Rolling SSA Oscillator [LuxAlgo]The Rolling SSA Oscillator indicator is a cycle-analysis tool that utilizes Singular Spectrum Analysis (SSA) to decompose price action into its most significant periodic components, providing a real-time view of underlying market rhythms. Unlike traditional lagging oscillators, this script uses eigendecomposition to isolate dominant trends and noise-reduced oscillations for better market timing.
🔶 USAGE
The indicator provides two primary components derived from the price's spectral signature: a Long-Term Periodic component and a Short-Term Periodic component. These can be used to identify trend direction, cyclical reversals, and momentum exhaustion.
🔹 Trading Signals
Trend Direction: When the Long-Term Periodic component (solid line) is above the zero level and colored green, the primary underlying cycle is in an upward phase. Conversely, a red line below zero indicates a downward phase.
Cycle Crosses: Traders can look for the Short-Term Periodic component (dotted line) crossing the Long-Term component or the zero line to anticipate shorter-term shifts in momentum.
Normalization: When the "Normalize" setting is enabled, the components are scaled relative to their combined absolute magnitude. This is particularly useful for identifying extreme cycle peaks regardless of absolute price volatility.
🔶 DETAILS
Singular Spectrum Analysis (SSA) is a powerful non-parametric technique used in time-series analysis. This indicator implements a rolling version of SSA through the following mathematical steps:
Embedding: The price data is mapped into a trajectory matrix using the "Window" length defined in the settings.
Decomposition: A covariance matrix is computed, followed by eigendecomposition to find the eigenvalues and eigenvectors.
Grouping & Reconstruction: The eigenvectors are sorted by their energy (eigenvalues). The script specifically reconstructs the first two components to form the Long-Term trend/cycle and the subsequent two components to form the Short-Term cycle.
Because this script uses matrix.eigenvalues() and matrix.eigenvectors() , it requires significant computation. The "Window" input determines the "resolution" of the cycles; a larger window can capture longer-term rhythms but increases the lag and computational load.
🔶 SETTINGS
Window: Controls the embedding dimension (L). This defines the maximum cycle length the indicator can effectively resolve.
Long Term Periodic: Toggles the visibility of the primary trend-following cycle (Components 1 & 2).
Short Term Periodic: Toggles the visibility of the faster, more reactive cycle (Components 3 & 4).
Normalize: If enabled, adjusts the output so that the combined amplitude of both components stays within a consistent range, making it easier to spot cyclical extremes.
🔹 Dashboard
Dashboard: Toggles the on-screen statistics table.
Position: Determines where the dashboard is displayed (Top Right, Bottom Right, or Bottom Left).
Size: Adjusts the text size within the dashboard.
Indicator
Singular Spectrum Decomposition [LuxAlgo]The Singular Spectrum Decomposition indicator is a powerful analytical tool that decomposes price action into distinct, interpretable components—Trend, Periodic cycles, and Noise—using the Singular Spectrum Analysis (SSA) methodology.
It provides traders with a clear view of underlying market structures and offers a jump-free, extrapolated trend forecast based on Linear Recurrence Relations (LRR).
Warning: This decomposition is displayed retrospectively ; historical values observed are subject to repainting .
🔶 USAGE
The indicator operates by analyzing a specific window of recent price data to extract its most significant internal dynamics. It splits the "messy" raw price into four visual layers:
Trend (Overlay): The primary low-frequency component, plotted directly on the price chart. This represents the core directional bias of the asset.
Long Term Periodic (P1): The most dominant cyclical component, typically representing major swings or seasonalities.
Short Term Periodic (P2): The second most dominant cycle, capturing faster oscillations and intermediate pullbacks.
Noise: The high-frequency residual data that lacks a consistent pattern, useful for identifying market volatility or "washout" periods.
🔹 Cycle Exhaustion (P1/P2 Extremes)
Traders can monitor the separate indicator pane to identify when cyclical components (P1 and P2) reach historical extremes. When the Long Term Periodic (P1) line begins to curve back toward the zero line after a prolonged extension, it often signals "cycle exhaustion," suggesting that the current swing is losing momentum and a reversal or consolidation may be imminent.
🔹 Trend-Forecast Confluence & Mean Reversion
The dashed Trend extrapolation acts as a projected path for the market's core bias. If the current market price is significantly far from the solid Trend line while the forecast indicates a flattening or reversal, traders can look for mean-reversion opportunities. A price returning to a rising Trend forecast confirms the trend's strength, while a price crossing through a flat Trend forecast suggests a structural shift.
🔹 Timing Entries with Dashboard Metrics
The "Average Period" displayed on the dashboard provides a mathematical blueprint for entry timing. For example, if the Short Term (P2) Average Period is 20 bars, a trader might look for long entries approximately 10 bars after a peak (the expected trough). By aligning these peak-to-trough measurements with the Trend's direction, users can improve the precision of their entries within a trending market.
🔹 Filtering Fakeouts with the Noise Component
The Noise component helps distinguish between high-conviction moves and market "static." A sharp price breakout accompanied by a relatively flat Noise component suggests a sustainable, structurally supported move. Conversely, if a breakout occurs while the Noise component is spiking aggressively, it may indicate a "washout" or a liquidity-driven fakeout that lacks a fundamental trend shift.
🔶 DETAILS
The script implements a full SSA pipeline: Embedding (creating a trajectory matrix), Singular Value Decomposition (via eigendecomposition of the covariance matrix), and Diagonal Averaging (reconstructed the time series).
