Ehlers Adaptive Trend Indicator [Alpha Extract]Ehlers Adaptive Trend Indicator
The Ehlers Adaptive Trend Indicator combines Ehlers' advanced digital signal processing techniques with dynamic volatility bands to identify robust trend conditions and potential reversals. This powerful tool helps traders visualize trend strength, adaptive support/resistance levels, and momentum shifts across various market conditions.
🔶 CALCULATION
The indicator employs a sophisticated adaptive algorithm that responds to changing market conditions:
• Ehlers Filter : Calculates a weighted average based on momentum differences to create an adaptive trend baseline.
• Dynamic Bands : Volatility-adjusted bands that expand and contract based on recent price action.
• Trend Level : A dynamic support/resistance level that adapts to the current trend direction.
• Smoothed Volatility : Market volatility measured and smoothed to provide reliable band width.
Formula:
• Ehlers Basis = Weighted average of price, with weights determined by momentum differences
• Volatility = Standard deviation of price over Ehlers Length period
• Smoothed Volatility = EMA of volatility over Smoothing Length
• Upper Band = Ehlers Basis + Smoothed Volatility × Sensitivity
• Lower Band = Ehlers Basis - Smoothed Volatility × Sensitivity
• Trend Level = Adaptive support in uptrends, resistance in downtrends
🔶 DETAILS
Visual Features :
• Ehlers Basis Line (Yellow): The core adaptive trend reference that serves as the primary trend indicator.
• Trend Level Line (Dynamic Color): Changes between green (bullish) and red (bearish) based on the current trend state.
• Fill Areas : Transparent green fill during bullish trends and transparent red fill during bearish trends for clear visual identification.
• Bar Coloring : Optional price bar coloring that reflects the current trend direction for enhanced visualization.
Interpretation :
• **Bullish Signal**: Price crosses above the upper band, triggering a trend change with the Trend Level becoming dynamic support.
• **Bearish Signal**: Price drops below the lower band, confirming a trend change with the Trend Level becoming dynamic resistance.
• **Trend Continuation**: Trend Level rises in bullish markets and falls in bearish markets, providing adaptive trailing support/resistance.
🔶 EXAMPLES
The chart demonstrates:
• Bullish Trend Identification : When price breaks above the upper band, the indicator shifts to bullish mode with green trend level and fill.
• Bearish Trend Identification : When price falls below the lower band, the indicator shifts to bearish mode with red trend level and fill.
• Trend Persistence : Trend Level adapts to market movement, rising during uptrends to provide dynamic support and falling during downtrends to act as resistance.
Example Snapshots :
• During a strong uptrend, the Trend Level continuously adjusts upward, keeping traders in the trend while filtering out minor retracements.
• During trend reversals, clear color changes and Trend Level shifts provide early warning of potential direction changes.
🔶 SETTINGS
Customization Options :
• Ehlers Length (p1) (Default: 30): Controls the primary adaptive calculation period, balancing responsiveness with stability.
• Momentum Length (p2) (Default: 25): Determines the lag for momentum calculations used in the adaptive weighting.
• Smoothing Length (Default: 10): Adjusts the volatility smoothing period—higher values provide more stable bands.
• Sensitivity (Default: 1.0): Multiplier for band width—higher values increase distance between bands, lower values tighten them.
• Visual Settings : Customizable colors for bullish and bearish trends, basis line, and optional bar coloring.
The Ehlers Adaptive Trend Indicator combines John Ehlers' digital signal processing expertise with modern volatility analysis to create a robust trend-following system that adapts to changing market conditions, helping traders stay on the right side of the market.
Indicator
[GYTS] Ultimate Smoother (3-poles + 2 poles)Ultimate Smoother (3-pole)
🌸 Part of GoemonYae Trading System (GYTS) 🌸
🌸 --------- INTRODUCTION --------- 🌸
💮 Release of 3-Pole Ultimate Smoother
This indicator presents a new 3-pole version of John Ehlers' Ultimate Smoother (2024) . This results in an unconventional filter that exhibits effectively zero lag in practical trading applications, regardless of the set period. By using a 2-pole high-pass filter in its design, it responds to price direction changes on the same bar, while still allowing the user to control smoothness.
💮 What is the Ultimate Smoother?
The original Ultimate Smoother is a revolutionary filter designed by John Ehlers (2024) that smooths price data with virtually zero lag in the pass band. While conventional filters always introduce lag when removing market noise, the Ultimate Smoother maintains phase alignment at low frequencies while still providing excellent noise reduction.
💮 Mathematical Foundation
The Ultimate Smoother achieves its remarkable properties through a clever mathematical approach:
1. Instead of directly designing a low-pass filter (like traditional moving averages), it subtracts a high-pass filter from an all-pass filter (the original input data).
