GannLSVZO is a refined version of the Volume Zone Oscillator, enhanced by the implementation of the upgraded Discrete Fourier Transform, the Laplace Stieltjes Transform. Its primary function is to streamline price data and diminish market noise, thus offering a clearer and more precise reflection of price trends.
By combining the Laplace with Gann Swing Entries and Exits (orange X) and with Ehler's white noise histogram, users gain a comprehensive perspective on volume-related market conditions.
HOW TO USE THE INDICATOR:
The default period is 2 but can be adjusted after backtesting. (I suggest 5 VZO length and NoiceR max length 8 as-well)
The VZO points to a positive trend when it is rising above the 0% level, and a negative trend when it is falling below the 0% level. 0% level can be adjusted in setting by adjusting VzoDifference. Oscillations rising below 0% level or falling above 0% level result in a natural trend.
ORIGINALITY & USFULLNESS:
Personal combination of Gann swings and Laplace Stieltjes Transform of a price which results in less noise Volume Zone Oscillator.
The Laplace Stieltjes Transform is a mathematical technique that transforms discrete data from the time domain into its corresponding representation in the frequency domain. This process involves breaking down a signal into its individual frequency components, thereby exposing the amplitude and phase characteristics inherent in each frequency element.
This indicator utilizes the concept of Ehler's Universal Oscillator and displays a histogram, offering critical insights into the prevailing levels of market noise. The Ehler's Universal Oscillator is grounded in a statistical model that captures the erratic and unpredictable nature of market movements. Through the application of this principle, the histogram aids traders in pinpointing times when market volatility is either rising or subsiding.
The Gann swings and the Gan swing strategy is developed by meomeo105, this Gann high and low algorithm forms the basis of the EMA modification.
DETAILED DESCRIPTION:
My detailed description of the indicator and use cases which I find very valuable.
What is oscillator?
Oscillators are chart indicators that can assist a trader in determining overbought or oversold conditions in ranging (non-trending) markets.
What is volume zone oscillator?
Price Zone Oscillator measures if the most recent closing price is above or below the preceding closing price.
Volume Zone Oscillator is Volume multiplied by the 1 or -1 depending on the difference of the preceding 2 close prices and smoothed with Exponential moving Average.
What does this mean?
If the VZO is above 0 and VZO is rising. We have a bullish trend. Most likely.
If the VZO is below 0 and VZO is falling. We have a bearish trend. Most likely.
Rising means that VZO on close is higher than the previous day.
Falling means that VZO on close is lower than the previous day.
What if VZO is falling above 0 line?
It means we have a high probability of a bearish trend.
Thus the indicator returns 0 and Strategy closes all it's positions when falling above 0 (or rising bellow 0) and we combine higher and lower timeframes to gauge the trend.
What is approximation and smoothing?
They are mathematical concepts for making a discrete set of numbers a
continuous curved line.
Laplace Stieltjes Transform approximation of a close price are taken from aprox library.
Key Features:
You can tailor the Indicator/Strategy to your preferences with adjustable parameters such as VZO length, noise reduction settings, and smoothing length.
Volume Zone Oscillator (VZO) shows market sentiment with the VZO, enhanced with Exponential Moving Average (EMA) smoothing for clearer trend identification.
Noise Reduction leverages Euler's White noise capabilities for effective noise reduction in the VZO, providing a cleaner and more accurate representation of market dynamics.
Choose between the traditional Fast Laplace Stieltjes Transform (FLT) and the innovative Double Discrete Fourier Transform (DTF32) soothed price series to suit your analytical needs.
Use dynamic calculation of Laplace coefficient or the static one. You may modify those inputs and Strategy entries with Gann swings.
I suggest using "Close all" input False when fine-tuning Inputs for 1 TimeFrame. When you export data to Excel/Numbers/GSheets I suggest using "Close all" input as True, except for the lowest TimeFrame. I suggest using 100% equity as your default quantity for fine-tune purposes. I have to mention that 100% equity may lead to unrealistic backtesting results. Be avare. When backtesting for trading purposes use Contracts or USDT.
Fouriertransform
Gann + Laplace Smoothed Hybrid Volume Spread Analysis Indicator
This Indicator stands apart by integrating the principles of the upgraded Discrete Fourier Transform (DFT), the Laplace Stieltjes Transform and volume spread analysis, enhanced with a layer of Fourier smoothing to distill market noise and highlight trend directions with unprecedented clarity.
The length of EMA and Strategy Entries are modified with the Gann swings.
This smoothing process allows traders to discern the true underlying patterns in volume and price action, stripped of the distractions of short-term fluctuations and noise.
The core functionality of the GannLSHVSA revolves around the innovative combination of volume change analysis, spread determination (calculated from the open and close price difference), and the strategic use of the EMA (default 10) to fine-tune the analysis of spread by incorporating volume changes.
Trend direction is validated through a moving average (MA) of the histogram, which acts analogously to the Volume MA found in traditional volume indicators. This MA serves as a pivotal reference point, enabling traders to confidently engage with the market when the histogram's movement concurs with the trend direction, particularly when it crosses the Trend MA line, signalling optimal entry points.
It returns 0 when MA of the histogram and EMA of the Price Spread are not align.
WHAT IS GannLSHVSA INDICATOR:
The GannLSHVSA plots a positive trend when a positive Volume smoothed Spread and EMA of Volume smoothed price is above 0, and a negative when negative Volume smoothed Spread and EMA of Volume smoothed price is below 0. When this conditions are not met it plots 0.
ORIGINALITY & USEFULNESS:
The GannLSHVSA Strategy is unique because it applies upgraded DFT, the Laplace Stieltjes Transform for data smoothing, effectively filtering out the minor fluctuations and leaving traders with a clear picture of the market's true movements. The DFT's ability to break down market signals into constituent frequencies offers a granular view of market dynamics, highlighting the amplitude and phase of each frequency component. This, combined with the strategic application of Ehler's Universal Oscillator principles via a histogram, furnishes traders with a nuanced understanding of market volatility and noise levels, thereby facilitating more informed trading decisions. The Gann swing strategy is developed by meomeo105, this Gann high and low algorithm forms the basis of the EMA modification.
DETAILED DESCRIPTION:
My detailed description of the indicator and use cases which I find very valuable.
What is the meaning of price spread?
In finance, a spread refers to the difference between two prices, rates, or yields. One of the most common types is the bid-ask spread, which refers to the gap between the bid (from buyers) and the ask (from sellers) prices of a security or asset.
We are going to use Open-Close spread.
What is Volume spread analysis?
Volume spread analysis (VSA) is a method of technical analysis that compares the volume per candle, range spread, and closing price to determine price direction.
What does this mean?
We need to have a positive Volume Price Spread and a positive Moving average of Volume price spread for a positive trend. OR via versa a negative Volume Price Spread and a negative Moving average of Volume price spread for a negative trend.
What if we have a positive Volume Price Spread and a negative Moving average of Volume Price Spread?
It results in a neutral, not trending price action.
Thus the Indicator/Strategy returns 0 and Closes all long and short positions.
I suggest using "Close all" input False when fine-tuning Inputs for 1 TimeFrame. When you export data to Excel/Numbers/GSheets I suggest using "Close all" input as True, except for the lowest TimeFrame. I suggest using 100% equity as your default quantity for fine-tune purposes. I have to mention that 100% equity may lead to unrealistic backtesting results. Be avare. When backtesting for trading purposes use Contracts or USDT.
6 days ago
Release Notes
Harmonic Rolling VWAP (Zeiierman)█ Overview
The Harmonic Rolling VWAP (Zeiierman) indicator combines the concept of the Rolling Volume Weighted Average Price (VWAP) with advanced harmonic analysis using Discrete Fourier Transform (DFT). This innovative indicator aims to provide traders with a dynamic view of price action, capturing both the volume-weighted price and underlying harmonic patterns. By leveraging this combination, traders can gain deeper insights into market trends and potential reversal points.
█ How It Works
The Harmonic Rolling VWAP calculates the rolling VWAP over a specified window of bars, giving more weight to periods with higher trading volume. This VWAP is then subjected to harmonic analysis using the Discrete Fourier Transform (DFT), which decomposes the VWAP into its frequency components.
Key Components:
Rolling VWAP (RVWAP): A moving average that gives more weight to higher volume periods, calculated over a user-defined window.
True Range (TR): Measures volatility by comparing the current high and low prices, considering the previous close price.
Discrete Fourier Transform (DFT): Analyzes the harmonic patterns within the RVWAP by decomposing it into its frequency components.
Standard Deviation Bands: These bands provide a visual representation of price volatility around the RVWAP, helping traders identify potential overbought or oversold conditions.
█ How to Use
Identify Trends: The RVWAP line helps in identifying the underlying trend by smoothing out short-term price fluctuations and focusing on volume-weighted prices.
Assess Volatility: The standard deviation bands around the RVWAP give a clear view of price volatility, helping traders identify potential breakout or breakdown points.
Find Entry and Exit Points: Traders can look for entries when the price is near the lower bands in an uptrend or near the upper bands in a downtrend. Exits can be considered when the price approaches the opposite bands or shows harmonic divergence.
█ Settings
VWAP Source: Defines the price data used for VWAP calculations. The source input defines the price data used for calculations. This setting affects the VWAP calculations and the resulting bands.
Window: Sets the number of bars used for the rolling calculations. The window input sets the number of bars used for the rolling calculations. A larger window smooths the VWAP and standard deviation bands, making the indicator less sensitive to short-term price fluctuations. A smaller window makes the indicator more responsive to recent price changes.
-----------------
Disclaimer
The information contained in my Scripts/Indicators/Ideas/Algos/Systems does not constitute financial advice or a solicitation to buy or sell any securities of any type. I will not accept liability for any loss or damage, including without limitation any loss of profit, which may arise directly or indirectly from the use of or reliance on such information.
All investments involve risk, and the past performance of a security, industry, sector, market, financial product, trading strategy, backtest, or individual's trading does not guarantee future results or returns. Investors are fully responsible for any investment decisions they make. Such decisions should be based solely on an evaluation of their financial circumstances, investment objectives, risk tolerance, and liquidity needs.
My Scripts/Indicators/Ideas/Algos/Systems are only for educational purposes!
Fourier Adjusted Average True Range [BackQuant]Fourier Adjusted Average True Range
1. Conceptual Foundation and Innovation
The FA-ATR leverages the principles of Fourier analysis to dissect market prices into their constituent cyclical components. By applying Fourier Transform to the price data, the FA-ATR captures the dominant cycles and trends which are often obscured in noisy market data. This integration allows the FA-ATR to adapt its readings based on underlying market dynamics, offering a refined view of volatility that is sensitive to both market direction and momentum.
2. Technical Composition and Calculation
The core of the FA-ATR involves calculating the traditional ATR, which measures market volatility by decomposing the entire range of price movements. The FA-ATR extends this by incorporating a Fourier Transform of price data to assess cyclical patterns over a user-defined period 'N'. This process synthesizes both the magnitude of price changes and their rhythmic occurrences, resulting in a more comprehensive volatility indicator.
Fourier Transform Application: The Fourier series is calculated using price data to identify the fundamental frequency of market movements. This frequency helps in adjusting the ATR to reflect more accurately the current market conditions.
Dynamic Adjustment: The ATR is then adjusted by the magnitude of the dominant cycle from the Fourier analysis, enhancing or reducing the ATR value based on the intensity and phase of market cycles.
3. Features and User Inputs
Customizability: Traders can modify the Fourier period, ATR period, and the multiplication factor to suit different trading styles and market environments.
Visualization : The FA-ATR can be plotted directly on the chart, providing a visual representation of volatility. Additionally, the option to paint candles according to the trend direction enhances the usability and interpretative ease of the indicator.
Confluence with Moving Averages: Optionally, a moving average of the FA-ATR can be displayed, serving as a confluence factor for confirming trends or potential reversals.
4. Practical Applications
The FA-ATR is particularly useful in markets characterized by periodic fluctuations or those that exhibit strong cyclical trends. Traders can utilize this indicator to:
Adjust Stop-Loss Orders: More accurately set stop-loss orders based on a volatility measure that accounts for cyclical market changes.
Trend Confirmation: Use the FA-ATR to confirm trend strength and sustainability, helping to avoid false signals often encountered in volatile markets.
Strategic Entry and Exit: The indicator's responsiveness to changing market dynamics makes it an excellent tool for planning entries and exits in a trend-following or a breakout trading strategy.
5. Advantages and Strategic Value
By integrating Fourier analysis, the FA-ATR provides a volatility measure that is both adaptive and anticipatory, giving traders a forward-looking tool that adjusts to changes before they become apparent through traditional indicators. This anticipatory feature makes it an invaluable asset for traders looking to gain an edge in fast-paced and rapidly changing market conditions.
6. Summary and Usage Tips
The Fourier Adjusted Average True Range is a cutting-edge development in technical analysis, offering traders an enhanced tool for assessing market volatility with increased accuracy and responsiveness. Its ability to adapt to the market's cyclical nature makes it particularly useful for those trading in highly volatile or cyclically influenced markets.
Traders are encouraged to integrate the FA-ATR into their trading systems as a supplementary tool to improve risk management and decision-making accuracy, thereby potentially increasing the effectiveness of their trading strategies.
INDEX:BTCUSD
INDEX:ETHUSD
BINANCE:SOLUSD
Fourier Smoothed Hybrid Volume Spread AnalysisIndicator id:
USER;91bdff47320b4284a375f428f683b21e
(only relevant to those that use API requests)
MEANINGFUL DESCRIPTION:
The Fourier Smoothed Hybrid Volume Spread Analysis (FSHVSA) indicator is an innovative trading tool designed to fuse volume analysis with trend detection capabilities, offering traders a comprehensive view of market dynamics.
This indicator stands apart by integrating the principles of the Discrete Fourier Transform (DFT) and volume spread analysis, enhanced with a layer of Fourier smoothing to distill market noise and highlight trend directions with unprecedented clarity.
This smoothing process allows traders to discern the true underlying patterns in volume and price action, stripped of the distractions of short-term fluctuations and noise.
The core functionality of the FSHVSA revolves around the innovative combination of volume change analysis, spread determination (calculated from the open and close price difference), and the strategic use of the EMA (default 10) to fine-tune the analysis of spread by incorporating volume changes.
Trend direction is validated through a moving average (MA) of the histogram, which acts analogously to the Volume MA found in traditional volume indicators. This MA serves as a pivotal reference point, enabling traders to confidently engage with the market when the histogram's movement concurs with the trend direction, particularly when it crosses the Trend MA line, signalling optimal entry points.
It returns 0 when MA of the histogram and EMA of the Price Spread are not align.
HOW TO USE THE INDICATOR:
The FSHVSA plots a positive trend when a positive Volume smoothed Spread and EMA of Volume smoothed price is above 0, and a negative when negative Volume smoothed Spread and EMA of Volume smoothed price is below 0. When this conditions are not met it plots 0.
ORIGINALITY & USEFULNESS:
The FSHVSA is unique because it applies DFT for data smoothing, effectively filtering out the minor fluctuations and leaving traders with a clear picture of the market's true movements. The DFT's ability to break down market signals into constituent frequencies offers a granular view of market dynamics, highlighting the amplitude and phase of each frequency component. This, combined with the strategic application of Ehler's Universal Oscillator principles via a histogram, furnishes traders with a nuanced understanding of market volatility and noise levels, thereby facilitating more informed trading decisions.
DETAILED DESCRIPTION:
My detailed description of the indicator and use cases which I find very valuable.
What is the meaning of price spread?
In finance, a spread refers to the difference between two prices, rates, or yields. One of the most common types is the bid-ask spread, which refers to the gap between the bid (from buyers) and the ask (from sellers) prices of a security or asset.
We are going to use Open-Close spread.
What is Volume spread analysis?
Volume spread analysis (VSA) is a method of technical analysis that compares the volume per candle, range spread, and closing price to determine price direction.
What does this mean?
We need to have a positive Volume Price Spread and a positive Moving average of Volume price spread for a positive trend. OR via versa a negative Volume Price Spread and a negative Moving average of Volume price spread for a negative trend.
What if we have a positive Volume Price Spread and a negative Moving average of Volume Price Spread ?
It results in a neutral, not trending price action.
Thus the indicator returns 0.
In the next Image you can see that trend is negative on 4h, neutral on 12h and neutral on 1D. That means trend is negative .
I am sorry, the chart is a bit messy. The idea is to use the indicator over more than 1 Timeframe.
What is approximation and smoothing?
They are mathematical concepts for making a discrete set of numbers a
continuous curved line.
Fourier and Euler approximation of a spread are taken from aprox library.
Key Features:
Noise Reduction leverages Euler's White noise capabilities for effective Volume smoothing, providing a cleaner and more accurate representation of market dynamics.
Choose between the innovative Double Discrete Fourier Transform (DTF32) and Regular Open & Close price series.