🔹 Jump-Free Extrapolation
A common issue with LRR-based forecasts is a vertical "jump" at the connection point between historical data and the forecast. This tool solves this by calculating the relative deltas of the LRR projection and anchoring them to the final value of the smoothed SSA reconstruction. This ensures a seamless visual transition while maintaining the mathematical integrity of the projected trajectory.
🔹 Dashboard Metrics
The indicator includes a real-time dashboard that calculates the "Average Period" of the periodic components using zero-crossing detection. This allows traders to quantify the frequency of cycles (e.g., a 40-bar cycle vs. a 15-bar cycle) without manual measurement.
🔶 SETTINGS
Window Length (L): The embedding window. Larger values capture longer cycles and provide a smoother trend, but may increase lag in the decomposition.
Buffer Length (N): The number of recent bars used for the static decomposition.
Forecast Length: The number of bars to extrapolate the Trend component into the future.
Show Trend on Price: Toggles the visibility of the reconstructed trend line on the main chart.
Show Periodic/Noise: Toggles the visibility of the individual sub-components in the indicator pane.
Show Extrapolation: Enables or disables the dashed forecast line for the trend.
Dashboard Settings: Controls the visibility, position, and size of the metrics table.
Indicator
Harmonic Frequency Visualizer [BackQuant]Harmonic Frequency Visualizer
Overview
Harmonic Frequency Visualizer is a cycle-analysis and cross-asset resonance tool that uses a simplified Discrete Fourier Transform (DFT) to measure how strongly specific cycle periods are present in price. It is not a “trend indicator” and it is not trying to predict direction by itself. Its job is to quantify rhythm: which repeating periods (in bars) are currently dominant, whether those cycles are expanding or contracting (phase direction), and whether multiple instruments are sharing the same dominant periods at the same time (resonance).
This indicator has two main output modes:
Spectrum : a frequency “snapshot” showing amplitude at each tested period for up to five instruments.
Spectrogram : a history heatmap showing how the spectrum evolves through time (for the chart instrument).
Spectrum
Spectrogram
On top of that, it produces a Dominant Cycle Oscillator derived from the dominant cycle’s phase, which gives a continuous cycle position metric (peak/trough style zones) without repainting.
This is designed for traders who want cycle context the same way they want volatility context: not as a magic signal, but as structure.
What “frequency” and “cycles” mean in trading terms
A cycle period (say 21 bars) means: “a repeating pattern that tends to complete one full oscillation every 21 bars.” If price contains such a pattern, the DFT will detect a strong correlation between price and a 21-bar sine/cosine wave.
Markets do not have perfectly stable periodic motion, but they often show:
Mean-reverting swings around value.
Trend pulses with pullback cadence.
Volatility clustering that creates rhythmic expansions and contractions.
Cycle tools are trying to measure those repeating components, and DFT is the standard mathematical way to do it.
Where DFT comes from (the core idea)
The Discrete Fourier Transform comes from Fourier analysis, a foundational signal processing concept:
Fourier’s idea : any sufficiently well-behaved signal can be expressed as a sum of sine and cosine waves at different frequencies, each with:
An amplitude (how strong that wave is).
A phase (where you are within the wave cycle).
In continuous math you get the Fourier Transform. In sampled data (like candles) you use the Discrete Fourier Transform. It converts a time series (price over time) into a frequency description (strength of different cycles).
In markets:
Time domain: candles and price series.
Frequency domain: cycle periods and their strengths.
Why sine and cosine, not just sine
A sine wave alone cannot represent every phase alignment cleanly. DFT uses both cosine and sine components because together they form an orthogonal basis that can represent any phase shift.
You can think of it like this:
Cosine component captures “in-phase” alignment with the cycle.
Sine component captures “quadrature” (90-degree shifted) alignment.
Combining them gives full information: amplitude + phase.
Mathematically, a single frequency component can be written as:
A * cos(ωt + φ)
But DFT estimates A and φ by separately accumulating cosine and sine projections.
How this script implements the DFT (and what it is actually measuring)
This is not a full-spectrum FFT across every frequency. It is a targeted DFT across a fixed set of cycle periods:
Tested periods
The script tests 8 predefined periods:
5, 8, 13, 21, 34, 55, 89, 120
These are Fibonacci-like cycle candidates commonly used in cycle/market structure work. The point is not that Fibonacci is magic. The point is that these represent a reasonable spread from short to long rhythms without needing hundreds of frequencies (which would be heavy in Pine).
Normalization step (important)
Before computing the DFT, the script normalizes the series:
mn = SMA(src, lookback)
sd = stdev(src, lookback)
norm = (src - mn) / sd (if sd != 0)
Why normalize:
DFT amplitude depends on the scale of the input series.
If you compare BTC and TLT raw prices, the magnitude is meaningless.
Z-score normalization makes amplitude more comparable across instruments and regimes.
So the spectrum is measuring “cyclical structure in standardized deviations,” not raw dollars.
Projection onto cosine and sine
For each tested period P:
ω = 2π / P (angular frequency for that period)
Compute:
- sCos = Σ(norm * cos(ωk))
- sSin = Σ(norm * sin(ωk))
Interpretation:
You are correlating the last window of normalized price with a cosine wave of period P.
And also correlating it with a sine wave of period P.
If the price has a strong P-bar rhythm, these sums grow in magnitude.
Window length detail
The script uses:
window = min(lookback - 1, 99)
So even if lookback is 200, the internal DFT accumulation caps at 100 bars for performance stability. This is a deliberate trade: stable computation in Pine, while still letting you define normalization lookback and overall context.