2. At very low frequencies, the high-pass filter contributes almost nothing, so the output closely matches the input in both amplitude and phase.
3. At higher frequencies, the high-pass filter's response increasingly matches the input data, resulting in cancellation through subtraction.
The 3-pole version extends this principle by using a higher-order high-pass filter, requiring additional coefficients and handling more terms in the numerator of the transfer function.
🌸 --------- USAGE GUIDE --------- 🌸
💮 Period Parameter Behaviour
The period parameter in the 3-pole Ultimate Smoother works somewhat counterintuitively:
- Longer periods: Result in less smooth, but more responsive following of the price. The filter output more closely tracks the input data.
- Shorter periods: Produce smoother output but may exhibit overshooting (extrapolating price movement) for larger movements.
This is different from most filters where longer periods typically produce smoother outputs with more lag.
💮 When to Choose 3-Pole vs. 2-Pole
- Choose the 3-pole version when you need zero-lag but want to control the smoothness
- Choose the 2-pole version when you are okay with some lag with the benefit of more smoothness.
🌸 --------- ACKNOWLEDGEMENTS --------- 🌸
This indicator builds upon the pioneering work of John Ehlers, particularly from his article April 2024 edition of TASC's Traders' Tips . The original version is published on PulseWire by @PineCodersTASC .
This 3-pole extension was developed by @GoemonYae . Feedback is highly appreciated! Indicator
Weighted Fourier Transform: Spectral Gating & Main Frequency🙏🏻 This drop has 2 purposes:
1) to inform every1 who'd ever see it that Weighted Fourier Tranform does exist, while being available nowhere online, not even in papers, yet there's nothing incredibly complicated about it, and it can/should be used in certain cases;
2) to show PulseWire users how they can use it now in dem endevours, to show em what spectral filtering is, and what can they do with all of it in diy mode.
... so we gonna have 2 sections in the description
Section 1: Weighted Fourier Transform
It's quite easy to include weights in Fourier analysis: you just premultiply each datapoint by its corresponding weight -> feed to direct Fourier Transform, and then divide by weights after inverse Fourier transform. Alternatevely, in direct transform you just multiply contributions of each data point to the real and imaginary parts of the Fourier transform by corresponding weights (in accumulation phase), and in inverse transform you divide by weights instead during the accumulation phase. Everything else stays the same just like in non-weighted version.
If you're from the first target group let's say, you prolly know a thing or deux about how to code & about Fourier Transform, so you can just check lines of code to see the implementation of Weighted Discrete version of Fourier Transform, and port it to to any technology you desire. Pine Script is a developing technology that is incredibly comfortable in use for quant-related tasks and anything involving time series in general. While also using Python for research and C++ for development, every time I can do what I want in Pine Script, I reach for it and never touch matlab, python, R, or anything else.
Weighted version allows you to explicetly include order/time information into the operation, which is essential with every time series, although not widely used in mainstream just as many other obvious and right things. If you think deeply, you'll understand that you can apply a usual non-weighted Fourier to any 2d+ data you can (even if none of these dimensions represent time), because this is a geometric tool in essence. By applying linearly decaying weights inside Fourier transform, you're explicetly saying, "one of these dimensions is Time, and weights represent the order". And obviously you can combine multiple weightings, eg time and another characteristic of each datum, allows you to include another non-spatial dimension in your model.
By doing that, on properly processed (not only stationary but Also centered around zero data), you can get some interesting results that you won't be able to recreate without weights:
^^ A sine wave, centered around zero, period of 16. Gray line made by: DWFT (direct weighted Fourier transform) -> spectral gating -> IWFT (inverse weighted Fourier transform) -> plotting the last value of gated reconstructed data, all applied to expanding window. Look how precisely it follows the original data (the sine wave) with no lag at all. This can't be done by using non-weighted version of Fourier transform.
^^ spectral filtering applied to the whole dataset, calculated on the latest data update
And you should never forget about Fast Fourier Transform, tho it needs recursion...
Section 2: About use cases for quant trading, about this particular implementaion in Pine Script 6 (currently the latest version as of Friday 13, December 2k24).
Given the current state of things, we have certain limits on matrix size on PulseWire (and we need big dope matrixes to calculate polynomial regression -> detrend & center our data before Fourier), and recursion is not yet available in Pine Script, so the script works on short datasets only, and requires some time.
A note on detrending. For quality results, Fourier Transform should be applied to not only stationary but also centered around zero data. The rightest way to do detrending of time series
is to fit Cumulative Weighted Moving Polynomial Regression (known as WLSMA in some narrow circles xD) and calculate the deltas between datapoint at time t and this wonderful fit at time t. That's exactly what you see on the main chart of script description: notice the distances between chart and WLSMA, now look lower and see how it matches the distances between zero and purple line in WFT study. Using residuals of one regression fit of the whole dataset makes less sense in time series context, we break some 'time' and order rules in a way, tho not many understand/cares abouit it in mainstream quant industry.
