WLSMA: fast approximation🙏🏻 Sup TV & @alexgrover
O(N) algocomplexity, just one loop inside. No, you can't do O(1) @ updates in moving window mode, only expanding window will allow that.
Now I have time series & stats models of my own creation, nowhere else available, just TV and my github for now, ain’t no legacy academic industry I always have fun about, but back in 2k20 when I consciously ain’t known much about quant, I remember seeing post by @alexgrover recreating Moving Regression Endpoint dropped on price chart (called LSMA here) as a linear filter combination of filters (yea yeah DSP terms) as 3WMA - 2SMA. Now it’s my time to do smth alike aye?
...
This script is remake of my 1st degree WLSMA via linear filter combo. It’s much faster, we aint calculate moving regression per se, we just match its freq response. You can see it on the screen (WLSMAfa) almost perfectly matching the original one (WLSMA).
...
While humans like to overfit, I fw generalizations. So your lovely WMA is actually just one case of a more general weight pattern: pow(len - i, e), where pow is the power function and e is the exponent itself. So:
- If e = 0, then we have SMA (every number in 0th power is one)
- If e = 1, we get WMA
- If e = 2, we get quadratic weights.
We can recreate WLSMA freq response then by combining 2 filters with e = 1 and e = 2.
This is still an approximation, even tho enormously precise for the tasks you’ve shared with me. Due to the non-linear nature of the thing it’s all we can do, and as window size grows, even this small discrepancy converges with true WLSMA value, so we’re all good. Pls don’t try to model this 0.00xxxx discrepancy, it’s not natural.
...
DSP approach is unnatural for prices, but you can put this thing on volume delta and be happy, or on other metrics of yours, if for some reason u dont wanna estimate thresholds by fitting a distro.
All good TV
∞
P.S.: strangely, the first script made & dropped in the location in Saint P where my actual quant way has started ~5 years ago xD, very thankful
Dsp
[GYTS] Filters ToolkitFilters Toolkit indicator
🌸 Part of GoemonYae Trading System (GYTS) 🌸
🌸 --------- 1. INTRODUCTION --------- 🌸
💮 Overview
The GYTS Filters Toolkit indicator is an advanced, interactive interface built atop the high‐performance, curated functions provided by the FiltersToolkit library . It allows traders to experiment with different combinations of filtering methods -— from smoothing low-pass filters to aggressive detrenders. With this toolkit, you can build custom indicators tailored to your specific trading strategy, whether you're looking for trend following, mean reversion, or cycle identification approaches.
🌸 --------- 2. FILTER METHODS AND TYPES --------- 🌸
💮 Filter categories
The available filters fall into four main categories, each marked with a distinct symbol:
🌗 Low Pass Filters (Smoothers)
These filters attenuate high-frequency components (noise) while allowing low-frequency components (trends) to pass through. Examples include:
Ultimate Smoother
Super Smoother (2-pole and 3-pole variants)
MESA Adaptive Moving Average (MAMA) and Following Adaptive Moving Average (FAMA)
BiQuad Low Pass Filter
ADXvma (Adaptive Directional Volatility Moving Average)
A2RMA (Adaptive Autonomous Recursive Moving Average)
Low pass filters are displayed on the price chart by default, as they follow the overall price movement. If they are combined with a high-pass or bandpass filter, they will be displayed in the subgraph.
🌓 High Pass Filters (Detrenders)
These filters do the opposite of low pass filters - they remove low-frequency components (trends) while allowing high-frequency components to pass through. Examples include:
Butterworth High Pass Filter
BiQuad High Pass Filter
High pass filters are displayed as oscillators in the subgraph below the price chart, as they fluctuate around a zero line.
🌑 Band Pass Filters (Cycle Isolators)
These filters combine aspects of both low and high pass filters, isolating specific frequency ranges while attenuating both higher and lower frequencies. Examples include:
Ehlers Bandpass Filter
Cyber Cycle
Relative Vigor Index (RVI)
BiQuad Bandpass Filter
Band pass filters are also displayed as oscillators in a separate panel.
🔮 Predictive Filter
Voss Predictive Filter: A special filter that attempts to predict future values of band-limited signals (only to be used as post-filter). Keep its prediction horizon short (1–3 bars) for reasonable accuracy.
Note that the the library contains elaborate documentation and source material of each filter.
🌸 --------- 3. INDICATOR FEATURES --------- 🌸
💮 Multi-filter configuration
One of the most powerful aspects of this indicator is the ability to configure multiple filters. compare them and observe their combined effects. There are four primary filters, each with its own parameter settings.
💮 Post-filtering
Process a filter’s output through an additional filter by enabling the post-filter option. This creates a filter chain where the output of one filter becomes the input to another. Some powerful combinations include:
Ultimate Smoother → MAMA: Creates an adaptive smoothing effect that responds well to market changes, good for trend-following strategies
Butterworth → Super Smoother → Butterworth: Produces a well-behaved oscillator with minimal phase distortion, John Ehlers also calls a "roofing filter". Great for identifying overbought/oversold conditions with minimal lag.
A bandpass filter → Voss Prediction filter: Attempts to predict future movements of cyclical components, handy to find peaks and troughs of the market cycle.
💮 Aggregate filters
Arguably the coolest feature: aggregating filters allow you to combine multiple filters with different weights. Important notes about aggregation:
You can only aggregate filters that appear on the same chart (price chart or oscillator panel).
The weights are automatically normalised, so only their relative values matter
Setting a weight to 0 (zero) excludes that filter from the aggregation
Filters don't need to be visibly displayed to be included in aggregation
💮 Rich visualisation & alerts
The indicator intelligently determines whether a filter is displayed on the price chart or in the subgraph (as an oscillator) based on its characteristics.
Dynamic colour palettes, adjustable line widths, transparency, and custom fill between any of enabled filters or between oscillators and the zero-line.
A clear legend showing which filters are active and how they're configured
Alerts for direction changes and crossovers of all filters
🌸 --------- 4. ACKNOWLEDGEMENTS --------- 🌸
This toolkit builds on the work of numerous pioneers in technical analysis and digital signal processing:
John Ehlers, whose groundbreaking research forms the foundation of many filters.
Robert Bristow-Johnson for the BiQuad filter formulations.
The TradingView community, especially @The_Peaceful_Lizard, @alexgrover, and others mentioned in the code of the library.
Everyone who has provided feedback, testing and support!
[GYTS] FiltersToolkit LibraryFiltersToolkit Library
🌸 Part of GoemonYae Trading System (GYTS) 🌸
🌸 --------- 1. INTRODUCTION --------- 🌸
💮 What Does This Library Contain?
This library is a curated collection of high-performance digital signal processing (DSP) filters and auxiliary functions designed specifically for financial time series analysis. It includes a shortlist of our favourite and best performing filters — each rigorously tested and selected for their responsiveness, minimal lag and robustness in diverse market conditions. These tools form an integral part of the GoemonYae Trading System (GYTS), chosen for their unique characteristics in handling market data.
The library contains two main categories:
1. Smoothing filters (low-pass filters and moving averages) for e.g. denoising, trend following
2. Detrending tools (high-pass and band-pass filters, known as "oscillators") for e.g. mean reversion
This collection is finely tuned for practical trading applications and is therefore not meant to be exhaustive. However, will continue to expand as we discover and validate new filtering techniques. I welcome collaboration and suggestions for novel approaches.
🌸 ——— 2. ADDED VALUE ——— 🌸
💮 Unified syntax and comprehensive documentation
The FiltersToolkit Library brings together a wide array of valuable filters under a unified, intuitive syntax. Each function is thoroughly documented, with clear explanations and academic sources that underline the mathematical rigour behind the methods. This level of documentation not only facilitates integration into trading strategies but also helps underlying the underlying concepts and rationale.