Amplitude computation
Once sCos and sSin are computed:
raw magnitude = sqrt(sCos² + sSin²)
This is the length of the vector (sCos, sSin). That vector length is the standard way to combine the orthogonal components into one strength metric.
Then it scales it into a 0–100 “display amplitude”:
amp = sqrt(sCos² + sSin²) / lookback * 100 * sensitivity
amp is capped to 100
So:
Higher amplitude means stronger alignment with that cycle period.
Sensitivity is a user control to amplify or damp the display scaling.
Important: amplitude here is not a probability, and it is not guaranteed “signal quality.” It is a standardized “how much of that cycle exists in the recent window” metric.
Phase computation
Phase is computed using atan2(sSin, sCos). That matters because:
A simple atan(sin/cos) fails in different quadrants.
atan2 correctly resolves the angle from -π to +π.
Phase tells you where you are within the cycle:
Two cycles can have same amplitude but opposite phase.
Phase is what lets you infer “approaching peak vs trough” behavior.
Dominant cycle selection
The script chooses the dominant cycle as the period with the highest amplitude among the tested periods:
domIdx = argmax(amp )
domAmp = max amplitude
domPhase = phase at domIdx
This dominant cycle is used for:
Spectrogram history matrix (chart symbol).
Dominant cycle oscillator.
Data window outputs (dominant period, oscillator value).
Spectrum View: what you see and how to read it
In Spectrum mode, the indicator draws a frequency snapshot for up to five instruments. Each instrument gets a spectrum line (or bars/area depending on style) plotted across the 8 periods on the x-axis, with amplitude (0–100) on the y-axis.
X-axis meaning
Each x position corresponds to a period (5 → 120 bars). You are not looking at “frequency in Hz.” You are looking at “period in bars,” which is more intuitive in trading.
Y-axis meaning
Amplitude is a scaled measure of how strongly that period is present in the recent normalized data. Higher means stronger.
Plot styles
Waveform: connects amplitude points into a continuous shape, best for seeing spectrum shape.
Bars: draws vertical bars per period, best for quick comparison.
Area: similar to waveform but filled toward baseline for emphasis.
Dominant peaks and phase direction labels
The script highlights dominant cycles per symbol (if enabled):
If max amplitude > 20, it labels that peak with the symbol name.
If Show Phase Direction is enabled, it appends ▲ or ▼.
Phase direction logic:
rising = sin(phase) < 0
▲ means cycle is in a “rising” phase segment
▼ means cycle is in a “falling” phase segment
This is not “price will rise now.” It is “the dominant cycle’s instantaneous phase suggests you are on the upward vs downward half of that oscillation.” In real markets, you use this as context, not as a standalone trade trigger.
It also draws small ▲/▼ markers on secondary peaks (amp > 15) to show phase direction of other meaningful cycles, giving you a richer picture than “one dominant period.”
Resonance Zones: cross-asset harmonic alignment
Resonance is where this tool becomes more than a single-chart curiosity.
What resonance means here
A resonance zone is flagged when at least 3 out of 5 instruments have strong amplitude at the same tested period. Mechanically:
For each period i:
- Count instruments with amp > 30
- If count >= 3, mark resonance at that period
When resonance is detected:
A vertical highlight box is drawn behind that period.
A ⚡ marker is printed at the top.
Interpretation:
Multiple assets are expressing a similar cycle length at the same time.
This can indicate macro rhythm, shared liquidity timing, or cross-market synchronization.
This is especially useful when your instrument set includes:
Rates proxy (TLT), commodities (oil, gold), and crypto indices.
You can visually spot when markets are “vibrating” together at a shared period.
Resonance is not automatically bullish or bearish. It is telling you “cycle length agreement,” which can help with timing models and contextual trade planning.
Spectrogram View: frequency over time
Spectrum mode is a snapshot. Spectrogram mode adds time evolution.
What a spectrogram is
A spectrogram is a 2D heatmap where:
Rows = different periods (frequency bands).
Columns = time history (bars ago → now).
Color = amplitude strength.
This allows you to see:
Which cycles are persistent vs fleeting.
When dominant cycle shifts occur (energy moves from one period to another).
Cycle regime transitions (short cycles dominating in chop vs longer cycles dominating in trend).
How the script builds the spectrogram matrix
It maintains a matrix with:
NUM_PERIODS rows (8 periods)
histBars columns (history length)
Each bar:
Remove the oldest column.
Append the newest amplitude array from chartSpec.
So the spectrogram is always a rolling history of the chart symbol’s cycle amplitudes. It does not attempt to store five symbols (too heavy), it focuses on the active chart for time evolution.
Heat coloring
Amplitude values map to a custom gradient:
Low = dark blue
Mid = blue/cyan to orange
High = yellow
This makes dominant energy bands visually obvious. A stable bright band means persistent cycle dominance.
Dominant Cycle Oscillator: phase mapped to a 0–100 oscillator
The oscillator is derived from the dominant cycle phase (chart symbol):
oscRaw = cos(domPhase)
oscValue = 50 + 50 * oscRaw (maps -1..1 into 0..100)
Interpretation:
When cos(phase) ≈ +1, oscillator near 100 (cycle peak zone).
When cos(phase) ≈ -1, oscillator near 0 (cycle trough zone).
Midline 50 corresponds to the quarter-cycle transition points.
It also colors the oscillator by phase direction:
oscRising = sin(domPhase) < 0
Rising phase = green-ish
Falling phase = red-ish
This gives you a clean timing reference:
The dominant period tells you the cycle length.