Two ways of using the script:
Spectral Gating aka Spectral filtering. Frequency domain filtering is quite responsive and for a greater computational cost does not introduce a lag the way it works with time-domain filtering. Works this way: direct Fourier transform your data to get frequency & phase info -> compute power spectrum out of it -> zero out all dem freqs that ain't hit your threshold -> inverse Fourier tranform what's left -> repeat at each datapoint plotting the very first value of reconstructed array*. With this you can watch for zero crossings to make appropriate trading decisions.
^^ plot Freq pass to use the script this way, use Level setting to control the intensity of gating. These 3 only available values: -1, 0 and 1, are the general & natural ones.
* if you turn on labels in script's style settings, you see the gray dots perfectly fitting your data. They get recalculated (for the whole dataset) at each update. You call it repainting, this is for analytical & aesthetic purposes. Included for demonstration only.
Finding main/dominant frequency & period. You can use it to set up Length for your other studies, and for analytical purposes simply to understand the periodicity of your data.
^^ plot main frequency/main period to use the script this way. On the screenshot, you can see the script applied to sine wave of period 16, notice how many datapoints it took the algo to figure out the signal's period quite good in expanding window mode
Now what's the next step? You can try applying signal windowing techniques to make it all less data-driven but your ego-driven, make a weighted periodogram or autocorrelogram (check Wiener-Khinchin Theorem ), and maybe whole shiny spectrogram?
... you decide, choice is yours,
The butterfly reflect the doors ...
∞ Indicator
MathTransformLibrary "MathTransform"
Auxiliary functions for transforming data using mathematical and statistical methods
scaler_zscore(x, lookback_window)
Calculates Z-Score normalization of a series.
Parameters:
x (float) : : floating point series to normalize
lookback_window (int) : : lookback period for calculating mean and standard deviation
Returns: Z-Score normalized series
scaler_min_max(x, lookback_window, min_val, max_val, empiric_min, empiric_max, empiric_mid)
Performs Min-Max scaling of a series within a given window, user-defined bounds, and optional midpoint
Parameters:
x (float) : : floating point series to transform
lookback_window (int) : : int : optional lookback window size to consider for scaling.
min_val (float) : : float : minimum value of the scaled range. Default is 0.0.
max_val (float) : : float : maximum value of the scaled range. Default is 1.0.
empiric_min (float) : : float : user-defined minimum value of the input data. This means that the output could exceed the `min_val` bound if there is data in `x` lesser than `empiric_min`. If na, it's calculated from `x` and `lookback_window`.
empiric_max (float) : : float : user-defined maximum value of the input data. This means that the output could exceed the `max_val` bound if there is data in `x` greater than `empiric_max`. If na, it's calculated from `x` and `lookback_window`.
empiric_mid (float) : : float : user-defined midpoint value of the input data. If na, it's calculated from `empiric_min` and `empiric_max`.
Returns: rescaled series
log(x, base)
Applies logarithmic transformation to a value, base can be user-defined.
Parameters:
x (float) : : floating point value to transform
base (float) : : logarithmic base, must be greater than 0
Returns: logarithm of the value to the given base, if x <= 0, returns logarithm of 1 to the given base
exp(x, base)
Applies exponential transformation to a value, base can be user-defined.
Parameters:
x (float) : : floating point value to transform
base (float) : : base of the exponentiation, must be greater than 0
Returns: the result of raising the base to the power of the value
power(x, exponent)
Applies power transformation to a value, exponent can be user-defined.
Parameters:
x (float) : : floating point value to transform
exponent (float) : : exponent for the transformation
Returns: the value raised to the given exponent, preserving the sign of the original value
tanh(x, scale)
The hyperbolic tangent is the ratio of the hyperbolic sine and hyperbolic cosine. It limits an output to a range of −1 to 1.
Parameters:
x (float) : : floating point series
scale (float)
sigmoid(x, scale, offset)
Applies the sigmoid function to a series.
Parameters:
x (float) : : floating point series to transform
scale (float) : : scaling factor for the sigmoid function
offset (float) : : offset for the sigmoid function
Returns: transformed series using the sigmoid function
sigmoid_double(x, scale, offset)
Applies a double sigmoid function to a series, handling positive and negative values differently.
Parameters:
x (float) : : floating point series to transform
scale (float) : : scaling factor for the sigmoid function
offset (float) : : offset for the sigmoid function
Returns: transformed series using the double sigmoid function
logistic_decay(a, b, c, t)
Calculates logistic decay based on given parameters.