💮 Optimised performance and readability
The code prioritizes computational efficiency while maintaining readability. Key optimizations include:
- Minimizing redundant calculations in recursive filters
- Smart coefficient caching
- Efficient state management
- Vectorized operations where applicable
💮 Enhanced functionality and flexibility
Some filters in this library introduce extended functionality beyond the original publications. For instance, the MESA Adaptive Moving Average (MAMA) and Ehlers’ Combined Bandpass Filter incorporate multiple variations found in the literature, thereby providing traders with flexible tools that can be fine-tuned to different market conditions.
🌸 ——— 3. THE FILTERS ——— 🌸
💮 Hilbert Transform Function
This function implements the Hilbert Transform as utilised by John Ehlers. It converts a real-valued time series into its analytic signal, enabling the extraction of instantaneous phase and frequency information—an essential step in adaptive filtering.
Source: John Ehlers - "Rocket Science for Traders" (2001), "TASC 2001 V. 19:9", "Cybernetic Analysis for Stocks and Futures" (2004)
💮 Homodyne Discriminator
By leveraging the Hilbert Transform, this function computes the dominant cycle period through a Homodyne Discriminator. It extracts the in-phase and quadrature components of the signal, facilitating a robust estimation of the underlying cycle characteristics.
Source: John Ehlers - "Rocket Science for Traders" (2001), "TASC 2001 V. 19:9", "Cybernetic Analysis for Stocks and Futures" (2004)
💮 MESA Adaptive Moving Average (MAMA)
An advanced dual-stage adaptive moving average, this function outputs both the MAMA and its companion FAMA. It combines adaptive alpha computation with elements from Kaufman’s Adaptive Moving Average (KAMA) to provide a responsive and reliable trend indicator.
Source: John Ehlers - "Rocket Science for Traders" (2001), "TASC 2001 V. 19:9", "Cybernetic Analysis for Stocks and Futures" (2004)
💮 BiQuad Filters
A family of second-order recursive filters offering exceptional control over frequency response:
- High-pass filter for detrending
- Low-pass filter for smooth trend following
- Band-pass filter for cycle isolation
The quality factor (Q) parameter allows fine-tuning of the resonance characteristics, making these filters highly adaptable to different market conditions.
Source: Robert Bristow-Johnson's Audio EQ Cookbook, implemented by @The_Peaceful_Lizard
💮 Relative Vigor Index (RVI)
This filter evaluates the strength of a trend by comparing the closing price to the trading range. Operating similarly to a band-pass filter, the RVI provides insights into market momentum and potential reversals.
Source: John Ehlers – “Cybernetic Analysis for Stocks and Futures” (2004)
💮 Cyber Cycle
The Cyber Cycle filter emphasises market cycles by smoothing out noise and highlighting the dominant cyclical behaviour. It is particularly useful for detecting trend reversals and cyclical patterns in the price data.
Source: John Ehlers – “Cybernetic Analysis for Stocks and Futures” (2004)
💮 Butterworth High Pass Filter
Inspired by the classical Butterworth design, this filter achieves a maximally flat magnitude response in the passband while effectively removing low-frequency trends. Its design minimises phase distortion, which is vital for accurate signal interpretation.
Source: John Ehlers – “Cybernetic Analysis for Stocks and Futures” (2004)
💮 2-Pole SuperSmoother
Employing a two-pole design, the SuperSmoother filter reduces high-frequency noise with minimal lag. It is engineered to preserve trend integrity while offering a smooth output even in noisy market conditions.
Source: John Ehlers – “Cybernetic Analysis for Stocks and Futures” (2004)
💮 3-Pole SuperSmoother
An extension of the 2-pole design, the 3-pole SuperSmoother further attenuates high-frequency noise. Its additional pole delivers enhanced smoothing at the cost of slightly increased lag.
Source: John Ehlers – “Cybernetic Analysis for Stocks and Futures” (2004)
💮 Adaptive Directional Volatility Moving Average (ADXVma)
This adaptive moving average adjusts its smoothing factor based on directional volatility. By combining true range and directional movement measurements, it remains exceptionally flat during ranging markets and responsive during directional moves.
Source: Various implementations across platforms, unified and optimized
💮 Ehlers Combined Bandpass Filter with Automated Gain Control (AGC)
This sophisticated filter merges a highpass pre-processing stage with a bandpass filter. An integrated Automated Gain Control normalises the output to a consistent range, while offering both regular and truncated recursive formulations to manage lag.
Source: John F. Ehlers – “Truncated Indicators” (2020), “Cycle Analytics for Traders” (2013)
💮 Voss Predictive Filter
A forward-looking filter that predicts future values of a band-limited signal in real time. By utilising multiple time-delayed feedback terms, it provides anticipatory coupling and delivers a short-term predictive signal.
Source: John Ehlers - "A Peek Into The Future" (TASC 2019-08)
💮 Adaptive Autonomous Recursive Moving Average (A2RMA)
This filter dynamically adjusts its smoothing through an adaptive mechanism based on an efficiency ratio and a dynamic threshold. A double application of an adaptive moving average ensures both responsiveness and stability in volatile and ranging markets alike. Very flat response when properly tuned.
Source: @alexgrover (2019)
💮 Ultimate Smoother (2-Pole)
The Ultimate Smoother filter is engineered to achieve near-zero lag in its passband by subtracting a high-pass response from an all-pass response. This creates a filter that maintains signal fidelity at low frequencies while effectively filtering higher frequencies at the expense of slight overshooting.
Source: John Ehlers - TASC 2024-04 "The Ultimate Smoother"
Note: This library is actively maintained and enhanced. Suggestions for additional filters or improvements are welcome through the usual channels. The source code contains a list of tested filters that did not make it into the curated collection.
PaddingThe Padding library is a comprehensive and flexible toolkit designed to extend time series data within TradingView, making it an indispensable resource for advanced signal processing tasks such as FFT, filtering, convolution, and wavelet analysis. At its core, the library addresses the common challenge of edge effects by "padding" your data—that is, by appending additional data points beyond the natural boundaries of your original dataset. This extension not only mitigates the distortions that can occur at the endpoints but also helps to maintain the integrity of various transformations and calculations performed on the series. The library accomplishes this while preserving the ordering of your data, ensuring that the most recent point always resides at index 0.
Central to the functionality of this library are two key enumerations: Direction and PaddingType. The Direction enum determines where the padding will be applied. You can choose to extend the data in the forward direction (ahead of the current values), in the backward direction (behind the current values), or in both directions simultaneously. The PaddingType enum defines the specific method used for extending the data. The library supports several methods—including symmetric, reflect, periodic, antisymmetric, antireflect, smooth, constant, and zero padding—each of which has been implemented to suit different analytical scenarios. For instance, symmetric padding mirrors the original data across its boundaries, while reflect padding continues the trend by reflecting around endpoint values. Periodic padding repeats the data, and antisymmetric padding mirrors the data with alternating signs to counterbalance it. The antireflect and smooth methods take into account the derivatives of your data, thereby extending the series in a way that preserves or smoothly continues these derivative values. Constant and zero padding simply extend the series using fixed endpoint values or zeros. Together, these enums allow you to fine-tune how your data is extended, ensuring that the padding method aligns with the specific requirements of your analysis.
The library is designed to work with both single variable inputs and array inputs. When using array-based methods—particularly with the antireflect and smooth padding types—please note that the implementation intentionally discards the last data point as a result of the delta computation process. This behavior is an important consideration when integrating the library into your TradingView studies, as it affects the overall data length of the padded series. Despite this, the library’s structure and documentation make it straightforward to incorporate into your existing scripts. You simply provide your data source, define the length of your data window, and select the desired padding type and direction, along with any optional parameters to control the extent of the padding (using both_period, forward_period, or backward_period).