The oscillator tells you where you are within that cycle.
It is not forecasting price. It is telling you the current phase position of the strongest detected cycle component.
Alerts and practical timing usage
Alerts are based on the oscillator:
Cross above 80: dominant cycle entering peak zone.
Cross below 20: dominant cycle entering trough zone.
Cross 50: midline cross (phase transition).
In practice, you use these as “timing context” alerts, for example:
If your trend model is bullish and cycle oscillator enters trough zone, it can hint at a favorable pullback timing window.
If you are mean-reversion trading and cycle peak zone aligns with resistance, that confluence matters.
Again: cycle timing needs structure confirmation. The oscillator alone is not a trade system.
Multi-instrument design and non-repaint behavior
The indicator requests five external instruments via request.security. It uses:
close with lookahead_on
This forces the data to be “previous confirmed close” so the spectral calculations do not repaint intra-bar. That matters because cycle measures can change drastically within a bar if you let them use live values.
So:
Spectra for external symbols are based on confirmed historical closes.
Chart symbol spectrogram and oscillator are also stable in the sense they depend on confirmed series values (dominant phase updates bar-to-bar).
Key parameters and how they change behavior
Analysis Lookback
Affects normalization and the DFT window cap:
Higher lookback stabilizes mean/stdev normalization and reduces random shifts.
Lower lookback makes the tool more reactive but more prone to regime noise.
Because the inner DFT accumulation caps at 100 bars, very high lookback mostly affects normalization rather than the raw projection length.
Sensitivity
Scales displayed amplitude:
Higher sensitivity makes peaks stand out more.
Lower sensitivity compresses amplitude.
It is a display control, not a physics constant.
View Mode
Spectrum: cross-asset snapshot comparison, resonance detection.
Spectrogram: time evolution of cycle energy for chart symbol.
Show Phase Direction
Adds ▲/▼ markers derived from sin(phase). Useful for quick cycle position intuition, but do not treat ▲ as “buy.”
Show Resonance Zones
Marks periods where many instruments share strong energy. Useful for macro rhythm alignment.
Highlight Dominant Cycles
Labels peaks. If you disable it, the chart becomes cleaner but less informative.
Spectrogram History
Controls how many columns are stored. Higher makes a longer heatmap but costs more drawing.
Limitations and what not to assume
This tool is honest DSP applied to market data, but market data is not a stationary sine wave generator. Key limitations:
Cycles drift. Dominant period can shift as regime changes.
The tool only tests 8 candidate periods. If the true dominant period is 30, it will express as energy near 34 or distributed across neighbors.
Normalization helps comparability, but does not make amplitude “absolute truth.”
DFT assumes a stable frequency over the window. Markets often violate that.
Phase-based oscillators are timing aids, not predictors.
This is why the indicator is best used as:
Context for entries/exits, not a standalone system.
A way to see when cycle energy concentrates or disperses.
A way to detect when multiple markets share a timing rhythm.
How to use it properly (workflows)
1) Cycle regime identification
If short periods (5–13) dominate, market is often choppy, reactive, and mean-reverting.
If mid periods (21–55) dominate, market often shows swing structure.
If long periods (89–120) dominate, market can be in slower macro drift, trend legs, or compressed volatility regimes.
2) Timing layer for an existing strategy
Use your trend model to decide direction.
Use dominant cycle oscillator to decide timing within that direction.
Use spectrogram to avoid trading when dominant period is unstable or flipping rapidly.
3) Cross-asset confirmation
If you see resonance at a period, watch whether your main instrument is also showing strength there.
Resonance can justify holding a cycle-based timing thesis with more confidence because it is not isolated.
4) Expectation management
If the spectrum is flat (no peaks above threshold), that is information:
No clean dominant cycle, randomness dominates.
Cycle-based timing will be unreliable.
Summary
Harmonic Frequency Visualizer uses a targeted Discrete Fourier Transform across predefined cycle periods to measure amplitude and phase of cyclical components in price. It supports multi-instrument spectrum comparison, resonance detection when several markets share strong energy at the same periods, and a spectrogram heatmap for the chart instrument showing how cycle dominance evolves over time. A dominant cycle oscillator maps phase into a 0–100 timing readout with alerts for peak/trough/midline transitions. It is a cycle context engine designed to complement trend, structure, and risk models, not replace them. Indicator
SpectrumLibrary "Spectrum"
This library includes spectrum analysis tools such as the Fast Fourier Transform (FFT).
method toComplex(data, polar)
Creates an array of complex type objects from a float type array.
Namespace types: array
Parameters:
data (array) : The float type array of input data.
polar (bool) : Initialization coordinates; the default is false (cartesian).
Returns: The complex type array of converted data.
method sAdd(data, value, end, start, step)
Performs scalar addition of a given float type array and a simple float value.
Namespace types: array
Parameters:
data (array) : The float type array of input data.
value (float) : The simple float type value to be added.
end (int) : The last index of the input array (exclusive) on which the operation is performed.
start (int) : The first index of the input array (inclusive) on which the operation is performed; the default value is 0.
step (int) : The step by which the function iterates over the input data array between the specified boundaries; the default value is 1.
Returns: The modified input array.
method sMult(data, value, end, start, step)
Performs scalar multiplication of a given float type array and a simple float value.
