NW Volume Profile - Kernel-Smoothed [Dots3Red]📊 NW VOLUME PROFILE - KERNEL-SMOOTHED
A volume profile answers a different question than a normal chart. Instead of "how much traded today," it asks "how much traded at each price." This version applies Nadaraya-Watson kernel smoothing to that profile before reading any level off it — turning a jagged, noisy histogram into the actual underlying distribution of where volume concentrated.
🎯 WHY THIS MATTERS
A raw volume profile is built from independent price bins — each one only knows its own volume, nothing about its neighbors. That makes it noisy: a single oversized candle can create a spike that looks like an important level but is really just where one bar happened to land. Reading real structure off a raw histogram means squinting past that noise.
This script smooths the profile before drawing anything. Every bin's displayed value becomes a weighted average of its neighborhood — nearby bins count heavily, distant bins barely at all, following a Gaussian curve. The lumps from individual candles melt away, and what's left is the true shape of the distribution that was underneath the noise the whole time. All the levels described below — POC, Value Area, HVN, LVN — are read from that smoothed curve, not the raw one.
🧮 HOW THE SMOOTHING WORKS
Each price bin's raw volume gets replaced by:
smoothed(i) = Σⱼ w(i,j) · raw / Σⱼ w(i,j)
where w(i,j) is a Gaussian weight based on how many bins apart i and j are, controlled by the Bandwidth setting. A small bandwidth stays close to the raw histogram; a large one produces one broad, simplified hump. This is genuine kernel regression applied across the price axis, not a moving average or a visual blur — it's the same mathematical technique used in the smoothed lines several Dots3Red scripts already use for slope/trend estimation, applied here to a distribution instead of a time series.
Toggle "Show Raw Histogram Behind" to see the original jagged bars faintly displayed underneath the smoothed profile — a direct before/after comparison on your own chart.
📏 WHAT EACH LEVEL MEANS
🟡 POC (Point of Control) — the single price with the highest smoothed volume. The market's center of gravity for the current window; price tends to be pulled back toward it.
🔵 Value Area — the price region around the POC containing a configurable share of total volume (default 70%). Price trading inside it is trading at a level the market recently agreed was fair — chop and rotation are common here. Price breaking out of it is the market rejecting that agreement, which is often when moves extend rather than stall.
🟢 HVN (High Volume Node) — a secondary local peak in the smoothed distribution. Acts like a sticky zone; price tends to slow down or pause when revisiting one.
🔴 LVN (Low Volume Node) — a local trough where very little volume ever traded. Acts like a thin spot; price tends to move through it quickly rather than lingering, since few positions were ever opened there.
HVN and LVN are drawn as full-width dotted lines across the chart (not just labels at the profile edge), specifically so they stay visible and trackable even after price has moved well away from where the profile itself was drawn.
🧭 HOW TO USE
👀 Start with where price sits relative to the Value Area. Inside it: expect rotation and two-way trade. Outside it: the move has already broken from recent consensus, which historically has more follow-through than reversion.
🧲 Treat POC as a magnet, not a wall. It is the level most likely to be revisited, not a guaranteed reversal point. How price behaves when it gets there — accepted or rejected — is the actual signal, not the level itself.
🐌 Expect hesitation at HVNs. A move approaching an HVN from your prior window is approaching a zone where the market has previously done a lot of business — some slowing or consolidation there is common.
⚡ Expect speed through LVNs. A thin zone with very little historical volume tends to get crossed quickly rather than acting as support or resistance. If price is moving toward one, a fast move through it before finding real support/resistance at the next node is a reasonable expectation.
🔧 Adjust Bandwidth to match what you're looking for. A tighter bandwidth reveals more granular structure (closer to raw); a wider one collapses the profile into its dominant, unmistakable levels. There's no universally correct setting — it depends on whether you want detail or clarity.
💡 EXAMPLE
Say the profile shows POC at 61,200, a Value Area from 60,400 to 62,100, and an LVN line sitting at 59,800. Price later drops to 60,450 — right at the edge of the Value Area. Two distinct scenarios are now readable from the profile: if price holds and turns back up, the 61,200 POC above is the natural target the market has repeatedly gravitated toward. If instead price breaks below 60,400, the empty LVN at 59,800 offers little historical volume to slow the decline — a fast move through that zone before finding the next real level is the more likely path. Same chart, two different expectations, both read directly off the same profile without any additional indicator.
⚙️ SETTINGS
📊 Profile
• Lookback (bars) — size of the rolling window the profile is built from
• Price Bins — vertical resolution of the profile
• Body Volume Only — distribute volume across the candle body instead of the full high-low range
🧮 Kernel Smoothing
• Bandwidth — width of the Gaussian kernel in bin units; controls detail vs. simplification
📏 Levels
• Value Area % — share of total volume the Value Area is expanded to contain
• Node Detection Leg — how many neighboring bins define a local peak/trough
• LVN Max Ratio of POC — how thin a trough must be, relative to POC, to count as an LVN
🎨 Visualization
• Show Raw Histogram Behind, POC Line, Value Area, HVN/LVN Marks — each independently toggleable
• Profile Width — how far the profile extends horizontally
🖥️ Dashboard
• Show/hide, position — displays current POC, Value Area bounds, node counts, and the active window/bandwidth settings
📝 NOTES
This profile is a rolling window — its levels update as the window slides forward with each new bar, which is expected behavior for a volume profile rather than a repainting signal (nothing appears and then vanishes; the underlying window is simply moving). Thin-volume symbols will produce a ragged profile regardless of smoothing settings — this tool is most informative on liquid instruments with consistent volume.
⚠️ DISCLAIMER
This is an analytical and visualization tool. It does not generate trade signals and does not constitute financial advice. Historical volume concentration at a given level does not guarantee how price will behave there in the future. Indicator
Machine Learning Price Bands Kernel Regression SignalsOVERVIEW
Every "AI band" on this platform draws two lines and asserts them. None of them can tell you how often price actually stays inside.
This one can — because it is built on a method that comes with a MATHEMATICAL COVERAGE GUARANTEE, and then it CHECKS WHETHER IT KEPT THE PROMISE, live, on your chart:
Coverage (empirical vs nominal) 89.1% vs 90% n = 20,266
Is the miss REAL? -0.9 pp z = -4.3 (real)
Verdict undercovering — real, but small
That is not a band. That is a prediction interval that has been audited, and it is the whole reason this tool exists.
It is a research and framing tool. NOT a strategy, NOT a signal service, NOT a validated edge.
THE MACHINE LEARNING, SPELLED OUT — no buzzwords, here is the actual model
1. NADARAYA-WATSON KERNEL REGRESSION. Non-parametric: no functional form is assumed, the data chooses the shape. Each past bar votes on the current estimate with a Gaussian weight that decays with distance. This is the same estimator Lo, Mamaysky and Wang used in the Journal of Finance to make chart-pattern recognition objective. It is real machine learning, and it is sixty years old.
The kernel here is CAUSAL. It only ever looks backwards. A centred kernel — the kind most "Nadaraya-Watson envelope" scripts use — peeks at bars that have not happened yet, and that is why their historical fit looks so much better than their live one.
2. BANDWIDTH BY PREDICTIVE MODEL SELECTION. The bandwidth h is the only real parameter, and it is not a magic number: several candidates are run in parallel and scored on their ROLLING ONE-STEP-AHEAD SQUARED ERROR. The winner is used. That is honest model selection — the criterion you would use to choose any forecaster — rather than a knob you turn until the chart looks nice.
3. CONFORMAL PREDICTION INTERVALS. The half-width is the (1-alpha) empirical quantile of the recent ABSOLUTE one-step-ahead errors. Under exchangeability this carries a FINITE-SAMPLE coverage guarantee, with NO distributional assumption at all: no normality, no GARCH, no volatility model. The model's own recent mistakes size the band — which is why it widens when the model starts being WRONG, not merely when price starts moving.
4. ADAPTIVE CONFORMAL INFERENCE — Gibbs and Candes, NeurIPS 2021.
Here is the problem with plain conformal prediction on markets, stated plainly: its guarantee holds under EXCHANGEABILITY, and financial returns are the textbook counterexample. Volatility CLUSTERS. So a residual quantile computed over a trailing window is always a step behind, the band is too narrow exactly when it matters, and the misses bunch together. Coverage lands quietly under nominal. Measured live on NIFTY futures before this was added: 89.1% against a nominal 90%, on the 1m, the 3m and the 1h, every one of them roughly four standard errors below target. Not a bug. The assumption breaking.
ACI makes the miscoverage level a LEARNED parameter:
alpha(t+1) = alpha(t) + gamma * (alpha - err(t))
Miss the interval and alpha falls, so the quantile rises and the band WIDENS. Cover it and alpha creeps back, so the band TIGHTENS. Long-run coverage provably converges to the target IRRESPECTIVE OF THE DATA GENERATING PROCESS — no exchangeability assumption anywhere.
A band that notices it is undercovering and fixes itself. Watch the alpha row: where it settles BELOW nominal is a direct measurement of how badly exchangeability fails on your instrument.
MEASURED, ON THE SAME INSTRUMENT, BEFORE AND AFTER:
timeframe plain conformal with ACI
1m 89.1% 90.1%
3m 89.1% 90.1%
5m 89.1% 90.1%
15m 90.1% 90.1%
1h 89.1% 90.1%
(nominal 90%)
Five timeframes, a four-standard-error undercoverage on four of them, closed. The binomial test now returns "calibrated — within sampling noise" and means it. That is not a backtest of a trading rule. That is a mathematical promise being kept, and being checked.
5. NORMALISED NONCONFORMITY — Papadopoulos et al. (2008), Lei et al. (2018).
The plain score |price - fit| is a SCALAR, which means the band is THE SAME WIDTH in a dead tape and in a crash. It therefore OVERCOVERS in calm and UNDERCOVERS in chaos — and the single marginal coverage figure is the average of those two errors, looking correct while being wrong in both directions.
Normalising divides each residual by a local scale estimate before taking the quantile, and multiplies it back when drawing:
score = |price - fit| / sigma band = fit +/- q * sigma
The band now scales with LOCAL DIFFICULTY — and note it is the MODEL'S difficulty, not the market's volatility. Related, but not the same thing, and the first one is what a prediction interval is actually about.
6. THE COVERAGE AUDIT. A guarantee you do not verify is just a claim.
TWO QUESTIONS ABOUT THE COVERAGE, AND THE PANEL ANSWERS BOTH
IS THE MISS REAL? That is a binomial z-test and it needs no tolerance at all. Each bar is a Bernoulli trial with p = nominal, so the standard error of the observed coverage is sqrt(p(1-p)/n).