Mathematical equations presented in Pinescript:
Fourier of the real (x axis) discrete:
x_0 = array.get(x, 0) + array.get(x, 1) + array.get(x, 2)
x_1 = array.get(x, 0) + array.get(x, 1) * math.cos( -2 * math.pi * _dir / 3 ) - array.get(y, 1) * math.sin( -2 * math.pi * _dir / 3 ) + array.get(x, 2) * math.cos( -4 * math.pi * _dir / 3 ) - array.get(y, 2) * math.sin( -4 * math.pi * _dir / 3 )
x_2 = array.get(x, 0) + array.get(x, 1) * math.cos( -4 * math.pi * _dir / 3 ) - array.get(y, 1) * math.sin( -4 * math.pi * _dir / 3 ) + array.get(x, 2) * math.cos( -8 * math.pi * _dir / 3 ) - array.get(y, 2) * math.sin( -8 * math.pi * _dir / 3 )
Fourier of the imaginary (y axis) discrete:
y_0 = array.get(x, 0) + array.get(x, 1) + array.get(x, 2)
y_1 = array.get(x, 0) + array.get(x, 1) * math.sin( -2 * math.pi * _dir / 3 ) + array.get(y, 1) * math.cos( -2 * math.pi * _dir / 3 ) + array.get(x, 2) * math.sin( -4 * math.pi * _dir / 3 ) + array.get(y, 2) * math.cos( -4 * math.pi * _dir / 3 )
y_2 = array.get(x, 0) + array.get(x, 1) * math.sin( -4 * math.pi * _dir / 3 ) + array.get(y, 1) * math.cos( -4 * math.pi * _dir / 3 ) + array.get(x, 2) * math.sin( -8 * math.pi * _dir / 3 ) + array.get(y, 2) * math.cos( -8 * math.pi * _dir / 3 )
Euler's Smooth with Discrete Furrier approximated Volume.
a = math.sqrt(2) * math.pi / _devided
b = math.cos(math.sqrt(2) * 180 / _devided)
c2 = 2 * math.pow(a, 2) * b
c3 = math.pow(a, 4)
c1 = 1 - 2 * math.pow(a, 2) * math.cos(b) + math.pow(a, 4)
filt := na(filt ) ? 0 : c1 * (w + nz(w )) / 2.0 + c2 * nz(filt ) + c3 * nz(filt )
Usecase:
First option:
Leverage the script to identify Bullish and Bearish trends, shown with green and red triangle.
Combine Different Timeframes to accurately determine market trend.
Second option:
Pull the data with API sockets to automate your trading journey.
plot(close, title="ClosePrice", display=display.status_line)
plot(open, title="OpenPrice", display=display.status_line)
plot(greencon ? 1 : redcon ? -1 : 0, title="position", display=display.status_line)
Use ClosePrice, OpenPrice and "position" titles to easily read and backtest your strategy utilising more than 1 Time Frame.
Indicator id:
USER;91bdff47320b4284a375f428f683b21e
(only relevant to those that use API requests)
Discrete Fourier Transformed Money Flow IndexThe Discrete Fourier Transform Money Flow Index indicator integrates the Money Flow Index (MFI) with Discrete Fourier Transform (credit to author wbburgin - May 26 2023 ) smoothing to offer a refined and smoothed depiction of the MFI's underlying trend. The MFI is calculated using the formula: MFI = 100 - (100 / (1 + MR)), where a high MFI value indicates robust buying pressure (signaling an overbought condition), and a low MFI value indicates substantial selling pressure (signaling an oversold condition).
Why is the DFT and MFI combined?
The aim of this combination between DFT and MFI is to effectively filter out short-term fluctuations and noise, enabling a clearer assessment of the overall trend. This smoothing process enhances the reliability of the MFI by emphasizing dominant and sustained buying or selling pressures. This script executes a full DFT but only uses filtering from one frequency component. The choice to focus on the magnitude at index 0 is significant as it captures the dominant or fundamental frequency in the data. By analyzing this primary cyclic behavior, we can identify recurring patterns and potential turning points more easily. This streamlined approach simplifies interpretation and enhances efficiency by reducing complexity associated with multiple frequency components. Overall, focusing on the dominant frequency and applying it to the MFI provides a concise and actionable assessment of the underlying data.
Note: The FMFI indicator provides both smoothed and non-smoothed versions of the MFI, with the option to toggle the original non-smoothed MFI on or off in the settings.
Application
FMFI functions as a trend-following indicator. Bullish trends are denoted by the color white, while bearish trends are represented by the color purple. Circles plotted on the FMFI indicate regular bull and bear signals. Additionally, red arrows indicate a strong negative trend, while green arrows indicate a strong positive trend. These arrows are calculated based on the presence of regular bull and bear signals within overbought and oversold zones. To enhance its effectiveness, it is recommended to combine this indicator with other complementary technical analysis tools and integrate it into a comprehensive trading strategy. Traders are encouraged to explore a wide range of settings and timeframes to align the indicator with their unique trading preferences and adapt it to the current market conditions. By doing so, traders can optimize the indicator's performance and increase their potential for successful trading outcomes.
Utility
Traders and investors can employ this indicator to enhance their trend-following strategies. The white-colored components of the FMFI can help identify potential buying zones, while the purple-colored components can assist in identifying potential selling points. The red and green arrows can be used to pinpoint moments of strong bull or bear momentum, allowing traders to position themselves advantageously in their trading activities. Please note that future performance of any trading strategy is fundamentally unknowable, and past results do not guarantee future performance.
Discrete Fourier Transform Overlay [wbburgin]The discrete Fourier transform (DFT) overlay uses a discrete Fourier transform algorithm to identify trend direction. This is a simpler interpretation that only uses the magnitude of the first frequency component obtained from the DFT algorithm, but can be useful for visualization purposes. I haven't seen many Fourier scripts on TradingView that actually have the magnitude plotted on the chart (some have lines, for instance, but that makes it difficult to look into the past or to see previous lines).
About the Discrete Fourier Transform
The DFT is a mathematical transformation that decomposes a time-domain signal into its constituent frequency components. By applying the DFT to OHLC data, we can interpret the periodicities and trends present in the market. I've designed the overlay so that you can choose your source for the Fourier transform, as well as the length.
Settings and Configuration
The "Fourier Period" is the transform length of the DFT algorithm. This input indicates the number of data points considered for the DFT calculation. For example, if this input is set to 20, the DFT will be performed on the most recent 20 data points of the input series. The transform length affects the resolution and accuracy of the frequency analysis. A shorter transform length may provide a broader frequency range but with less detail, while a longer transform length can provide finer frequency resolution but may be computationally more intensive (I recommend using under 100 - anything above that might take too much time to load on the platform).
The "Fourier X Series" is the source you want the Fourier transform to be applied to. I have it set in default to the close.
"Kernel Smoothing" is the bar-start of the rational quadratic kernel used to smooth the frequency component. Think of it just like a normal moving average if you are unfamiliar with the concept, it functions similarly to the "length" value of a moving average.
Quinn-Fernandes Fourier Transform of Filtered Price [Loxx]Down the Rabbit Hole We Go: A Deep Dive into the Mysteries of Quinn-Fernandes Fast Fourier Transform and Hodrick-Prescott Filtering
In the ever-evolving landscape of financial markets, the ability to accurately identify and exploit underlying market patterns is of paramount importance. As market participants continuously search for innovative tools to gain an edge in their trading and investment strategies, advanced mathematical techniques, such as the Quinn-Fernandes Fourier Transform and the Hodrick-Prescott Filter, have emerged as powerful analytical tools. This comprehensive analysis aims to delve into the rich history and theoretical foundations of these techniques, exploring their applications in financial time series analysis, particularly in the context of a sophisticated trading indicator. Furthermore, we will critically assess the limitations and challenges associated with these transformative tools, while offering practical insights and recommendations for overcoming these hurdles to maximize their potential in the financial domain.
Our investigation will begin with a comprehensive examination of the origins and development of both the Quinn-Fernandes Fourier Transform and the Hodrick-Prescott Filter. We will trace their roots from classical Fourier analysis and time series smoothing to their modern-day adaptive iterations. We will elucidate the key concepts and mathematical underpinnings of these techniques and demonstrate how they are synergistically used in the context of the trading indicator under study.
As we progress, we will carefully consider the potential drawbacks and challenges associated with using the Quinn-Fernandes Fourier Transform and the Hodrick-Prescott Filter as integral components of a trading indicator. By providing a critical evaluation of their computational complexity, sensitivity to input parameters, assumptions about data stationarity, performance in noisy environments, and their nature as lagging indicators, we aim to offer a balanced and comprehensive understanding of these powerful analytical tools.
In conclusion, this in-depth analysis of the Quinn-Fernandes Fourier Transform and the Hodrick-Prescott Filter aims to provide a solid foundation for financial market participants seeking to harness the potential of these advanced techniques in their trading and investment strategies. By shedding light on their history, applications, and limitations, we hope to equip traders and investors with the knowledge and insights necessary to make informed decisions and, ultimately, achieve greater success in the highly competitive world of finance.
█ Fourier Transform and Hodrick-Prescott Filter in Financial Time Series Analysis
Financial time series analysis plays a crucial role in making informed decisions about investments and trading strategies. Among the various methods used in this domain, the Fourier Transform and the Hodrick-Prescott (HP) Filter have emerged as powerful techniques for processing and analyzing financial data. This section aims to provide a comprehensive understanding of these two methodologies, their significance in financial time series analysis, and their combined application to enhance trading strategies.
█ The Quinn-Fernandes Fourier Transform: History, Applications, and Use in Financial Time Series Analysis
The Quinn-Fernandes Fourier Transform is an advanced spectral estimation technique developed by John J. Quinn and Mauricio A. Fernandes in the early 1990s. It builds upon the classical Fourier Transform by introducing an adaptive approach that improves the identification of dominant frequencies in noisy signals. This section will explore the history of the Quinn-Fernandes Fourier Transform, its applications in various domains, and its specific use in financial time series analysis.
History of the Quinn-Fernandes Fourier Transform
The Quinn-Fernandes Fourier Transform was introduced in a 1993 paper titled "The Application of Adaptive Estimation to the Interpolation of Missing Values in Noisy Signals." In this paper, Quinn and Fernandes developed an adaptive spectral estimation algorithm to address the limitations of the classical Fourier Transform when analyzing noisy signals.
The classical Fourier Transform is a powerful mathematical tool that decomposes a function or a time series into a sum of sinusoids, making it easier to identify underlying patterns and trends. However, its performance can be negatively impacted by noise and missing data points, leading to inaccurate frequency identification.
Quinn and Fernandes sought to address these issues by developing an adaptive algorithm that could more accurately identify the dominant frequencies in a noisy signal, even when data points were missing. This adaptive algorithm, now known as the Quinn-Fernandes Fourier Transform, employs an iterative approach to refine the frequency estimates, ultimately resulting in improved spectral estimation.
Applications of the Quinn-Fernandes Fourier Transform
The Quinn-Fernandes Fourier Transform has found applications in various fields, including signal processing, telecommunications, geophysics, and biomedical engineering. Its ability to accurately identify dominant frequencies in noisy signals makes it a valuable tool for analyzing and interpreting data in these domains.
For example, in telecommunications, the Quinn-Fernandes Fourier Transform can be used to analyze the performance of communication systems and identify interference patterns. In geophysics, it can help detect and analyze seismic signals and vibrations, leading to improved understanding of geological processes. In biomedical engineering, the technique can be employed to analyze physiological signals, such as electrocardiograms, leading to more accurate diagnoses and better patient care.
Use of the Quinn-Fernandes Fourier Transform in Financial Time Series Analysis
In financial time series analysis, the Quinn-Fernandes Fourier Transform can be a powerful tool for isolating the dominant cycles and frequencies in asset price data. By more accurately identifying these critical cycles, traders can better understand the underlying dynamics of financial markets and develop more effective trading strategies.
The Quinn-Fernandes Fourier Transform is used in conjunction with the Hodrick-Prescott Filter, a technique that separates the underlying trend from the cyclical component in a time series. By first applying the Hodrick-Prescott Filter to the financial data, short-term fluctuations and noise are removed, resulting in a smoothed representation of the underlying trend. This smoothed data is then subjected to the Quinn-Fernandes Fourier Transform, allowing for more accurate identification of the dominant cycles and frequencies in the asset price data.
By employing the Quinn-Fernandes Fourier Transform in this manner, traders can gain a deeper understanding of the underlying dynamics of financial time series and develop more effective trading strategies. The enhanced knowledge of market cycles and frequencies can lead to improved risk management and ultimately, better investment performance.
The Quinn-Fernandes Fourier Transform is an advanced spectral estimation technique that has proven valuable in various domains, including financial time series analysis. Its adaptive approach to frequency identification addresses the limitations of the classical Fourier Transform when analyzing noisy signals, leading to more accurate and reliable analysis. By employing the Quinn-Fernandes Fourier Transform in financial time series analysis, traders can gain a deeper understanding of the underlying financial instrument.
Drawbacks to the Quinn-Fernandes algorithm
While the Quinn-Fernandes Fourier Transform is an effective tool for identifying dominant cycles and frequencies in financial time series, it is not without its drawbacks. Some of the limitations and challenges associated with this indicator include:
1. Computational complexity: The adaptive nature of the Quinn-Fernandes Fourier Transform requires iterative calculations, which can lead to increased computational complexity. This can be particularly challenging when analyzing large datasets or when the indicator is used in real-time trading environments.
2. Sensitivity to input parameters: The performance of the Quinn-Fernandes Fourier Transform is dependent on the choice of input parameters, such as the number of harmonic periods, frequency tolerance, and Hodrick-Prescott filter settings. Choosing inappropriate parameter values can lead to inaccurate frequency identification or reduced performance. Finding the optimal parameter settings can be challenging, and may require trial and error or a more sophisticated optimization process.
3. Assumption of stationary data: The Quinn-Fernandes Fourier Transform assumes that the underlying data is stationary, meaning that its statistical properties do not change over time. However, financial time series data is often non-stationary, with changing trends and volatility. This can limit the effectiveness of the indicator and may require additional preprocessing steps, such as detrending or differencing, to ensure the data meets the assumptions of the algorithm.
4. Limitations in noisy environments: Although the Quinn-Fernandes Fourier Transform is designed to handle noisy signals, its performance may still be negatively impacted by significant noise levels. In such cases, the identification of dominant frequencies may become less reliable, leading to suboptimal trading signals or strategies.
5. Lagging indicator: As with many technical analysis tools, the Quinn-Fernandes Fourier Transform is a lagging indicator, meaning that it is based on past data. While it can provide valuable insights into historical market dynamics, its ability to predict future price movements may be limited. This can result in false signals or late entries and exits, potentially reducing the effectiveness of trading strategies based on this indicator.
Despite these drawbacks, the Quinn-Fernandes Fourier Transform remains a valuable tool for financial time series analysis when used appropriately. By being aware of its limitations and adjusting input parameters or preprocessing steps as needed, traders can still benefit from its ability to identify dominant cycles and frequencies in financial data, and use this information to inform their trading strategies.
█ Deep-dive into the Hodrick-Prescott Fitler
The Hodrick-Prescott (HP) filter is a statistical tool used in economics and finance to separate a time series into two components: a trend component and a cyclical component. It is a powerful tool for identifying long-term trends in economic and financial data and is widely used by economists, central banks, and financial institutions around the world.
The HP filter was first introduced in the 1990s by economists Robert Hodrick and Edward Prescott. It is a simple, two-parameter filter that separates a time series into a trend component and a cyclical component. The trend component represents the long-term behavior of the data, while the cyclical component captures the shorter-term fluctuations around the trend.
The HP filter works by minimizing the following objective function:
Minimize: (Sum of Squared Deviations) + λ (Sum of Squared Second Differences)
Where:
1. The first term represents the deviation of the data from the trend.
2. The second term represents the smoothness of the trend.
3. λ is a smoothing parameter that determines the degree of smoothness of the trend.
The smoothing parameter λ is typically set to a value between 100 and 1600, depending on the frequency of the data. Higher values of λ lead to a smoother trend, while lower values lead to a more volatile trend.
The HP filter has several advantages over other smoothing techniques. It is a non-parametric method, meaning that it does not make any assumptions about the underlying distribution of the data. It also allows for easy comparison of trends across different time series and can be used with data of any frequency.
Another significant advantage of the HP Filter is its ability to adapt to changes in the underlying trend. This feature makes it particularly well-suited for analyzing financial time series, which often exhibit non-stationary behavior. By employing the HP Filter to smooth financial data, traders can more accurately identify and analyze the long-term trends that drive asset prices, ultimately leading to better-informed investment decisions.
However, the HP filter also has some limitations. It assumes that the trend is a smooth function, which may not be the case in some situations. It can also be sensitive to changes in the smoothing parameter λ, which may result in different trends for the same data. Additionally, the filter may produce unrealistic trends for very short time series.
Despite these limitations, the HP filter remains a valuable tool for analyzing economic and financial data. It is widely used by central banks and financial institutions to monitor long-term trends in the economy, and it can be used to identify turning points in the business cycle. The filter can also be used to analyze asset prices, exchange rates, and other financial variables.
The Hodrick-Prescott filter is a powerful tool for analyzing economic and financial data. It separates a time series into a trend component and a cyclical component, allowing for easy identification of long-term trends and turning points in the business cycle. While it has some limitations, it remains a valuable tool for economists, central banks, and financial institutions around the world.
█ Combined Application of Fourier Transform and Hodrick-Prescott Filter
The integration of the Fourier Transform and the Hodrick-Prescott Filter in financial time series analysis can offer several benefits. By first applying the HP Filter to the financial data, traders can remove short-term fluctuations and noise, effectively isolating the underlying trend. This smoothed data can then be subjected to the Fourier Transform, allowing for the identification of dominant cycles and frequencies with greater precision.