Namespace types: array
Parameters:
data (array) : The float type array of input data.
value (float) : The simple float type value to be added.
end (int) : The last index of the input array (exclusive) on which the operation is performed.
start (int) : The first index of the input array (inclusive) on which the operation is performed; the default value is 0.
step (int) : The step by which the function iterates over the input data array between the specified boundaries; the default value is 1.
Returns: The modified input array.
method eMult(data, data02, end, start, step)
Performs elementwise multiplication of two given complex type arrays.
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : the first complex type array of input data.
data02 (array type from RezzaHmt/Complex/1) : The second complex type array of input data.
end (int) : The last index of the input arrays (exclusive) on which the operation is performed.
start (int) : The first index of the input arrays (inclusive) on which the operation is performed; the default value is 0.
step (int) : The step by which the function iterates over the input data array between the specified boundaries; the default value is 1.
Returns: The modified first input array.
method eCon(data, end, start, step)
Performs elementwise conjugation on a given complex type array.
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : The complex type array of input data.
end (int) : The last index of the input array (exclusive) on which the operation is performed.
start (int) : The first index of the input array (inclusive) on which the operation is performed; the default value is 0.
step (int) : The step by which the function iterates over the input data array between the specified boundaries; the default value is 1.
Returns: The modified input array.
method zeros(length)
Creates a complex type array of zeros.
Namespace types: series int, simple int, input int, const int
Parameters:
length (int) : The size of array to be created.
method bitReverse(data)
Rearranges a complex type array based on the bit-reverse permutations of its size after zero-padding.
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : The complex type array of input data.
Returns: The modified input array.
method R2FFT(data, inverse)
Calculates Fourier Transform of a time series using Cooley-Tukey Radix-2 Decimation in Time FFT algorithm, wikipedia.org
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : The complex type array of input data.
inverse (int) : Set to -1 for FFT and to 1 for iFFT.
Returns: The modified input array containing the FFT result.
method LBFFT(data, inverse)
Calculates Fourier Transform of a time series using Leo Bluestein's FFT algorithm, wikipedia.org This function is nearly 4 times slower than the R2FFT function in practice.
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : The complex type array of input data.
inverse (int) : Set to -1 for FFT and to 1 for iFFT.
Returns: The modified input array containing the FFT result.
method DFT(data, inverse)
This is the original DFT algorithm. It is not suggested to be used regularly.
Namespace types: array
Parameters:
data (array type from RezzaHmt/Complex/1) : The complex type array of input data.
inverse (int) : Set to -1 for DFT and to 1 for iDFT.
Returns: The complex type array of DFT result. Library
[Excalibur] Ehlers AutoCorrelation Periodogram ModifiedKeep your coins folks, I don't need them, don't want them. If you wish be generous, I do hope that charitable peoples worldwide with surplus food stocks may consider stocking local food banks before stuffing monetary bank vaults, for the crusade of remedying the needs of less than fortunate children, parents, elderly, homeless veterans, and everyone else who deserves nutritional sustenance for the soul.
DEDICATION:
This script is dedicated to the memory of Nikolai Dmitriyevich Kondratiev (Никола́й Дми́триевич Кондра́тьев) as tribute for being a pioneering economist and statistician, paving the way for modern econometrics by advocation of rigorous and empirical methodologies. One of his most substantial contributions to the study of business cycle theory include a revolutionary hypothesis recognizing the existence of dynamic cycle-like phenomenon inherent to economies that are characterized by distinct phases of expansion, stagnation, recession and recovery, what we now know as "Kondratiev Waves" (K-waves). Kondratiev was one of the first economists to recognize the vital significance of applying quantitative analysis on empirical data to evaluate economic dynamics by means of statistical methods. His understanding was that conceptual models alone were insufficient to adequately interpret real-world economic conditions, and that sophisticated analysis was necessary to better comprehend the nature of trending/cycling economic behaviors. Additionally, he recognized prosperous economic cycles were predominantly driven by a combination of technological innovations and infrastructure investments that resulted in profound implications for economic growth and development.
I will mention this... nation's economies MUST be supported and defended to continuously evolve incrementally in order to flourish in perpetuity OR suffer through eras with lasting ramifications of societal stagnation and implosion.
Analogous to the realm of economics, aperiodic cycles/frequencies, both enduring and ephemeral, do exist in all facets of life, every second of every day. To name a few that any blind man can naturally see are: heartbeat (cardiac cycles), respiration rates, circadian rhythms of sleep, powerful magnetic solar cycles, seasonal cycles, lunar cycles, weather patterns, vegetative growth cycles, and ocean waves. Do not pretend for one second that these basic aforementioned examples do not affect business cycle fluctuations in minuscule and monumental ways hour to hour, day to day, season to season, year to year, and decade to decade in every nation on the planet. Kondratiev's original seminal theories in macroeconomics from nearly a century ago have proven remarkably prescient with many of his antiquated elementary observations/notions/hypotheses in macroeconomics being scholastically studied and topically researched further. Therefore, I am compelled to honor and recognize his statistical insight and foresight.
If only.. Kondratiev could hold a pocket sized computer in the cup of both hands bearing the PulseWire logo and platform services, I truly believe he would be amazed in marvelous delight with a GARGANTUAN smile on his face.
INTRODUCTION:
Firstly, this is NOT technically speaking an indicator like most others. I would describe it as an advanced cycle period detector to obtain market data spectral estimates with low latency and moderate frequency resolution. Developers can take advantage of this detector by creating scripts that utilize a "Dominant Cycle Source" input to adaptively govern algorithms. Be forewarned, I would only recommend this for advanced developers, not novice code dabbling. Although, there is some Pine wizardry introduced here for novice Pine enthusiasts to witness and learn from. AI did describe the code into one super-crunched sentence as, "a rare feat of exceptionally formatted code masterfully balancing visual clarity, precision, and complexity to provide immense educational value for both programming newcomers and expert Pine coders alike."