Parameters:
a (float) : : parameter affecting the steepness of the curve
b (float) : : parameter affecting the direction of the decay
c (float) : : the upper bound of the function's output
t (float) : : time variable
Returns: value of the logistic decay function at time t Library
Reflex & Trendflex█ OVERVIEW
Reflex and Trendflex are zero-lag oscillators that decompose price into independent cycle and trend components using SuperSmoother filtering. These indicators isolate each component separately, providing clearer identification of cyclical reversals (Reflex) versus trending movements (Trendflex).
Based on Dr. John F. Ehlers' "Reflex: A New Zero-Lag Indicator" article (February 2020, TASC), both oscillators use normalized slope deviation analysis to minimize lag while maintaining signal clarity. The SuperSmoother filter removes high-frequency noise, then deviations from linear regression (Reflex) or current value (Trendflex) are measured and normalized by RMS for consistent amplitude across instruments and timeframes.
█ CONCEPTS
SuperSmoother Filter
Both oscillators begin with a two-pole Butterworth low-pass filter that smooths price data without the excessive lag of simple moving averages. The filter uses exponential decay coefficients and cosine modulation based on the cutoff period, providing aggressive smoothing while preserving signal timing.
Reflex: Cycle Component
Reflex isolates cyclical price behavior by measuring deviation from a linear regression line fitted through the SuperSmoother output. For each bar, the filter calculates a linear slope over the lookback period, then sums how much the smoothed price deviates from this trendline. These deviations represent pure cyclical movement - price oscillations around the dominant trend. The result is normalized by RMS (root mean square) to produce consistent amplitude regardless of volatility or timeframe.
Trendflex: Trend Component
Trendflex extracts trending behavior by measuring cumulative deviation from the current SuperSmoother value. Instead of comparing to a regression line, it simply sums the differences between the current smoothed value and all past values in the period. This captures sustained directional movement rather than oscillations. Like Reflex, normalization by RMS ensures comparable readings across different instruments.
RMS Normalization
Both oscillators normalize their raw deviation measurements using an exponentially weighted RMS calculation: `rms = 0.04 * deviation² + 0.96 * rms `. This adaptive normalization ensures the oscillator amplitude remains stable as volatility changes, making threshold levels meaningful across different market conditions.
█ INTERPRETATION
Reflex (Cycle Component)
Oscillates around zero representing cyclical price behavior isolated from trend:
• Above zero : Price is in upward phase of cycle
• Below zero : Price is in downward phase of cycle
• Zero crossings : Potential cycle reversal points
• Extremes : Indicate stretched cyclical condition, often precede mean reversion
Best used for identifying cyclical turning points in ranging or oscillating markets. More sensitive to reversals than Trendflex.
Trendflex (Trend Component)
Oscillates around zero representing trending behavior isolated from cycles:
• Above zero : Sustained upward trend
• Below zero : Sustained downward trend
• Zero crossings : Trend direction changes
• Magnitude : Strength of trend (larger absolute values = stronger trend)
Best used for confirming trend direction and identifying trend exhaustion. Less noisy than Reflex due to focus on directional movement rather than oscillations.
Combined Analysis
Using both oscillators together provides powerful signal confirmation:
• Both positive: Strong uptrend with positive cycle phase (high probability long setup)
• Both negative: Strong downtrend with negative cycle phase (high probability short setup)
• Divergent signals: Conflicting cycle and trend (choppy conditions, reduce position size)
• Reflex reversal with Trendflex agreement: Cyclical turn within established trend (entry/exit timing)
Dynamic Thresholds
Threshold bands identify statistically significant oscillator readings that warrant attention:
• Breach above +threshold : Strong bullish cycle (Reflex) or trend (Trendflex) behavior - potential overbought condition
• Breach below -threshold : Strong bearish cycle or trend behavior - potential oversold condition
• Return inside thresholds : Signal strength normalizing, potential reversal or consolidation ahead
• Threshold compression : During low volatility, thresholds narrow (especially with StdDev mode), making breaches more frequent
• Threshold expansion : During high volatility, thresholds widen, filtering out minor oscillations
Combine threshold breaches with zero-line position for stronger signals:
• Threshold breach + zero-line cross = high-conviction signal
• Threshold breach without zero-line support = monitor for confirmation
Alert Conditions
Six built-in alerts trigger on bar close (no repainting):
• Above +Threshold : Oscillator crossed above positive threshold (strong bullish behavior)
• Below -Threshold : Oscillator crossed below negative threshold (strong bearish behavior)
• Reflex Above Zero : Reflex crossed above zero (bullish cycle phase)
• Reflex Below Zero : Reflex crossed below zero (bearish cycle phase)
• Trendflex Above Zero : Trendflex crossed above zero (bullish trend shift)
• Trendflex Below Zero : Trendflex crossed below zero (bearish trend shift)
█ SETTINGS & PARAMETER TUNING
Oscillator Settings
• Source : Price series to decompose
• Reflex Period (5-50): SuperSmoother period for cycle component. Lower values increase responsiveness to cyclical turns but add noise. Default 20.