IS THE MISS BIG ENOUGH TO CARE ABOUT? That is a judgement, and you set it.
These are NOT the same question, and conflating them is how a band gets waved through as "calibrated". Measured live on NIFTY futures: at n = 20,266 the standard error is 0.21 pp, so an empirical coverage of 89.1% against a nominal 90% is a 0.9 pp miss — FOUR STANDARD ERRORS. Unmistakably real. Arguably too small to trade differently. A 5 pp tolerance called that "calibrated", which was the headline row of the script asserting the one thing the script exists to verify, and asserting it wrongly.
The panel now reports the size of the miss, its significance, and a verdict that distinguishes "within sampling noise" from "real, but small" from "MISCALIBRATED — do not trust the band". You get to decide which of those matters to you, and you get the numbers to decide with.
AND THEN THE ROW NOBODY HAS: CONDITIONAL COVERAGE
Coverage 90.0% vs 90% n = 20,178
calm / normal / turbulent 96.4% · 90.1% · 83.2%
A single marginal number can read a perfect 90% while the interval covers 96% of quiet bars and 83% of violent ones. Ninety per cent is then the AVERAGE OF TWO ERRORS — it looks right while being wrong in both directions, and it is wrong in the direction that costs you money exactly when it costs you money.
Exact conditional coverage is provably impossible without strong assumptions. But you can always MEASURE it, and almost nobody does. Bars are split into calm, normal and turbulent thirds by the percentile rank of ATR, and coverage is scored inside each. If the three numbers fan apart, the band is not breathing — and the normalised score is what closes the gap.
Turn the normalised score off and watch those three fan out. That is the demonstration.
FADE OR FOLLOW? THE TOOL DOES NOT PRETEND TO KNOW
Price leaving a 90% interval is statistically unusual. Whether to FADE it (an outlier, so bet on reversion) or FOLLOW it (the model has broken, so bet on the new regime) is an EMPIRICAL question, and the honest answer is often neither.
So both are logged, both are graded, and BOTH ARE TESTED AGAINST EACH OTHER.
That last part matters more than it sounds. Knowing that fading beats an unconditional control, and that following also beats an unconditional control, does not answer the question a trader is actually asking at a band break — which of the two should I do? They are mutually exclusive responses to the SAME event. So they are run head to head with a Welch t-test, and the answer is allowed to be:
FADE or FOLLOW? NEITHER — the break does not tell you which
If the difference does not clear the noise, then on this instrument the break carries no directional information, and saying so IS the finding. A tool that cannot report its own failure is an advertisement, not a measurement.
And the chart agrees with the panel. An unproven direction is still drawn — it is arithmetic, and you may want it — but it is drawn MUTED and labelled "(not proven)". It used to print "Follow the break" in full colour while the panel directly beneath it said "neither proven". The paint has to agree with the code.
THE ANTI-BIAS GUARDS
ENTRY IS THE CLOSE, for the event and for the control alike. A band break is a SIGNAL, not a fill. Entering at the band — a better price — while the control enters at the close hands every signal a free head start and manufactures an edge out of nothing.
THE CONTROL IS DIRECTION-MATCHED. In a downtrend there are more break-downs than break-ups, so FOLLOW skews short and FADE skews long. A direction-skewed event set measured against a 50/50 control inherits the index drift for free and calls it an edge. Longs are compared only with control longs, shorts only with control shorts, and the control is blended back using the events' OWN direction mix.
IDENTICAL GEOMETRY. Every event and every control trade uses the same stop and the same R multiple, so the comparison is apples to apples.
Both barriers on one bar: the STOP is assumed first — conservative, and the only assumption that cannot flatter the result. Unresolved trades at the horizon are marked to market, not booked as losses. Nothing is marked proven below t = 1.96.
NON-REPAINT
The kernel is causal, the bandwidth is chosen on past error only, the interval is built from past residuals, and coverage is scored by asking whether the actual close landed inside the interval that was published BEFORE it. Everything is computed on confirmed bars. Nothing is drawn and then moved.
WHY THESE PARTS ARE ONE TOOL
The regression gives the trend. Without the interval, a band is a guess. Without model selection, the bandwidth is a knob you turn until you like the picture. Without the coverage audit, a conformal interval is an unverified promise. And without the signal calibration, "price left the band" is folklore. Each piece is worthless alone — which is exactly why they ship together.
DATA AND SCOPE
Any symbol, any timeframe. ATR-normalised throughout. No volume required.
EXPORTS (Data Window — consume from other scripts via input.source())
EXP_Fit, EXP_Upper, EXP_Lower, EXP_Bandwidth, EXP_Coverage, EXP_Miscal, EXP_Signal, EXP_Entry, EXP_Stop, EXP_Target
CONCEPT CREDIT
Nadaraya-Watson kernel regression — E. A. Nadaraya and G. S. Watson (1964). Its use for technical pattern recognition in finance — Andrew W. Lo, Harry Mamaysky and Jiang Wang, "Foundations of Technical Analysis", Journal of Finance 55(4), 2000. Conformal prediction — Vladimir Vovk, Alexander Gammerman and Glenn Shafer; the split/inductive form used here follows Papadopoulos et al. and Lei et al. Triple-barrier forward labelling — Marcos Lopez de Prado. Welch's t-test — B. L. Welch. ATR — J. Welles Wilder.
The causal-kernel implementation, the parallel bandwidth selection, the live coverage audit, the binomial calibration test and the fade-versus-follow head-to-head are the author's own. Clean-room implementation; no third-party Pine code is reused. Not affiliated with, nor endorsed by, any of the above.
HONESTY AND LIMITATIONS
Conformal coverage is guaranteed under EXCHANGEABILITY. Financial returns are NOT exchangeable — volatility clusters, regimes shift — so the guarantee is approximate in practice. THAT IS PRECISELY WHY THE COVERAGE IS AUDITED LIVE INSTEAD OF ASSUMED. When empirical coverage drifts from nominal you are watching the assumption break, in real time, and you should believe what you see rather than the label.
A prediction interval says where price is LIKELY TO BE. It says nothing about DIRECTION, and it is not a forecast. Coverage being correct does not make band breaks tradeable — those are two different claims, and the tool tests them separately for exactly that reason.
Calibration figures are IN-SAMPLE, with no costs or slippage, and use overlapping windows. A proven in-sample edge is NOT a guarantee out-of-sample. Nothing here predicts price.
DISCLAIMER
Research and educational tool only. NOT financial advice, NOT a recommendation, and NO guarantee of results. Entry, stop and target output is arithmetic, not advice. Trading carries risk of loss. Test out-of-sample and make your own decisions. The author accepts no liability for any use. Indicator
Indicator
Nadaraya-Watson Envelope [Gabremoku]Nadaraya-Watson Envelope
This indicator builds a non-repainting Nadaraya-Watson envelope using a one-sided Gaussian kernel, so every value is computed from the current bar and past bars only. The goal is to provide a smoother adaptive baseline than a standard moving average while keeping the script operationally honest and suitable for live use.
What makes this script different:
- The central basis is a kernel-weighted Nadaraya-Watson estimate, not a classic SMA/EMA baseline.
- The main envelope is not built from standard deviation by default. It uses kernel-weighted mean absolute deviation (MAD), which is generally less sensitive to single-bar outliers and often produces a more stable channel.
- Standard deviation bands can still be enabled as an optional overlay, so users can compare MAD-based and Stdev-based dispersion around the same kernel basis.
- Signal logic is configurable. Breakout labels can be triggered by close crossing the band, wick piercing the band, or full body breakout, which makes the visual behavior easier to align with the trader’s interpretation.
How it works:
The script applies Gaussian weights to past bars inside the selected window. More recent bars receive the highest weight, while older bars progressively contribute less. The Bandwidth input controls how fast those weights decay. In practice, the effective lookback is usually much shorter than the full Window setting when Bandwidth is low. A practical rule of thumb is that the effective lookback is about 3 × Bandwidth bars, capped by the Window value.
The indicator computes:
1. A kernel-weighted mean, used as the Nadaraya-Watson basis.
2. A kernel-weighted MAD, used as the primary envelope width.
3. An optional kernel-weighted standard deviation, displayed only when the comparison bands are enabled.
The upper and lower MAD bands are then filled with a gradient that increases in strength as price moves away from the basis toward the envelope edges. This makes the visual intensity reflect displacement magnitude, not just bullish or bearish direction.
Compression logic:
The compression zone is based on min-max normalization of envelope width over a lookback period. This is not a statistical percentile rank. A threshold of 0.15 means the current envelope width is near the lower end of the observed width range over the selected compression lookback.
Signal modes:
- Close Cross: triggers only when the close crosses a band.
- Wick Pierce: triggers when the candle’s high or low exceeds a band.
- Body Breakout: triggers when the candle body exceeds a band.
Use Wick Pierce if you want signal labels to match the visible moment where candles extend outside the envelope.
How to use it:
- Use the basis as an adaptive trend reference.
- Use the MAD envelope to judge whether price is stretched relative to recent kernel-weighted behavior.
- Watch compression zones for narrow-range conditions that may precede expansion.
- Compare MAD and Stdev bands when you want to evaluate whether recent volatility is dominated by isolated spikes or by broader dispersion.
Practical notes:
- This script is non-repainting by construction because it does not use centered calculations or future bars.
- Low Bandwidth values create a more reactive basis and shorter effective memory.
- High Bandwidth values create a smoother basis and wider historical influence.
- Increasing Window far beyond roughly 3 × Bandwidth usually has little additional effect.
- Signal labels are state-machine filtered, so they are designed to mark sequence transitions rather than every repeated touch outside the bands.
This indicator is intended as a visual decision-support tool, not as a standalone trading system. It helps traders study adaptive trend, envelope displacement, compression, and breakout structure in a cleaner way than a standard volatility channel. Indicator
Aegis Kinetic Trend Matrix [wjdtks255]Aegis Kinetic Trend Matrix
■ OVERVIEW
The Aegis Kinetic Trend Matrix is a professional-grade trend-following framework designed to unify macroeconomic bias filters, micro-execution entry triggers, and volatility boundaries into a single, cohesive candle-overlay system.
By integrating three robust open-source concepts—CM_EMA Trend Bars, HalfTrend, and the Nadaraya-Watson Envelope (NWE)—this system provides traders with a multi-layered filtration process to capture structural market swings with precision.