By combining these two powerful techniques, traders can gain a more comprehensive understanding of the underlying dynamics of financial time series. This enhanced knowledge can lead to the development of more effective trading strategies, better risk management, and ultimately, improved investment performance.
The Fourier Transform and the Hodrick-Prescott Filter are powerful tools for financial time series analysis. Each technique offers unique benefits, with the Fourier Transform being adept at identifying dominant cycles and frequencies, and the HP Filter excelling at isolating long-term trends from short-term noise. By combining these methodologies, traders can develop a deeper understanding of the underlying dynamics of financial time series, leading to more informed investment decisions and improved trading strategies. As the financial markets continue to evolve, the combined application of these techniques will undoubtedly remain an essential aspect of modern financial analysis.
█ Features
Endpointed and Non-repainting
This is an endpointed and non-repainting indicator. These are crucial factors that contribute to its usefulness and reliability in trading and investment strategies. Let us break down these concepts and discuss why they matter in the context of a financial indicator.
1. Endpoint nature: An endpoint indicator uses the most recent data points to calculate its values, ensuring that the output is timely and reflective of the current market conditions. This is in contrast to non-endpoint indicators, which may use earlier data points in their calculations, potentially leading to less timely or less relevant results. By utilizing the most recent data available, the endpoint nature of this indicator ensures that it remains up-to-date and relevant, providing traders and investors with valuable and actionable insights into the market dynamics.
2. Non-repainting characteristic: A non-repainting indicator is one that does not change its values or signals after they have been generated. This means that once a signal or a value has been plotted on the chart, it will remain there, and future data will not affect it. This is crucial for traders and investors, as it offers a sense of consistency and certainty when making decisions based on the indicator's output.
Repainting indicators, on the other hand, can change their values or signals as new data comes in, effectively "repainting" the past. This can be problematic for several reasons:
a. Misleading results: Repainting indicators can create the illusion of a highly accurate or successful trading system when backtesting, as the indicator may adapt its past signals to fit the historical price data. This can lead to overly optimistic performance results that may not hold up in real-time trading.
b. Decision-making uncertainty: When an indicator repaints, it becomes challenging for traders and investors to trust its signals, as the signal that prompted a trade may change or disappear after the fact. This can create confusion and indecision, making it difficult to execute a consistent trading strategy.
The endpoint and non-repainting characteristics of this indicator contribute to its overall reliability and effectiveness as a tool for trading and investment decision-making. By providing timely and consistent information, this indicator helps traders and investors make well-informed decisions that are less likely to be influenced by misleading or shifting data.
Inputs
Source: This input determines the source of the price data to be used for the calculations. Users can select from options like closing price, opening price, high, low, etc., based on their preferences. Changing the source of the price data (e.g., from closing price to opening price) will alter the base data used for calculations, which may lead to different patterns and cycles being identified.
Calculation Bars: This input represents the number of past bars used for the calculation. A higher value will use more historical data for the analysis, while a lower value will focus on more recent price data. Increasing the number of past bars used for calculation will incorporate more historical data into the analysis. This may lead to a more comprehensive understanding of long-term trends but could also result in a slower response to recent price changes. Decreasing this value will focus more on recent data, potentially making the indicator more responsive to short-term fluctuations.
Harmonic Period: This input represents the harmonic period, which is the number of harmonics used in the Fourier Transform. A higher value will result in more harmonics being used, potentially capturing more complex cycles in the price data. Increasing the harmonic period will include more harmonics in the Fourier Transform, potentially capturing more complex cycles in the price data. However, this may also introduce more noise and make it harder to identify clear patterns. Decreasing this value will focus on simpler cycles and may make the analysis clearer, but it might miss out on more complex patterns.
Frequency Tolerance: This input represents the frequency tolerance, which determines how close the frequencies of the harmonics must be to be considered part of the same cycle. A higher value will allow for more variation between harmonics, while a lower value will require the frequencies to be more similar. Increasing the frequency tolerance will allow for more variation between harmonics, potentially capturing a broader range of cycles. However, this may also introduce noise and make it more difficult to identify clear patterns. Decreasing this value will require the frequencies to be more similar, potentially making the analysis clearer, but it might miss out on some cycles.
Number of Bars to Render: This input determines the number of bars to render on the chart. A higher value will result in more historical data being displayed, but it may also slow down the computation due to the increased amount of data being processed. Increasing the number of bars to render on the chart will display more historical data, providing a broader context for the analysis. However, this may also slow down the computation due to the increased amount of data being processed. Decreasing this value will speed up the computation, but it will provide less historical context for the analysis.
Smoothing Mode: This input allows the user to choose between two smoothing modes for the source price data: no smoothing or Hodrick-Prescott (HP) smoothing. The choice depends on the user's preference for how the price data should be processed before the Fourier Transform is applied. Choosing between no smoothing and Hodrick-Prescott (HP) smoothing will affect the preprocessing of the price data. Using HP smoothing will remove some of the short-term fluctuations from the data, potentially making the analysis clearer and more focused on longer-term trends. Not using smoothing will retain the original price fluctuations, which may provide more detail but also introduce noise into the analysis.
Hodrick-Prescott Filter Period: This input represents the Hodrick-Prescott filter period, which is used if the user chooses to apply HP smoothing to the price data. A higher value will result in a smoother curve, while a lower value will retain more of the original price fluctuations. Increasing the Hodrick-Prescott filter period will result in a smoother curve for the price data, emphasizing longer-term trends and minimizing short-term fluctuations. Decreasing this value will retain more of the original price fluctuations, potentially providing more detail but also introducing noise into the analysis.
Alets and signals
This indicator featues alerts, signals and bar coloring. You have to option to turn these on/off in the settings menu.
Maximum Bars Restriction
This indicator requires a large amount of processing power to render on the chart. To reduce overhead, the setting "Number of Bars to Render" is set to 500 bars. You can adjust this to you liking.
█ Related Indicators and Libraries
Goertzel Cycle Composite Wave
Goertzel Browser
Fourier Spectrometer of Price w/ Extrapolation Forecast
Fourier Extrapolator of 'Caterpillar' SSA of Price
Normalized, Variety, Fast Fourier Transform Explorer
Real-Fast Fourier Transform of Price Oscillator
Real-Fast Fourier Transform of Price w/ Linear Regression
Fourier Extrapolation of Variety Moving Averages
Fourier Extrapolator of Variety RSI w/ Bollinger Bands
Fourier Extrapolator of Price w/ Projection Forecast
Fourier Extrapolator of Price
STD-Stepped Fast Cosine Transform Moving Average
Variety RSI of Fast Discrete Cosine Transform
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Goertzel Cycle Composite Wave [Loxx]As the financial markets become increasingly complex and data-driven, traders and analysts must leverage powerful tools to gain insights and make informed decisions. One such tool is the Goertzel Cycle Composite Wave indicator, a sophisticated technical analysis indicator that helps identify cyclical patterns in financial data. This powerful tool is capable of detecting cyclical patterns in financial data, helping traders to make better predictions and optimize their trading strategies. With its unique combination of mathematical algorithms and advanced charting capabilities, this indicator has the potential to revolutionize the way we approach financial modeling and trading.
*** To decrease the load time of this indicator, only XX many bars back will render to the chart. You can control this value with the setting "Number of Bars to Render". This doesn't have anything to do with repainting or the indicator being endpointed***
█ Brief Overview of the Goertzel Cycle Composite Wave
The Goertzel Cycle Composite Wave is a sophisticated technical analysis tool that utilizes the Goertzel algorithm to analyze and visualize cyclical components within a financial time series. By identifying these cycles and their characteristics, the indicator aims to provide valuable insights into the market's underlying price movements, which could potentially be used for making informed trading decisions.
The Goertzel Cycle Composite Wave is considered a non-repainting and endpointed indicator. This means that once a value has been calculated for a specific bar, that value will not change in subsequent bars, and the indicator is designed to have a clear start and end point. This is an important characteristic for indicators used in technical analysis, as it allows traders to make informed decisions based on historical data without the risk of hindsight bias or future changes in the indicator's values. This means traders can use this indicator trading purposes.
The repainting version of this indicator with forecasting, cycle selection/elimination options, and data output table can be found here:
Goertzel Browser
The primary purpose of this indicator is to:
1. Detect and analyze the dominant cycles present in the price data.
2. Reconstruct and visualize the composite wave based on the detected cycles.
To achieve this, the indicator performs several tasks:
1. Detrending the price data: The indicator preprocesses the price data using various detrending techniques, such as Hodrick-Prescott filters, zero-lag moving averages, and linear regression, to remove the underlying trend and focus on the cyclical components.
2. Applying the Goertzel algorithm: The indicator applies the Goertzel algorithm to the detrended price data, identifying the dominant cycles and their characteristics, such as amplitude, phase, and cycle strength.
3. Constructing the composite wave: The indicator reconstructs the composite wave by combining the detected cycles, either by using a user-defined list of cycles or by selecting the top N cycles based on their amplitude or cycle strength.
4. Visualizing the composite wave: The indicator plots the composite wave, using solid lines for the cycles. The color of the lines indicates whether the wave is increasing or decreasing.
This indicator is a powerful tool that employs the Goertzel algorithm to analyze and visualize the cyclical components within a financial time series. By providing insights into the underlying price movements, the indicator aims to assist traders in making more informed decisions.
█ What is the Goertzel Algorithm?
The Goertzel algorithm, named after Gerald Goertzel, is a digital signal processing technique that is used to efficiently compute individual terms of the Discrete Fourier Transform (DFT). It was first introduced in 1958, and since then, it has found various applications in the fields of engineering, mathematics, and physics.
The Goertzel algorithm is primarily used to detect specific frequency components within a digital signal, making it particularly useful in applications where only a few frequency components are of interest. The algorithm is computationally efficient, as it requires fewer calculations than the Fast Fourier Transform (FFT) when detecting a small number of frequency components. This efficiency makes the Goertzel algorithm a popular choice in applications such as:
1. Telecommunications: The Goertzel algorithm is used for decoding Dual-Tone Multi-Frequency (DTMF) signals, which are the tones generated when pressing buttons on a telephone keypad. By identifying specific frequency components, the algorithm can accurately determine which button has been pressed.
2. Audio processing: The algorithm can be used to detect specific pitches or harmonics in an audio signal, making it useful in applications like pitch detection and tuning musical instruments.
3. Vibration analysis: In the field of mechanical engineering, the Goertzel algorithm can be applied to analyze vibrations in rotating machinery, helping to identify faulty components or signs of wear.
4. Power system analysis: The algorithm can be used to measure harmonic content in power systems, allowing engineers to assess power quality and detect potential issues.
The Goertzel algorithm is used in these applications because it offers several advantages over other methods, such as the FFT:
1. Computational efficiency: The Goertzel algorithm requires fewer calculations when detecting a small number of frequency components, making it more computationally efficient than the FFT in these cases.
2. Real-time analysis: The algorithm can be implemented in a streaming fashion, allowing for real-time analysis of signals, which is crucial in applications like telecommunications and audio processing.
3. Memory efficiency: The Goertzel algorithm requires less memory than the FFT, as it only computes the frequency components of interest.
4. Precision: The algorithm is less susceptible to numerical errors compared to the FFT, ensuring more accurate results in applications where precision is essential.
The Goertzel algorithm is an efficient digital signal processing technique that is primarily used to detect specific frequency components within a signal. Its computational efficiency, real-time capabilities, and precision make it an attractive choice for various applications, including telecommunications, audio processing, vibration analysis, and power system analysis. The algorithm has been widely adopted since its introduction in 1958 and continues to be an essential tool in the fields of engineering, mathematics, and physics.
█ Goertzel Algorithm in Quantitative Finance: In-Depth Analysis and Applications
The Goertzel algorithm, initially designed for signal processing in telecommunications, has gained significant traction in the financial industry due to its efficient frequency detection capabilities. In quantitative finance, the Goertzel algorithm has been utilized for uncovering hidden market cycles, developing data-driven trading strategies, and optimizing risk management. This section delves deeper into the applications of the Goertzel algorithm in finance, particularly within the context of quantitative trading and analysis.
Unveiling Hidden Market Cycles:
Market cycles are prevalent in financial markets and arise from various factors, such as economic conditions, investor psychology, and market participant behavior. The Goertzel algorithm's ability to detect and isolate specific frequencies in price data helps trader analysts identify hidden market cycles that may otherwise go unnoticed. By examining the amplitude, phase, and periodicity of each cycle, traders can better understand the underlying market structure and dynamics, enabling them to develop more informed and effective trading strategies.
Developing Quantitative Trading Strategies:
The Goertzel algorithm's versatility allows traders to incorporate its insights into a wide range of trading strategies. By identifying the dominant market cycles in a financial instrument's price data, traders can create data-driven strategies that capitalize on the cyclical nature of markets.
For instance, a trader may develop a mean-reversion strategy that takes advantage of the identified cycles. By establishing positions when the price deviates from the predicted cycle, the trader can profit from the subsequent reversion to the cycle's mean. Similarly, a momentum-based strategy could be designed to exploit the persistence of a dominant cycle by entering positions that align with the cycle's direction.
Enhancing Risk Management:
The Goertzel algorithm plays a vital role in risk management for quantitative strategies. By analyzing the cyclical components of a financial instrument's price data, traders can gain insights into the potential risks associated with their trading strategies.
By monitoring the amplitude and phase of dominant cycles, a trader can detect changes in market dynamics that may pose risks to their positions. For example, a sudden increase in amplitude may indicate heightened volatility, prompting the trader to adjust position sizing or employ hedging techniques to protect their portfolio. Additionally, changes in phase alignment could signal a potential shift in market sentiment, necessitating adjustments to the trading strategy.
Expanding Quantitative Toolkits:
Traders can augment the Goertzel algorithm's insights by combining it with other quantitative techniques, creating a more comprehensive and sophisticated analysis framework. For example, machine learning algorithms, such as neural networks or support vector machines, could be trained on features extracted from the Goertzel algorithm to predict future price movements more accurately.
Furthermore, the Goertzel algorithm can be integrated with other technical analysis tools, such as moving averages or oscillators, to enhance their effectiveness. By applying these tools to the identified cycles, traders can generate more robust and reliable trading signals.
The Goertzel algorithm offers invaluable benefits to quantitative finance practitioners by uncovering hidden market cycles, aiding in the development of data-driven trading strategies, and improving risk management. By leveraging the insights provided by the Goertzel algorithm and integrating it with other quantitative techniques, traders can gain a deeper understanding of market dynamics and devise more effective trading strategies.
█ Indicator Inputs
src: This is the source data for the analysis, typically the closing price of the financial instrument.
detrendornot: This input determines the method used for detrending the source data. Detrending is the process of removing the underlying trend from the data to focus on the cyclical components.
The available options are:
hpsmthdt: Detrend using Hodrick-Prescott filter centered moving average.
zlagsmthdt: Detrend using zero-lag moving average centered moving average.
logZlagRegression: Detrend using logarithmic zero-lag linear regression.
hpsmth: Detrend using Hodrick-Prescott filter.
zlagsmth: Detrend using zero-lag moving average.
DT_HPper1 and DT_HPper2: These inputs define the period range for the Hodrick-Prescott filter centered moving average when detrendornot is set to hpsmthdt.
DT_ZLper1 and DT_ZLper2: These inputs define the period range for the zero-lag moving average centered moving average when detrendornot is set to zlagsmthdt.
DT_RegZLsmoothPer: This input defines the period for the zero-lag moving average used in logarithmic zero-lag linear regression when detrendornot is set to logZlagRegression.
HPsmoothPer: This input defines the period for the Hodrick-Prescott filter when detrendornot is set to hpsmth.
ZLMAsmoothPer: This input defines the period for the zero-lag moving average when detrendornot is set to zlagsmth.
MaxPer: This input sets the maximum period for the Goertzel algorithm to search for cycles.
squaredAmp: This boolean input determines whether the amplitude should be squared in the Goertzel algorithm.
useAddition: This boolean input determines whether the Goertzel algorithm should use addition for combining the cycles.
useCosine: This boolean input determines whether the Goertzel algorithm should use cosine waves instead of sine waves.
UseCycleStrength: This boolean input determines whether the Goertzel algorithm should compute the cycle strength, which is a normalized measure of the cycle's amplitude.
WindowSizePast: These inputs define the window size for the composite wave.
FilterBartels: This boolean input determines whether Bartel's test should be applied to filter out non-significant cycles.
BartNoCycles: This input sets the number of cycles to be used in Bartel's test.
BartSmoothPer: This input sets the period for the moving average used in Bartel's test.
BartSigLimit: This input sets the significance limit for Bartel's test, below which cycles are considered insignificant.
SortBartels: This boolean input determines whether the cycles should be sorted by their Bartel's test results.
StartAtCycle: This input determines the starting index for selecting the top N cycles when UseCycleList is set to false. This allows you to skip a certain number of cycles from the top before selecting the desired number of cycles.
UseTopCycles: This input sets the number of top cycles to use for constructing the composite wave when UseCycleList is set to false. The cycles are ranked based on their amplitudes or cycle strengths, depending on the UseCycleStrength input.
SubtractNoise: This boolean input determines whether to subtract the noise (remaining cycles) from the composite wave. If set to true, the composite wave will only include the top N cycles specified by UseTopCycles.
█ Exploring Auxiliary Functions
The following functions demonstrate advanced techniques for analyzing financial markets, including zero-lag moving averages, Bartels probability, detrending, and Hodrick-Prescott filtering. This section examines each function in detail, explaining their purpose, methodology, and applications in finance. We will examine how each function contributes to the overall performance and effectiveness of the indicator and how they work together to create a powerful analytical tool.