Understand all of the above aforementioned? Buckle up and proceed for a lengthy read of verbose complexity...
This is my enhanced and heavily modified version of autocorrelation periodogram (ACP) for Pine Script v5.0. It was originally devised by the mathemagician John Ehlers for detecting dominant cycles (frequencies) in an asset's price action. I have been sitting on code similar to this for a long time, but I decided to unleash the advanced code with my fashion. Originally Ehlers released this with multiple versions, one in a 2016 TASC article and the other in his last published 2013 book "Cycle Analytics for Traders", chapter 8. He wasn't joking about "concepts of advanced technical trading" and ACP is nowhere near to his most intimidating and ingenious calculations in code. I will say the book goes into many finer details about the original periodogram, so if you wish to delve into even more elaborate info regarding Ehlers' original ACP form AND how you may adapt algorithms, you'll have to obtain one. Note to reader, comparing Ehlers' original code to my chimeric code embracing the "Power of Pine", you will notice they have little resemblance.
What you see is a new species of autocorrelation periodogram combining Ehlers' innovation with my fascinations of what ACP could be in a Pine package. One other intention of this script's code is to pay homage to Ehlers' lifelong works. Like Kondratiev, Ehlers is also a hardcore cycle enthusiast. I intend to carry on the fire Ehlers envisioned and I believe that is literally displayed here as a pleasant "fiery" example endowed with Pine. With that said, I tried to make the code as computationally efficient as possible, without going into dozens of more crazy lines of code to speed things up even more. There's also a few creative modifications I made by making alterations to the originating formulas that I felt were improvements, one of them being lag reduction. By recently questioning every single thing I thought I knew about ACP, combined with the accumulation of my current knowledge base, this is the innovative revision I came up with. I could have improved it more but decided not to mind thrash too many TV members, maybe later...
I am now confident Pine should have adequate overhead left over to attach various indicators to the dominant cycle via input.source(). TV, I apologize in advance if in the future a server cluster combusts into a raging inferno... Coders, be fully prepared to build entire algorithms from pure raw code, because not all of the built-in Pine functions fully support dynamic periods (e.g. length=ANYTHING). Many of them do, as this was requested and granted a while ago, but some functions are just inherently finicky due to implementation combinations and MUST be emulated via raw code. I would imagine some comprehensive library or numerous authored scripts have portions of raw code for Pine built-ins some where on TV if you look diligently enough.
Notice: Unfortunately, I will not provide any integration support into member's projects at all. I have my own projects that require way too much of my day already. While I was refactoring my life (forgoing many other "important" endeavors) in the early half of 2023, I primarily focused on this code over and over in my surplus time. During that same time I was working on other innovations that are far above and beyond what this code is. I hope you understand.
The best way programmatically may be to incorporate this code into your private Pine project directly, after brutal testing of course, but that may be too challenging for many in early development. Being able to see the periodogram is also beneficial, so input sourcing may be the "better" avenue to tether portions of the dominant cycle to algorithms. Unique indication being able to utilize the dominantCycle may be advantageous when tethering this script to those algorithms. The easiest way is to manually set your indicators to what ACP recognizes as the dominant cycle, but that's actually not considered dynamic real time adaption of an indicator. Different indicators may need a proportion of the dominantCycle, say half it's value, while others may need the full value of it. That's up to you to figure that out in practice. Sourcing one or more custom indicators dynamically to one detector's dominantCycle may require code like this: `int sourceDC = int(math.max(6, math.min(49, input.source(close, "Dominant Cycle Source"))))`. Keep in mind, some algos can use a float, while algos with a for loop require an integer.
I have witnessed a few attempts by talented TV members for a Pine based autocorrelation periodogram, but not in this caliber. Trust me, coding ACP is no ordinary task to accomplish in Pine and modifying it blessed with applicable improvements is even more challenging. For over 4 years, I have been slowly improving this code here and there randomly. It is beautiful just like a real flame, but... this one can still burn you! My mind was fried to charcoal black a few times wrestling with it in the distant past. My very first attempt at translating ACP was a month long endeavor because PSv3 simply didn't have arrays back then. Anyways, this is ACP with a newer engine, I hope you enjoy it. Any TV subscriber can utilize this code as they please. If you are capable of sufficiently using it properly, please use it wisely with intended good will. That is all I beg of you.
Lastly, you now see how I have rasterized my Pine with Ehlers' swami-like tech. Yep, this whole time I have been using hline() since PSv3, not plot(). Evidently, plot() still has a deficiency limited to only 32 plots when it comes to creating intense eye candy indicators, the last I checked. The use of hline() is the optimal choice for rasterizing Ehlers styled heatmaps. This does only contain two color schemes of the many I have formerly created, but that's all that is essentially needed for this gizmo. Anything else is generally for a spectacle or seeing how brutal Pine can be color treated. The real hurdle is being able to manipulate colors dynamically with Merlin like capabilities from multiple algo results. That's the true challenging part of these heatmap contraptions to obtain multi-colored "predator vision" level indication. You now have basic hline() food for thought empowerment to wield as you can imaginatively dream in Pine projects.