In practical application, the Padding library enables you to extend historical data beyond its original range in a controlled and predictable manner. This is particularly useful when preparing datasets for further signal processing, as it helps to reduce artifacts that can otherwise compromise the results of your analytical routines. Whether you are an experienced Pine Script developer or a trader exploring advanced data analysis techniques, this library offers a robust solution that enhances the reliability and accuracy of your studies by ensuring your algorithms operate on a more complete and well-prepared dataset.
Library "Padding"
A comprehensive library for padding time series data with various methods. Supports both single variable and array inputs, with flexible padding directions and periods. Designed for signal processing applications including FFT, filtering, convolution, and wavelets. All methods maintain data ordering with most recent point at index 0.
symmetric(source, series_length, direction, both_period, forward_period, backward_period)
Applies symmetric padding by mirroring the input data across boundaries
Parameters:
source (float) : Input value to pad from
series_length (int) : Length of the data window
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to series_length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to series_length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with symmetric padding applied
method symmetric(source, direction, both_period, forward_period, backward_period)
Applies symmetric padding to an array by mirroring the data across boundaries
Namespace types: array
Parameters:
source (array) : Array of values to pad
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to array length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to array length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with symmetric padding applied
reflect(source, series_length, direction, both_period, forward_period, backward_period)
Applies reflect padding by continuing trends through reflection around endpoint values
Parameters:
source (float) : Input value to pad from
series_length (int) : Length of the data window
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to series_length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to series_length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with reflect padding applied
method reflect(source, direction, both_period, forward_period, backward_period)
Applies reflect padding to an array by continuing trends through reflection around endpoint values
Namespace types: array
Parameters:
source (array) : Array of values to pad
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to array length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to array length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with reflect padding applied
periodic(source, series_length, direction, both_period, forward_period, backward_period)
Applies periodic padding by repeating the input data
Parameters:
source (float) : Input value to pad from
series_length (int) : Length of the data window
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to series_length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to series_length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with periodic padding applied
method periodic(source, direction, both_period, forward_period, backward_period)
Applies periodic padding to an array by repeating the data
Namespace types: array
Parameters:
source (array) : Array of values to pad
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to array length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to array length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with periodic padding applied
antisymmetric(source, series_length, direction, both_period, forward_period, backward_period)
Applies antisymmetric padding by mirroring data and alternating signs
Parameters:
source (float) : Input value to pad from
series_length (int) : Length of the data window
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to series_length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to series_length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with antisymmetric padding applied
method antisymmetric(source, direction, both_period, forward_period, backward_period)
Applies antisymmetric padding to an array by mirroring data and alternating signs
Namespace types: array
Parameters:
source (array) : Array of values to pad
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to array length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to array length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with antisymmetric padding applied
antireflect(source, series_length, direction, both_period, forward_period, backward_period)
Applies antireflect padding by reflecting around endpoints while preserving derivatives
Parameters:
source (float) : Input value to pad from
series_length (int) : Length of the data window
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to series_length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to series_length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with antireflect padding applied
method antireflect(source, direction, both_period, forward_period, backward_period)
Applies antireflect padding to an array by reflecting around endpoints while preserving derivatives
Namespace types: array
Parameters:
source (array) : Array of values to pad
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to array length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to array length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with antireflect padding applied. Note: Last data point is lost when using array input
smooth(source, series_length, direction, both_period, forward_period, backward_period)
Applies smooth padding by extending with constant derivatives from endpoints
Parameters:
source (float) : Input value to pad from
series_length (int) : Length of the data window
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to series_length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to series_length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with smooth padding applied
method smooth(source, direction, both_period, forward_period, backward_period)
Applies smooth padding to an array by extending with constant derivatives from endpoints
Namespace types: array
Parameters:
source (array) : Array of values to pad
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to array length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to array length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with smooth padding applied. Note: Last data point is lost when using array input
constant(source, series_length, direction, both_period, forward_period, backward_period)
Applies constant padding by extending endpoint values
Parameters:
source (float) : Input value to pad from
series_length (int) : Length of the data window
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to series_length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to series_length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with constant padding applied
method constant(source, direction, both_period, forward_period, backward_period)
Applies constant padding to an array by extending endpoint values
Namespace types: array
Parameters:
source (array) : Array of values to pad
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to array length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to array length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with constant padding applied
zero(source, series_length, direction, both_period, forward_period, backward_period)
Applies zero padding by extending with zeros
Parameters:
source (float) : Input value to pad from
series_length (int) : Length of the data window
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to series_length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to series_length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with zero padding applied
method zero(source, direction, both_period, forward_period, backward_period)
Applies zero padding to an array by extending with zeros
Namespace types: array
Parameters:
source (array) : Array of values to pad
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to array length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to array length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with zero padding applied
pad_data(source, series_length, padding_type, direction, both_period, forward_period, backward_period)
Generic padding function that applies specified padding type to input data
Parameters:
source (float) : Input value to pad from
series_length (int) : Length of the data window
padding_type (series PaddingType) : Type of padding to apply (see PaddingType enum)
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to series_length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to series_length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with specified padding applied
method pad_data(source, padding_type, direction, both_period, forward_period, backward_period)
Generic padding function that applies specified padding type to array input
Namespace types: array
Parameters:
source (array) : Array of values to pad
padding_type (series PaddingType) : Type of padding to apply (see PaddingType enum)
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to array length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to array length if not specified
Returns: Array ordered with most recent point at index 0, containing original data with specified padding applied. Note: Last data point is lost when using antireflect or smooth padding types
make_padded_data(source, series_length, padding_type, direction, both_period, forward_period, backward_period)
Creates a window-based padded data series that updates with each new value. WARNING: Function must be called on every bar for consistency. Do not use in scopes where it may not execute on every bar.
Parameters:
source (float) : Input value to pad from
series_length (int) : Length of the data window
padding_type (series PaddingType) : Type of padding to apply (see PaddingType enum)
direction (series Direction) : Direction to apply padding
both_period (int) : Optional - periods to pad in both directions. Overrides forward_period and backward_period if specified
forward_period (int) : Optional - periods to pad forward. Defaults to series_length if not specified
backward_period (int) : Optional - periods to pad backward. Defaults to series_length if not specified
Returns: Array ordered with most recent point at index 0, containing windowed data with specified padding applied
6 Band Parametric EQThis indicator implements a complete parametric equalizer on any data source using high-pass and low-pass filters, high and low shelving filters, and six fully configurable bell filters. Each filter stage features standard audio DSP controls including frequency, Q factor, and gain where applicable. While parametric EQ is typically used for audio processing, this implementation raises questions about the nature of filtering in technical analysis. Why stop at simple moving averages when you can shape your signal's frequency response with surgical precision? The answer may reveal more about our assumptions than our indicators.
Filter Types and Parameters
High-Pass Filter:
A high-pass filter attenuates frequency components below its cutoff frequency while passing higher frequencies. The Q parameter controls resonance at the cutoff point, with higher values creating more pronounced peaks.
Low-Pass Filter:
The low-pass filter does the opposite - it attenuates frequencies above the cutoff while passing lower frequencies. Like the high-pass, its Q parameter affects the resonance at the cutoff frequency.
High/Low Shelf Filters:
Shelf filters boost or cut all frequencies above (high shelf) or below (low shelf) the target frequency. The slope parameter determines the steepness of the transition around the target frequency , with a value of 1.0 creating a gentle slope and lower values making the transition more abrupt. The gain parameter sets the amount of boost or cut in decibels.
Bell Filters:
Bell (or peaking) filters create a boost or cut centered around a specific frequency. A bell filter's frequency parameter determines the center point of the effect, while Q controls the width of the affected frequency range - higher Q values create a narrower bandwidth. The gain parameter defines the amount of boost or cut in decibels.