• Trendflex Period (5-50): SuperSmoother period for trend component. Lower values respond faster to trend changes. Default 20.
Display Settings
• Reflex/Trendflex Display : Toggle visibility and customize colors for each oscillator independently
• Zero Line : Reference line showing neutral oscillator position
Dynamic Thresholds
Optional significance bands that identify when oscillator readings indicate strong cyclical or trending behavior:
• Threshold Mode : Choose calculation method based on market characteristics
- MAD (Median Absolute Deviation) : Outlier-resistant, best for markets with occasional spikes (default)
- Standard Deviation : Volatility-sensitive, adapts quickly to regime changes
- Percentile Rank : Fixed probability bands (e.g., 90% = only 10% of values exceed threshold)
• Apply To : Select which oscillator (Reflex or Trendflex) to calculate thresholds for
• Period (2-200): Lookback window for threshold calculation. Default 50.
• Multiplier (k) : Scaling factor for MAD/StdDev modes. Higher values = fewer threshold breaches (default 1.5)
• Percentile (%) : For Percentile mode only. Higher percentile = more selective threshold (default 90%)
Parameter Interactions
• Shorter periods make both oscillators more sensitive but noisier
• Reflex typically more volatile than Trendflex at same period settings
• For ranging markets: shorter Reflex period (10-15) captures swings better
• For trending markets: shorter Trendflex period (10-15) follows trend shifts faster
█ LIMITATIONS
Inherent Characteristics
• Near-zero lag, not zero-lag : Despite the name, some lag remains from SuperSmoother filtering
• Normalization artifacts : RMS normalization can produce unusual readings during volatility regime changes
• Period dependency : Oscillator characteristics change significantly with different period settings - no "correct" universal parameter
Market Conditions to Avoid
• Very low volatility : Normalization amplifies noise in quiet markets, producing false signals
• Sudden gaps : SuperSmoother assumes continuous data; large gaps disrupt filter continuity requiring bars to stabilize
• Micro timeframes : Sub-minute charts contain microstructure noise that overwhelms signal quality
Parameter Selection Pitfalls
• Matching periods to dominant cycle : If period doesn't align with actual market cycle period, signals degrade
• Threshold over-tuning : Optimizing threshold parameters for past data often fails forward - use conservative defaults
• Ignoring component differences : Reflex and Trendflex measure different aspects - don't expect identical behavior
█ NOTES
Credits
These indicators are based on Dr. John F. Ehlers' "Reflex: A New Zero-Lag Indicator" published in the February 2020 issue of Technical Analysis of Stocks & Commodities (TASC) magazine. The article introduces a novel approach to isolating cycle and trend components using SuperSmoother filtering combined with normalized deviation analysis.
For those interested in the underlying mathematics and DSP concepts:
• Ehlers, J.F. (February 2020). "Reflex: A New Zero-Lag Indicator" - Technical Analysis of Stocks & Commodities magazine
• Ehlers, J.F. (2001). Rocket Science for Traders: Digital Signal Processing Applications . John Wiley & Sons
• Various TASC articles by John Ehlers on SuperSmoother filters and oscillator design
by ♚@e2e4 Indicator
Fourier series Model Of The Market█ OVERVIEW
The Fourier Series Model of the Market (FSMM) decomposes price action into harmonic components using bandpass filtering, then reconstructs a composite wave weighted by rolling energy ratios. This approach isolates cyclical market behavior at multiple frequencies, emphasizing dominant cycles for cleaner signal generation. The energy-adaptive weighting is the key differentiator from simple harmonic summation: cycles that dominate current price action contribute more to the output.
Based on Fourier analysis principles applied to financial markets, the indicator extracts harmonics (fundamental, 2nd, 3rd, etc.) using second-order IIR bandpass filters, then weights each harmonic's contribution by its relative energy compared to adjacent harmonics. This energy-adaptive weighting naturally emphasizes the cycles that are most prominent in current market conditions.
█ CONCEPTS
Fourier Decomposition
Fourier analysis represents any periodic signal as a sum of sine waves at different frequencies. In market analysis, price action can be decomposed into a fundamental cycle (the base period) plus harmonics at integer multiples of that frequency (period/2, period/3, etc.). Each harmonic captures oscillations at a specific frequency band, and their sum reconstructs the original cyclical behavior.