■ KEY FEATURES
CM_EMA Trend Bars: Dynamically shifts candlestick colors based on a 34-period EMA algorithm to isolate core macro direction and eliminate market noise.
HalfTrend Execution Spine: High-precision trailing anchor that tracks micro-trend pivots, offering distinct, instant BUY and SELL execution labels.
Nadaraya-Watson Envelope: Uses non-parametric kernel regression bounds to highlight overextended pricing and filter volatility exhaustion zones at major structural extremes.
■ 개요 (Korean)
Aegis Kinetic Trend Matrix는 거시적 추세 필터링, 미세 타점 포착, 그리고 변동성의 한계 구간을 단 하나의 캔들 오버레이 시스템으로 결합한 하이브리드 트레이딩 프레임워크입니다.
CM_EMA 트렌드 바, 하프트렌드(HalfTrend), 나다라야-왓슨 엔벨로프(NWE) 시스템을 유기적으로 결합하여, 거친 시장 소음을 여과하고 구조적 변곡점을 정밀하게 잡아내도록 설계되었습니다.
■ 핵심 기능
CM_EMA 트렌드 바: 34선 기준 가격 배열에 따라 캔들 색상을 직관적으로 변경하여 시장의 대추세 방향성을 명확히 정의합니다.
하프트렌드 실행 축: 단기적인 마이크로 추세 전환을 정밀하게 추적하며, 즉각적인 BUY/SELL 진입 라벨을 제공합니다.
Nadaraya-Watson 엔벨로프: 커널 회귀 분석 기반의 동적 밴드를 통해 가격의 과도한 이격을 감지하고 추세적 극한 구간의 반전 포인트를 필터링합니다.
■ Credits & Acknowledgments
This indicator is a combined integration based on public domain open-source works. Special credits and gratitude go to the original authors of CM_EMA, HalfTrend, and Nadaraya-Watson Envelope (AlexGrozav) for sharing their invaluable source code with the global community. Indicator
Adaptive Lorentzian Classification [Quantum Algo]Quantum ML Engine — Adaptive Lorentzian Classification
█ OVERVIEW
Quantum ML Engine is a machine-learning classifier that predicts the direction of price over a configurable horizon using an Approximate Nearest Neighbors (ANN) search across historical feature vectors. Instead of relying on a single oscillator, it compares the current bar's "fingerprint" — a vector of up to six normalized features — against thousands of past bars, finds the most similar market conditions, and lets those historical outcomes vote on what is likely to happen next.
By default the engine measures similarity with Lorentzian distance, log(1 + |Δ|), rather than Euclidean distance. Market data is heavily distorted around major events (CPI prints, FOMC, black swans), and Lorentzian distance naturally compresses these outliers — analogous to how mass warps space-time — so a single extreme bar cannot dominate the neighbor selection.
This is an original, fully self-contained implementation written from scratch with zero library imports. The concept of applying Lorentzian distance to kNN classification on charts was pioneered in the open-source work of @jdehorty (Machine Learning: Lorentzian Classification), building on earlier kNN studies by @capissimo. Full credit to both for the foundational research. This script does not reuse their code; it re-derives the approach independently and extends it in the ways described below.
█ WHAT IS DIFFERENT IN THIS IMPLEMENTATION
1 — Time-aligned training set
Each training sample pairs the feature vector recorded AT a given bar with the realized outcome over the following H bars. Features and labels are stored on the same time axis, so the classifier learns from correctly matched cause-and-effect pairs. There is no lookahead: a sample only enters the training set once its outcome is fully realized.
2 — ATR neutral-zone labeling
Historical moves smaller than a configurable multiple of ATR are labeled NEUTRAL instead of long/short. Sideways noise therefore never teaches the model a false directional lesson. Set the multiplier to 0 to disable.
3 — Six engineered features with importance weights
RSI, WaveTrend, CCI, ADX, MFI (volume flow) and Fisher Transform, each normalized to a common 0–1 scale. Every feature slot has its own weight input, so you can tell the engine which dimensions matter more for your market without removing features entirely.
4 — Four selectable distance metrics
Lorentzian (default), Manhattan, Euclidean, and a 50/50 Lorentzian-Manhattan Hybrid. Switching metrics changes the geometry of the neighborhood and is a powerful tuning lever per asset class.
5 — Distance-weighted voting with a confidence score
Closer neighbors vote louder (weight = 1 / (1 + distance)). The agreement between neighbors is expressed as a 0–100% confidence value printed on every bar, and a minimum-confidence gate suppresses low-conviction signals entirely.
6 — Adaptive K
The neighbor count automatically shrinks (up to 40%) when volatility ranks high over the last 100 bars, making the model more reactive in fast markets, and expands back in quiet regimes for stability. Can be disabled for a fixed K.
7 — Sliding training window
The engine always trains on the most recent N bars rather than the oldest bars in chart history, so the model reflects current market structure.
8 — Configurable prediction horizon
The training/holding horizon is an input (1–20 bars) instead of a hardcoded constant.
9 — Three exit modes
Fixed-horizon exits, dynamic kernel-slope exits, and an optional ATR trailing stop with the stop level plotted on the chart.
10 — Higher-timeframe confluence filter
Optionally require price to be above (longs) or below (shorts) an EMA on a higher timeframe of your choice.
█ HOW IT WORKS
1. On every bar, six features are computed and normalized.
2. The bar's feature vector is compared against samples inside the sliding training window, sampled with a minimum chronological spacing (default 4 bars) so neighbors come from distinct market episodes rather than one cluster.
3. A monotonic distance threshold maintains a stable pool of approximate nearest neighbors; when the pool exceeds K, the threshold resets to the 75th-percentile distance, allowing genuinely closer samples to rotate in over time.
4. Neighbors vote long / short / neutral, weighted by proximity. The weighted sum becomes the prediction; the degree of agreement becomes the confidence.
5. The raw signal is then passed through optional filters: volatility regime (recent ATR vs long-run ATR), trend regime (EMA separation normalized by ATR), ADX, EMA/SMA trend, higher-timeframe trend, and a Nadaraya-Watson kernel regression filter (rational quadratic estimate with a Gaussian crossover mode for smoother color transitions).
6. Entries print only when the ML signal, the confidence gate, and all enabled filters agree.
█ SETTINGS GUIDE
General — source, training window size, prediction horizon, neutral-zone width.
ML Engine — K, adaptive K toggle, chronological spacing, distance metric, distance weighting, minimum confidence.
Feature Engineering — feature type, parameters and weight for each of the six slots.
Filters — volatility, regime, ADX, EMA/SMA, higher-timeframe confluence.
Kernel — lookback, relative weighting, regression level, lag, smoothing mode.
Exits — fixed vs dynamic exits, ATR trailing stop and multiplier.
Display — bar colors, prediction labels (value + confidence), dashboard, color compression.
█ DASHBOARD
The on-chart panel shows the live signal, prediction confidence, current adaptive K, volatility and trend regime states, kernel bias, and a calibration win-rate. The calibration statistic simply checks whether price moved in the predicted direction over the horizon after each signal. It exists ONLY to give feedback while tuning features — it is not a backtest, includes no costs or risk management, and must not be treated as a performance claim.
█ USAGE NOTES
— Works on any symbol and timeframe; intraday (15m–4H) and daily charts are typical starting points. Crypto, FX, indices and equities all behave differently — retune the features and metric per market.
— Higher minimum confidence = fewer but more selective signals. Raising chronological spacing diversifies neighbors on lower timeframes.
— Signals are evaluated on bar close. Like any bar-close logic, the in-progress bar can change until it closes.
— Best used as a confluence layer inside a complete trading plan with your own risk management, not as a standalone buy/sell system.
█ CREDITS
Concept inspiration: @jdehorty (Machine Learning: Lorentzian Classification) and @capissimo (kNN implementations). This script is an independent, original implementation with the extensions listed above.
█ DISCLAIMER
This script is provided for educational and informational purposes only. It is not financial advice, and past behavior — including the on-chart calibration statistics — does not guarantee future results. Trading involves substantial risk of loss. Always do your own research and manage risk responsibly. Indicator
Regression Trend Reversal Signals & Forecasts [AlgoAlpha]🟠 OVERVIEW
Regression Trend Reversal Signals & Forecasts combines multiple regression methods into a single trend and reversal framework. It allows traders to choose between Linear Regression, Theil-Sen Regression, LOESS smoothing, Nadaraya-Watson smoothing, Polynomial Regression, and a Kalman Filter to estimate the underlying price path.
The selected regression line acts as the center of a dynamic channel. The channel width is based on the standard deviation of the distance between price and the regression line, allowing it to adapt to changing market conditions.
The script also identifies potential reversal conditions when price extends beyond the channel and then shows signs of rejection. In addition, it can project the current regression slope forward to provide a simple forecast of the current trend path.
🟠 CONCEPTS
Regression Line — A statistical estimate of the underlying price trend. Different methods can be selected, ranging from straight-line regressions to adaptive smoothing techniques.
Theil-Sen Regression — A robust regression method that uses median slopes from all point pairs, reducing the influence of outlier price spikes.
LOESS Regression — A locally weighted regression that fits nearby observations more heavily than distant observations to create a smooth trend curve.
Nadaraya-Watson Smoothing — A kernel-weighted averaging method that estimates trend by assigning larger weights to recent observations.
Kalman Filter — A recursive estimation method that continuously updates the trend estimate as new prices arrive.
Polynomial Regression — A curved regression model that can capture non-linear trend structures using higher-order polynomial functions.
Regression Deviation Bands — Channel boundaries calculated from the standard deviation of price relative to the regression line.
Regression Slope — The rate of change of the regression estimate used to determine trend direction and forecast projections.
🟠 FEATURES
Regression Channel — Dynamic bands expand and contract based on how far price deviates from the regression line.
Trend Flip Signals — Generates directional markers when the regression trend changes from rising to falling or from falling to rising.
Reversal Signals — Marks potential bullish and bearish reversals when price extends beyond the channel and begins rejecting those extremes.
Forecast Projection — Extends the current regression slope into future bars and optionally displays projected channel boundaries.
🟠 HOW TO USE
Select a regression method that matches the market behavior you want to analyze. Linear and Theil-Sen are suited to directional trends, while LOESS, Nadaraya-Watson, and Kalman provide smoother adaptive estimates.
Use the regression line as the primary trend reference. Rising regression values indicate strengthening conditions, while falling values indicate weakening conditions.
Monitor the channel boundaries for extended price movement away from the regression estimate.
Watch for bullish reversal markers below the lower band after downside extensions and bearish reversal markers above the upper band after upside extensions.