Zero-Lag Moving Average:
The zero-lag moving average function is designed to minimize the lag typically associated with moving averages. This is achieved through a two-step weighted linear regression process that emphasizes more recent data points. The function calculates a linearly weighted moving average (LWMA) on the input data and then applies another LWMA on the result. By doing this, the function creates a moving average that closely follows the price action, reducing the lag and improving the responsiveness of the indicator.
The zero-lag moving average function is used in the indicator to provide a responsive, low-lag smoothing of the input data. This function helps reduce the noise and fluctuations in the data, making it easier to identify and analyze underlying trends and patterns. By minimizing the lag associated with traditional moving averages, this function allows the indicator to react more quickly to changes in market conditions, providing timely signals and improving the overall effectiveness of the indicator.
Bartels Probability:
The Bartels probability function calculates the probability of a given cycle being significant in a time series. It uses a mathematical test called the Bartels test to assess the significance of cycles detected in the data. The function calculates coefficients for each detected cycle and computes an average amplitude and an expected amplitude. By comparing these values, the Bartels probability is derived, indicating the likelihood of a cycle's significance. This information can help in identifying and analyzing dominant cycles in financial markets.
The Bartels probability function is incorporated into the indicator to assess the significance of detected cycles in the input data. By calculating the Bartels probability for each cycle, the indicator can prioritize the most significant cycles and focus on the market dynamics that are most relevant to the current trading environment. This function enhances the indicator's ability to identify dominant market cycles, improving its predictive power and aiding in the development of effective trading strategies.
Detrend Logarithmic Zero-Lag Regression:
The detrend logarithmic zero-lag regression function is used for detrending data while minimizing lag. It combines a zero-lag moving average with a linear regression detrending method. The function first calculates the zero-lag moving average of the logarithm of input data and then applies a linear regression to remove the trend. By detrending the data, the function isolates the cyclical components, making it easier to analyze and interpret the underlying market dynamics.
The detrend logarithmic zero-lag regression function is used in the indicator to isolate the cyclical components of the input data. By detrending the data, the function enables the indicator to focus on the cyclical movements in the market, making it easier to analyze and interpret market dynamics. This function is essential for identifying cyclical patterns and understanding the interactions between different market cycles, which can inform trading decisions and enhance overall market understanding.
Bartels Cycle Significance Test:
The Bartels cycle significance test is a function that combines the Bartels probability function and the detrend logarithmic zero-lag regression function to assess the significance of detected cycles. The function calculates the Bartels probability for each cycle and stores the results in an array. By analyzing the probability values, traders and analysts can identify the most significant cycles in the data, which can be used to develop trading strategies and improve market understanding.
The Bartels cycle significance test function is integrated into the indicator to provide a comprehensive analysis of the significance of detected cycles. By combining the Bartels probability function and the detrend logarithmic zero-lag regression function, this test evaluates the significance of each cycle and stores the results in an array. The indicator can then use this information to prioritize the most significant cycles and focus on the most relevant market dynamics. This function enhances the indicator's ability to identify and analyze dominant market cycles, providing valuable insights for trading and market analysis.
Hodrick-Prescott Filter:
The Hodrick-Prescott filter is a popular technique used to separate the trend and cyclical components of a time series. The function applies a smoothing parameter to the input data and calculates a smoothed series using a two-sided filter. This smoothed series represents the trend component, which can be subtracted from the original data to obtain the cyclical component. The Hodrick-Prescott filter is commonly used in economics and finance to analyze economic data and financial market trends.
The Hodrick-Prescott filter is incorporated into the indicator to separate the trend and cyclical components of the input data. By applying the filter to the data, the indicator can isolate the trend component, which can be used to analyze long-term market trends and inform trading decisions. Additionally, the cyclical component can be used to identify shorter-term market dynamics and provide insights into potential trading opportunities. The inclusion of the Hodrick-Prescott filter adds another layer of analysis to the indicator, making it more versatile and comprehensive.
Detrending Options: Detrend Centered Moving Average:
The detrend centered moving average function provides different detrending methods, including the Hodrick-Prescott filter and the zero-lag moving average, based on the selected detrending method. The function calculates two sets of smoothed values using the chosen method and subtracts one set from the other to obtain a detrended series. By offering multiple detrending options, this function allows traders and analysts to select the most appropriate method for their specific needs and preferences.
The detrend centered moving average function is integrated into the indicator to provide users with multiple detrending options, including the Hodrick-Prescott filter and the zero-lag moving average. By offering multiple detrending methods, the indicator allows users to customize the analysis to their specific needs and preferences, enhancing the indicator's overall utility and adaptability. This function ensures that the indicator can cater to a wide range of trading styles and objectives, making it a valuable tool for a diverse group of market participants.
The auxiliary functions functions discussed in this section demonstrate the power and versatility of mathematical techniques in analyzing financial markets. By understanding and implementing these functions, traders and analysts can gain valuable insights into market dynamics, improve their trading strategies, and make more informed decisions. The combination of zero-lag moving averages, Bartels probability, detrending methods, and the Hodrick-Prescott filter provides a comprehensive toolkit for analyzing and interpreting financial data. The integration of advanced functions in a financial indicator creates a powerful and versatile analytical tool that can provide valuable insights into financial markets. By combining the zero-lag moving average,
█ In-Depth Analysis of the Goertzel Cycle Composite Wave Code
The Goertzel Cycle Composite Wave code is an implementation of the Goertzel Algorithm, an efficient technique to perform spectral analysis on a signal. The code is designed to detect and analyze dominant cycles within a given financial market data set. This section will provide an extremely detailed explanation of the code, its structure, functions, and intended purpose.
Function signature and input parameters:
The Goertzel Cycle Composite Wave function accepts numerous input parameters for customization, including source data (src), the current bar (forBar), sample size (samplesize), period (per), squared amplitude flag (squaredAmp), addition flag (useAddition), cosine flag (useCosine), cycle strength flag (UseCycleStrength), past sizes (WindowSizePast), Bartels filter flag (FilterBartels), Bartels-related parameters (BartNoCycles, BartSmoothPer, BartSigLimit), sorting flag (SortBartels), and output buffers (goeWorkPast, cyclebuffer, amplitudebuffer, phasebuffer, cycleBartelsBuffer).
Initializing variables and arrays:
The code initializes several float arrays (goeWork1, goeWork2, goeWork3, goeWork4) with the same length as twice the period (2 * per). These arrays store intermediate results during the execution of the algorithm.
Preprocessing input data:
The input data (src) undergoes preprocessing to remove linear trends. This step enhances the algorithm's ability to focus on cyclical components in the data. The linear trend is calculated by finding the slope between the first and last values of the input data within the sample.
Iterative calculation of Goertzel coefficients:
The core of the Goertzel Cycle Composite Wave algorithm lies in the iterative calculation of Goertzel coefficients for each frequency bin. These coefficients represent the spectral content of the input data at different frequencies. The code iterates through the range of frequencies, calculating the Goertzel coefficients using a nested loop structure.
Cycle strength computation:
The code calculates the cycle strength based on the Goertzel coefficients. This is an optional step, controlled by the UseCycleStrength flag. The cycle strength provides information on the relative influence of each cycle on the data per bar, considering both amplitude and cycle length. The algorithm computes the cycle strength either by squaring the amplitude (controlled by squaredAmp flag) or using the actual amplitude values.
Phase calculation:
The Goertzel Cycle Composite Wave code computes the phase of each cycle, which represents the position of the cycle within the input data. The phase is calculated using the arctangent function (math.atan) based on the ratio of the imaginary and real components of the Goertzel coefficients.
Peak detection and cycle extraction:
The algorithm performs peak detection on the computed amplitudes or cycle strengths to identify dominant cycles. It stores the detected cycles in the cyclebuffer array, along with their corresponding amplitudes and phases in the amplitudebuffer and phasebuffer arrays, respectively.
Sorting cycles by amplitude or cycle strength:
The code sorts the detected cycles based on their amplitude or cycle strength in descending order. This allows the algorithm to prioritize cycles with the most significant impact on the input data.
Bartels cycle significance test:
If the FilterBartels flag is set, the code performs a Bartels cycle significance test on the detected cycles. This test determines the statistical significance of each cycle and filters out the insignificant cycles. The significant cycles are stored in the cycleBartelsBuffer array. If the SortBartels flag is set, the code sorts the significant cycles based on their Bartels significance values.
Waveform calculation:
The Goertzel Cycle Composite Wave code calculates the waveform of the significant cycles for specified time windows. The windows are defined by the WindowSizePast parameters, respectively. The algorithm uses either cosine or sine functions (controlled by the useCosine flag) to calculate the waveforms for each cycle. The useAddition flag determines whether the waveforms should be added or subtracted.
Storing waveforms in a matrix:
The calculated waveforms for the cycle is stored in the matrix - goeWorkPast. This matrix holds the waveforms for the specified time windows. Each row in the matrix represents a time window position, and each column corresponds to a cycle.
Returning the number of cycles:
The Goertzel Cycle Composite Wave function returns the total number of detected cycles (number_of_cycles) after processing the input data. This information can be used to further analyze the results or to visualize the detected cycles.
The Goertzel Cycle Composite Wave code is a comprehensive implementation of the Goertzel Algorithm, specifically designed for detecting and analyzing dominant cycles within financial market data. The code offers a high level of customization, allowing users to fine-tune the algorithm based on their specific needs. The Goertzel Cycle Composite Wave's combination of preprocessing, iterative calculations, cycle extraction, sorting, significance testing, and waveform calculation makes it a powerful tool for understanding cyclical components in financial data.
█ Generating and Visualizing Composite Waveform
The indicator calculates and visualizes the composite waveform for specified time windows based on the detected cycles. Here's a detailed explanation of this process:
Updating WindowSizePast:
The WindowSizePast is updated to ensure they are at least twice the MaxPer (maximum period).
Initializing matrices and arrays:
The matrix goeWorkPast is initialized to store the Goertzel results for specified time windows. Multiple arrays are also initialized to store cycle, amplitude, phase, and Bartels information.
Preparing the source data (srcVal) array:
The source data is copied into an array, srcVal, and detrended using one of the selected methods (hpsmthdt, zlagsmthdt, logZlagRegression, hpsmth, or zlagsmth).
Goertzel function call:
The Goertzel function is called to analyze the detrended source data and extract cycle information. The output, number_of_cycles, contains the number of detected cycles.
Initializing arrays for waveforms:
The goertzel array is initialized to store the endpoint Goertzel.
Calculating composite waveform (goertzel array):
The composite waveform is calculated by summing the selected cycles (either from the user-defined cycle list or the top cycles) and optionally subtracting the noise component.
Drawing composite waveform (pvlines):
The composite waveform is drawn on the chart using solid lines. The color of the lines is determined by the direction of the waveform (green for upward, red for downward).
To summarize, this indicator generates a composite waveform based on the detected cycles in the financial data. It calculates the composite waveforms and visualizes them on the chart using colored lines.
█ Enhancing the Goertzel Algorithm-Based Script for Financial Modeling and Trading
The Goertzel algorithm-based script for detecting dominant cycles in financial data is a powerful tool for financial modeling and trading. It provides valuable insights into the past behavior of these cycles. However, as with any algorithm, there is always room for improvement. This section discusses potential enhancements to the existing script to make it even more robust and versatile for financial modeling, general trading, advanced trading, and high-frequency finance trading.
Enhancements for Financial Modeling
Data preprocessing: One way to improve the script's performance for financial modeling is to introduce more advanced data preprocessing techniques. This could include removing outliers, handling missing data, and normalizing the data to ensure consistent and accurate results.
Additional detrending and smoothing methods: Incorporating more sophisticated detrending and smoothing techniques, such as wavelet transform or empirical mode decomposition, can help improve the script's ability to accurately identify cycles and trends in the data.
Machine learning integration: Integrating machine learning techniques, such as artificial neural networks or support vector machines, can help enhance the script's predictive capabilities, leading to more accurate financial models.
Enhancements for General and Advanced Trading
Customizable indicator integration: Allowing users to integrate their own technical indicators can help improve the script's effectiveness for both general and advanced trading. By enabling the combination of the dominant cycle information with other technical analysis tools, traders can develop more comprehensive trading strategies.
Risk management and position sizing: Incorporating risk management and position sizing functionality into the script can help traders better manage their trades and control potential losses. This can be achieved by calculating the optimal position size based on the user's risk tolerance and account size.
Multi-timeframe analysis: Enhancing the script to perform multi-timeframe analysis can provide traders with a more holistic view of market trends and cycles. By identifying dominant cycles on different timeframes, traders can gain insights into the potential confluence of cycles and make better-informed trading decisions.
Enhancements for High-Frequency Finance Trading
Algorithm optimization: To ensure the script's suitability for high-frequency finance trading, optimizing the algorithm for faster execution is crucial. This can be achieved by employing efficient data structures and refining the calculation methods to minimize computational complexity.
Real-time data streaming: Integrating real-time data streaming capabilities into the script can help high-frequency traders react to market changes more quickly. By continuously updating the cycle information based on real-time market data, traders can adapt their strategies accordingly and capitalize on short-term market fluctuations.
Order execution and trade management: To fully leverage the script's capabilities for high-frequency trading, implementing functionality for automated order execution and trade management is essential. This can include features such as stop-loss and take-profit orders, trailing stops, and automated trade exit strategies.
While the existing Goertzel algorithm-based script is a valuable tool for detecting dominant cycles in financial data, there are several potential enhancements that can make it even more powerful for financial modeling, general trading, advanced trading, and high-frequency finance trading. By incorporating these improvements, the script can become a more versatile and effective tool for traders and financial analysts alike.
█ Understanding the Limitations of the Goertzel Algorithm
While the Goertzel algorithm-based script for detecting dominant cycles in financial data provides valuable insights, it is important to be aware of its limitations and drawbacks. Some of the key drawbacks of this indicator are:
Lagging nature:
As with many other technical indicators, the Goertzel algorithm-based script can suffer from lagging effects, meaning that it may not immediately react to real-time market changes. This lag can lead to late entries and exits, potentially resulting in reduced profitability or increased losses.
Parameter sensitivity:
The performance of the script can be sensitive to the chosen parameters, such as the detrending methods, smoothing techniques, and cycle detection settings. Improper parameter selection may lead to inaccurate cycle detection or increased false signals, which can negatively impact trading performance.
Complexity:
The Goertzel algorithm itself is relatively complex, making it difficult for novice traders or those unfamiliar with the concept of cycle analysis to fully understand and effectively utilize the script. This complexity can also make it challenging to optimize the script for specific trading styles or market conditions.
Overfitting risk:
As with any data-driven approach, there is a risk of overfitting when using the Goertzel algorithm-based script. Overfitting occurs when a model becomes too specific to the historical data it was trained on, leading to poor performance on new, unseen data. This can result in misleading signals and reduced trading performance.
Limited applicability:
The Goertzel algorithm-based script may not be suitable for all markets, trading styles, or timeframes. Its effectiveness in detecting cycles may be limited in certain market conditions, such as during periods of extreme volatility or low liquidity.
While the Goertzel algorithm-based script offers valuable insights into dominant cycles in financial data, it is essential to consider its drawbacks and limitations when incorporating it into a trading strategy. Traders should always use the script in conjunction with other technical and fundamental analysis tools, as well as proper risk management, to make well-informed trading decisions.
█ Interpreting Results
The Goertzel Cycle Composite Wave indicator can be interpreted by analyzing the plotted lines. The indicator plots two lines: composite waves. The composite wave represents the composite wave of the price data.
The composite wave line displays a solid line, with green indicating a bullish trend and red indicating a bearish trend.
Interpreting the Goertzel Cycle Composite Wave indicator involves identifying the trend of the composite wave lines and matching them with the corresponding bullish or bearish color.
█ Conclusion
The Goertzel Cycle Composite Wave indicator is a powerful tool for identifying and analyzing cyclical patterns in financial markets. Its ability to detect multiple cycles of varying frequencies and strengths make it a valuable addition to any trader's technical analysis toolkit. However, it is important to keep in mind that the Goertzel Cycle Composite Wave indicator should be used in conjunction with other technical analysis tools and fundamental analysis to achieve the best results. With continued refinement and development, the Goertzel Cycle Composite Wave indicator has the potential to become a highly effective tool for financial modeling, general trading, advanced trading, and high-frequency finance trading. Its accuracy and versatility make it a promising candidate for further research and development.
█ Footnotes
What is the Bartels Test for Cycle Significance?
The Bartels Cycle Significance Test is a statistical method that determines whether the peaks and troughs of a time series are statistically significant. The test is named after its inventor, George Bartels, who developed it in the mid-20th century.
The Bartels test is designed to analyze the cyclical components of a time series, which can help traders and analysts identify trends and cycles in financial markets. The test calculates a Bartels statistic, which measures the degree of non-randomness or autocorrelation in the time series.
The Bartels statistic is calculated by first splitting the time series into two halves and calculating the range of the peaks and troughs in each half. The test then compares these ranges using a t-test, which measures the significance of the difference between the two ranges.
If the Bartels statistic is greater than a critical value, it indicates that the peaks and troughs in the time series are non-random and that there is a significant cyclical component to the data. Conversely, if the Bartels statistic is less than the critical value, it suggests that the peaks and troughs are random and that there is no significant cyclical component.