All filters run in series, processing the signal in this order: high-pass → low shelf → bell filters → high shelf → low-pass. Each stage can be independently enabled or bypassed.
The frequency parameter for all filters represents the period length of the targeted frequency component. Lower values target higher frequencies and vice versa. All gain values are in decibels, where positive values boost and negative values cut.
The 6-Band Parametric EQ combines these filters into a comprehensive frequency shaping tool. Just as audio engineers use parametric EQs to sculpt sound, this indicator lets you shape market data's frequency components with surgical precision. But beyond its technical implementation, this indicator serves as a thought experiment about the nature of filtering in technical analysis. While traditional indicators often rely on simple moving averages or single-frequency filters, the parametric EQ takes this concept to its logical extreme - offering complete control over the frequency domain of price action. Whether this level of filtering precision is useful for analysis is perhaps less important than what it reveals about our assumptions regarding market data and its frequency components.
FIR Low Pass Filter Suite (FIR)The FIR Low Pass Filter Suite is an advanced signal processing indicator that applies finite impulse response (FIR) filtering techniques to price data. At its core, the indicator uses windowed-sinc filtering, which provides optimal frequency response characteristics for separating trend from noise in financial data.
The indicator offers multiple window functions including Kaiser, Kaiser-Bessel Derived (KBD), Hann, Hamming, Blackman, Triangular, and Lanczos. Each window type provides different trade-offs between main-lobe width and side-lobe attenuation, allowing users to fine-tune the frequency response characteristics of the filter. The Kaiser and KBD windows provide additional control through an alpha parameter that adjusts the shape of the window function.
A key feature is the ability to operate in either linear or logarithmic space. Logarithmic filtering can be particularly appropriate for financial data due to the multiplicative nature of price movements. The indicator includes an envelope system that can adaptively calculate bands around the filtered price using either arithmetic or geometric deviation, with separate controls for upper and lower bands to account for the asymmetric nature of market movements.
The implementation handles edge effects through proper initialization and offers both centered and forward-only filtering modes. Centered mode provides zero phase distortion but introduces lag, while forward-only mode operates causally with no lag but introduces some phase distortion. All calculations are performed using vectorized operations for efficiency, with carefully designed state management to handle the filter's warm-up period.
Visual feedback is provided through customizable color gradients that can reflect the current trend direction, with optional glow effects and background fills to enhance visibility. The indicator maintains high numerical precision throughout its calculations while providing smooth, artifact-free output suitable for both analysis and visualization.
Lanczos CandlesThis indicator reconstructs price action using Lanczos resampling, incorporating lower timeframe data to create a more detailed representation of market movements. Traditional candle aggregation on higher timeframes tends to lose some price action detail - this indicator attempts to preserve more of that information through mathematical resampling.
The indicator samples price data from a lower timeframe and uses the Lanczos algorithm, a mathematical method commonly used in signal processing and image resampling, to reconstruct the price series at the chart's timeframe. The process helps maintain price movements that might otherwise be smoothed out in regular candle aggregation.
The main settings allow you to select the source timeframe for sampling, adjust the Lanczos filter width to balance smoothness versus detail preservation, and optionally enable Heikin Ashi calculation. The filter width parameter (default: 3) affects how aggressive the smoothing is - higher values produce smoother results while lower values retain more of the original variation.
This approach can be useful for technical analysis when you want to work with higher timeframes while maintaining awareness of significant price movements that occurred within those candles. The optional Heikin Ashi mode can help visualize trends in the resampled data.
The indicator works best when there's a clear ratio between your chart timeframe and the source timeframe (for example, using 1-minute data to build 5-minute candles).
Sinc MAKaiser Windowed Sinc Moving Average Indicator
The Kaiser Windowed Sinc Moving Average is an advanced technical indicator that combines the sinc function with the Kaiser window to create a highly customizable finite impulse response (FIR) filter for financial time series analysis.
Sinc Function: The Ideal Low-Pass Filter
At the core of this indicator is the sinc function, which represents the impulse response of an ideal low-pass filter. In signal processing and technical analysis, the sinc function is crucial because it allows for the creation of filters with precise frequency cutoff characteristics. When applied to financial data, this means the ability to separate long-term trends from short-term fluctuations with remarkable accuracy.
The primary advantage of using a sinc-based filter is the independent control over two critical parameters: the cutoff frequency and the number of samples used. The cutoff frequency, analogous to the "length" in traditional moving averages, determines which price movements are considered significant (low frequency) and which are treated as noise (high frequency). By adjusting the cutoff, analysts can fine-tune the filter to respond to specific market cycles or timeframes of interest.
The number of samples used in the filter doesn't affect the cutoff frequency but instead influences the filter's accuracy and steepness. Increasing the sample size results in a better approximation of the ideal low-pass filter, leading to sharper transitions between passed and attenuated frequencies. This allows for more precise trend identification and noise reduction without changing the fundamental frequency response characteristics.
Kaiser Window: Optimizing the Sinc Filter
While the sinc function provides excellent frequency domain characteristics, it has infinite length in the time domain, which is impractical for real-world applications. This is where the Kaiser window comes into play. By applying the Kaiser window to the sinc function, we create a finite-length filter that approximates the ideal response while minimizing unwanted oscillations (known as the Gibbs phenomenon) in the frequency domain.
The Kaiser window introduces an additional parameter, alpha, which controls the trade-off between the main-lobe width and side-lobe levels in the frequency response. This parameter allows users to fine-tune the filter's behavior, balancing between sharp cutoffs and minimal ripple effects.
Customizable Parameters
The Kaiser Windowed Sinc Moving Average offers several key parameters for customization:
Cutoff: Controls the filter's cutoff frequency, determining the divide between trends and noise.
Length: Sets the number of samples used in the FIR filter calculation, affecting the filter's accuracy and computational complexity.
Alpha: Influences the shape of the Kaiser window, allowing for fine-tuning of the filter's frequency response characteristics.
Centered and Non-Centered Modes
The indicator provides two operational modes:
Non-Centered (Real-time) Mode: Uses half of the windowed sinc function, suitable for real-time analysis and current market conditions.
Centered Mode: Utilizes the full windowed sinc function, resulting in a zero-phase filter. This mode introduces a delay but offers the most accurate trend identification for historical analysis.
Visualization Features
To enhance the analytical value of the indicator, several visualization options are included:
Gradient Coloring: Offers a range of color schemes to represent trend direction and strength.
Glow Effect: An optional visual enhancement for improved line visibility.
Background Fill: Highlights the area between the moving average and price, aiding in trend visualization.
Applications in Technical Analysis
The Kaiser Windowed Sinc Moving Average is particularly useful for precise trend identification, cycle analysis, and noise reduction in financial time series. Its ability to create custom low-pass filters with independent control over cutoff and filter accuracy makes it a powerful tool for analyzing various market conditions and timeframes.
Compared to traditional moving averages, this indicator offers superior frequency response characteristics and reduced lag in trend identification when properly tuned. It provides greater flexibility in filter design, allowing analysts to create moving averages tailored to specific trading strategies or market behaviors.
Conclusion
The Kaiser Windowed Sinc Moving Average represents an advanced approach to price smoothing and trend identification in technical analysis. By making the ideal low-pass filter characteristics of the sinc function practically applicable through Kaiser windowing, this indicator provides traders and analysts with a sophisticated tool for examining price trends and cycles.
Its implementation in Pine Script contributes to the TradingView community by making advanced signal processing techniques accessible for experimentation and further development in technical analysis. This indicator serves not only as a practical tool for market analysis but also as an educational resource for those interested in the intersection of signal processing and financial markets.