PERIODOGRAM UTILITY IN REAL WORLD SCENARIOS:
This code is a testament to the abilities that have yet to be fully realized with indication advancements. Periodograms, spectrograms, and heatmaps are a powerful tool with real-world applications in various fields such as financial markets, electrical engineering, astronomy, seismology, and neuro/medical applications. For instance, among these diverse fields, it may help traders and investors identify market cycles/periodicities in financial markets, support engineers in optimizing electrical or acoustic systems, aid astronomers in understanding celestial object attributes, assist seismologists with predicting earthquake risks, help medical researchers with neurological disorder identification, and detection of asymptomatic cardiovascular clotting in the vaxxed via full body thermography. In either field of study, technologies in likeness to periodograms may very well provide us with a better sliver of analysis beyond what was ever formerly invented. Periodograms can identify dominant cycles and frequency components in data, which may provide valuable insights and possibly provide better-informed decisions. By utilizing periodograms within aspects of market analytics, individuals and organizations can potentially refrain from making blinded decisions and leverage data-driven insights instead.
PERIODOGRAM INTERPRETATION:
The periodogram renders the power spectrum of a signal, with the y-axis representing the periodicity (frequencies/wavelengths) and the x-axis representing time. The y-axis is divided into periods, with each elevation representing a period. In this periodogram, the y-axis ranges from 6 at the very bottom to 49 at the top, with intermediate values in between, all indicating the power of the corresponding frequency component by color. The higher the position occurs on the y-axis, the longer the period or lower the frequency. The x-axis of the periodogram represents time and is divided into equal intervals, with each vertical column on the axis corresponding to the time interval when the signal was measured. The most recent values/colors are on the right side.
The intensity of the colors on the periodogram indicate the power level of the corresponding frequency or period. The fire color scheme is distinctly like the heat intensity from any casual flame witnessed in a small fire from a lighter, match, or camp fire. The most intense power would be indicated by the brightest of yellow, while the lowest power would be indicated by the darkest shade of red or just black. By analyzing the pattern of colors across different periods, one may gain insights into the dominant frequency components of the signal and visually identify recurring cycles/patterns of periodicity.
SETTINGS CONFIGURATIONS BRIEFLY EXPLAINED:
Source Options: These settings allow you to choose the data source for the analysis. Using the `Source` selection, you may tether to additional data streams (e.g. close, hlcc4, hl2), which also may include samples from any other indicator. For example, this could be my "Chirped Sine Wave Generator" script found in my member profile. By using the `SineWave` selection, you may analyze a theoretical sinusoidal wave with a user-defined period, something already incorporated into the code. The `SineWave` will be displayed over top of the periodogram.
Roofing Filter Options: These inputs control the range of the passband for ACP to analyze. Ehlers had two versions of his highpass filters for his releases, so I included an option for you to see the obvious difference when performing a comparison of both. You may choose between 1st and 2nd order high-pass filters.
Spectral Controls: These settings control the core functionality of the spectral analysis results. You can adjust the autocorrelation lag, adjust the level of smoothing for Fourier coefficients, and control the contrast/behavior of the heatmap displaying the power spectra. I provided two color schemes by checking or unchecking a checkbox.
Dominant Cycle Options: These settings allow you to customize the various types of dominant cycle values. You can choose between floating-point and integer values, and select the rounding method used to derive the final dominantCycle values. Also, you may control the level of smoothing applied to the dominant cycle values.
DOMINANT CYCLE VALUE SELECTIONS:
External to the acs() function, the code takes a dominant cycle value returned from acs() and changes its numeric form based on a specified type and form chosen within the indicator settings. The dominant cycle value can be represented as an integer or a decimal number, depending on the attached algorithm's requirements. For example, FIR filters will require an integer while many IIR filters can use a float. The float forms can be either rounded, smoothed, or floored. If the resulting value is desired to be an integer, it can be rounded up/down or just be in an integer form, depending on how your algorithm may utilize it.
AUTOCORRELATION SPECTRUM FUNCTION BASICALLY EXPLAINED:
In the beginning of the acs() code, the population of caches for precalculated angular frequency factors and smoothing coefficients occur. By precalculating these factors/coefs only once and then storing them in an array, the indicator can save time and computational resources when performing subsequent calculations that require them later.
In the following code block, the "Calculate AutoCorrelations" is calculated for each period within the passband width. The calculation involves numerous summations of values extracted from the roofing filter. Finally, a correlation values array is populated with the resulting values, which are normalized correlation coefficients.
Moving on to the next block of code, labeled "Decompose Fourier Components", Fourier decomposition is performed on the autocorrelation coefficients. It iterates this time through the applicable period range of 6 to 49, calculating the real and imaginary parts of the Fourier components. Frequencies 6 to 49 are the primary focus of interest for this periodogram. Using the precalculated angular frequency factors, the resulting real and imaginary parts are then utilized to calculate the spectral Fourier components, which are stored in an array for later use.
The next section of code smooths the noise ridden Fourier components between the periods of 6 and 49 with a selected filter. This species also employs numerous SuperSmoothers to condition noisy Fourier components. One of the big differences is Ehlers' versions used basic EMAs in this section of code. I decided to add SuperSmoothers.
The final sections of the acs() code determines the peak power component for normalization and then computes the dominant cycle period from the smoothed Fourier components. It first identifies a single spectral component with the highest power value and then assigns it as the peak power. Next, it normalizes the spectral components using the peak power value as a denominator. It then calculates the average dominant cycle period from the normalized spectral components using Ehlers' "Center of Gravity" calculation. Finally, the function returns the dominant cycle period along with the normalized spectral components for later external use to plot the periodogram.