The Bartels Cycle Significance Test is particularly useful in financial analysis because it can help traders and analysts identify significant cycles in asset prices, which can in turn inform investment decisions. However, it is important to note that the test is not perfect and can produce false signals in certain situations, particularly in noisy or volatile markets. Therefore, it is always recommended to use the test in conjunction with other technical and fundamental indicators to confirm trends and cycles.
Deep-dive into the Hodrick-Prescott Fitler
The Hodrick-Prescott (HP) filter is a statistical tool used in economics and finance to separate a time series into two components: a trend component and a cyclical component. It is a powerful tool for identifying long-term trends in economic and financial data and is widely used by economists, central banks, and financial institutions around the world.
The HP filter was first introduced in the 1990s by economists Robert Hodrick and Edward Prescott. It is a simple, two-parameter filter that separates a time series into a trend component and a cyclical component. The trend component represents the long-term behavior of the data, while the cyclical component captures the shorter-term fluctuations around the trend.
The HP filter works by minimizing the following objective function:
Minimize: (Sum of Squared Deviations) + λ (Sum of Squared Second Differences)
Where:
1. The first term represents the deviation of the data from the trend.
2. The second term represents the smoothness of the trend.
3. λ is a smoothing parameter that determines the degree of smoothness of the trend.
The smoothing parameter λ is typically set to a value between 100 and 1600, depending on the frequency of the data. Higher values of λ lead to a smoother trend, while lower values lead to a more volatile trend.
The HP filter has several advantages over other smoothing techniques. It is a non-parametric method, meaning that it does not make any assumptions about the underlying distribution of the data. It also allows for easy comparison of trends across different time series and can be used with data of any frequency.
However, the HP filter also has some limitations. It assumes that the trend is a smooth function, which may not be the case in some situations. It can also be sensitive to changes in the smoothing parameter λ, which may result in different trends for the same data. Additionally, the filter may produce unrealistic trends for very short time series.
Despite these limitations, the HP filter remains a valuable tool for analyzing economic and financial data. It is widely used by central banks and financial institutions to monitor long-term trends in the economy, and it can be used to identify turning points in the business cycle. The filter can also be used to analyze asset prices, exchange rates, and other financial variables.
The Hodrick-Prescott filter is a powerful tool for analyzing economic and financial data. It separates a time series into a trend component and a cyclical component, allowing for easy identification of long-term trends and turning points in the business cycle. While it has some limitations, it remains a valuable tool for economists, central banks, and financial institutions around the world.
Goertzel Browser [Loxx]As the financial markets become increasingly complex and data-driven, traders and analysts must leverage powerful tools to gain insights and make informed decisions. One such tool is the Goertzel Browser indicator, a sophisticated technical analysis indicator that helps identify cyclical patterns in financial data. This powerful tool is capable of detecting cyclical patterns in financial data, helping traders to make better predictions and optimize their trading strategies. With its unique combination of mathematical algorithms and advanced charting capabilities, this indicator has the potential to revolutionize the way we approach financial modeling and trading.
█ Brief Overview of the Goertzel Browser
The Goertzel Browser is a sophisticated technical analysis tool that utilizes the Goertzel algorithm to analyze and visualize cyclical components within a financial time series. By identifying these cycles and their characteristics, the indicator aims to provide valuable insights into the market's underlying price movements, which could potentially be used for making informed trading decisions.
The primary purpose of this indicator is to:
1. Detect and analyze the dominant cycles present in the price data.
2. Reconstruct and visualize the composite wave based on the detected cycles.
3. Project the composite wave into the future, providing a potential roadmap for upcoming price movements.
To achieve this, the indicator performs several tasks:
1. Detrending the price data: The indicator preprocesses the price data using various detrending techniques, such as Hodrick-Prescott filters, zero-lag moving averages, and linear regression, to remove the underlying trend and focus on the cyclical components.
2. Applying the Goertzel algorithm: The indicator applies the Goertzel algorithm to the detrended price data, identifying the dominant cycles and their characteristics, such as amplitude, phase, and cycle strength.
3. Constructing the composite wave: The indicator reconstructs the composite wave by combining the detected cycles, either by using a user-defined list of cycles or by selecting the top N cycles based on their amplitude or cycle strength.
4. Visualizing the composite wave: The indicator plots the composite wave, using solid lines for the past and dotted lines for the future projections. The color of the lines indicates whether the wave is increasing or decreasing.
5. Displaying cycle information: The indicator provides a table that displays detailed information about the detected cycles, including their rank, period, Bartel's test results, amplitude, and phase.
This indicator is a powerful tool that employs the Goertzel algorithm to analyze and visualize the cyclical components within a financial time series. By providing insights into the underlying price movements and their potential future trajectory, the indicator aims to assist traders in making more informed decisions.
█ What is the Goertzel Algorithm?
The Goertzel algorithm, named after Gerald Goertzel, is a digital signal processing technique that is used to efficiently compute individual terms of the Discrete Fourier Transform (DFT). It was first introduced in 1958, and since then, it has found various applications in the fields of engineering, mathematics, and physics.
The Goertzel algorithm is primarily used to detect specific frequency components within a digital signal, making it particularly useful in applications where only a few frequency components are of interest. The algorithm is computationally efficient, as it requires fewer calculations than the Fast Fourier Transform (FFT) when detecting a small number of frequency components. This efficiency makes the Goertzel algorithm a popular choice in applications such as:
1. Telecommunications: The Goertzel algorithm is used for decoding Dual-Tone Multi-Frequency (DTMF) signals, which are the tones generated when pressing buttons on a telephone keypad. By identifying specific frequency components, the algorithm can accurately determine which button has been pressed.
2. Audio processing: The algorithm can be used to detect specific pitches or harmonics in an audio signal, making it useful in applications like pitch detection and tuning musical instruments.
3. Vibration analysis: In the field of mechanical engineering, the Goertzel algorithm can be applied to analyze vibrations in rotating machinery, helping to identify faulty components or signs of wear.
4. Power system analysis: The algorithm can be used to measure harmonic content in power systems, allowing engineers to assess power quality and detect potential issues.
The Goertzel algorithm is used in these applications because it offers several advantages over other methods, such as the FFT:
1. Computational efficiency: The Goertzel algorithm requires fewer calculations when detecting a small number of frequency components, making it more computationally efficient than the FFT in these cases.
2. Real-time analysis: The algorithm can be implemented in a streaming fashion, allowing for real-time analysis of signals, which is crucial in applications like telecommunications and audio processing.
3. Memory efficiency: The Goertzel algorithm requires less memory than the FFT, as it only computes the frequency components of interest.
4. Precision: The algorithm is less susceptible to numerical errors compared to the FFT, ensuring more accurate results in applications where precision is essential.
The Goertzel algorithm is an efficient digital signal processing technique that is primarily used to detect specific frequency components within a signal. Its computational efficiency, real-time capabilities, and precision make it an attractive choice for various applications, including telecommunications, audio processing, vibration analysis, and power system analysis. The algorithm has been widely adopted since its introduction in 1958 and continues to be an essential tool in the fields of engineering, mathematics, and physics.
█ Goertzel Algorithm in Quantitative Finance: In-Depth Analysis and Applications
The Goertzel algorithm, initially designed for signal processing in telecommunications, has gained significant traction in the financial industry due to its efficient frequency detection capabilities. In quantitative finance, the Goertzel algorithm has been utilized for uncovering hidden market cycles, developing data-driven trading strategies, and optimizing risk management. This section delves deeper into the applications of the Goertzel algorithm in finance, particularly within the context of quantitative trading and analysis.
Unveiling Hidden Market Cycles:
Market cycles are prevalent in financial markets and arise from various factors, such as economic conditions, investor psychology, and market participant behavior. The Goertzel algorithm's ability to detect and isolate specific frequencies in price data helps trader analysts identify hidden market cycles that may otherwise go unnoticed. By examining the amplitude, phase, and periodicity of each cycle, traders can better understand the underlying market structure and dynamics, enabling them to develop more informed and effective trading strategies.
Developing Quantitative Trading Strategies:
The Goertzel algorithm's versatility allows traders to incorporate its insights into a wide range of trading strategies. By identifying the dominant market cycles in a financial instrument's price data, traders can create data-driven strategies that capitalize on the cyclical nature of markets.
For instance, a trader may develop a mean-reversion strategy that takes advantage of the identified cycles. By establishing positions when the price deviates from the predicted cycle, the trader can profit from the subsequent reversion to the cycle's mean. Similarly, a momentum-based strategy could be designed to exploit the persistence of a dominant cycle by entering positions that align with the cycle's direction.
Enhancing Risk Management:
The Goertzel algorithm plays a vital role in risk management for quantitative strategies. By analyzing the cyclical components of a financial instrument's price data, traders can gain insights into the potential risks associated with their trading strategies.
By monitoring the amplitude and phase of dominant cycles, a trader can detect changes in market dynamics that may pose risks to their positions. For example, a sudden increase in amplitude may indicate heightened volatility, prompting the trader to adjust position sizing or employ hedging techniques to protect their portfolio. Additionally, changes in phase alignment could signal a potential shift in market sentiment, necessitating adjustments to the trading strategy.
Expanding Quantitative Toolkits:
Traders can augment the Goertzel algorithm's insights by combining it with other quantitative techniques, creating a more comprehensive and sophisticated analysis framework. For example, machine learning algorithms, such as neural networks or support vector machines, could be trained on features extracted from the Goertzel algorithm to predict future price movements more accurately.
Furthermore, the Goertzel algorithm can be integrated with other technical analysis tools, such as moving averages or oscillators, to enhance their effectiveness. By applying these tools to the identified cycles, traders can generate more robust and reliable trading signals.
The Goertzel algorithm offers invaluable benefits to quantitative finance practitioners by uncovering hidden market cycles, aiding in the development of data-driven trading strategies, and improving risk management. By leveraging the insights provided by the Goertzel algorithm and integrating it with other quantitative techniques, traders can gain a deeper understanding of market dynamics and devise more effective trading strategies.
█ Indicator Inputs
src: This is the source data for the analysis, typically the closing price of the financial instrument.
detrendornot: This input determines the method used for detrending the source data. Detrending is the process of removing the underlying trend from the data to focus on the cyclical components.
The available options are:
hpsmthdt: Detrend using Hodrick-Prescott filter centered moving average.
zlagsmthdt: Detrend using zero-lag moving average centered moving average.
logZlagRegression: Detrend using logarithmic zero-lag linear regression.
hpsmth: Detrend using Hodrick-Prescott filter.
zlagsmth: Detrend using zero-lag moving average.
DT_HPper1 and DT_HPper2: These inputs define the period range for the Hodrick-Prescott filter centered moving average when detrendornot is set to hpsmthdt.
DT_ZLper1 and DT_ZLper2: These inputs define the period range for the zero-lag moving average centered moving average when detrendornot is set to zlagsmthdt.
DT_RegZLsmoothPer: This input defines the period for the zero-lag moving average used in logarithmic zero-lag linear regression when detrendornot is set to logZlagRegression.
HPsmoothPer: This input defines the period for the Hodrick-Prescott filter when detrendornot is set to hpsmth.
ZLMAsmoothPer: This input defines the period for the zero-lag moving average when detrendornot is set to zlagsmth.
MaxPer: This input sets the maximum period for the Goertzel algorithm to search for cycles.
squaredAmp: This boolean input determines whether the amplitude should be squared in the Goertzel algorithm.
useAddition: This boolean input determines whether the Goertzel algorithm should use addition for combining the cycles.
useCosine: This boolean input determines whether the Goertzel algorithm should use cosine waves instead of sine waves.
UseCycleStrength: This boolean input determines whether the Goertzel algorithm should compute the cycle strength, which is a normalized measure of the cycle's amplitude.
WindowSizePast and WindowSizeFuture: These inputs define the window size for past and future projections of the composite wave.
FilterBartels: This boolean input determines whether Bartel's test should be applied to filter out non-significant cycles.
BartNoCycles: This input sets the number of cycles to be used in Bartel's test.
BartSmoothPer: This input sets the period for the moving average used in Bartel's test.
BartSigLimit: This input sets the significance limit for Bartel's test, below which cycles are considered insignificant.
SortBartels: This boolean input determines whether the cycles should be sorted by their Bartel's test results.
UseCycleList: This boolean input determines whether a user-defined list of cycles should be used for constructing the composite wave. If set to false, the top N cycles will be used.
Cycle1, Cycle2, Cycle3, Cycle4, and Cycle5: These inputs define the user-defined list of cycles when 'UseCycleList' is set to true. If using a user-defined list, each of these inputs represents the period of a specific cycle to include in the composite wave.
StartAtCycle: This input determines the starting index for selecting the top N cycles when UseCycleList is set to false. This allows you to skip a certain number of cycles from the top before selecting the desired number of cycles.
UseTopCycles: This input sets the number of top cycles to use for constructing the composite wave when UseCycleList is set to false. The cycles are ranked based on their amplitudes or cycle strengths, depending on the UseCycleStrength input.
SubtractNoise: This boolean input determines whether to subtract the noise (remaining cycles) from the composite wave. If set to true, the composite wave will only include the top N cycles specified by UseTopCycles.
█ Exploring Auxiliary Functions
The following functions demonstrate advanced techniques for analyzing financial markets, including zero-lag moving averages, Bartels probability, detrending, and Hodrick-Prescott filtering. This section examines each function in detail, explaining their purpose, methodology, and applications in finance. We will examine how each function contributes to the overall performance and effectiveness of the indicator and how they work together to create a powerful analytical tool.
Zero-Lag Moving Average:
The zero-lag moving average function is designed to minimize the lag typically associated with moving averages. This is achieved through a two-step weighted linear regression process that emphasizes more recent data points. The function calculates a linearly weighted moving average (LWMA) on the input data and then applies another LWMA on the result. By doing this, the function creates a moving average that closely follows the price action, reducing the lag and improving the responsiveness of the indicator.
The zero-lag moving average function is used in the indicator to provide a responsive, low-lag smoothing of the input data. This function helps reduce the noise and fluctuations in the data, making it easier to identify and analyze underlying trends and patterns. By minimizing the lag associated with traditional moving averages, this function allows the indicator to react more quickly to changes in market conditions, providing timely signals and improving the overall effectiveness of the indicator.
Bartels Probability:
The Bartels probability function calculates the probability of a given cycle being significant in a time series. It uses a mathematical test called the Bartels test to assess the significance of cycles detected in the data. The function calculates coefficients for each detected cycle and computes an average amplitude and an expected amplitude. By comparing these values, the Bartels probability is derived, indicating the likelihood of a cycle's significance. This information can help in identifying and analyzing dominant cycles in financial markets.
The Bartels probability function is incorporated into the indicator to assess the significance of detected cycles in the input data. By calculating the Bartels probability for each cycle, the indicator can prioritize the most significant cycles and focus on the market dynamics that are most relevant to the current trading environment. This function enhances the indicator's ability to identify dominant market cycles, improving its predictive power and aiding in the development of effective trading strategies.
Detrend Logarithmic Zero-Lag Regression:
The detrend logarithmic zero-lag regression function is used for detrending data while minimizing lag. It combines a zero-lag moving average with a linear regression detrending method. The function first calculates the zero-lag moving average of the logarithm of input data and then applies a linear regression to remove the trend. By detrending the data, the function isolates the cyclical components, making it easier to analyze and interpret the underlying market dynamics.
The detrend logarithmic zero-lag regression function is used in the indicator to isolate the cyclical components of the input data. By detrending the data, the function enables the indicator to focus on the cyclical movements in the market, making it easier to analyze and interpret market dynamics. This function is essential for identifying cyclical patterns and understanding the interactions between different market cycles, which can inform trading decisions and enhance overall market understanding.
Bartels Cycle Significance Test:
The Bartels cycle significance test is a function that combines the Bartels probability function and the detrend logarithmic zero-lag regression function to assess the significance of detected cycles. The function calculates the Bartels probability for each cycle and stores the results in an array. By analyzing the probability values, traders and analysts can identify the most significant cycles in the data, which can be used to develop trading strategies and improve market understanding.
The Bartels cycle significance test function is integrated into the indicator to provide a comprehensive analysis of the significance of detected cycles. By combining the Bartels probability function and the detrend logarithmic zero-lag regression function, this test evaluates the significance of each cycle and stores the results in an array. The indicator can then use this information to prioritize the most significant cycles and focus on the most relevant market dynamics. This function enhances the indicator's ability to identify and analyze dominant market cycles, providing valuable insights for trading and market analysis.
Hodrick-Prescott Filter:
The Hodrick-Prescott filter is a popular technique used to separate the trend and cyclical components of a time series. The function applies a smoothing parameter to the input data and calculates a smoothed series using a two-sided filter. This smoothed series represents the trend component, which can be subtracted from the original data to obtain the cyclical component. The Hodrick-Prescott filter is commonly used in economics and finance to analyze economic data and financial market trends.
The Hodrick-Prescott filter is incorporated into the indicator to separate the trend and cyclical components of the input data. By applying the filter to the data, the indicator can isolate the trend component, which can be used to analyze long-term market trends and inform trading decisions. Additionally, the cyclical component can be used to identify shorter-term market dynamics and provide insights into potential trading opportunities. The inclusion of the Hodrick-Prescott filter adds another layer of analysis to the indicator, making it more versatile and comprehensive.