Use trend flip signals as confirmation that the regression slope has changed direction.
Compare price location within the channel to gauge whether price is trading near trend equilibrium or at an extreme deviation.
Use the forecast projection as a continuation estimate of the current regression slope rather than a prediction of future market behavior.
🟠 CONCLUSION
Regression Reversal Signals combines multiple regression techniques, adaptive deviation channels, reversal detection, and forward projections into a single framework. By allowing traders to switch between several trend estimation methods, it provides different perspectives on trend structure and price deviation. The indicator helps identify trend direction, potential reversals, and areas where price has moved unusually far from its estimated path. Indicator
Nadaraya-Watson Regression Liquidity Sweeps [AlgoAlpha]🟠 OVERVIEW
This script combines Nadaraya-Watson regression, momentum analysis, and liquidity level tracking into a single workflow. It measures the slope of a smoothed price regression curve, converts that slope into a normalized oscillator, and uses momentum shifts to identify areas where liquidity may be resting.
The oscillator is built from the rate of change of the Nadaraya-Watson estimate rather than price itself. This allows momentum transitions to be measured relative to the underlying regression trend. When momentum weakens after an extended move, the script records swing-based liquidity levels that can later be swept by price.
A volatility-adjusted Nadaraya-Watson band is also displayed on the chart. This provides context for trend direction, momentum strength, and potential rebound conditions around the regression value.
🟠 CONCEPTS
Nadaraya-Watson Regression — A kernel-based smoothing method that estimates an underlying price curve by weighting nearby historical data more heavily than distant data.
Normalized Regression Slope — The change in the Nadaraya-Watson estimate divided by its recent standard deviation, allowing momentum strength to be compared across different market conditions.
Liquidity Sweep Level — A horizontal level created from a swing high or swing low when momentum begins to weaken, representing an area that may later attract price.
Oscillator Signal Line — An EMA of the normalized oscillator used to identify momentum crossovers and momentum phase changes.
Rebound Condition — A signal generated when price moves back through the Nadaraya-Watson value while oscillator direction remains aligned with the prevailing momentum bias.
🟠 FEATURES
Normalized Nadaraya-Watson Oscillator — Measures momentum using the slope of a smoothed regression curve.
Liquidity Sweep Detection — Creates liquidity levels when bullish or bearish momentum begins to weaken.
Volatility-Adjusted Regression Band — Displays a dynamic overlay around the Nadaraya-Watson estimate using smoothed ATR values.
Momentum Weakening Signals — Marks locations where oscillator momentum begins to lose strength against the current directional bias.
Rebound Signals — Highlights situations where price reclaims or loses the regression value while momentum remains aligned with trend direction.
🟠 HOW TO USE
Monitor the oscillator relative to its signal line to identify momentum shifts and changes in directional bias.
Watch for newly created liquidity levels after momentum weakening events, as these levels may become future sweep targets.
Use sweeps of upper or lower liquidity levels to identify areas where price has taken resting liquidity.
Look for bullish rebound signals when price reclaims the regression value while bullish momentum remains active.
Look for bearish rebound signals when price loses the regression value while bearish momentum remains active.
Combine oscillator direction, liquidity levels, and regression band structure to build context around trend continuation or reversal scenarios.
🟠 CONCLUSION
The Nadaraya-Watson Regression Liquidity Sweeps indicator combines regression-based momentum analysis, volatility-adjusted trend structure, and liquidity level tracking. By linking momentum transitions to swing-derived liquidity zones, it helps identify where liquidity may be forming and when it has been swept. This provides traders with additional context for trend analysis, pullbacks, and potential reversal areas. Indicator
Iterative Epanechnikov ChannelThe Iterative Epanechnikov Channel is a kernel-based smoothing and state estimation framework that applies an Epanechnikov kernel regression to price data, combined with a rolling standard deviation envelope to construct adaptive dynamic channel boundaries.
Unlike kernels with infinite support that allow distant historical observations to influence the estimate, the Epanechnikov kernel uses a compact weighting structure that strictly limits influence to a finite window. This ensures that only relevant, recent price information contributes to the regression, resulting in a more localized and structurally responsive estimate of price behavior.
The resulting channel is structurally responsive while remaining statistically efficient, making it particularly effective for tracking short-to-intermediate trend development, identifying localized overextension, and adapting quickly to evolving conditions.
Its primary utility is as a state estimation and structural tracking tool for price behavior, rather than a persistence-weighted regime model.
TRADING USES
The Epanechnikov Channel is best interpreted as a localized structural filter; within a multi-model framework, it captures the immediate structural state of price, helping distinguish early transitions, active trends, and short-term disequilibrium relative to slower, memory-weighted estimators.
Trend Detection
The channel basis line (Epanechnikov smoothed price) provides a responsive representation of underlying market direction. Sustained movement above or below the basis reflects directional continuation, while rapid shifts in the basis can indicate emerging changes in trend.
Structural Responsiveness
Due to the compact support of the Epanechnikov kernel, only recent price data contributes to the estimate. This produces sharper turning points and faster adaptation to new information, allowing the channel to respond efficiently to changes in market structure.
Mean Reversion Context
Because the estimator is more localized, price interacts with the channel boundaries more frequently. These interactions represent short-term deviations from the estimated state:
- Upper band: localized overextension
- Lower band: localized underextension
This makes the channel well-suited for mean reversion frameworks and volatility-based entry timing.
State Estimation
The channel functions as a continuous estimator of market state:
- The basis represents the inferred local price state
- The envelope represents dynamic volatility dispersion around that state
Compared to heavy-tailed kernels, the Epanechnikov-based state estimate is more sensitive to current conditions and less influenced by distant history, providing a clearer view of present market structure.
Volatility & Risk Context
The rolling standard deviation envelope expands and contracts based on realized volatility, providing a contextual risk framework. Wider channels indicate increased uncertainty and dispersion, while tighter channels indicate compression and lower variance conditions.
THEORY
The Epanechnikov kernel is a quadratic, compact-support kernel used in Nadaraya–Watson nonparametric regression, introduced by V. A. Epanechnikov (1969, Non-Parametric Estimation of a Multivariate Probability Density, Theory of Probability & Its Applications) as the mean squared error–optimal bounded kernel; in this implementation it is applied causally (non-repainting) and centered at the current bar using only historical data, with the original startAtBar offset removed to maintain proper kernel alignment with the estimation point.
It is defined as:
K(u) = 3/4 (1 − u²), for |u| ≤ 1
Where:
---> u represents normalized distance from the current observation
---> ℓ (lookback) defines the window over which the kernel operates
Unlike Gaussian kernels, which apply exponentially decaying weights over an infinite range, the Epanechnikov kernel assigns zero weight to all observations outside its finite support. This produces a strictly localized estimator that is both computationally efficient and statistically optimal in a mean squared error sense among bounded kernels.
Because the kernel is centered on the current observation and evaluated using only past data, the implementation remains causal and non-repainting while preserving the essential structure of kernel regression.
The rolling standard deviation complements this by measuring dispersion around the estimated state, forming a volatility-adaptive envelope. Rather than acting as a strict statistical confidence interval, it provides a dynamic representation of market expansion and contraction. The Epanechnikov kernel is a localized smoothing estimator rather than a structural similarity model, thus dispersion is defined using price-based volatility rather than kernel-weighted variance, providing a stable and interpretable envelope consistent with its role as a reactive state estimator.
The iterative implementation processes data sequentially (bar-by-bar), ensuring computational efficiency and making the indicator suitable for real-time use without repainting.
CALIBRATION
Calibration determines the balance between responsiveness, noise, and structural clarity.
Length (Lookback)
- Lower (8–16): More responsive, increased sensitivity to short-term structure
- Medium (20–40): Balanced for swing trading and intermediate regimes
- Higher (50–64+): Smoother output, reduced noise, slower response to turning points
Smoothing Mode (Single vs Double Pass)
Controls the tradeoff between responsiveness and stability:
Single Pass:
- Pure Epanechnikov regression
- Maximum responsiveness
- Faster detection of structural changes
- Increased sensitivity to noise
Double Pass:
- Applies the kernel regression twice
- Reduces variance and smooths fluctuations
- Produces cleaner structural output
- Introduces additional lag
This parameter allows users to tune the indicator based on whether early signal detection or stability is preferred.
MARKET USAGE
Stock, Forex, Crypto, Commodities, and Indices. Indicator
Meridian Lens PRO🟦 Meridian Lens PRO is a multi-kernel trend indicator built on the KernelLens Nadaraya–Watson regression library (a_jabbaroff/KernelLens/1). Three independently configurable kernel lines — Fast, Medium, and Slow — cover the full reactivity spectrum from scalping to position trading, each accepting any of the eight kernel families and three filter modes exposed by the library. The visual layer applies volume-intensity-adaptive coloring, gradient-filled trailing bands, 3-layer neon glow signal arrows, and a theme-aware dashboard — all driven by a single theme selection from ten optical-brand palettes.
🟦 HOW IT WORKS
Meridian Lens PRO calls the KernelLens library's unified dispatcher (`kl.estimate`) three times per bar — once for each kernel line:
```
Fast = kl.estimate(type, src, bw=8, α, period, phase, filter)
Medium = kl.estimate(type, src, bw=16, α, period, phase, filter)
Slow = kl.estimate(type, src, bw=32, α, period, phase, filter)
```
Each line independently selects its kernel family (Rational Quadratic, Gaussian, Periodic, Locally Periodic, Epanechnikov, Tricube, Triangular, Cosine), its filter mode (No Filter / Smooth / Zero Lag), its bandwidth, shape α, period, phase, and line width. The library handles all weighted-sum computation, loop-depth selection, NA-safe iteration, and input validation internally.
The Medium line is the primary trend reference — it drives the trailing bands, the main signal arrows, the dashboard trend cell, and the direction variable that colors every visual component. The Fast line provides early-warning reactivity for short-term entry timing. The Slow line anchors the macro trend for crossover logic and confluence scoring.
🟦 KERNEL LIBRARY INTEGRATION
Meridian Lens imports the published KernelLens library and uses the following exports:
| Library Export | Used For |
|---|---|
| `kl.estimate()` | Unified dispatcher — routes to the correct kernel based on user's dropdown selection |
| `kl.trendState()` | Returns +1 / −1 / 0 for each kernel's slope — drives dashboard arrows and signal triggers |
| `kl.crossSignal()` | Detects Fast × Slow crossovers — drives the Cross row in the dashboard and crossover alerts |
The indicator does not reimplement any kernel math — all regression computation is delegated to the library, ensuring that every bug fix or optimization in the library automatically propagates to this indicator.