POST SCRIPT:
Concluding, I have to acknowledge a newly found analyst for assistance that I couldn't receive from anywhere else. For one, Claude doesn't know much about Pine, is unfortunately color blind, and can't even see the Pine reference, but it was able to intuitively shred my code with laser precise realizations. Not only that, formulating and reformulating my description needed crucial finesse applied to it, and I couldn't have provided what you have read here without that artificial insight. Finding the right order of words to convey the complexity of ACP and the elaborate accompanying content was a daunting task. No code in my life has ever absorbed so much time and hard fricking work, than what you witness here, an ACP gem cut pristinely. I'm unveiling my version of ACP for an empowering cause, in the hopes a future global army of code wielders will tether it to highly functional computational contraptions they might possess. Here is ACP fully blessed poetically with the "Power of Pine" in sublime code. ENJOY! Indicator
JohnEhlersFourierTransformLibrary "JohnEhlersFourierTransform"
Fourier Transform for Traders By John Ehlers, slightly modified to allow to inspect other than the 8-50 frequency spectrum.
reference:
www.mesasoftware.com
high_pass_filter(source) Detrended version of the data by High Pass Filtering with a 40 Period cutoff
Parameters:
source : float, data source.
Returns: float.
transformed_dft(source, start_frequency, end_frequency) DFT by John Elhers.
Parameters:
source : float, data source.
start_frequency : int, lower bound of the frequency window, must be a positive number >= 0, window must be less than or 30.
end_frequency : int, upper bound of the frequency window, must be a positive number >= 0, window must be less than or 30.
Returns: tuple with float, float array.
db_to_rgb(db, transparency) converts the frequency decibels to rgb.
Parameters:
db : float, decibels value.
transparency : float, transparency value.
Returns: color. Library
Indicator
[blackcat] L3 Ehlers Spectrogram IndicatorLevel: 3
Background
John F. Ehlers introuced Dominant Cycle and Spectrogram Indicator in Mar, 2008.
Function
In "Measuring Cycle Periods" in Mar, 2008, John Ehlers presented a very interesting technique of measuring dominant market cycle periods by means of multiple bandpass filtering. By utilizing an approach similar to audio equalizers, the signal (here, the price series) is fed into a set of simple second-order infinite impulse response bandpass filters. Filters are tuned to 8,9,10,...,50 periods. The filter with the highest output represents the dominant cycle. This script implements a high-pass filter and a six-tap low-pass Fir filter on input, then 42 parallel Iir band-pass filters. Finally, it plots the Ehlers spectrogram as a heat map.
Ehlers Spectrogram indicates market power status:
With high market power, the spectrogram become fuzzy;
With low market power, the spectrogram become distinct.
It also discloses market dominant cycles and subcycles, which indicates the major rhythm. Regarding the rhythm of the market, Chinese and Western cultures have reached a consensus: Dr. Ehlers can find the main harmonic components of the market through DSP analysis via spectrum; Master Zen compares the market rhythm to Bach’s fugue from a philosophical perspective, where there is a rhythm of life.
Finally, please allow me to quote Master Zen’s statement on market rhythm:
"The market is cruel. For those who try to violate the market rhythm, the market is their death place; the market is beautiful, and the market is Bach’s fugue. There is a rhythm of life. Rhythm is always the rhythm of the market. A market participant without a sense of rhythm is always torturing. Put aside your greed and fear and listen to the rhythm of the market. In the weekend, let go of everything, but listen to the rhythm of nature, the rhythm of life, the rhythm of music, and then come back to listen to the rhythm of the market. Dancing with the market, your greed and fear are peeled off one by one, you will become very bright."
Key Signal
DomCyc--> Ehlers Dominant Cycle
Plot2~Plot60--> Ehlers Spectrum Heat Map
Pros and Cons
100% John F. Ehlers definition translation, even variable names are the same. This help readers who would like to use pine to read his book.
Remarks
The 73th script for Blackcat1402 John F. Ehlers Week publication.
Readme
In real life, I am a prolific inventor. I have successfully applied for more than 60 international and regional patents in the past 12 years. But in the past two years or so, I have tried to transfer my creativity to the development of trading strategies. Tradingview is the ideal platform for me. I am selecting and contributing some of the hundreds of scripts to publish in Tradingview community. Welcome everyone to interact with me to discuss these interesting pine scripts.
The scripts posted are categorized into 5 levels according to my efforts or manhours put into these works.
Level 1 : interesting script snippets or distinctive improvement from classic indicators or strategy. Level 1 scripts can usually appear in more complex indicators as a function module or element.
Level 2 : composite indicator/strategy. By selecting or combining several independent or dependent functions or sub indicators in proper way, the composite script exhibits a resonance phenomenon which can filter out noise or fake trading signal to enhance trading confidence level.
Level 3 : comprehensive indicator/strategy. They are simple trading systems based on my strategies. They are commonly containing several or all of entry signal, close signal, stop loss, take profit, re-entry, risk management, and position sizing techniques. Even some interesting fundamental and mass psychological aspects are incorporated.
Level 4 : script snippets or functions that do not disclose source code. Interesting element that can reveal market laws and work as raw material for indicators and strategies. If you find Level 1~2 scripts are helpful, Level 4 is a private version that took me far more efforts to develop.
Level 5 : indicator/strategy that do not disclose source code. private version of Level 3 script with my accumulated script processing skills or a large number of custom functions. I had a private function library built in past two years. Level 5 scripts use many of them to achieve private trading strategy. Indicator
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