Related script:
Ehlers Band-Pass FilterHeyo,
This indicator is an original translation from Ehlers' book "Cycle Analytics for Traders Advanced".
First, I describe the indicator as usual and later you can find a very insightful quote of the book.
Key Features
Signal Line: Represents the output of the band-pass filter, highlighting the dominant cycle in the data.
Trigger Line: A leading indicator derived from the signal line, providing early signals for potential market reversals.
Dominant Cycle: Measures the dominant cycle period by counting the number of bars between zero crossings of the band-pass filter output.
Calculation:
The band-pass filter is implemented using a combination of high-pass and low-pass filters.
The filter's parameters, such as period and bandwidth, can be adjusted to tune the filter to specific market cycles.
The signal line is normalized using an Automatic Gain Control (AGC) to provide consistent amplitude regardless of price swings.
The trigger line is derived by applying a high-pass filter to the signal line, creating a leading
waveform.
Usage
The indicator is effective in identifying peaks and valleys in the market data.
It works best in cyclic market conditions and may produce false signals during trending periods.
The dominant cycle measurement helps traders understand the prevailing market cycle length, aiding in better decision-making.
Quoted from the Book
Band-Pass Filters
“A little of the data narrowly passed,” said Tom broadly.
Perhaps the least appreciated and most underutilized filter in technical analysis is the band-pass filter. The band-pass filter simultaneously diminishes the amplitude at low frequencies, qualifying it as a detrender, and diminishes the amplitude at high frequencies, qualifying it as a data smoother.
It passes only those frequency components from input to output in which the trader is interested. The filtering produced by a band-pass filter is superior because the rejection in the stop bands is related to its bandwidth. The degree of rejection of undesired frequency components is called selectivity. The band-stop filter is the dual of the band-pass filter. It rejects a band of frequency components as a notch at the output and passes all other frequency components virtually unattenuated. Since the bandwidth of the deep rejection in the notch is relatively narrow and since the spectrum of market cycles is relatively broad due to systemic noise, the band-stop filter has little application in trading.
Measuring the Cycle Period
The band-pass filter can be used as a relatively simple measurement of the dominant cycle.
A cycle is complete when the waveform crosses zero two times from the last zero crossing. Therefore, each successive zero crossing of the indicator marks a half cycle period. We can establish the dominant cycle period as twice the spacing between successive zero crossings.
When we measure the dominant cycle period this way, it is best to widen the pass band of the band-pass filter to avoid distorting the measurement simply due to the selectivity of the filter. Using an input bandwidth of 0.7 produces an octave-wide pass band. For example, if the center period of the filter is 20 and the relative bandwidth is 0.7, the bandwidth is 14. That means the pass band of the filter extends from 13-bar periods to 27-bar periods.
That is, roughly an octave exists because the longest period is twice the shortest period of the pass band. It is imperative that a high-pass filter is tuned one octave below the half-bandwidth edge of the band-pass filter to ensure a nominal zero mean of the filtered output. Without a zero mean, the zero crossings can have a substantial error.
Since the measurement of the dominant cycle can vary dramatically from zero crossing to zero
crossing, the code limits the change between measurements to be no more than 25 percent.
While measuring the changing dominant cycle period via zero crossings of the band-pass waveform is easy, it is not necessarily the most accurate method.
Best regards,
simwai
Good Luck with your trading! 🙌
Biquad Band Pass FilterThis indicator utilizes a biquad band pass filter to isolate and highlight a specific frequency band in price data, helping traders focus on price movements within a targeted frequency range.
The Length parameter determines the center frequency of the filter, affecting which frequency band is isolated. Adjusting this parameter allows you to focus on different parts of the price movement spectrum.
The Bandwidth (BW) controls the width of the frequency band in octaves. It represents the bandwidth between -3 dB frequencies for the band pass filter. A narrower bandwidth results in a more focused filtering effect, isolating a tighter range of frequencies.
Key Features of Biquad Filters
Biquad filters are a type of digital filter that provides a combination of low-pass, high-pass, band-pass, and notch filtering capabilities. In this implementation, the biquad filter is configured as a band pass filter, which allows frequencies within a specified band to pass while attenuating frequencies outside this band. This is particularly useful in trading to isolate specific price movements, making it easier to detect patterns and trends within a targeted frequency range.
Biquad filters are known for their smooth response and minimal phase distortion, making them ideal for technical analysis. The customizable length and bandwidth allow for flexible adaptation to different trading strategies and market conditions. Designed for real-time charting, the biquad filter operates efficiently without significant lag, ensuring timely analysis.
By incorporating this biquad band pass filter into your trading toolkit, you can enhance your chart analysis with clearer insights into specific frequency bands of price movements, leading to more informed trading decisions.
Biquad High Pass FilterThis indicator utilizes a biquad high pass filter to filter out low-frequency components from price data, helping traders focus on high-frequency movements and detect rapid changes in trends.
The Length parameter determines the cutoff frequency of the filter, affecting how quickly the filter responds to changes in price. A shorter length allows the filter to react more quickly to high-frequency movements.
The Q Factor controls the sharpness of the filter. A higher Q value results in a more precise filtering effect by narrowing the frequency band. However, be cautious when setting the Q factor too high, as it can induce resonance, amplifying certain frequencies and potentially making the filter less effective by introducing unwanted noise.
Key Features of Biquad Filters
Biquad filters are a type of digital filter that provides a combination of low-pass, high-pass, band-pass, and notch filtering capabilities. In this implementation, the biquad filter is configured as a high pass filter, which allows high-frequency signals to pass while attenuating lower-frequency components. This is particularly useful in trading to highlight rapid price movements, making it easier to spot short-term trends and patterns.
Biquad filters are known for their smooth response and minimal phase distortion, making them ideal for technical analysis. The customizable length and Q factor allow for flexible adaptation to different trading strategies and market conditions. Designed for real-time charting, the biquad filter operates efficiently without significant lag, ensuring timely analysis.
By incorporating this biquad high pass filter into your trading toolkit, you can enhance your chart analysis with clearer insights into rapid price movements, leading to more informed trading decisions.
Biquad Low Pass FilterThis indicator utilizes a biquad low pass filter to smooth out price data, helping traders identify trends and reduce noise in their analysis.
The Length parameter acts as the length of the moving average, determining the smoothness and responsiveness of the filter. Adjusting this parameter changes how quickly the filter reacts to price changes.
The Q Factor controls the sharpness of the filter. A higher Q value results in a narrower frequency band, enhancing the precision of the filter. However, be cautious when setting the Q factor too high, as it can induce resonance, amplifying certain frequencies and potentially making the filter less effective by introducing noise.
Key Features of Biquad Filters
Biquad filters are a type of digital filter that provides a combination of low-pass, high-pass, band-pass, and notch filtering capabilities. In this implementation, the biquad filter is configured as a low pass filter, which allows low-frequency signals to pass while attenuating higher-frequency noise. This is particularly useful in trading to smooth out price data, making it easier to spot underlying trends and patterns.
Biquad filters are known for their smooth response and minimal phase distortion, making them ideal for technical analysis. The customizable length and Q factor allow for flexible adaptation to different trading strategies and market conditions. Designed for real-time charting, the biquad filter operates efficiently without significant lag, ensuring timely analysis.
By incorporating this biquad low pass filter into your trading toolkit, you can enhance your chart analysis with clearer insights into price movements, leading to more informed trading decisions.
STD-Filtered, Variety FIR Digital Filters w/ ATR Bands [Loxx]STD-Filtered, Variety FIR Digital Filters w/ ATR Bands is a FIR Digital Filter indicator with ATR bands. This indicator contains 12 different digital filters. Some of these have already been covered by indicators that I've recently posted. The difference here is that this indicator has ATR bands, allows for frequency filtering, adds a frequency multiplier, and attempts show causality by lagging price input by 1/2 the period input during final application of weights. Period is restricted to even numbers.