🟦 THREE KERNEL LINES
**Fast Kernel** — The most reactive line. Default bandwidth 8, No Filter. Designed for scalping and short-term entry timing. Flips direction frequently on noisy charts — its signal markers are OFF by default to avoid visual clutter.
**Medium Kernel** — The primary trend reference. Default bandwidth 16, Smooth filter. Drives the trailing bands, the main 3-layer glow signal arrows, the dashboard Trend cell, and the direction variable that colors every visual component. This is the indicator's core signal.
**Slow Kernel** — The macro trend anchor. Default bandwidth 32, Smooth filter. Provides structural support for crossover logic (Fast × Slow) and triple-line confluence scoring. Its signal markers are ON by default because Slow flips are rare and meaningful.
Each kernel group exposes: Show toggle, Kernel Type dropdown (8 families), Bandwidth, Shape α (RQ only), Period (Periodic / Locally Periodic only), Phase (non-repainting offset), Filter (None / Smooth / Zero Lag), and Line Width.
🟦 NON-REPAINTING BEHAVIOR
Meridian Lens inherits non-repainting behavior directly from the KernelLens library's `_phase` parameter. Each kernel line has its own Phase input (default: 2), which shifts the kernel center into the past by that many bars.
- Phase = 0 — live estimate, flickers on the current bar (real-time only; history is immutable)
- Phase = 1 — 1-bar lag, non-repainting once the bar is confirmed
- Phase = 2 — recommended balance between freshness and stability (default)
- Phase = 3+ — extra stability for swing and position trading
Historical repainting never occurs at any phase value. The library contains no `request.security` calls, no lookahead, and no array rotation that could leak future data. Every historical bar's plotted value is final once confirmed.
🟦 SIGNAL SYSTEM
The indicator produces three tiers of trend-flip signals, each visually distinct:
**Medium Signals (Primary)** — 3-layer neon glow arrows rendered when the Medium kernel's direction flips. The outer halo is large and 80% transparent, the middle layer is normal-sized and 50% transparent, and the core arrow is small and fully opaque — creating a luminous halo effect on dark charts. Controlled by the "Glow Effect" toggle.
**Slow Signals** — Minimal tiny arrows (40% transparent) that fire when the Slow kernel flips direction. ON by default — these mark rare, meaningful macro trend changes.
**Fast Signals** — Minimal tiny arrows (40% transparent) that fire when the Fast kernel flips direction. OFF by default — enable for early-warning entry timing on lower timeframes.
🟦 VISUAL PIPELINE
**Volume-Intensity Adaptive Color** — The Medium line's transparency responds to the current volume reading. High volume = bright line (volume-confirmed trend), low volume = dim line (low-conviction drift). Uses a 33-bar HMA-smoothed normalized volume metric. Disable for a fixed 50% transparency.
**Trailing Bands** — Gradient-filled bands on the bullish/bearish side of the Medium line. Band width is driven by the rolling 100-bar average candle body size multiplied by a configurable distance factor (default: 2.0×). Bull bands fill below the Medium line during uptrends, bear bands fill above during downtrends.
**Theme System** — Ten cohesive palettes drive every visual component:
| Theme | Bull | Bear |
|---|---|---|
| Prism | Forest green | Crimson red |
| Focus | Cyan steel | Deep orange |
| Solar | Warm amber | Indigo red |
| Frost | Sky blue | Soft lavender |
| Laser | Neon lime | Hot crimson |
| Aurora | Bright gold | Scarlet |
| Plasma | Electric aqua | Magenta |
| Bloom | Mint green | Hot pink |
| Eclipse | Deep navy | Dark crimson |
| Carbon | Near-black | Silver grey |
🟦 PRO DASHBOARD
A 2-column, 11-row theme-aware status panel that updates only on the last bar (zero historical overhead). Supports Dark and Light display modes with configurable position and text size.
| Row | Label | Content |
|---|---|---|
| Header | MERIDIAN LENS | DARK / LIGHT |
| Theme | Theme | Active palette name |
| Kernel | Kernel | Medium kernel type |
| Divider | KERNELS | — |
| Fast | Fast | ▲/▼ + price value (bull/bear colored) |
| Medium | Medium | ▲/▼ + price value (bull/bear colored) |
| Slow | Slow | ▲/▼ + price value (bull/bear colored) |
| Divider | SIGNALS | — |
| Trend | Trend | ▲ BULL / ▼ BEAR |
| Cross | Cross | ↑ UP / ↓ DOWN / — |
| Strength | Strength | ▰▰▰ TRIPLE / ▰▰▱ STRONG / ▰▱▱ WEAK / ▱▱▱ NEUTRAL |
**Confluence Strength** — Counts how many of the three kernels (Fast, Medium, Slow) have their trend aligned with the Medium's direction. Score 3 = TRIPLE BULL/BEAR, 2 = STRONG, 1 = WEAK, 0 = NEUTRAL.
🟦 ALERT CONDITIONS
Six opt-in alert conditions, each gated by its own toggle:
| Alert | Fires When |
|---|---|
| Bull Crossover | Fast line crosses above Slow line |
| Bear Crossover | Fast line crosses below Slow line |
| Trend Up | Medium kernel trend flips to rising |
| Trend Down | Medium kernel trend flips to falling |
| Triple Bullish | Fast > Medium > Slow AND Medium rising |
| Triple Bearish | Fast < Medium < Slow AND Medium falling |
All alerts use `alertcondition()` for maximum compatibility with PulseWire's alert system including webhooks.
🟦 RECOMMENDED PRESETS
| Style | Fast bw | Med bw | Slow bw | Phase | Med Filter | Chart |
|---|---|---|---|---|---|---|
| Scalper | 4–8 | 8–16 | 16–32 | 1 | No Filter | 1m–5m |
| Day Trader | 8–12 | 14–24 | 24–48 | 2 | Smooth | 15m–1h |
| Swing | 16–24 | 24–40 | 48–80 | 2 | Smooth | 4h–1D |
| Position | 24–48 | 40–80 | 80–200 | 3 | Smooth | 1D–1W |
🟦 COMPATIBILITY
- Pine Script v6
- All exchanges, all asset classes (crypto, forex, equities, commodities)
- All timeframes (1 minute through Monthly)
- No exchange-specific logic — fully deterministic
🟦 TECHNICAL NOTES
- **Library dependency** — `import a_jabbaroff/KernelLens/1` — all kernel regression math is delegated to the library
- **Plot budget** — 5 plots + 2 fills + 10 plotshapes = well under Pine's 64-plot limit
- **Table** — Single `var table` created once on `barstate.islast`, zero historical overhead
- **No persistent drawing objects** — no `box.new`, `label.new`, `line.new` — no garbage collection needed
- **Non-repainting** — inherits from the library's `_phase` parameter; no `request.security`, no lookahead
- **Volume-intensity** — uses HMA-smoothed normalized volume (33-bar window) for adaptive transparency
🟦 DISCLAIMER
Meridian Lens PRO is a technical analysis overlay indicator built on the KernelLens Nadaraya–Watson regression library. It is provided solely for educational and research purposes and does not constitute financial, investment, or trading advice.
Kernel regression is a local smoothing technique. It estimates the mean of a source series in the neighborhood of the current bar based on historical data, but it does not predict future prices, does not generate trading signals on its own, and does not guarantee the profitability of any strategy built on top of its output.
Past performance of any model does not guarantee future results. Markets contain systemic risks that cannot be eliminated by any amount of mathematical rigor. Responsibility for any trading decisions rests entirely with the user. Always apply sound capital management, conduct your own independent analysis, and never risk capital you are not prepared to lose.
The author assumes no liability for direct or indirect losses incurred through the use of Meridian Lens or the underlying KernelLens library.
Indicator
KernelLens🟦 KernelLens is a professional kernel regression library for Pine Script v6, providing eight mathematically rigorous Nadaraya–Watson estimators, a three-mode filter layer, a unified string dispatcher, and a suite of trading utilities — all built from the ground up on correct non-parametric statistics. Unlike existing Pine smoothing libraries — which inherit a decade-old loop-bound bug that silently reduces every kernel window to a handful of bars, regardless of the bandwidth parameter — KernelLens is built with auditable math, NA-safe iteration, input validation at every entry point, and academic references cited inline next to the formulas they describe.
The library integrates eight independent kernel families — Rational Quadratic, Gaussian, Periodic, Locally Periodic, Epanechnikov, Tricube, Triangular, and Cosine — behind a consistent API, with every raw estimator wrapped in a filter layer (None / Smooth / Zero Lag), a unified dispatcher for dropdown-driven kernel selection, and five utility exports covering slope detection, trend state, crossover signaling, residual confidence bands, and Silverman's rule-of-thumb bandwidth recommendation. Every public function validates its inputs, raises descriptive runtime errors on misuse, and returns `na` only when there is genuinely no data — never as a silent fallback.
🟦 MATHEMATICAL FOUNDATION
**The Nadaraya–Watson Estimator**
Given a source series `y_t` and a symmetric kernel `K` with scale parameter `ℓ` (the "bandwidth"), the Nadaraya–Watson estimator of the regression function `m(x) = E ` evaluated at the current bar is:
```
Σᵢ K(dᵢ / ℓ) · y_{t−i}
ŷ(t) = ───────────────────────
Σᵢ K(dᵢ / ℓ)
```
where `dᵢ` is the bar-distance from the kernel center and the sum runs over a finite window determined by the effective support of `K`.
The estimator is a locally weighted average: bars close to the kernel center contribute heavily, distant bars contribute proportionally less, and bars outside the support contribute nothing. It is asymptotically unbiased up to `O(ℓ²)` for twice-differentiable `m`, with variance of order `(n·ℓ)⁻¹` — the classical bias–variance trade-off that defines all non-parametric smoothers.
**Why Kernel Regression Beats Rolling Means**
A simple moving average gives every bar in the window the same weight. Kernel regression gives each bar a weight that decays smoothly with distance, producing:
- **Smoother output** — no step artifacts when bars enter / leave the window
- **Better bias control** — the peak of the kernel sits exactly on the point being estimated
- **Kernel-specific behavior** — compact-support kernels eliminate tail contamination entirely; Rational Quadratic's `α` parameter exposes multi-scale mixing; Periodic kernels resonate with known cycle lengths
The math has been the academic standard for non-parametric regression since Nadaraya (1964) and Watson (1964). KernelLens brings it to Pine Script v6 in its correct, bug-free form.