Detrending Options: Detrend Centered Moving Average:
The detrend centered moving average function provides different detrending methods, including the Hodrick-Prescott filter and the zero-lag moving average, based on the selected detrending method. The function calculates two sets of smoothed values using the chosen method and subtracts one set from the other to obtain a detrended series. By offering multiple detrending options, this function allows traders and analysts to select the most appropriate method for their specific needs and preferences.
The detrend centered moving average function is integrated into the indicator to provide users with multiple detrending options, including the Hodrick-Prescott filter and the zero-lag moving average. By offering multiple detrending methods, the indicator allows users to customize the analysis to their specific needs and preferences, enhancing the indicator's overall utility and adaptability. This function ensures that the indicator can cater to a wide range of trading styles and objectives, making it a valuable tool for a diverse group of market participants.
The auxiliary functions functions discussed in this section demonstrate the power and versatility of mathematical techniques in analyzing financial markets. By understanding and implementing these functions, traders and analysts can gain valuable insights into market dynamics, improve their trading strategies, and make more informed decisions. The combination of zero-lag moving averages, Bartels probability, detrending methods, and the Hodrick-Prescott filter provides a comprehensive toolkit for analyzing and interpreting financial data. The integration of advanced functions in a financial indicator creates a powerful and versatile analytical tool that can provide valuable insights into financial markets. By combining the zero-lag moving average,
█ In-Depth Analysis of the Goertzel Browser Code
The Goertzel Browser code is an implementation of the Goertzel Algorithm, an efficient technique to perform spectral analysis on a signal. The code is designed to detect and analyze dominant cycles within a given financial market data set. This section will provide an extremely detailed explanation of the code, its structure, functions, and intended purpose.
Function signature and input parameters:
The Goertzel Browser function accepts numerous input parameters for customization, including source data (src), the current bar (forBar), sample size (samplesize), period (per), squared amplitude flag (squaredAmp), addition flag (useAddition), cosine flag (useCosine), cycle strength flag (UseCycleStrength), past and future window sizes (WindowSizePast, WindowSizeFuture), Bartels filter flag (FilterBartels), Bartels-related parameters (BartNoCycles, BartSmoothPer, BartSigLimit), sorting flag (SortBartels), and output buffers (goeWorkPast, goeWorkFuture, cyclebuffer, amplitudebuffer, phasebuffer, cycleBartelsBuffer).
Initializing variables and arrays:
The code initializes several float arrays (goeWork1, goeWork2, goeWork3, goeWork4) with the same length as twice the period (2 * per). These arrays store intermediate results during the execution of the algorithm.
Preprocessing input data:
The input data (src) undergoes preprocessing to remove linear trends. This step enhances the algorithm's ability to focus on cyclical components in the data. The linear trend is calculated by finding the slope between the first and last values of the input data within the sample.
Iterative calculation of Goertzel coefficients:
The core of the Goertzel Browser algorithm lies in the iterative calculation of Goertzel coefficients for each frequency bin. These coefficients represent the spectral content of the input data at different frequencies. The code iterates through the range of frequencies, calculating the Goertzel coefficients using a nested loop structure.
Cycle strength computation:
The code calculates the cycle strength based on the Goertzel coefficients. This is an optional step, controlled by the UseCycleStrength flag. The cycle strength provides information on the relative influence of each cycle on the data per bar, considering both amplitude and cycle length. The algorithm computes the cycle strength either by squaring the amplitude (controlled by squaredAmp flag) or using the actual amplitude values.
Phase calculation:
The Goertzel Browser code computes the phase of each cycle, which represents the position of the cycle within the input data. The phase is calculated using the arctangent function (math.atan) based on the ratio of the imaginary and real components of the Goertzel coefficients.
Peak detection and cycle extraction:
The algorithm performs peak detection on the computed amplitudes or cycle strengths to identify dominant cycles. It stores the detected cycles in the cyclebuffer array, along with their corresponding amplitudes and phases in the amplitudebuffer and phasebuffer arrays, respectively.
Sorting cycles by amplitude or cycle strength:
The code sorts the detected cycles based on their amplitude or cycle strength in descending order. This allows the algorithm to prioritize cycles with the most significant impact on the input data.
Bartels cycle significance test:
If the FilterBartels flag is set, the code performs a Bartels cycle significance test on the detected cycles. This test determines the statistical significance of each cycle and filters out the insignificant cycles. The significant cycles are stored in the cycleBartelsBuffer array. If the SortBartels flag is set, the code sorts the significant cycles based on their Bartels significance values.
Waveform calculation:
The Goertzel Browser code calculates the waveform of the significant cycles for both past and future time windows. The past and future windows are defined by the WindowSizePast and WindowSizeFuture parameters, respectively. The algorithm uses either cosine or sine functions (controlled by the useCosine flag) to calculate the waveforms for each cycle. The useAddition flag determines whether the waveforms should be added or subtracted.
Storing waveforms in matrices:
The calculated waveforms for each cycle are stored in two matrices - goeWorkPast and goeWorkFuture. These matrices hold the waveforms for the past and future time windows, respectively. Each row in the matrices represents a time window position, and each column corresponds to a cycle.
Returning the number of cycles:
The Goertzel Browser function returns the total number of detected cycles (number_of_cycles) after processing the input data. This information can be used to further analyze the results or to visualize the detected cycles.
The Goertzel Browser code is a comprehensive implementation of the Goertzel Algorithm, specifically designed for detecting and analyzing dominant cycles within financial market data. The code offers a high level of customization, allowing users to fine-tune the algorithm based on their specific needs. The Goertzel Browser's combination of preprocessing, iterative calculations, cycle extraction, sorting, significance testing, and waveform calculation makes it a powerful tool for understanding cyclical components in financial data.
█ Generating and Visualizing Composite Waveform
The indicator calculates and visualizes the composite waveform for both past and future time windows based on the detected cycles. Here's a detailed explanation of this process:
Updating WindowSizePast and WindowSizeFuture:
The WindowSizePast and WindowSizeFuture are updated to ensure they are at least twice the MaxPer (maximum period).
Initializing matrices and arrays:
Two matrices, goeWorkPast and goeWorkFuture, are initialized to store the Goertzel results for past and future time windows. Multiple arrays are also initialized to store cycle, amplitude, phase, and Bartels information.
Preparing the source data (srcVal) array:
The source data is copied into an array, srcVal, and detrended using one of the selected methods (hpsmthdt, zlagsmthdt, logZlagRegression, hpsmth, or zlagsmth).
Goertzel function call:
The Goertzel function is called to analyze the detrended source data and extract cycle information. The output, number_of_cycles, contains the number of detected cycles.
Initializing arrays for past and future waveforms:
Three arrays, epgoertzel, goertzel, and goertzelFuture, are initialized to store the endpoint Goertzel, non-endpoint Goertzel, and future Goertzel projections, respectively.
Calculating composite waveform for past bars (goertzel array):
The past composite waveform is calculated by summing the selected cycles (either from the user-defined cycle list or the top cycles) and optionally subtracting the noise component.
Calculating composite waveform for future bars (goertzelFuture array):
The future composite waveform is calculated in a similar way as the past composite waveform.
Drawing past composite waveform (pvlines):
The past composite waveform is drawn on the chart using solid lines. The color of the lines is determined by the direction of the waveform (green for upward, red for downward).
Drawing future composite waveform (fvlines):
The future composite waveform is drawn on the chart using dotted lines. The color of the lines is determined by the direction of the waveform (fuchsia for upward, yellow for downward).
Displaying cycle information in a table (table3):
A table is created to display the cycle information, including the rank, period, Bartel value, amplitude (or cycle strength), and phase of each detected cycle.
Filling the table with cycle information:
The indicator iterates through the detected cycles and retrieves the relevant information (period, amplitude, phase, and Bartel value) from the corresponding arrays. It then fills the table with this information, displaying the values up to six decimal places.
To summarize, this indicator generates a composite waveform based on the detected cycles in the financial data. It calculates the composite waveforms for both past and future time windows and visualizes them on the chart using colored lines. Additionally, it displays detailed cycle information in a table, including the rank, period, Bartel value, amplitude (or cycle strength), and phase of each detected cycle.
█ Enhancing the Goertzel Algorithm-Based Script for Financial Modeling and Trading
The Goertzel algorithm-based script for detecting dominant cycles in financial data is a powerful tool for financial modeling and trading. It provides valuable insights into the past behavior of these cycles and potential future impact. However, as with any algorithm, there is always room for improvement. This section discusses potential enhancements to the existing script to make it even more robust and versatile for financial modeling, general trading, advanced trading, and high-frequency finance trading.
Enhancements for Financial Modeling
Data preprocessing: One way to improve the script's performance for financial modeling is to introduce more advanced data preprocessing techniques. This could include removing outliers, handling missing data, and normalizing the data to ensure consistent and accurate results.
Additional detrending and smoothing methods: Incorporating more sophisticated detrending and smoothing techniques, such as wavelet transform or empirical mode decomposition, can help improve the script's ability to accurately identify cycles and trends in the data.
Machine learning integration: Integrating machine learning techniques, such as artificial neural networks or support vector machines, can help enhance the script's predictive capabilities, leading to more accurate financial models.
Enhancements for General and Advanced Trading
Customizable indicator integration: Allowing users to integrate their own technical indicators can help improve the script's effectiveness for both general and advanced trading. By enabling the combination of the dominant cycle information with other technical analysis tools, traders can develop more comprehensive trading strategies.
Risk management and position sizing: Incorporating risk management and position sizing functionality into the script can help traders better manage their trades and control potential losses. This can be achieved by calculating the optimal position size based on the user's risk tolerance and account size.
Multi-timeframe analysis: Enhancing the script to perform multi-timeframe analysis can provide traders with a more holistic view of market trends and cycles. By identifying dominant cycles on different timeframes, traders can gain insights into the potential confluence of cycles and make better-informed trading decisions.
Enhancements for High-Frequency Finance Trading
Algorithm optimization: To ensure the script's suitability for high-frequency finance trading, optimizing the algorithm for faster execution is crucial. This can be achieved by employing efficient data structures and refining the calculation methods to minimize computational complexity.
Real-time data streaming: Integrating real-time data streaming capabilities into the script can help high-frequency traders react to market changes more quickly. By continuously updating the cycle information based on real-time market data, traders can adapt their strategies accordingly and capitalize on short-term market fluctuations.
Order execution and trade management: To fully leverage the script's capabilities for high-frequency trading, implementing functionality for automated order execution and trade management is essential. This can include features such as stop-loss and take-profit orders, trailing stops, and automated trade exit strategies.
While the existing Goertzel algorithm-based script is a valuable tool for detecting dominant cycles in financial data, there are several potential enhancements that can make it even more powerful for financial modeling, general trading, advanced trading, and high-frequency finance trading. By incorporating these improvements, the script can become a more versatile and effective tool for traders and financial analysts alike.
█ Understanding the Limitations of the Goertzel Algorithm
While the Goertzel algorithm-based script for detecting dominant cycles in financial data provides valuable insights, it is important to be aware of its limitations and drawbacks. Some of the key drawbacks of this indicator are:
Lagging nature:
As with many other technical indicators, the Goertzel algorithm-based script can suffer from lagging effects, meaning that it may not immediately react to real-time market changes. This lag can lead to late entries and exits, potentially resulting in reduced profitability or increased losses.
Parameter sensitivity:
The performance of the script can be sensitive to the chosen parameters, such as the detrending methods, smoothing techniques, and cycle detection settings. Improper parameter selection may lead to inaccurate cycle detection or increased false signals, which can negatively impact trading performance.
Complexity:
The Goertzel algorithm itself is relatively complex, making it difficult for novice traders or those unfamiliar with the concept of cycle analysis to fully understand and effectively utilize the script. This complexity can also make it challenging to optimize the script for specific trading styles or market conditions.
Overfitting risk:
As with any data-driven approach, there is a risk of overfitting when using the Goertzel algorithm-based script. Overfitting occurs when a model becomes too specific to the historical data it was trained on, leading to poor performance on new, unseen data. This can result in misleading signals and reduced trading performance.
No guarantee of future performance: While the script can provide insights into past cycles and potential future trends, it is important to remember that past performance does not guarantee future results. Market conditions can change, and relying solely on the script's predictions without considering other factors may lead to poor trading decisions.
Limited applicability: The Goertzel algorithm-based script may not be suitable for all markets, trading styles, or timeframes. Its effectiveness in detecting cycles may be limited in certain market conditions, such as during periods of extreme volatility or low liquidity.
While the Goertzel algorithm-based script offers valuable insights into dominant cycles in financial data, it is essential to consider its drawbacks and limitations when incorporating it into a trading strategy. Traders should always use the script in conjunction with other technical and fundamental analysis tools, as well as proper risk management, to make well-informed trading decisions.
█ Interpreting Results
The Goertzel Browser indicator can be interpreted by analyzing the plotted lines and the table presented alongside them. The indicator plots two lines: past and future composite waves. The past composite wave represents the composite wave of the past price data, and the future composite wave represents the projected composite wave for the next period.
The past composite wave line displays a solid line, with green indicating a bullish trend and red indicating a bearish trend. On the other hand, the future composite wave line is a dotted line with fuchsia indicating a bullish trend and yellow indicating a bearish trend.
The table presented alongside the indicator shows the top cycles with their corresponding rank, period, Bartels, amplitude or cycle strength, and phase. The amplitude is a measure of the strength of the cycle, while the phase is the position of the cycle within the data series.
Interpreting the Goertzel Browser indicator involves identifying the trend of the past and future composite wave lines and matching them with the corresponding bullish or bearish color. Additionally, traders can identify the top cycles with the highest amplitude or cycle strength and utilize them in conjunction with other technical indicators and fundamental analysis for trading decisions.
This indicator is considered a repainting indicator because the value of the indicator is calculated based on the past price data. As new price data becomes available, the indicator's value is recalculated, potentially causing the indicator's past values to change. This can create a false impression of the indicator's performance, as it may appear to have provided a profitable trading signal in the past when, in fact, that signal did not exist at the time.
The Goertzel indicator is also non-endpointed, meaning that it is not calculated up to the current bar or candle. Instead, it uses a fixed amount of historical data to calculate its values, which can make it difficult to use for real-time trading decisions. For example, if the indicator uses 100 bars of historical data to make its calculations, it cannot provide a signal until the current bar has closed and become part of the historical data. This can result in missed trading opportunities or delayed signals.
█ Conclusion
The Goertzel Browser indicator is a powerful tool for identifying and analyzing cyclical patterns in financial markets. Its ability to detect multiple cycles of varying frequencies and strengths make it a valuable addition to any trader's technical analysis toolkit. However, it is important to keep in mind that the Goertzel Browser indicator should be used in conjunction with other technical analysis tools and fundamental analysis to achieve the best results. With continued refinement and development, the Goertzel Browser indicator has the potential to become a highly effective tool for financial modeling, general trading, advanced trading, and high-frequency finance trading. Its accuracy and versatility make it a promising candidate for further research and development.
█ Footnotes
What is the Bartels Test for Cycle Significance?
The Bartels Cycle Significance Test is a statistical method that determines whether the peaks and troughs of a time series are statistically significant. The test is named after its inventor, George Bartels, who developed it in the mid-20th century.
The Bartels test is designed to analyze the cyclical components of a time series, which can help traders and analysts identify trends and cycles in financial markets. The test calculates a Bartels statistic, which measures the degree of non-randomness or autocorrelation in the time series.
The Bartels statistic is calculated by first splitting the time series into two halves and calculating the range of the peaks and troughs in each half. The test then compares these ranges using a t-test, which measures the significance of the difference between the two ranges.
If the Bartels statistic is greater than a critical value, it indicates that the peaks and troughs in the time series are non-random and that there is a significant cyclical component to the data. Conversely, if the Bartels statistic is less than the critical value, it suggests that the peaks and troughs are random and that there is no significant cyclical component.
The Bartels Cycle Significance Test is particularly useful in financial analysis because it can help traders and analysts identify significant cycles in asset prices, which can in turn inform investment decisions. However, it is important to note that the test is not perfect and can produce false signals in certain situations, particularly in noisy or volatile markets. Therefore, it is always recommended to use the test in conjunction with other technical and fundamental indicators to confirm trends and cycles.
Deep-dive into the Hodrick-Prescott Fitler
The Hodrick-Prescott (HP) filter is a statistical tool used in economics and finance to separate a time series into two components: a trend component and a cyclical component. It is a powerful tool for identifying long-term trends in economic and financial data and is widely used by economists, central banks, and financial institutions around the world.
The HP filter was first introduced in the 1990s by economists Robert Hodrick and Edward Prescott. It is a simple, two-parameter filter that separates a time series into a trend component and a cyclical component. The trend component represents the long-term behavior of the data, while the cyclical component captures the shorter-term fluctuations around the trend.
The HP filter works by minimizing the following objective function:
Minimize: (Sum of Squared Deviations) + λ (Sum of Squared Second Differences)
Where:
The first term represents the deviation of the data from the trend.
The second term represents the smoothness of the trend.
λ is a smoothing parameter that determines the degree of smoothness of the trend.
The smoothing parameter λ is typically set to a value between 100 and 1600, depending on the frequency of the data. Higher values of λ lead to a smoother trend, while lower values lead to a more volatile trend.
The HP filter has several advantages over other smoothing techniques. It is a non-parametric method, meaning that it does not make any assumptions about the underlying distribution of the data. It also allows for easy comparison of trends across different time series and can be used with data of any frequency.