The 3 most important parameters are the frequency cutoff, the filter window type and the "causal" parameter.
Included filter types:
- Hamming
- Hanning
- Blackman
- Blackman Harris
- Blackman Nutall
- Nutall
- Bartlet Zero End Points
- Bartlet Hann
- Hann
- Sine
- Lanczos
- Flat Top
Frequency cutoff can vary between 0 and 0.5. General rule is that the greater the cutoff is the "faster" the filter is, and the smaller the cutoff is the smoother the filter is.
You can read more about discrete-time signal processing and some of the windowing functions in this indicator here:
Window function
Window Functions and Their Applications in Signal Processing
What are FIR Filters?
In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier Transform. Several window functions can be defined, based on a constant (rectangular window), B-splines, other polynomials, sinusoids, cosine-sums, adjustable, hybrid, and other types. The windowing operation consists of multipying the given sampled signal by the window function. For trading purposes, these FIR filters act as advanced weighted moving averages.
A finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely (usually decaying).
The impulse response (that is, the output in response to a Kronecker delta input) of an Nth-order discrete-time FIR filter lasts exactly {\displaystyle N+1}N+1 samples (from first nonzero element through last nonzero element) before it then settles to zero.
FIR filters can be discrete-time or continuous-time, and digital or analog.
A FIR filter is (similar to, or) just a weighted moving average filter, where (unlike a typical equally weighted moving average filter) the weights of each delay tap are not constrained to be identical or even of the same sign. By changing various values in the array of weights (the impulse response, or time shifted and sampled version of the same), the frequency response of a FIR filter can be completely changed.
An FIR filter simply CONVOLVES the input time series (price data) with its IMPULSE RESPONSE. The impulse response is just a set of weights (or "coefficients") that multiply each data point. Then you just add up all the products and divide by the sum of the weights and that is it; e.g., for a 10-bar SMA you just add up 10 bars of price data (each multiplied by 1) and divide by 10. For a weighted-MA you add up the product of the price data with triangular-number weights and divide by the total weight.
What is a Standard Deviation Filter?
If price or output or both don't move more than the (standard deviation) * multiplier then the trend stays the previous bar trend. This will appear on the chart as "stepping" of the moving average line. This works similar to Super Trend or Parabolic SAR but is a more naive technique of filtering.
Included
Bar coloring
Loxx's Expanded Source Types
Signals
Alerts
Related indicators
STD/C-Filtered, N-Order Power-of-Cosine FIR Filter
STD/C-Filtered, Power-of-Cosine FIR Filter
STD/C-Filtered, Truncated Taylor Family FIR Filter
STD/Clutter-Filtered, Variety FIR Filters
STD/Clutter-Filtered, Kaiser Window FIR Digital Filter
Goertzel Cycle Period [Loxx]Goertzel Cycle Period is an indicator that uses Goertzel algorithm to extract the cycle period of ticker's price input to then be injected into advanced, adaptive indicators and technical analysis algorithms.
The following information is extracted from: "MESA vs Goertzel-DFT, 2003 by Dennis Meyers"
Background
MESA which stands for Maximum Entropy Spectral Analysis is a widely used mathematical technique designed to find the frequencies present in data. MESA was developed by J.P Burg for his Ph.D dissertation at Stanford University in 1975. The use of the MESA technique for stocks has been written about in many articles and has been popularized as a trading technique by John Ehlers.
The Fourier Transform is a mathematical technique named after the famed French mathematician Jean Baptiste Joseph Fourier 1768-1830. In its digital form, namely the discrete-time Fourier Transform (DFT) series, is a widely used mathematical technique to find the frequencies of discrete time sampled data. The use of the DFT has been written about in many articles in this magazine (see references section).
Today, both MESA and DFT are widely used in science and engineering in digital signal processing. The application of MESA and Fourier mathematical techniques are prevalent in our everyday life from everything from television to cell phones to wireless internet to satellite communications.
MESA Advantages & Disadvantage
MESA is a mathematical technique that calculates the frequencies of a time series from the autoregressive coefficients of the time series. We have all heard of regression. The simplest regression is the straight line regression of price against time where price(t) = a+b*t and where a and b are calculated such that the square of the distance between price and the best fit straight line is minimized (also called least squares fitting). With autoregression we attempt to predict tomorrows price by a linear combination of M past prices.
One of the major advantages of MESA is that the frequency examined is not constrained to multiples of 1/N (1/N is equal to the DFT frequency spacing and N is equal to the number of sample points). For instance with the DFT and N data points we can only look a frequencies of 1/N, 2/N, Ö.., 0.5. With MESA we can examine any frequency band within that range and any frequency spacing between i/N and (i+1)/N . For example, if we had 100 bars of price data, we might be interested in looking for all cycles between 3 bars per cycle and 30 bars/ cycle only and with a frequency spacing of 0.5 bars/cycle. DFT would examine all bars per cycle of between 2 and 50 with a frequency spacing constrained to 1/100.
Another of the major advantages of MESA is that the dominant spectral (frequency) peaks of the price series, if they exist, can be identified with fewer samples than the DFT technique. For instance if we had a 10 bar price period and a high signal to noise ratio we could accurately identify this period with 40 data samples using the MESA technique. This same resolution might take 128 samples for the DFT. One major disadvantage of the MESA technique is that with low signal to noise ratios, that is below 6db (signal amplitude/noise amplitude < 2), the ability of MESA to find the dominant frequency peaks is severely diminished.(see Kay, Ref 10, p 437). With noisy price series this disadvantage can become a real problem. Another disadvantage of MESA is that when the dominant frequencies are found another procedure has to be used to get the amplitude and phases of these found frequencies. This two stage process can make MESA much slower than the DFT and FFT . The FFT stands for Fast Fourier Transform. The Fast Fourier Transform(FFT) is a computationally efficient algorithm which is a designed to rapidly evaluate the DFT. We will show in examples below the comparisons between the DFT & MESA using constructed signals with various noise levels.
DFT Advantages and Disadvantages.
The mathematical technique called the DFT takes a discrete time series(price) of N equally spaced samples and transforms or converts this time series through a mathematical operation into set of N complex numbers defined in what is called the frequency domain. Why would we what to do that? Well it turns out that we can do all kinds of neat analysis tricks in the frequency domain which are just to hard to do, computationally wise, with the original price series in the time domain. If we make the assumption that the price series we are examining is made up of signals of various frequencies plus noise, than in the frequency domain we can easily filter out the frequencies we have no interest in and minimize the noise in the data. We could then transform the resultant back into the time domain and produce a filtered price series that hopefully would be easier to trade. The advantages of the DFT and itís fast computation algorithm the FFT, are that it is extremely fast in calculating the frequencies of the input price series. In addition it can determine frequency peaks for very noisy price series even when the signal amplitude is less than the noise amplitude. One of the disadvantages of the FFT is that straight line, parabolic trends and edge effects in the price series can distort the frequency spectrum. In addition, end effects in the price series can distort the frequency spectrum. Another disadvantage of the FFT is that it needs a lot more data than MESA for spectral resolution. However this disadvantage has largely been nullified by the speed of today's computers.
Goertzel algorithm attempts to resolve these problems...
What is the Goertzel algorithm?
The Goertzel algorithm is a technique in digital signal processing (DSP) for efficient evaluation of the individual terms of the discrete Fourier transform (DFT). It is useful in certain practical applications, such as recognition of dual-tone multi-frequency signaling (DTMF) tones produced by the push buttons of the keypad of a traditional analog telephone. The algorithm was first described by Gerald Goertzel in 1958.