🟦 THE EIGHT KERNELS
All eight kernels implement the Nadaraya–Watson weighting scheme. They differ in support (compact versus infinite), smoothness (how many times differentiable), and how weight decays with distance.
| # | Kernel | Formula | Support | Smoothness | Character |
|---|---|---|---|---|---|
| 1 | **Rational Quadratic** | `(1 + d² / (2·α·ℓ²))^(−α)` | ℝ | C∞ | Multi-scale mixer — `α` controls stretch versus wiggle |
| 2 | **Gaussian (RBF)** | `exp(−d² / (2·ℓ²))` | ℝ | C∞ | The canonical smoother — smoothest possible with L² optimality |
| 3 | **Periodic** | `exp(−2·sin²(π·d/p) / ℓ²)` | ℝ | C∞ | Resonates with repetition distance `p` — ideal for cycles |
| 4 | **Locally Periodic** | Periodic · Gaussian | ℝ | C∞ | Seasonal patterns that slowly drift with trend |
| 5 | **Epanechnikov** | `(3/4)(1 − u²) · 𝟙{|u|≤1}` | | C⁰ | Asymptotically MSE-optimal (Watson 1964) — no tail contamination |
| 6 | **Tricube** | `(70/81)(1 − \|u\|³)³ · 𝟙{|u|≤1}` | | C² | The LOWESS standard — near-Gaussian with compact support |
| 7 | **Triangular** | `(1 − \|u\|) · 𝟙{|u|≤1}` | | C⁰ | Simplest non-uniform kernel — fastest to compute |
| 8 | **Cosine** | `(π/4)·cos(π·u/2) · 𝟙{|u|≤1}` | | C¹ | Raised-cosine taper — smoother boundary than Epanechnikov |
where `u = d/ℓ` and `𝟙` is the indicator function.
**Infinite-Support vs Compact-Support — Why Both Matter**
| | Infinite Support (RQ, Gauss, Periodic, LocPeriodic) | Compact Support (Epa, Tricube, Triangular, Cosine) |
|---|---|---|
| **Tail weight** | Never exactly zero | Exactly zero beyond ±ℓ |
| **Loop depth** | `3·ℓ` (3-σ cutoff, ≈99.7% mass) | Exactly `ℓ` |
| **Bar contamination** | Distant bars still pull the estimate a tiny amount | Distant bars cannot affect the estimate at all |
| **Best for** | Smooth trends, Gaussian-process intuition | Robust regression, outlier resistance |
KernelLens picks the correct loop depth automatically based on kernel family: `_depthInfinite` for Gaussian-family kernels, `_depthCompact` for bounded kernels, `_depthPeriodic` for Periodic (which must span enough cycles to reach stable weights).
**Why Eight, Not Four**
Most Pine kernel libraries ship only the four kernels from MacKay's Gaussian process tutorial. KernelLens adds the four compact-support classical kernels because:
- **Epanechnikov** minimises asymptotic mean squared error among all non-negative kernels of bounded support (Watson 1964) — it is the MSE-optimal baseline against which all other kernels are measured
- **Tricube** is the kernel used by LOWESS (Cleveland 1979), the de-facto standard for robust locally weighted scatterplot smoothing
- **Triangular** is the cheapest non-uniform compact kernel — useful when loop-budget matters on intraday charts with huge dataset size
- **Cosine** is C¹-continuous at the support boundary, unlike Epanechnikov's C⁰ discontinuity, producing visibly smoother transitions at kernel edges
Adding them makes the library an academically complete toolkit, not just a Pine port of one tutorial.
🟦 FILTER LAYER — NONE / SMOOTH / ZERO LAG
Every kernel export accepts a `_filter` parameter with three valid values. The filter layer is implemented identically across all eight kernels, so switching kernel families does not change filter behavior.
**"No Filter" — Single-Pass Raw Estimate**
```
ŷ = K(y)
```
One Nadaraya–Watson pass over the source. Cheapest mode, most reactive, fully represents the underlying kernel. Use this when you want the kernel's raw behavior with no additional smoothing or lag correction.
**"Smooth" — Double-Pass Estimate**
```
ŷ = K(K(y))
```
The kernel is applied once to the source, then applied again to its own output using the same bandwidth and the same parameters. The result is a more strongly smoothed curve at the cost of one extra loop pass per bar.
This is mathematically equivalent to convolving the kernel with itself — the effective kernel is wider and flatter, pulling longer-range context into each estimate without requiring the user to double the bandwidth.
**"Zero Lag" — Ehlers De-Lagged Estimate**
```
ŷ = 2·K(y) − K(K(y))
```
The ZLEMA identity from Ehlers (*Rocket Science for Traders*, 2000): subtract the smoothing lag from the raw estimate, effectively shifting the output back in time to match the source more closely.
The intuition: `K(y)` lags `y` by some amount; `K(K(y))` lags `K(y)` by the same amount; so `K(y) − K(K(y))` is an estimate of the lag itself, and adding it back to `K(y)` cancels out. The result tracks the source more tightly than either pass alone, at the cost of slightly noisier turning points.
**Lazy Evaluation — No Wasted Cycles**
In `"No Filter"` mode, the second pass is skipped entirely — it never runs. The filter branch uses an `if` block (not a ternary), so Pine's short-circuit semantics prevent the unused computation. A single kernel call costs one pass; `"Smooth"` or `"Zero Lag"` costs two. You only pay for what you use.
🟦 KERNEL CENTER OFFSET — THE `_phase` PARAMETER
Every KernelLens kernel takes a `_phase` parameter that shifts the kernel center into the past by `_phase` bars. It is the library's non-repainting knob.
**_phase = 0 — Live Estimate**
The kernel is centered on the current bar. The most recent price has maximum weight, and the estimate is as fresh as possible. Suitable for live signal generation, but the most recent bar can re-evaluate as it develops within its interval — standard Pine real-time behavior.
**_phase > 0 — Non-Repainting Historical Estimate**
The kernel center is moved `_phase` bars into the past. The estimate becomes the smoothed value *at that historical bar*, not the current bar. Once the bar at `bar_index − _phase` is fully confirmed (`barstate.isconfirmed`), its estimate cannot change again.
This is the standard trick for publishing kernel indicators that do not repaint: you get a stable, historically accurate curve at the cost of shifting the entire output `_phase` bars to the right on the chart. A `_phase = 25` call gives a curve that lags live price by 25 bars but is guaranteed stable for every past bar.
**Why It Belongs in the Library, Not the Caller**
Pushing `_phase` into the kernel's own loop is not the same as evaluating the kernel at a shifted source (`K(src )`). Shifting the source just uses a stale input with a current-bar-centered kernel, which still produces a fresh estimate of a stale series. KernelLens's `_phase` genuinely moves the kernel center, producing a historical-bar estimate that computes over the correct surrounding window.
🟦 NON-REPAINTING BEHAVIOR
Repainting is the single most-asked question about any Pine indicator, and the single most common source of silent failure when a retail trader moves from backtest to live. A strategy that looks flawless on historical bars and then bleeds money the moment it is deployed is almost always suffering from some form of repainting. KernelLens is engineered from first principles to eliminate every class of repainting by construction — not by patching symptoms, but by removing the dependencies that cause repainting in the first place.
**The Two Forms of Repainting**
| Form | Symptom | Typical Cause |
|---|---|---|
| **Historical repainting** | A bar that was closed days or weeks ago silently changes its plotted value when the chart is refreshed or scrolled | `request.security()` with `lookahead = barmerge.lookahead_on`, un-gated higher-timeframe data, or incorrect array rotation that reads into future bars |
| **Real-time repainting** | The plotted value on the live (current developing) bar flickers tick-by-tick as new price ticks arrive, then freezes at a final value when the bar closes | The indicator reads `close ` (or any current-bar value) inside a weighted sum — the current-bar weight changes every tick |
KernelLens avoids the first kind **entirely and unconditionally**: the library contains no `request.security` calls, no higher-timeframe lookups, no `lookahead_on` usage, and no array rotation that could leak future bars into the window. Every historical bar plotted by any KernelLens kernel is computed exclusively from bars that existed at the time that bar was closed. The plotted history is immutable.
Real-time repainting is controlled explicitly by the `_phase` parameter — it is the user's choice whether to accept tick-by-tick flicker on the live bar in exchange for zero lag (`_phase = 0`) or to eliminate the flicker entirely at the cost of a small fixed lag (`_phase ≥ 1`).
**Why Kernel Regression Normally Repaints (And How KernelLens Stops It)**
A traditional Nadaraya–Watson call centered on the current bar evaluates:
```
ŷ(t) = Σᵢ K(dᵢ/ℓ) · y_{t−i} for i = 0 … depth
```
On the live bar, the term `y_{t−0} = close ` is the current real-time price — which changes on every tick. Every tick moves the weighted sum, every tick moves the estimate, and the trader watching the chart sees the kernel plot flicker as the bar develops. The historical bars (where `close ` for that past bar is now fixed) are stable, but the live plot is unstable.
KernelLens's `_phase` parameter shifts the loop so the kernel runs over `i = _phase … _phase + depth`. With `_phase = 2`:
```
ŷ(t) = Σᵢ K((i−2)/ℓ) · y_{t−i} for i = 2 … 2 + depth
```
The sum no longer touches `close ` or `close ` — every bar it reads is already confirmed and cannot change. The live-bar kernel output is therefore identical from the first tick of the bar to the last tick of the bar, and identical again when the bar finally closes. There is no flicker and nothing to repaint.
**The Lag / Stability Trade-Off**
| `_phase` | Lag on Live Bar | Live-Bar Flicker | Historical Repainting | Best For |
|---|---|---|---|---|
| **0** | 0 bars | Yes (real-time only; history is stable) | None | Scalping, academic research, calibration |
| **1** | 1 bar | None | None | Fast day-trading; minimum acceptable lag for a live trading desk |
| **2** | 2 bars | None | None | Default for most users — the sweet spot between freshness and stability |
| **3** | 3 bars | None | None | Swing trading — extra margin against false flickers from erratic ticks |
| **5+** | 5+ bars | None | None | Position trading, long-term chart analysis, published signal marks |
Even at `_phase = 0`, **historical repainting never occurs** — only the live bar flickers during its own development. Once a bar closes, its plotted value is final; scrolling away and back, refreshing the chart, or re-opening PulseWire will never change that historical plot. The flicker is exclusively a live-bar tick-by-tick phenomenon.