However, the HP filter also has some limitations. It assumes that the trend is a smooth function, which may not be the case in some situations. It can also be sensitive to changes in the smoothing parameter λ, which may result in different trends for the same data. Additionally, the filter may produce unrealistic trends for very short time series.
Despite these limitations, the HP filter remains a valuable tool for analyzing economic and financial data. It is widely used by central banks and financial institutions to monitor long-term trends in the economy, and it can be used to identify turning points in the business cycle. The filter can also be used to analyze asset prices, exchange rates, and other financial variables.
The Hodrick-Prescott filter is a powerful tool for analyzing economic and financial data. It separates a time series into a trend component and a cyclical component, allowing for easy identification of long-term trends and turning points in the business cycle. While it has some limitations, it remains a valuable tool for economists, central banks, and financial institutions around the world.
Fourier Spectrometer of Price w/ Extrapolation Forecast [Loxx]Fourier Spectrometer of Price w/ Extrapolation Forecast is a forecasting indicator that forecasts the sinusoidal frequency of input price. This method uses Linear Regression with a Fast Fourier Transform function for the forecast and is different from previous forecasting methods I've posted. Dotted lines are the forecast frequencies. You can change the UI colors and line widths. This comes with 8 frequencies out of the box. Instead of drawing sinusoidal manually on your charts, you can use this instead. This will render better results than eyeballing the Sine Wave that folks use for trading. this is the real math that automates that process.
Each signal line can be shown as a linear superposition of periodic (sinusoidal) components with different periods (frequencies) and amplitudes. Roughly, the indicator shows those components. It strongly depends on the probing window and changes (recalculates) after each tick; e.g., you can see the set of frequencies showing whether the signal is fast or slow-changing, etc. Sometimes only a small number of leading / strongest components (e.g., 3) can extrapolate the signal quite well.
Related Indicators
Fourier Extrapolator of 'Caterpillar' SSA of Price
Real-Fast Fourier Transform of Price w/ Linear Regression
Fourier Extrapolator of Price w/ Projection Forecast
Itakura-Saito Autoregressive Extrapolation of Price
Helme-Nikias Weighted Burg AR-SE Extra. of Price
***The period parameter doesn't correspond to how many bars back the drawing begins. Lines re rendered according to skipping mechanism due to TradingView limitations.
Fourier Extrapolator of 'Caterpillar' SSA of Price [Loxx]Fourier Extrapolator of 'Caterpillar' SSA of Price is a forecasting indicator that applies Singular Spectrum Analysis to input price and then injects that transformed value into the Quinn-Fernandes Fourier Transform algorithm to generate a price forecast. The indicator plots two curves: the green/red curve indicates modeled past values and the yellow/fuchsia dotted curve indicates the future extrapolated values.
What is the Fourier Transform Extrapolator of price?
Fourier Extrapolator of Price is a multi-harmonic (or multi-tone) trigonometric model of a price series xi, i=1..n, is given by:
xi = m + Sum( a*Cos(w*i) + b*Sin(w*i), h=1..H )
Where:
xi - past price at i-th bar, total n past prices;
m - bias;
a and b - scaling coefficients of harmonics;
w - frequency of a harmonic ;
h - harmonic number;
H - total number of fitted harmonics.
Fitting this model means finding m, a, b, and w that make the modeled values to be close to real values. Finding the harmonic frequencies w is the most difficult part of fitting a trigonometric model. In the case of a Fourier series, these frequencies are set at 2*pi*h/n. But, the Fourier series extrapolation means simply repeating the n past prices into the future.
Quinn-Fernandes algorithm find sthe harmonic frequencies. It fits harmonics of the trigonometric series one by one until the specified total number of harmonics H is reached. After fitting a new harmonic , the coded algorithm computes the residue between the updated model and the real values and fits a new harmonic to the residue.
see here: A Fast Efficient Technique for the Estimation of Frequency , B. G. Quinn and J. M. Fernandes, Biometrika, Vol. 78, No. 3 (Sep., 1991), pp . 489-497 (9 pages) Published By: Oxford University Press
Fourier Transform Extrapolator of Price inputs are as follows:
npast - number of past bars, to which trigonometric series is fitted;
nharm - total number of harmonics in model;
frqtol - tolerance of frequency calculations.
What is Singular Spectrum Analysis ( SSA )?
Singular spectrum analysis ( SSA ) is a technique of time series analysis and forecasting. It combines elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. SSA aims at decomposing the original series into a sum of a small number of interpretable components such as a slowly varying trend, oscillatory components and a ‘structureless’ noise. It is based on the singular value decomposition ( SVD ) of a specific matrix constructed upon the time series. Neither a parametric model nor stationarity-type conditions have to be assumed for the time series. This makes SSA a model-free method and hence enables SSA to have a very wide range of applicability.
For our purposes here, we are only concerned with the "Caterpillar" SSA . This methodology was developed in the former Soviet Union independently (the ‘iron curtain effect’) of the mainstream SSA . The main difference between the main-stream SSA and the "Caterpillar" SSA is not in the algorithmic details but rather in the assumptions and in the emphasis in the study of SSA properties. To apply the mainstream SSA , one often needs to assume some kind of stationarity of the time series and think in terms of the "signal plus noise" model (where the noise is often assumed to be ‘red’). In the "Caterpillar" SSA , the main methodological stress is on separability (of one component of the series from another one) and neither the assumption of stationarity nor the model in the form "signal plus noise" are required.
"Caterpillar" SSA
The basic "Caterpillar" SSA algorithm for analyzing one-dimensional time series consists of:
Transformation of the one-dimensional time series to the trajectory matrix by means of a delay procedure (this gives the name to the whole technique);
Singular Value Decomposition of the trajectory matrix;
Reconstruction of the original time series based on a number of selected eigenvectors.
This decomposition initializes forecasting procedures for both the original time series and its components. The method can be naturally extended to multidimensional time series and to image processing.
The method is a powerful and useful tool of time series analysis in meteorology, hydrology, geophysics, climatology and, according to our experience, in economics, biology, physics, medicine and other sciences; that is, where short and long, one-dimensional and multidimensional, stationary and non-stationary, almost deterministic and noisy time series are to be analyzed.
"Caterpillar" SSA inputs are as follows:
lag - How much lag to introduce into the SSA algorithm, the higher this number the slower the process and smoother the signal
ncomp - Number of Computations or cycles of of the SSA algorithm; the higher the slower
ssapernorm - SSA Period Normalization
numbars =- number of past bars, to which SSA is fitted
Included:
Bar coloring
Alerts
Signals
Loxx's Expanded Source Types
Related Fourier Transform Indicators
Real-Fast Fourier Transform of Price w/ Linear Regression
Fourier Extrapolator of Variety RSI w/ Bollinger Bands
Fourier Extrapolator of Price w/ Projection Forecast
Related Projection Forecast Indicators
Itakura-Saito Autoregressive Extrapolation of Price
Helme-Nikias Weighted Burg AR-SE Extra. of Price
Related SSA Indicators
End-pointed SSA of FDASMA
End-pointed SSA of Williams %R
Normalized, Variety, Fast Fourier Transform Explorer [Loxx]Normalized, Variety, Fast Fourier Transform Explorer demonstrates Real, Cosine, and Sine Fast Fourier Transform algorithms. This indicator can be used as a rule of thumb but shouldn't be used in trading.
What is the Discrete Fourier Transform?
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.
What is the Complex Fast Fourier Transform?
The complex Fast Fourier Transform algorithm transforms N real or complex numbers into another N complex numbers. The complex FFT transforms a real or complex signal x in the time domain into a complex two-sided spectrum X in the frequency domain. You must remember that zero frequency corresponds to n = 0, positive frequencies 0 < f < f_c correspond to values 1 ≤ n ≤ N/2 −1, while negative frequencies −fc < f < 0 correspond to N/2 +1 ≤ n ≤ N −1. The value n = N/2 corresponds to both f = f_c and f = −f_c. f_c is the critical or Nyquist frequency with f_c = 1/(2*T) or half the sampling frequency. The first harmonic X corresponds to the frequency 1/(N*T).
The complex FFT requires the list of values (resolution, or N) to be a power 2. If the input size if not a power of 2, then the input data will be padded with zeros to fit the size of the closest power of 2 upward.
What is Real-Fast Fourier Transform?
Has conditions similar to the complex Fast Fourier Transform value, except that the input data must be purely real. If the time series data has the basic type complex64, only the real parts of the complex numbers are used for the calculation. The imaginary parts are silently discarded.
What is the Real-Fast Fourier Transform?
In many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry
X(N-k)=X(k)
and efficient FFT algorithms have been designed for this situation (see e.g. Sorensen, 1987). One approach consists of taking an ordinary algorithm (e.g. Cooley–Tukey) and removing the redundant parts of the computation, saving roughly a factor of two in time and memory. Alternatively, it is possible to express an even-length real-input DFT as a complex DFT of half the length (whose real and imaginary parts are the even/odd elements of the original real data), followed by O(N) post-processing operations.
It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartley transform (DHT), but it was subsequently argued that a specialized real-input DFT algorithm (FFT) can typically be found that requires fewer operations than the corresponding DHT algorithm (FHT) for the same number of inputs. Bruun's algorithm (above) is another method that was initially proposed to take advantage of real inputs, but it has not proved popular.
There are further FFT specializations for the cases of real data that have even/odd symmetry, in which case one can gain another factor of roughly two in time and memory and the DFT becomes the discrete cosine/sine transform(s) (DCT/DST). Instead of directly modifying an FFT algorithm for these cases, DCTs/DSTs can also be computed via FFTs of real data combined with O(N) pre- and post-processing.
What is the Discrete Cosine Transform?
A discrete cosine transform ( DCT ) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT , first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images (such as JPEG and HEIF, where small high-frequency components can be discarded), digital video (such as MPEG and H.26x), digital audio (such as Dolby Digital, MP3 and AAC ), digital television (such as SDTV, HDTV and VOD ), digital radio (such as AAC+ and DAB+), and speech coding (such as AAC-LD, Siren and Opus). DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations.
The use of cosine rather than sine functions is critical for compression, since it turns out (as described below) that fewer cosine functions are needed to approximate a typical signal, whereas for differential equations the cosines express a particular choice of boundary conditions. In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. The DCTs are generally related to Fourier Series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier Series coefficients of only periodically extended sequences. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common.
The most common variant of discrete cosine transform is the type-II DCT , which is often called simply "the DCT". This was the original DCT as first proposed by Ahmed. Its inverse, the type-III DCT , is correspondingly often called simply "the inverse DCT" or "the IDCT". Two related transforms are the discrete sine transform ( DST ), which is equivalent to a DFT of real and odd functions, and the modified discrete cosine transform (MDCT), which is based on a DCT of overlapping data. Multidimensional DCTs ( MD DCTs) are developed to extend the concept of DCT to MD signals. There are several algorithms to compute MD DCT . A variety of fast algorithms have been developed to reduce the computational complexity of implementing DCT . One of these is the integer DCT (IntDCT), an integer approximation of the standard DCT ,: ix, xiii, 1, 141–304 used in several ISO /IEC and ITU-T international standards.
What is the Discrete Sine Transform?
In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with odd symmetry (since the Fourier transform of a real and odd function is imaginary and odd), where in some variants the input and/or output data are shifted by half a sample.
A family of transforms composed of sine and sine hyperbolic functions exists. These transforms are made based on the natural vibration of thin square plates with different boundary conditions.
The DST is related to the discrete cosine transform (DCT), which is equivalent to a DFT of real and even functions. See the DCT article for a general discussion of how the boundary conditions relate the various DCT and DST types. Generally, the DST is derived from the DCT by replacing the Neumann condition at x=0 with a Dirichlet condition. Both the DCT and the DST were described by Nasir Ahmed T. Natarajan and K.R. Rao in 1974. The type-I DST (DST-I) was later described by Anil K. Jain in 1976, and the type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978.
Notable settings
windowper = period for calculation, restricted to powers of 2: "16", "32", "64", "128", "256", "512", "1024", "2048", this reason for this is FFT is an algorithm that computes DFT (Discrete Fourier Transform) in a fast way, generally in 𝑂(𝑁⋅log2(𝑁)) instead of 𝑂(𝑁2). To achieve this the input matrix has to be a power of 2 but many FFT algorithm can handle any size of input since the matrix can be zero-padded. For our purposes here, we stick to powers of 2 to keep this fast and neat. read more about this here: Cooley–Tukey FFT algorithm
SS = smoothing count, this smoothing happens after the first FCT regular pass. this zeros out frequencies from the previously calculated values above SS count. the lower this number, the smoother the output, it works opposite from other smoothing periods
Fmin1 = zeroes out frequencies not passing this test for min value
Fmax1 = zeroes out frequencies not passing this test for max value
barsback = moves the window backward
Inverse = whether or not you wish to invert the FFT after first pass calculation
Related indicators
Real-Fast Fourier Transform of Price Oscillator
STD-Stepped Fast Cosine Transform Moving Average
Real-Fast Fourier Transform of Price w/ Linear Regression
Variety RSI of Fast Discrete Cosine Transform
Additional reading
A Fast Computational Algorithm for the Discrete Cosine Transform by Chen et al.
Practical Fast 1-D DCT Algorithms With 11 Multiplications by Loeffler et al.
Cooley–Tukey FFT algorithm
Ahmed, Nasir (January 1991). "How I Came Up With the Discrete Cosine Transform". Digital Signal Processing. 1 (1): 4–5. doi:10.1016/1051-2004(91)90086-Z.
DCT-History - How I Came Up With The Discrete Cosine Transform
Comparative Analysis for Discrete Sine Transform as a suitable method for noise estimation
STD-Stepped Fast Cosine Transform Moving Average [Loxx]STD-Stepped Fast Cosine Transform Moving Average is an experimental moving average that uses Fast Cosine Transform to calculate a moving average. This indicator has standard deviation stepping in order to smooth the trend by weeding out low volatility movements.
What is the Discrete Cosine Transform?
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images (such as JPEG and HEIF, where small high-frequency components can be discarded), digital video (such as MPEG and H.26x), digital audio (such as Dolby Digital, MP3 and AAC), digital television (such as SDTV, HDTV and VOD), digital radio (such as AAC+ and DAB+), and speech coding (such as AAC-LD, Siren and Opus). DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations.
The use of cosine rather than sine functions is critical for compression, since it turns out (as described below) that fewer cosine functions are needed to approximate a typical signal, whereas for differential equations the cosines express a particular choice of boundary conditions. In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. The DCTs are generally related to Fourier Series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier Series coefficients of only periodically extended sequences. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common.
The most common variant of discrete cosine transform is the type-II DCT, which is often called simply "the DCT". This was the original DCT as first proposed by Ahmed. Its inverse, the type-III DCT, is correspondingly often called simply "the inverse DCT" or "the IDCT". Two related transforms are the discrete sine transform (DST), which is equivalent to a DFT of real and odd functions, and the modified discrete cosine transform (MDCT), which is based on a DCT of overlapping data. Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT to MD signals. There are several algorithms to compute MD DCT. A variety of fast algorithms have been developed to reduce the computational complexity of implementing DCT. One of these is the integer DCT (IntDCT), an integer approximation of the standard DCT, : ix, xiii, 1, 141–304 used in several ISO/IEC and ITU-T international standards.
Notable settings
windowper = period for calculation, restricted to powers of 2: "16", "32", "64", "128", "256", "512", "1024", "2048", this reason for this is FFT is an algorithm that computes DFT (Discrete Fourier Transform) in a fast way, generally in 𝑂(𝑁⋅log2(𝑁)) instead of 𝑂(𝑁2). To achieve this the input matrix has to be a power of 2 but many FFT algorithm can handle any size of input since the matrix can be zero-padded. For our purposes here, we stick to powers of 2 to keep this fast and neat. read more about this here: Cooley–Tukey FFT algorithm
smthper = smoothing count, this smoothing happens after the first FCT regular pass. this zeros out frequencies from the previously calculated values above SS count. the lower this number, the smoother the output, it works opposite from other smoothing periods
Included
Alerts
Signals
Loxx's Expanded Source Types
Additional reading
A Fast Computational Algorithm for the Discrete Cosine Transform by Chen et al.
Practical Fast 1-D DCT Algorithms With 11 Multiplications by Loeffler et al.
Cooley–Tukey FFT algorithm
Real-Fast Fourier Transform of Price Oscillator [Loxx]Real-Fast Fourier Transform Oscillator is a simple Real-Fast Fourier Transform Oscillator. You have the option to turn on inverse filter as well as min/max filters to fine tune the oscillator. This oscillator is normalized by default. This indicator is to demonstrate how one can easily turn the RFFT algorithm into an oscillator..
What is the Discrete Fourier Transform?
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.
What is the Complex Fast Fourier Transform?
The complex Fast Fourier Transform algorithm transforms N real or complex numbers into another N complex numbers. The complex FFT transforms a real or complex signal x in the time domain into a complex two-sided spectrum X in the frequency domain. You must remember that zero frequency corresponds to n = 0, positive frequencies 0 < f < f_c correspond to values 1 ≤ n ≤ N/2 −1, while negative frequencies −fc < f < 0 correspond to N/2 +1 ≤ n ≤ N −1. The value n = N/2 corresponds to both f = f_c and f = −f_c. f_c is the critical or Nyquist frequency with f_c = 1/(2*T) or half the sampling frequency. The first harmonic X corresponds to the frequency 1/(N*T).