Bandpass Filtering
Each harmonic is extracted using a second-order IIR (Infinite Impulse Response) bandpass filter tuned to that harmonic's frequency. The filter isolates price activity within a narrow frequency range while rejecting both higher-frequency noise and lower-frequency trend drift. Before filtering, the source is debiased via 2-bar momentum to remove DC offset, ensuring each bandpass operates around true zero.
Energy-Weighted Reconstruction
Rather than simply summing all harmonics equally, FSMM weights each harmonic by its rolling energy relative to the previous harmonic. The energy score combines the current harmonic value with its rate of change, so it reflects both amplitude and momentum. Higher harmonics that hold comparatively more energy therefore contribute more to the composite wave, while weaker harmonics fade out. This adaptive weighting allows the model to respond to changing market cyclicality.
Quadrature Component (Rate of Change)
The rate of change output represents the 90°-phase-shifted (quadrature) component of the wave. When the wave is at zero and rising, the rate of change is at maximum positive. This provides complementary information about cycle phase and can be used for timing entries relative to cycle position.
█ INTERPRETATION
Wave Output
The composite wave oscillates around zero, representing the sum of all extracted harmonic components weighted by energy:
• Above zero : Net bullish cyclical momentum across harmonics
• Below zero : Net bearish cyclical momentum across harmonics
• Zero crossings : Cycle phase transitions - potential reversal points
• Wave amplitude : Strength of cyclical behavior; larger swings indicate cleaner cycles
Rate of Change
The quadrature component (90° phase-shifted) provides cycle phase information:
• Maximum rate of change : Wave is near zero and accelerating - early cycle phase
• Zero rate of change : Wave is at peak or trough - cycle extremes
• Rate/Wave divergence : When wave makes new highs/lows but rate of change does not confirm (lower momentum), suggests cycle exhaustion or impending phase shift
Combined Analysis
• Wave crossing above zero with positive rate of change: Strong bullish cycle initiation
• Wave crossing below zero with negative rate of change: Strong bearish cycle initiation
• Wave at extreme with rate of change reversing: Potential cycle peak/trough
Threshold Bands
When enabled, threshold bands define statistically significant wave deviations:
• Breach above +threshold : Unusually strong bullish cyclical behavior
• Breach below -threshold : Unusually strong bearish cyclical behavior
• Return inside thresholds : Normalizing behavior, potential mean reversion ahead
Alert Conditions
Four built-in alerts trigger on bar close (no repainting):
• Above +Threshold : Strong bullish cycle behavior
• Below -Threshold : Strong bearish cycle behavior
• Above Zero : Bullish cycle phase shift
• Below Zero : Bearish cycle phase shift
█ SETTINGS & PARAMETER TUNING
Fourier Series Model
• Source : Price series to decompose into harmonic components.
• Period (6-100): Base period for the fundamental harmonic. Higher harmonics divide this period (harmonic 2 = period/2, harmonic 3 = period/3). Match to the dominant market cycle for best results. Default 20.
• Bandwidth (0.05-0.5): Bandpass filter selectivity. Lower values create narrower passbands that isolate harmonics more precisely but may miss slightly off-frequency cycles. Higher values capture broader ranges but reduce harmonic separation. Default 0.1 balances precision and robustness.
• Harmonics (1-20): Number of harmonic components to extract. More harmonics capture finer cyclical detail but increase computation. For most applications, 3-5 harmonics suffice. The fundamental alone (1 harmonic) functions as a simple bandpass filter.
Display Settings
• Wave Outputs : Toggle visibility and color of the composite Fourier wave.
• Rate of Change : Toggle visibility and color of the quadrature component (90° phase-shifted wave).
• Zero Line : Reference line for oscillator neutrality.
Diagnostics - Dynamic Thresholds
Optional significance bands that identify when wave readings indicate strong cyclical behavior:
• Dynamic Threshold : Toggle threshold bands and set colors.
• Threshold Mode : Select calculation method:
- MAD (Median Absolute Deviation) : Robust, outlier-resistant measure using k * MAD where MAD ≈ 0.6745 * stdev.
- Standard Deviation : Volatility-sensitive, calculated as k * stdev of wave over the lookback period.
- Percentile Rank : Fixed probability bands using percentile of |wave| (90% means only 10% of values exceed threshold).
• Period (2-200): Lookback for threshold calculations. Default 50.
• Multiplier (k) : Scaling for MAD/Standard Deviation modes. Default 1.5.
• Percentile (%) (0-100): For Percentile Rank mode only. Default 90%.
Parameter Interactions
• Shorter periods respond faster to cycle changes but may capture noise.
• Lower bandwidth + more harmonics = more precise decomposition but requires accurate period setting.
• Higher bandwidth is more forgiving of period mismatches.
• For strongly trending markets, restrict harmonics to 1-2 so the model tracks the dominant cycle with fewer higher-frequency components.