Like the DFT, the Goertzel algorithm analyses one selectable frequency component from a discrete signal. Unlike direct DFT calculations, the Goertzel algorithm applies a single real-valued coefficient at each iteration, using real-valued arithmetic for real-valued input sequences. For covering a full spectrum, the Goertzel algorithm has a higher order of complexity than fast Fourier transform (FFT) algorithms, but for computing a small number of selected frequency components, it is more numerically efficient. The simple structure of the Goertzel algorithm makes it well suited to small processors and embedded applications.
The main calculation in the Goertzel algorithm has the form of a digital filter, and for this reason the algorithm is often called a Goertzel filter
Where is Goertzel algorithm used?
This package contains the advanced mathematical technique called the Goertzel algorithm for discrete Fourier transforms. This mathematical technique is currently used in today's space-age satellite and communication applications and is applied here to stock and futures trading.
While the mathematical technique called the Goertzel algorithm is unknown to many, this algorithm is used everyday without even knowing it. When you press a cell phone button have you ever wondered how the telephone company knows what button tone you pushed? The answer is the Goertzel algorithm. This algorithm is built into tiny integrated circuits and immediately detects which of the 12 button tones(frequencies) you pushed.
Future Additions:
Bartels test for cycle significance, testing output cycles for utility
Hodrick Prescott Detrending, smoothing
Zero-Lag Regression Detrending, smoothing
High-pass or Double WMA filtering of source input price data
References:
1. Burg, J. P., ëMaximum Entropy Spectral Analysisî, Ph.D. dissertation, Stanford University, Stanford, CA. May 1975.
2. Kay, Steven M., ìModern Spectral Estimationî, Prentice Hall, 1988
3. Marple, Lawrence S. Jr., ìDigital Spectral Analysis With Applicationsî, Prentice Hall, 1987
4. Press, William H., et al, ìNumerical Receipts in C++: the Art of Scientific Computingî,
Cambridge Press, 2002.
5. Oppenheim, A, Schafer, R. and Buck, J., ìDiscrete Time Signal Processingî, Prentice Hall,
1996, pp663-634
6. Proakis, J. and Manolakis, D. ìDigital Signal Processing-Principles, Algorithms and
Applicationsî, Prentice Hall, 1996., pp480-481
7. Goertzel, G., ìAn Algorithm for he evaluation of finite trigonometric seriesî American Math
Month, Vol 65, 1958 pp34-35.
loxxfsrrdspfiltsLibrary "loxxfsrrdspfilts"
loxxfsrrdspfilts : FATL, SATL, RFTL, & RSTL Digital Signal Filters
fatl(src)
fatl
Parameters:
src : float
Returns: result float
rftl(src)
rftl
Parameters:
src : float
Returns: result float
satl(src)
satl
Parameters:
src : float
Returns: result float
rstl(src)
rstl
Parameters:
src : float
Returns: result float
FATL, SATL, RFTL, & RSTL Digital Signal Filter Smoother [Loxx]FATL, SATL, RFTL, & RSTL Digital Signal Filter (DSP) Smoother is is a baseline indicator with DSP processed source inputs
What are digital indicators: distinctions from standard tools, types of filters.
To date, dozens of technical analysis indicators have been developed: trend instruments, oscillators, etc. Most of them use the method of averaging historical data, which is considered crude. But there is another group of tools - digital indicators developed on the basis of mathematical methods of spectral analysis. Their formula allows the trader to filter price noise accurately and exclude occasional surges, making the forecast more effective in comparison with conventional indicators. In this review, you will learn about their distinctions, advantages, types of digital indicators and examples of strategies based on them.
Two non-standard strategies based on digital indicators
Basic technical analysis indicators built into most platforms are based on mathematical formulas. These formulas are a reflection of market behavior in past periods. In other words, these indicators are built based on patterns that were discovered as a result of statistical analysis, which allows one to predict further trend movement to some extent. But there is also a group of indicators called digital indicators. They are developed using mathematical analysis and are an algorithmic spectral system called ATCF (Adaptive Trend & Cycles Following). In this article, I will tell you more about the components of this system, describe the differences between digital and regular indicators, and give examples of 2 strategies with indicator templates.
ATCF - Market Spectrum Analysis Method
There is a theory according to which the market is chaotic and unpredictable, i.e. it cannot be accurately analyzed. After all, no one can tell how traders will react to certain news, or whether some large investor will want to play against the market like George Soros did with the Bank of England. But there is another theory: many general market trends are logical, and have a rationale, causes and effects. The economy is undulating, which means it can be described by mathematical methods.
Digital indicators are defined as a group of algorithms for assessing the market situation, which are based exclusively on mathematical methods. They differ from standard indicators by the form of analysis display. They display certain values: price, smoothed price, volumes. Many standard indicators are built on the basis of filtering the minute significant price fluctuations with the help of moving averages and their variations. But we can hardly call the MA a good filter, because digital indicators that use spectral filters make it possible to do a more accurate calculation.
Simply put, digital indicators are technical analysis tools in which spectral filters are used to filter out price noise instead of moving averages.
The display of traditional indicators is lines, areas, and channels. Digital indicators can be displayed both in the form of lines and in digital form (a set of numbers in columns, any data in a text field, etc.). The digital display of the data is more like an additional source of statistics; for trading, a standard visual linear chart view is used.
All digital models belong to the category of spectral analysis of the market situation. In conventional technical indicators, price indications are averaged over a fixed period of time, which gives a rather rough result. The use of spectral analysis allows us to increase trading efficiency due to the fact that digital indicators use a statistical data set of past periods, which is converted into a “frequency” of the market (period of fluctuations).
Fourier theory provides the following spectral ranging of the trend duration:
low frequency range (0-4) - a reflection of a long trend of 2 months or more
medium frequency range (5-40) - the trend lasts 10-60 days, thus it is referred to as a correction
high frequency range (41-130) - price noise that lasts for several days
The ATCF algorithm is built on the basis of spectral analysis and includes a set of indicators created using digital filters. Its consists of indicators and filters:
FATL: Built on the basis of a low-frequency digital trend filter
SATL: Built on the basis of a low-frequency digital trend filter of a different order
RFTL: High frequency trend line
RSTL: Low frequency trend line
Inclucded:
4 DSP filters
Bar coloring
Keltner channels with variety ranges and smoothing functions
Bollinger bands
40 Smoothing filters
33 souce types
Variable channels
Filter impulse & step responsesA simple utility tool to examine a filter's step & impulse responses.
By default you can see LSMA's responses.
How to use:
1. Insert your filter to "f(input)" function inside the code;
2. Let this tool help you to make your own filters.
I been seeing people dropping snaps with this stuff but NEVER, NO1 actually dropped the tool itself (4 real?).
Well here is it, for you.
Almost forgot, adjust "Position" parameter to make plots seen. Try to zoom out, and +-100
Detrended Synthetic Price [CC]The Detrended Synthetic Price was created by Bill Mars and this indicator is another undiscovered gem that I have found very useful. He obviously took inspiration from John Ehlers in the creation of this indicator and I would describe this indicator as a combination of a MACD and Ehlers work. This indicator is extremely smooth and gives very clear buy and sell signals. Let me know how this indicator works for you. I have included strong buy and sell signals in addition to normal ones so strong signals are darker in color and normal signals are lighter in color. Buy when the line turns green and sell when it turns red.
Let me know if there are any other indicators or scripts you would like to see me publish!
AnalysisInterpolationLoessLibrary "AnalysisInterpolationLoess"
LOESS, local weighted Smoothing function.
loess(sample_x, sample_y, point_span) LOESS, local weighted Smoothing function.
Parameters:
sample_x : int array, x values.
sample_y : float array, y values.
point_span : int, local point interval span.
aloess(sample_x, sample_y, point_span) aLOESS, adaptive local weighted Smoothing function.