The complex FFT requires the list of values (resolution, or N) to be a power 2. If the input size if not a power of 2, then the input data will be padded with zeros to fit the size of the closest power of 2 upward.
What is Real-Fast Fourier Transform?
Has conditions similar to the complex Fast Fourier Transform value, except that the input data must be purely real. If the time series data has the basic type complex64, only the real parts of the complex numbers are used for the calculation. The imaginary parts are silently discarded.
Included
Moving window from Last Bar setting. You can lock the oscillator in place on the current bar by adding 1 every time a new bar appears in the Last Bar Setting
Real-Fast Fourier Transform of Price w/ Linear Regression [Loxx]Real-Fast Fourier Transform of Price w/ Linear Regression is a indicator that implements a Real-Fast Fourier Transform on Price and modifies the output by a measure of Linear Regression. The solid line is the Linear Regression Trend of the windowed data, The green/red line is the Real FFT of price.
What is the Discrete Fourier Transform?
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.
What is the Complex Fast Fourier Transform?
The complex Fast Fourier Transform algorithm transforms N real or complex numbers into another N complex numbers. The complex FFT transforms a real or complex signal x in the time domain into a complex two-sided spectrum X in the frequency domain. You must remember that zero frequency corresponds to n = 0, positive frequencies 0 < f < f_c correspond to values 1 ≤ n ≤ N/2 −1, while negative frequencies −fc < f < 0 correspond to N/2 +1 ≤ n ≤ N −1. The value n = N/2 corresponds to both f = f_c and f = −f_c. f_c is the critical or Nyquist frequency with f_c = 1/(2*T) or half the sampling frequency. The first harmonic X corresponds to the frequency 1/(N*T).
The complex FFT requires the list of values (resolution, or N) to be a power 2. If the input size if not a power of 2, then the input data will be padded with zeros to fit the size of the closest power of 2 upward.
What is Real-Fast Fourier Transform?
Has conditions similar to the complex Fast Fourier Transform value, except that the input data must be purely real. If the time series data has the basic type complex64, only the real parts of the complex numbers are used for the calculation. The imaginary parts are silently discarded.
Inputs:
src = source price
uselreg = whether you wish to modify output with linear regression calculation
Windowin = windowing period, restricted to powers of 2: "4", "8", "16", "32", "64", "128", "256", "512", "1024", "2048"
Treshold = to modified power output to fine tune signal
dtrendper = adjust regression calculation
barsback = move window backward from bar 0
mutebars = mute bar coloring for the range
Further reading:
Real-valued Fast Fourier Transform Algorithms IEEE Transactions on Acoustics, Speech, and Signal Processing, June 1987
Related indicators utilizing Fourier Transform
Fourier Extrapolator of Variety RSI w/ Bollinger Bands
Fourier Extrapolation of Variety Moving Averages
Fourier Extrapolator of Price w/ Projection Forecast
Variety RSI of Fast Discrete Cosine Transform [Loxx]Variety RSI of Fast Discrete Cosine Transform is an RSI indicator with 7 types of RSI that is calculated on the Fast Discrete Cosine Transform of source. The source inputs are 33 different source types from Loxx's Expanded Source Types.
What is Discrete Cosine Transform?
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images (such as JPEG and HEIF, where small high-frequency components can be discarded), digital video (such as MPEG and H.26x), digital audio (such as Dolby Digital, MP3 and AAC), digital television (such as SDTV, HDTV and VOD), digital radio (such as AAC+ and DAB+), and speech coding (such as AAC-LD, Siren and Opus). DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations.
Fast Discrete Cosine Transform
The algorithm performs a fast cosine transform of the real function defined by nn samples on the real axis.
Depending on the passed parameters, it can be executed both direct and inverse conversion.
Input parameters:
tnn - Number of function values minus one. Should be 1024 degree of two. The algorithm does not check correct value passed.
a - array of Real 1025 Function values.
InverseFCT - the direction of the transformation. True if reverse, False if direct.
Output parameters: a - the result of the transformation. For more details, see description on the site. www.alglib.net
Included:
7 types of RSI
33 source inputs from Loxx's Expanded Source Types
2 types of signals
Alerts
Fourier Extrapolation of Variety Moving Averages [Loxx]Fourier Extrapolation of Variety Moving Averages is a Fourier Extrapolation (forecasting) indicator that has for inputs 38 different types of moving averages along with 33 different types of sources for those moving averages. This is a forecasting indicator of the selected moving average of the selected price of the underlying ticker. This indicator will repaint, so past signals are only as valid as the current bar. This indicator allows for up to 1500 bars between past bars and future projection bars. If the indicator won't load on your chart. check the error message for details on how to fix that, but you must ensure that past bars + futures bars is equal to or less than 1500.
Fourier Extrapolation using the Quinn-Fernandes algorithm is one of several (5-10) methods of signals forecasting that I'l be demonstrating in Pine Script.
What is Fourier Extrapolation?
This indicator uses a multi-harmonic (or multi-tone) trigonometric model of a price series xi, i=1..n, is given by:
xi = m + Sum( a*Cos(w*i) + b*Sin(w*i), h=1..H )
Where:
xi - past price at i-th bar, total n past prices;
m - bias;
a and b - scaling coefficients of harmonics;
w - frequency of a harmonic ;
h - harmonic number;
H - total number of fitted harmonics.
Fitting this model means finding m, a, b, and w that make the modeled values to be close to real values. Finding the harmonic frequencies w is the most difficult part of fitting a trigonometric model. In the case of a Fourier series, these frequencies are set at 2*pi*h/n. But, the Fourier series extrapolation means simply repeating the n past prices into the future.
This indicator uses the Quinn-Fernandes algorithm to find the harmonic frequencies. It fits harmonics of the trigonometric series one by one until the specified total number of harmonics H is reached. After fitting a new harmonic , the coded algorithm computes the residue between the updated model and the real values and fits a new harmonic to the residue.
see here: A Fast Efficient Technique for the Estimation of Frequency , B. G. Quinn and J. M. Fernandes, Biometrika, Vol. 78, No. 3 (Sep., 1991), pp . 489-497 (9 pages) Published By: Oxford University Press
The indicator has the following input parameters:
src - input source
npast - number of past bars, to which trigonometric series is fitted;
Nfut - number of predicted future bars;
nharm - total number of harmonics in model;
frqtol - tolerance of frequency calculations.
Included:
Loxx's Expanded Source Types
Loxx's Moving Averages
Other indicators using this same method
Fourier Extrapolator of Variety RSI w/ Bollinger Bands
Fourier Extrapolator of Price w/ Projection Forecast
Fourier Extrapolator of Price
Loxx's Moving Averages: Detailed explanation of moving averages inside this indicator
Loxx's Expanded Source Types: Detailed explanation of source types used in this indicator
Fourier Extrapolator of Variety RSI w/ Bollinger Bands [Loxx]Fourier Extrapolator of Variety RSI w/ Bollinger Bands is an RSI indicator that shows the original RSI, the Fourier Extrapolation of RSI in the past, and then the projection of the Fourier Extrapolated RSI for the future. This indicator has 8 different types of RSI including a new type of RSI called T3 RSI. The purpose of this indicator is to demonstrate the Fourier Extrapolation method used to model past data and to predict future price movements. This indicator will repaint. If you wish to use this for trading, then make sure to take a screenshot of the indicator when you enter the trade to save your analysis. This is the first of a series of forecasting indicators that can be used in trading. Due to how this indicator draws on the screen, you must choose values of npast and nfut that are equal to or less than 200. this is due to restrictions by TradingView and Pine Script in only allowing 500 lines on the screen at a time. Enjoy!
What is Fourier Extrapolation?
This indicator uses a multi-harmonic (or multi-tone) trigonometric model of a price series xi, i=1..n, is given by:
xi = m + Sum( a*Cos(w*i) + b*Sin(w*i), h=1..H )
Where:
xi - past price at i-th bar, total n past prices;
m - bias;
a and b - scaling coefficients of harmonics;
w - frequency of a harmonic ;
h - harmonic number;
H - total number of fitted harmonics.
Fitting this model means finding m, a, b, and w that make the modeled values to be close to real values. Finding the harmonic frequencies w is the most difficult part of fitting a trigonometric model. In the case of a Fourier series, these frequencies are set at 2*pi*h/n. But, the Fourier series extrapolation means simply repeating the n past prices into the future.
This indicator uses the Quinn-Fernandes algorithm to find the harmonic frequencies. It fits harmonics of the trigonometric series one by one until the specified total number of harmonics H is reached. After fitting a new harmonic , the coded algorithm computes the residue between the updated model and the real values and fits a new harmonic to the residue.
see here: A Fast Efficient Technique for the Estimation of Frequency , B. G. Quinn and J. M. Fernandes, Biometrika, Vol. 78, No. 3 (Sep., 1991), pp . 489-497 (9 pages) Published By: Oxford University Press
The indicator has the following input parameters:
src - input source
npast - number of past bars, to which trigonometric series is fitted;
Nfut - number of predicted future bars;
nharm - total number of harmonics in model;
frqtol - tolerance of frequency calculations.
Included:
Loxx's Expanded Source Types
Loxx's Variety RSI
Other indicators using this same method
Fourier Extrapolator of Price w/ Projection Forecast
Fourier Extrapolator of Price
Fourier Extrapolator of Price w/ Projection Forecast [Loxx]Due to popular demand, I'm pusblishing Fourier Extrapolator of Price w/ Projection Forecast.. As stated in it's twin indicator, this one is also multi-harmonic (or multi-tone) trigonometric model of a price series xi, i=1..n, is given by:
xi = m + Sum( a*Cos(w*i) + b*Sin(w*i), h=1..H )
Where:
xi - past price at i-th bar, total n past prices;
m - bias;
a and b - scaling coefficients of harmonics;
w - frequency of a harmonic ;
h - harmonic number;
H - total number of fitted harmonics.
Fitting this model means finding m, a, b, and w that make the modeled values to be close to real values. Finding the harmonic frequencies w is the most difficult part of fitting a trigonometric model. In the case of a Fourier series, these frequencies are set at 2*pi*h/n. But, the Fourier series extrapolation means simply repeating the n past prices into the future.
This indicator uses the Quinn-Fernandes algorithm to find the harmonic frequencies. It fits harmonics of the trigonometric series one by one until the specified total number of harmonics H is reached. After fitting a new harmonic , the coded algorithm computes the residue between the updated model and the real values and fits a new harmonic to the residue.
see here: A Fast Efficient Technique for the Estimation of Frequency , B. G. Quinn and J. M. Fernandes, Biometrika, Vol. 78, No. 3 (Sep., 1991), pp . 489-497 (9 pages) Published By: Oxford University Press
The indicator has the following input parameters:
src - input source
npast - number of past bars, to which trigonometric series is fitted;
Nfut - number of predicted future bars;
nharm - total number of harmonics in model;
frqtol - tolerance of frequency calculations.
The indicator plots two curves: the green/red curve indicates modeled past values and the yellow/fuchsia curve indicates the modeled future values.
The purpose of this indicator is to showcase the Fourier Extrapolator method to be used in future indicators.
Fourier Extrapolator of Price [Loxx]Fourier Extrapolator of Price is a multi-harmonic (or multi-tone) trigonometric model of a price series xi, i=1..n, is given by:
xi = m + Sum( a *Cos(w *i) + b *Sin(w *i), h=1..H )
Where:
xi - past price at i-th bar, total n past prices;
m - bias;
a and b - scaling coefficients of harmonics;
w - frequency of a harmonic;
h - harmonic number;
H - total number of fitted harmonics.
Fitting this model means finding m, a , b , and w that make the modeled values to be close to real values. Finding the harmonic frequencies w is the most difficult part of fitting a trigonometric model. In the case of a Fourier series, these frequencies are set at 2*pi*h/n. But, the Fourier series extrapolation means simply repeating the n past prices into the future.
This indicator uses the Quinn-Fernandes algorithm to find the harmonic frequencies. It fits harmonics of the trigonometric series one by one until the specified total number of harmonics H is reached. After fitting a new harmonic, the coded algorithm computes the residue between the updated model and the real values and fits a new harmonic to the residue.
see here: A Fast Efficient Technique for the Estimation of Frequency , B. G. Quinn and J. M. Fernandes, Biometrika, Vol. 78, No. 3 (Sep., 1991), pp. 489-497 (9 pages) Published By: Oxford University Press
The indicator has the following input parameters:
src - input source
npast - number of past bars, to which trigonometric series is fitted;
nharm - total number of harmonics in model;
frqtol - tolerance of frequency calculations.
The indicator plots the modeled past values
The purpose of this indicator is to showcase the Fourier Extrapolator method to be used in future indicators. While this method can also prediction future price movements, for our purpose here we will avoid doing.
Fast Fourier Transform (FFT) FilterDear friends!
I'm happy to present an implementation of the Fast Fourier Transform (FFT) algorithm. The script uses the FFT procedure to decompose the input time series into its cyclical constituents, in other words, its frequency components , and convert it back to the time domain with modified frequency content, that is, to filter it.
Input Description and Usage
Source and Length :
Indicates where the data comes from and the size of the lookback window used to build the dataset.
Standardize Input Dataset :
If enabled, the dataset is preprocessed by subtracting its mean and normalizing the result by the standard deviation, which is sometimes useful when analyzing seasonalities. This procedure is not recommended when using the FFT filter for smoothing (see below), as it will not preserve the average of the dataset.
Show Frequency-Domain Power Spectrum :
When enabled, the results of Fourier analysis (for the last price bar!) are plotted as a frequency-domain power spectrum , where “power” is a measure of the significance of the component in the dataset. In the spectrum, lower frequencies (longer cycles) are on the right, higher frequencies are on the left. The graph does not display the 0th component, which contains only information about the mean value. Frequency components that are allowed to pass through the filter (see below) are highlighted in magenta .
Dominant Cycles, Rows :
If this option is activated, the periods and relative powers of several dominant cyclical components that is, those that have a higher power, are listed in the table. The number of the component in the power spectrum (N) is shown in the first column. The number of rows in the table is defined by the user.
Show Inverse Fourier Transform (Filtered) :
When enabled, the reconstructed and filtered time-domain dataset (for the last price bar!) is displayed.
Apply FFT Filter in a Moving Window :
When enabled, the FFT filter with the same parameters is applied to each bar. The last data point of the reconstructed and filtered dataset is used to build a new time series. For example, by getting rid of high-frequency noise, the FFT filter can make the data smoother. By removing slowly evolving low-frequency components (including non-periodic constituents), one can reveal and analyze shorter cycles. Since filtering is done in real-time in a moving window (similar to the moving average), the modified data can potentially be used as part of a strategy and be subjected to other technical indicators.
Lowest Allowed N :
Indicates the number of the lowest frequency component used in the reconstructed time series.
Highest Allowed N :
Indicates the number of the highest frequency component used in the reconstructed time series.
Filtering Time Range block:
Specifies the time range over which real-time FFT filtering is applied. The reason for the presence of this block is that the FFT procedure is relatively computationally intensive. Therefore, the script execution may encounter the time limit imposed by TradingView when all historical bars are processed.
As always, I look forward to your feedback!
Also, leave a comment if you'd be interested in the tutorial on how to use this tool and/or in seeing the FFT filter in a strategy.
Fourier Analysis and Filtering [tbiktag]This tool uses Fourier transform to decompose the input time series into its periodic constituents and seasonalities , in other words, its frequency components . It also can reconstruct the time-domain data while using only the frequency components within a user-defined range (band-pass filtering). Thereby, this tool can reveal the cyclical characteristics of the studied market.
USAGE
The source and the size of the input data can be chosen by the “ Dataset Source ” and “ Dataset Size ” options. Price, volume, or some technical indicator (e.g., RSI, MACD, etc.) can serve as a source of the input data.
“ Action ” defines the type of the plot that will be displayed. Two options are available:
- Fourier Analysis
If selected, the frequency spectrum of the squares of the Fourier coefficient magnitudes is displayed. The zero-frequency component is on the right. Since the magnitudes of half of the coefficients are repeated, the graph displays only half of the frequency components.
The squared magnitude of a given frequency component is a measure of its power , that is, its contribution to the total variance of the dataset. Thus, by analyzing the frequency-domain spectrum, one can identify the most prominent seasonalities and then visualize them by using the " Band-pass Filtering " option (see below). Note that the zero component stores information about the amount of data, so it is naturally higher when the data is not centered at zero.
By activating the " Info about Frequency Component " option, the user can display information about the power and frequency of the selected Fourier component.
- Band-pass Filter
This option reconstructs and plots the dataset in the time domain, blocking frequency components outside of the cutoff frequencies (defined by the input parameters “ Upper Cutoff ” and “ Lower Cutoff ” input parameters in the “ Band-pass Filter Properties ” section).
FURTHER READING
In general, Fourier analysis has a long history of attempted applications for analyzing price data and estimating market cycles. For example, see the paper by John Ehlers
www.mesasoftware.com
and also some tools available here on TradingView, such as:
“Function: Discrete Fourier Transform” by @RicardoSantos
“Fourier series Model Of The Market” by @e2e4mfck
“Ehlers Discrete Fourier Transform” by @cheatcountry
Thus, I tried to make this tool versatile and user-friendly so you all can experiment with your own analysis.
Enjoy and don't hesitate to leave your feedback in the comments below!