• For ranging/oscillating markets, more harmonics (4-6) capture complex cycles.
█ LIMITATIONS
Inherent Characteristics
• Period dependency : Effectiveness depends on correctly matching the Period parameter to actual market cycles. Use cycle measurement tools (autocorrelation, FFT, dominant cycle indicators) to identify appropriate periods.
• Stationarity assumption : The indicator assumes cycle frequencies remain relatively stable within the lookback window. Rapidly shifting dominant cycles (regime transitions) may produce inconsistent results until the buffer adapts.
• Filter lag : Despite bandpass design, some lag remains inherent to causal filtering. Higher harmonics have less lag but more noise sensitivity.
• Energy weighting artifacts : During regime changes when harmonic energy ratios shift rapidly, weighting may produce transient anomalies.
Market Conditions to Avoid
• Strong trending markets : Pure trends with no cyclicality produce weak, meandering signals. The indicator assumes cyclical market behavior.
• News events/gaps : Large discontinuities disrupt filter continuity. Requires 1-2 full periods to stabilize.
• Period mismatch : If the Period parameter doesn't match actual market cycles, harmonic extraction produces noise rather than signal.
Parameter Selection Pitfalls
• Too many harmonics : Beyond 5-6 harmonics, additional components often capture noise rather than meaningful cycles.
• Bandwidth too narrow : Very low bandwidth (< 0.05) requires extremely precise period matching; slight mismatches cause signal loss.
• Over-optimization : Perfect historical parameter fits typically fail forward. Use robust defaults across multiple instruments.
█ NOTES
Credits
This indicator applies Fourier analysis principles to financial market data, building on the extensive work of Dr. John F. Ehlers in applying digital signal processing to trading. The bandpass filter implementation and harmonic decomposition approach draw from DSP fundamentals as presented in Ehlers' publications.
For those interested in the underlying mathematics and DSP concepts:
• Ehlers, J.F. (2001). Rocket Science for Traders: Digital Signal Processing Applications . John Wiley & Sons.
• Ehlers, J.F. (2013). Cycle Analytics for Traders . John Wiley & Sons.
• Various TASC articles by John Ehlers on bandpass filters, cycle analysis, and harmonic decomposition.
by ♚@e2e4 Indicator
Voss Predictive Filter█ OVERVIEW
The Voss Predictive Filter (VPF) is a negative group delay (NGD) filter that anticipates cyclical price movement through phase compensation. The VPF isolates band-limited cyclical components via a bandpass filter, then applies negative group delay to shift the signal's phase forward, causing the output to lead the input by a fraction of the cycle period.
Based on Dr. John F. Ehlers' "Voss Predictive Filter" article in Technical Analysis of Stocks & Commodities (TASC) magazine, the VPF displays a predictive oscillator with optional dynamic threshold bands for identifying significant cycle behavior. The indicator is timeframe-agnostic - the mathematics work identically from tick charts to monthly bars, though shorter timeframes require more careful parameter selection due to noise.
█ CONCEPTS
Bandpass Filtering
A bandpass filter isolates price activity within a specific frequency range, removing both high-frequency noise and low-frequency trend drift. The VPF uses a second-order IIR (Infinite Impulse Response) bandpass filter characterized by the center frequency (the Bandpass Period input) and bandwidth. The center frequency determines which cycle period the filter emphasizes, while bandwidth controls the damping coefficient - how tightly the filter focuses around that frequency. Before filtering, the source is debiased via 2-bar momentum to remove DC offset, ensuring the filter operates around a true zero centerline.
Negative Group Delay Filtering
The predictive capability stems from negative group delay (NGD) - a filter characteristic where output appears to "lead" the input. Most causal filters introduce lag (positive group delay), but by combining the bandpass filter output with appropriately weighted past values, the VPF achieves negative group delay characteristics.
This is a universal NGD filter application for band-limited signals: the bandpass filter isolates the cyclical component of interest, then the NGD stage advances the phase within this limited frequency range to create an anticipatory output. This isn't statistical forecasting; it's phase compensation that shifts the signal's timing forward, causing peaks and troughs to appear before they occur in the bandpass output.
Negative Group Delay Stage
The NGD stage combines the current bandpass output with weighted historical values to produce an output that leads the input. By subtracting a weighted average of past deviations from a scaled version of the current filter value, the algorithm advances the signal's phase: peaks and zero-crossings in the voss output appear before the corresponding events in the bandpass filter.
The prediction order (`3 * Prediction Multiplier`) controls how many past values contribute to the phase advance. Higher orders provide smoother output but reduce the leading effect; lower orders maximize anticipation at the cost of stability.