Parameters:
sample_x : int array, x values.
sample_y : float array, y values.
point_span : int, local point interval span.
Ehlers Hilbert Transform [CC]The Hilbert Transform was created by John Ehlers (Stocks & Commodities V. 18:3 (16-27)) and this indicator can work pretty well as a trend confirmation. This essentially transforms the underlying price data into a soundwave and when you compare the two (blue is positive and red is negative) then it provides fairly clear buy and sell signals. Ehlers did warn in his original article that this indicator has a lag of 4 bars so you have to keep that in mind. I have found that this indicator works pretty well when you buy when the blue line goes over the red line and sell when the blue lines hits the zero line. You could also ignore the red line and buy when the blue line crosses over the zero line and sell when it crosses under. Let me know how you wind up using this indicator in your trading.
Let me know if there are any other scripts you would like to see me publish!
Autocorrelation PlotA tool to plot auto correlation of time series, this is useful in identifying periodicity in a time series or signal.
Due to the limits of Pine Script you'll need to add it multiple times if you want autocorrelation beyond 55 periods. I have added it 4 times here for 220 periods.
For more information on Autocorrelation see: en.wikipedia.org
Note: There are 1 bar gaps every 55 because I wanted the labels to remain every 5, but you don't have to have gaps....
FUNCTION: Goertzel algorithm -- DFT of a specific frequency binThis function implements the Goertzel algorithm (for integer N).
The Goertzel algorithm is a technique in digital signal processing (DSP) for efficient evaluation of the individual terms of the discrete Fourier transform (DFT).
In short, it measure the power of a specific frequency like one bin of a DFT, over a rolling window (N) of samples.
Here you see an input signal that changes frequency and amplitude (from 7 bars to 17). I am running the indicator 3 times to show it measuring both frequencies and one in between (13). You can see it very accurately measures the signals present and their power, but is noisy in the transition. Changing the block len will cause it to be more responsive but noisier.
Here is a picture of the same signal, but with white noise added.
If you have a cycle you think is present you could use this to test it, but the function is designed for integration in to more complicated scripts. I think power is best interrupted on a log scale.
Given a period (in bars or samples) and a block_len (N in Goertzel terminology) the function returns the Real (InPhase) and Quadrature (Imaginary) components of your signal as well as calculating the power and the instantaneous angle (in radians).
I hope this proves useful to the DSP folks here.
Filter Information Box - PineCoders FAQWhen designing filters it can be interesting to have information about their characteristics, which can be obtained from the set of filter coefficients (weights). The following script analyzes the impulse response of a filter in order to return the following information:
Lag
Smoothness via the Herfindahl index
Percentage Overshoot
Percentage Of Positive Weights
The script also attempts to determine the type of the analyzed filter, and will issue warnings when the filter shows signs of unwanted behavior.
DISPLAYED INFORMATION AND METHODS
The script displays one box on the chart containing two sections. The filter metrics section displays the following information:
- Lag : Measured in bars and calculated from the convolution between the filter's impulse response and a linearly increasing sequence of value 0,1,2,3... . This sequence resets when the impulse response crosses under/over 0.
- Herfindahl index : A measure of the filter's smoothness described by Valeriy Zakamulin. The Herfindahl index measures the concentration of the filter weights by summing the squared filter weights, with lower values suggesting a smoother filter. With normalized weights the minimum value of the Herfindahl index for low-pass filters is 1/N where N is the filter length.
- Percentage Overshoot : Defined as the maximum value of the filter step response, minus 1 multiplied by 100. Larger values suggest higher overshoots.
- Percentage Positive Weights : Percentage of filter weights greater than 0.
Each of these calculations is based on the filter's impulse response, with the impulse position controlled by the Impulse Position setting (its default is 1000). Make sure the number of inputs the filter uses is smaller than Impulse Position and that the number of bars on the chart is also greater than Impulse Position . In order for these metrics to be as accurate as possible, make sure the filter weights add up to 1 for low-pass and band-stop filters, and 0 for high-pass and band-pass filters.
The comments section displays information related to the type of filter analyzed. The detection algorithm is based on the metrics described above. The script can detect the following type of filters:
All-Pass
Low-Pass
High-Pass
Band-Pass
Band-Stop
It is assumed that the user is analyzing one of these types of filters. The comments box also displays various warnings. For example, a warning will be displayed when a low-pass/band-stop filter has a non-unity pass-band, and another is displayed if the filter overshoot is considered too important.
HOW TO SET THE SCRIPT UP
In order to use this script, the user must first enter the filter settings in the section provided for this purpose in the top section of the script. The filter to be analyzed must then be entered into the:
f(input)
function, where `input` is the filter's input source. By default, this function is a simple moving average of period length . Be sure to remove it.
If, for example, we wanted to analyze a Blackman filter, we would enter the following:
f(input)=>
pi = 3.14159,sum = 0.,sumw = 0.
for i = 0 to length-1
k = i/length
w = 0.42 - 0.5 * cos(2 * pi * k) + 0.08 * cos(4 * pi * k)
sumw := sumw + w
sum := sum + w*input
sum/sumw
EXAMPLES
In this section we will look at the information given by the script using various filters. The first filter we will showcase is the linearly weighted moving average (WMA) of period 9.
As we can see, its lag is 2.6667, which is indeed correct as the closed form of the lag of the WMA is equal to (period-1)/3 , which for period 9 gives (9-1)/3 which is approximately equal to 2.6667. The WMA does not have overshoots, this is shown by the the percentage overshoot value being equal to 0%. Finally, the percentage of positive weights is 100%, as the WMA does not possess negative weights.
Lets now analyze the Hull moving average of period 9. This moving average aims to provide a low-lag response.
Here we can see how the lag is way lower than that of the WMA. We can also see that the Herfindahl index is higher which indicates the WMA is smoother than the HMA. In order to reduce lag the HMA use negative weights, here 55% (as there are 45% of positive ones). The use of negative weights creates overshoots, we can see with the percentage overshoot being 26.6667%.
The WMA and HMA are both low-pass filters. In both cases the script correctly detected this information. Let's now analyze a simple high-pass filter, calculated as follows:
input - sma(input,length)
Most weights of a high-pass filters are negative, which is why the lag value is negative. This would suggest the indicator is able to predict future input values, which of course is not possible. In the case of high-pass filters, the Herfindahl index is greater than 0.5 and converges toward 1, with higher values of length . The comment box correctly detected the type of filter we were using.
Let's now test the script using the simple center of gravity bandpass filter calculated as follows:
wma(input,length) - sma(input,length)
The script correctly detected the type of filter we are using. Another type of filter that the script can detect is band-stop filters. A simple band-stop filter can be made as follows:
input - (wma(input,length) - sma(input,length))
The script correctly detect the type of filter. Like high-pass filters the Herfindahl index is greater than 0.5 and converges toward 1, with greater values of length . Finally the script can detect all-pass filters, which are filters that do not change the frequency content of the input.
WARNING COMMENTS
The script can give warning when certain filter characteristics are detected. One of them is non-unity pass-band for low-pass filters. This warning comment is displayed when the weights of the filter do not add up to 1. As an example, let's use the following function as a filter:
sum(input,length)
Here the filter pass-band has non unity, and the sum of the weights is equal to length . Therefore the script would display the following comments:
We can also see how the metrics go wild (note that no filter type is detected, as the detected filter could be of the wrong type). The comment mentioning the detection of high overshoot appears when the percentage overshoot is greater than 50%. For example if we use the following filter:
5*wma(input,length) - 4*sma(input,length)
The script would display the following comment:
We can indeed see high overshoots from the filter:
@alexgrover for PineCoders
Look first. Then leap.