**KernelLens as a Non-Repainting Primitive**
KernelLens exposes real-time flicker as an explicit, user-controlled trade-off rather than a hidden behavior. The caller picks any point on the spectrum from "fully live" (`_phase = 0`, maximum reactivity with tick-by-tick flicker) to "fully confirmed" (`_phase ≥ 1`, one or more bars of lag in exchange for a curve that never redraws) with a single integer parameter. Historical repainting — the dangerous form that silently rewrites past plots — is eliminated unconditionally regardless of `_phase`.
**How to Verify Non-Repainting Yourself**
Do not trust the word "non-repainting" from any library — always verify. KernelLens can be verified in about thirty seconds:
1. Load a chart with KernelLens on it using `_phase = 2` (or any value > 0).
2. Take a screenshot at any specific historical bar.
3. Scroll far to the left, refresh the chart, or reload the indicator.
4. Return to the same bar. The plotted value at that bar must be pixel-identical to the screenshot — because the computation on that bar used only the bars before it, which have not changed.
5. Repeat with `_phase = 0`. The historical bars must still be pixel-identical — only the live bar's plot can differ between observations, and only because the live bar's `close` is now a different number than it was when you took the screenshot.
For a stricter test, use PulseWire's **Bar Replay** mode. Enable Bar Replay, step forward one bar at a time, and watch the kernel plot on each newly-closed bar. With `_phase ≥ 1`, the value plotted on each newly-closed bar will exactly match what the indicator shows after you exit replay mode and view the same bar normally. This is the gold-standard test — Bar Replay reproduces live-bar tick arrival in a controlled way.
**Common Misconceptions**
> *"Any Pine indicator that uses `close` repaints."*
False. Using `close` on a confirmed bar does not repaint — the confirmed bar's close is locked. What can repaint is using `close` on the live bar, and only within that live bar's interval. KernelLens with `_phase > 0` never reads the live-bar close at all.
> *"`lookahead = barmerge.lookahead_on` is always wrong."*
Context-dependent. `lookahead_on` is used correctly in some multi-timeframe indicators to request a higher-TF value that is already settled on the lower TF. KernelLens does not use `request.security` at all, so this question does not apply — but for libraries that do, `lookahead_on` is only problematic when it leaks values from bars that were not yet closed at the lower-TF time of evaluation.
> *"Non-repainting means zero lag."*
False. Zero lag and non-repainting are orthogonal properties. KernelLens `_phase = 0` is zero lag with real-time flicker; `_phase = 2` is two-bar lag with no flicker. You can have any combination of the two, and the right choice depends on the trading style.
> *"The `FILTER_ZEROLAG` mode makes the indicator non-repainting."*
False. `FILTER_ZEROLAG` is an Ehlers-style de-lagging filter applied to the kernel output; it reduces the perceived lag of the estimate, but it does not affect whether the live bar flickers. Non-repainting is controlled exclusively by `_phase`. Choose `_phase` for repainting behavior, and `_filter` for smoothness / lag shape — they are independent knobs.
**When to Accept Real-Time Flicker (`_phase = 0`)**
Despite everything above, there are legitimate reasons to deliberately use `_phase = 0`:
- **Academic research and backtesting** — you want the kernel mathematics in its classical form, centered on the point being estimated, with no phase adjustment
- **Scalping on very short timeframes** — a 2-bar lag on a 1-minute chart is a 2-minute delay, which can matter when you are exiting within a 4-minute window
- **Visual calibration** — when you are choosing a bandwidth by eye, the live-bar flicker actually helps: you see how sensitive the curve is to each incoming tick, which is diagnostic information
- **Indicators that read the kernel output only on `barstate.isconfirmed`** — if your signal logic is gated by `if barstate.isconfirmed`, then live-bar flicker is invisible to your signal (it sees only the frozen close-of-bar value), and you can safely use `_phase = 0` with no practical consequence
For every other case — and especially for any live alert or automated trading system — use `_phase ≥ 1`. Two bars of lag on a clean, stable curve is almost always worth more than zero lag on a curve that redraws itself several times per bar.
🟦 UNIFIED DISPATCHER — `estimate()`
For indicators where the user picks a kernel from a dropdown, writing eight separate ternary branches is tedious and error-prone. KernelLens ships with a unified dispatcher that routes to the correct kernel based on a string argument:
```pine
import a_jabbaroff/KernelLens/1 as kl
line = kl.estimate(
kernelType = kl.KERNEL_GAUSS,
src = close,
bandwidth = 32,
shapeAlpha = 1.0,
period = 1,
phase = 2,
filter = kl.FILTER_SMOOTH)
```
The dispatcher forwards to the matching typed export, so there is no performance penalty versus calling the kernel directly — it is a compile-time routing pass. Unknown kernel names raise a descriptive `runtime.error` naming every valid alternative, so typos fail loudly instead of silently returning `na`.
**Public Constants**
KernelLens exposes its string constants so callers never type the magic values by hand:
| Constant | Value |
|---|---|
| `FILTER_NONE` | `"No Filter"` |
| `FILTER_SMOOTH` | `"Smooth"` |
| `FILTER_ZEROLAG` | `"Zero Lag"` |
| `KERNEL_RQ` | `"Rational Quadratic"` |
| `KERNEL_GAUSS` | `"Gaussian"` |
| `KERNEL_PERIODIC` | `"Periodic"` |
| `KERNEL_LOCPER` | `"Locally Periodic"` |
| `KERNEL_EPA` | `"Epanechnikov"` |
| `KERNEL_TRICUBE` | `"Tricube"` |
| `KERNEL_TRIANG` | `"Triangular"` |
| `KERNEL_COSINE` | `"Cosine"` |
Using the constants in your caller code means the Pine compiler — not a runtime string compare — catches typos at edit time.
🟦 UTILITY LAYER — FIVE PROFESSIONAL HELPERS
KernelLens ships with five utility exports that complement the core estimators. They are the functions you almost always write immediately after getting a smoothed line, factored out so you don't rewrite them in every indicator.
**`slope(estimate, step)` — Discrete First Derivative**
Returns `(y_t − y_{t−step}) / step`, the normalized rate of change over `step` bars. Use it to detect whether a kernel output is trending up, flat, or down — the foundation for any trend-following signal built on top of KernelLens.
```pine
rising = kl.slope(line, 3) > 0.0
```
**`trendState(estimate, step)` — Ternary Trend Indicator**
Returns `+1` if the estimate is rising, `−1` if falling, `0` if exactly flat over the window. A single-call replacement for hand-rolled `line > line ? 1 : line < line ? -1 : 0` ladders.
**`crossSignal(fast, slow)` — Bi-directional Crossover**
Returns `+1` on the bar where `fast` crosses above `slow` (bullish), `−1` on a bearish cross, and `0` otherwise. Built on `ta.crossover` / `ta.crossunder`, so the signal is non-repainting once the bar is confirmed.
**`confidenceBand(src, estimate, window)` — Residual Standard Deviation**
Computes the rolling standard deviation of `(src − estimate)` over a user-defined window. Use the return value as the half-width of a confidence band around the estimate:
```pine
est = kl.gaussian(close, 32, 2, kl.FILTER_SMOOTH)
sigma = kl.confidenceBand(close, est, 50)
upper = est + 1.96 * sigma
lower = est - 1.96 * sigma
```
This is a computationally cheap proxy for the full kernel-weighted local variance — ideal when you need visual bands without paying for a second weighted pass.
**`silvermanBandwidth(src, window)` — Optimal ℓ Suggestion**
Returns the Silverman rule-of-thumb bandwidth:
```
h ≈ 1.06 · σ · n^(−1/5)
```
where `σ` is the rolling standard deviation of the source and `n` is the window size. This is the classical starting point for Gaussian-family bandwidths in academic texts (Silverman 1986). Because Pine requires `simple int` for kernel bandwidth, the returned value is intended for diagnostic display — plot it, read it off the chart, then hard-code the rounded integer into the kernel call.
🟦 INPUT VALIDATION — FAIL LOUDLY, FAIL EARLY
Every public function in KernelLens validates its inputs through a set of internal `_assert*` helpers. Invalid arguments never produce silent `na` fallbacks or buried zero-divisions — they raise `runtime.error` with a descriptive message identifying the function, the parameter, and the expected range.
| Helper | Checks | Raises On |
|---|---|---|
| `_assertFilter` | Filter string is `FILTER_NONE`, `FILTER_SMOOTH`, or `FILTER_ZEROLAG` | Typos like `"No FIlter"` (capital I) — a bug that exists in at least one published kernel indicator |
| `_assertBandwidth` | Bandwidth is a strictly positive integer | Negative or zero bandwidth, which would cause division by zero or infinite loops |
| `_assertPeriod` | Period is a strictly positive integer | Zero period, which would cause `sin(π·d/0)` in Periodic kernels |
| `_assertAlpha` | Rational Quadratic shape parameter is strictly positive | Zero or negative `α`, which would invert the RQ formula |
Error messages are prefixed `KernelLens:` (or `KernelLens.:`) so they are easy to spot in the PulseWire runtime log. Every message names the parameter that failed, the value that was passed, and the set of valid alternatives — so a misconfigured chart tells you exactly what to fix.
🟦 LOOP DEPTH — THE BUG FIX THAT MOTIVATED KERNELLENS
The two most popular Pine kernel libraries on PulseWire share the same fatal bug: both compute their loop depth as
```pine
_size = array.size(array.from(_src))
```
where `array.from(_src)` creates a **one-element array containing the current value of `_src`**, so `_size` is always `1`. The loop then runs `for i = 0 to 1 + startAtBar`, effectively using only `startAtBar + 2` bars — completely ignoring the user's bandwidth. Every published kernel indicator built on those libraries inherits this silent miscalculation.
KernelLens replaces the broken helper with three explicit depth selectors:
| Helper | Depth | Used By |
|---|---|---|
| `_depthInfinite(bw)` | `max(bw · 3, 4)` | Gaussian, Rational Quadratic, Locally Periodic |
| `_depthCompact(bw)` | `max(bw, 4)` | Epanechnikov, Tricube, Triangular, Cosine |
| `_depthPeriodic(bw, p)` | `max(bw · 3, p · 10, 4)` | Periodic |
For Gaussian-family kernels, the `3·ℓ` cutoff captures approximately 99.7% of the kernel mass (the three-sigma rule). For compact-support kernels, the depth equals the bandwidth exactly — the loop terminates at the kernel's natural zero point. For Periodic kernels, the depth is the larger of the scale-based and cycle-based minima, so the loop always spans enough periods to produce a stable weighted average.