█ INTERPRETATION
Zero-Line Crossovers
Crossings above zero suggest bullish momentum in the filtered cycle; below zero suggests bearish momentum. Crossings from near-zero regions are most reliable, as extreme excursions need time to return to equilibrium.
Threshold Bands
Threshold bands define "significant" deviation. Breaches indicate unusually strong behavior and can serve as:
• Trend confirmation when aligned with price direction
• Overbought/oversold warnings at extremes
• Trade entry filters (requiring threshold breach in the intended direction)
Threshold Mode affects sensitivity: MAD (outlier-resistant), Standard Deviation (volatility-sensitive), Percentile Rank (fixed probability bands).
Alert Conditions
Four built-in alerts trigger on bar close (no repainting): Above +Threshold (strong bullish cycle), Below -Threshold (strong bearish cycle), Above Zero (bullish phase shift), Below Zero (bearish phase shift).
█ SETTINGS & PARAMETER TUNING
Voss Predictive Filter
• Source : Price series to filter.
• Bandpass Period (1-100): Primary tuning parameter determining which cycle length the filter emphasizes. Short periods (8-15) are more responsive but noisier; medium periods (16-30) balance responsiveness and smoothness; long periods (31-100) focus on longer cycles with more smoothing.
• Bandwidth (0.01-0.45): Controls filter selectivity. Narrow bandwidths (0.01-0.15) isolate specific cycle periods precisely; medium (0.16-0.30) tolerate cycle irregularity; wide (0.31-0.45) capture broader cycle ranges. Shorter periods pair well with narrower bandwidths.
• Prediction Multiplier (2-10): Controls how many past values contribute to the phase advance. Higher values provide smoother output but reduce the leading effect; lower values maximize anticipation at the cost of stability.
Display Settings
Control visibility and colors of the Voss output, bandpass filter, and zero reference lines.
Diagnostics - Dynamic Thresholds
Three methods identify significant signal deviation:
• MAD (Median Absolute Deviation) : Robust, outlier-resistant measure using `k * MAD` where `MAD ≈ 0.6745 * stdev`.
• Standard Deviation : Volatility-sensitive, calculated as `k * stdev` of Voss over the lookback period.
• Percentile Rank : Fixed probability bands using the percentile of |Voss| (e.g., 90% means only 10% of values exceed threshold).
Settings:
• Dynamic Threshold : Toggle threshold bands and set colors.
• Threshold Mode : Select MAD, Standard Deviation, or Percentile Rank.
• Period (2-200): Lookback for threshold calculations. Default 50.
• Multiplier (k) : Scaling for MAD/Standard Deviation modes. Default 1.5.
• Percentile (%) (0-100): For Percentile Rank mode only. Default 90%.
█ LIMITATIONS
Inherent Characteristics
• Residual lag : Despite negative group delay design, some lag remains relative to price action.
• Cyclical markets required : Performs best on instruments with clear cyclical components. Strongly trending markets with little cyclicality produce less useful signals.
• Signal interpretation : Absolute Voss values are instrument-specific. Always interpret relative to adaptive threshold bands, not fixed levels.
Market Conditions to Avoid
• Sudden news events/gaps : Major discontinuities disrupt cycle continuity, causing erratic signals. Requires 1-2 full cycle periods to re-stabilize.
• Low volume/illiquid markets : Sporadic trading produces false cycles from liquidity artifacts. Use only on actively traded instruments during liquid hours.
• Regime changes : During cyclical ↔ trending transitions, watch for persistent extremes without mean reversion, increasing price/indicator divergence, or unresolved threshold breaches.
Parameter Selection Pitfalls
• Mismatched period : If Bandpass Period doesn't match actual market cycles, the filter produces weak signals. Use cycle measurement tools (FFT, autocorrelation, Dominant Cycle) to identify appropriate periods first.
• Overoptimization : Perfect historical fits typically fail forward. Choose robust parameters that work across multiple instruments and timeframes.
█ NOTES
Credits
This indicator is based on concepts from Dr. John F. Ehlers' work on predictive filters and bandpass techniques for technical analysis. Dr. Ehlers has published extensively on applying digital signal processing methods to financial markets in Technical Analysis of Stocks & Commodities (TASC) magazine. His articles on bandpass filters and predictive techniques, particularly the Voss Predictive Filter concept, provided the theoretical foundation for this implementation.
For those interested in the underlying mathematics and DSP concepts:
• Ehlers, J.F. (2001). Rocket Science for Traders: Digital Signal Processing Applications . John Wiley & Sons.
• Various TASC articles by John Ehlers on bandpass filters, cycle analysis, and predictive filtering techniques.
• Ehlers, J.F. "Voss Predictive Filter" - Technical Analysis of Stocks & Commodities magazine.
by ♚@e2e4 Indicator