The loop counter `i` runs over bar offsets starting at `_phase`, every bar lookup is NA-checked before being incorporated into the sum, and the final `num / den` division is guarded against zero denominators. On a fresh chart, the kernel gracefully returns `na` for bars where the window extends past available history, rather than producing poisoned sums from implicit NA arithmetic.
🟦 API REFERENCE
**Core Kernel Estimators — Eight Exports**
| Export | Signature |
|---|---|
| `rationalQuadratic` | `(src, bandwidth, shapeAlpha, phase, filter) → float` |
| `gaussian` | `(src, bandwidth, phase, filter) → float` |
| `periodic` | `(src, bandwidth, period, phase, filter) → float` |
| `locallyPeriodic` | `(src, bandwidth, period, phase, filter) → float` |
| `epanechnikov` | `(src, bandwidth, phase, filter) → float` |
| `tricube` | `(src, bandwidth, phase, filter) → float` |
| `triangular` | `(src, bandwidth, phase, filter) → float` |
| `cosineKernel` | `(src, bandwidth, phase, filter) → float` |
**Unified Dispatcher**
| Export | Signature |
|---|---|
| `estimate` | `(kernelType, src, bandwidth, shapeAlpha, period, phase, filter) → float` |
**Utility Layer — Five Exports**
| Export | Signature |
|---|---|
| `slope` | `(estimate, step) → float` |
| `trendState` | `(estimate, step) → int` |
| `crossSignal` | `(fast, slow) → int` |
| `confidenceBand` | `(src, estimate, window) → float` |
| `silvermanBandwidth` | `(src, window) → float` |
**Parameter Types**
| Name | Pine Type | Description |
|---|---|---|
| `src` | `series float` | Source series (close, hl2, ohlc4, or any other price-derived series) |
| `bandwidth` | `simple int` | Kernel scale `ℓ`, must be `> 0` |
| `shapeAlpha` | `simple float` | Rational Quadratic shape parameter, must be `> 0` |
| `period` | `simple int` | Periodic repetition distance, must be `> 0` |
| `phase` | `simple int` | Kernel center offset in bars, must be `≥ 0` |
| `filter` | `simple string` | One of `FILTER_NONE`, `FILTER_SMOOTH`, `FILTER_ZEROLAG` |
| `kernelType` | `simple string` | One of the eight `KERNEL_*` constants |
| `step` | `simple int` | Finite-difference step for `slope` / `trendState`, must be `≥ 1` |
| `window` | `simple int` | Rolling window for `confidenceBand` / `silvermanBandwidth`, must be `≥ 2` |
🟦 USAGE EXAMPLES
**Minimal — One Gaussian Curve**
```pine
//@version=6
indicator("KernelLens — Gaussian Demo", overlay = true)
import a_jabbaroff/KernelLens/1 as kl
line = kl.gaussian(close, 32, 2, kl.FILTER_SMOOTH)
plot(line, "Gaussian", color = color.orange, linewidth = 2)
```
**Fast / Slow Crossover System**
```pine
//@version=6
indicator("KernelLens — RQ Crossover", overlay = true)
import a_jabbaroff/KernelLens/1 as kl
fast = kl.rationalQuadratic(close, 8, 1.0, 2, kl.FILTER_NONE)
slow = kl.rationalQuadratic(close, 32, 1.0, 2, kl.FILTER_SMOOTH)
cross = kl.crossSignal(fast, slow)
plot(fast, "Fast", color = color.aqua, linewidth = 2)
plot(slow, "Slow", color = color.orange, linewidth = 2)
plotshape(cross == 1, "Bull", location = location.belowbar,
color = color.lime, style = shape.triangleup, size = size.tiny)
plotshape(cross == -1, "Bear", location = location.abovebar,
color = color.red, style = shape.triangledown, size = size.tiny)
```
**Confidence Band Envelope**
```pine
//@version=6
indicator("KernelLens — Confidence Band", overlay = true)
import a_jabbaroff/KernelLens/1 as kl
est = kl.tricube(close, 48, 2, kl.FILTER_SMOOTH)
sigma = kl.confidenceBand(close, est, 50)
k = 1.96
upper = est + k * sigma
lower = est - k * sigma
plot(est, "Estimate", color = color.orange, linewidth = 2)
p1 = plot(upper, "+1.96σ", color = color.new(color.aqua, 70))
p2 = plot(lower, "−1.96σ", color = color.new(color.aqua, 70))
fill(p1, p2, color = color.new(color.aqua, 92))
```
**Dropdown-Driven Kernel Selection**
```pine
//@version=6
indicator("KernelLens — Dropdown", overlay = true)
import a_jabbaroff/KernelLens/1 as kl
kernelType = input.string(kl.KERNEL_GAUSS, "Kernel",
options = )
bandwidth = input.int(32, "Bandwidth", minval = 2)
alphaRQ = input.float(1.0,"RQ Alpha", minval = 0.01, step = 0.25)
period = input.int(20, "Period", minval = 1)
phase = input.int(2, "Phase", minval = 0)
filter = input.string(kl.FILTER_SMOOTH, "Filter",
options = )
line = kl.estimate(kernelType, close, bandwidth, alphaRQ, period, phase, filter)
plot(line, "KernelLens", color = color.orange, linewidth = 2)
```
🟦 TIMEFRAME PRESETS — BANDWIDTH BY STYLE
Kernel bandwidth is the single most important parameter. It controls the trade-off between reactivity (small `ℓ`, tight fit, noisier) and stability (large `ℓ`, smooth curve, slower to react). The presets below are tested starting points — adjust by ±25 % to taste.
---
**SCALPER — 1m / 3m / 5m**
| Parameter | Value |
|---|---|
| Bandwidth (ℓ) | 8 |
| Phase | 1 |
| Filter | `FILTER_NONE` |
| Best Kernel | Rational Quadratic or Gaussian |
| RQ shapeAlpha | 1.0 |
**Why:** Short bandwidth means the kernel reacts within a handful of bars. `FILTER_NONE` removes the double-pass lag, so the estimate tracks price as tightly as possible. Phase 1 keeps the estimate nearly live while still avoiding the current-bar tick noise.
---
**DAY TRADER — 15m / 30m / 1H**
| Parameter | Value |
|---|---|
| Bandwidth (ℓ) | 16 |
| Phase | 2 |
| Filter | `FILTER_SMOOTH` |
| Best Kernel | Gaussian or Tricube |
| RQ shapeAlpha | 1.0 |
**Why:** Balanced reactivity — the 16-bar Gaussian is the default Silverman range for intraday price data, and `FILTER_SMOOTH` removes most of the bar-to-bar chop without significantly increasing lag. Tricube provides near-identical behaviour with strict compact support and is preferred on noisy assets where outlier bars should not influence the curve.
---
**SWING TRADER — 4H / 1D**
| Parameter | Value |
|---|---|
| Bandwidth (ℓ) | 32 |
| Phase | 3 |
| Filter | `FILTER_SMOOTH` |
| Best Kernel | Rational Quadratic |
| RQ shapeAlpha | 2.0 |
**Why:** Swing trades need structural signals, not intraday noise. Rational Quadratic with `α = 2.0` mixes medium and long length scales, producing a curve that ignores transient spikes but catches genuine regime shifts. Phase 3 shifts the estimate three bars back so each swing decision is made against a fully confirmed kernel output.
---
**POSITION / LONG-TERM — 1D / 1W / 1M**
| Parameter | Value |
|---|---|
| Bandwidth (ℓ) | 64 |
| Phase | 5 |
| Filter | `FILTER_SMOOTH` or `FILTER_ZEROLAG` |
| Best Kernel | Gaussian or Locally Periodic |
| Period (if LP) | 52 (weekly cycle) |
**Why:** Position traders care about the macro trajectory. A Gaussian with ℓ = 64 produces a curve that only turns on genuine multi-month inflections. Locally Periodic with `period = 52` is the ideal choice when a clear seasonal cycle is present — it uses both the long-range Gaussian envelope and the 52-bar periodicity to highlight cycle turns that align with trend.
---
**RESEARCH — Academic / Backtest**
| Parameter | Value |
|---|---|
| Bandwidth (ℓ) | Compute via `silvermanBandwidth(src, 200)` |
| Phase | 0 |
| Filter | `FILTER_NONE` |
| Best Kernel | Epanechnikov |
**Why:** Epanechnikov is the MSE-optimal kernel; `FILTER_NONE` keeps the estimator in its classical single-pass form; `phase = 0` centers the kernel on the bar being evaluated. This is the configuration that matches the statistical literature exactly — use it when publishing research, running Monte-Carlo studies, or calibrating against reference implementations.
🟦 BANDWIDTH SELECTION
Bandwidth `ℓ` is the single most consequential choice in kernel regression. Too small and the estimate overfits local noise; too large and it flattens real structure. KernelLens exposes two helpers to support both manual and semi-automated bandwidth selection.
**Manual — Start with ℓ ≈ √n**
A practical starting point for financial time series: set `ℓ ≈ √window_of_interest`. If you care about 100-bar structure, try `ℓ = 10`. If you care about 400-bar structure, try `ℓ = 20`. Adjust by ±25 % based on how noisy the result looks.
**Silverman's Rule of Thumb**
The closed-form optimal bandwidth for Gaussian-family kernels under Gaussian source assumptions:
```
h ≈ 1.06 · σ · n^(−1/5)
```
Call `silvermanBandwidth(src, window)` to compute this value live. Because Pine requires `simple int` bandwidth at compile time, the returned value is for diagnostic use — plot it, read the stable value off the chart, then hard-code the rounded integer into your kernel calls.
**Leave-One-Out Cross-Validation (Manual)**
For academic rigor, compute the leave-one-out mean squared error for a range of bandwidths and pick the minimum. KernelLens does not automate this (it would require `series int` bandwidth, which Pine does not support inside kernel loops), but the formula is straightforward:
```
LOOCV(ℓ) = (1/n) · Σᵢ (yᵢ − ŷᵢ⁻ⁱ(ℓ))²
```
where `ŷᵢ⁻ⁱ` is the kernel estimate at bar `i` computed without including bar `i` in the sum. Evaluate offline, pick the minimum, hard-code the result.
🟦 FILTER SELECTION — WHEN TO USE EACH
| Filter | Best For | Avoid When |
|---|---|---|
| `FILTER_NONE` | Live signal generation, research / calibration, compact-support kernels on noisy data | Choppy markets where you need extra smoothing |
| `FILTER_SMOOTH` | Swing and position trades, confidence band midlines, most day-trading setups | Scalping — the double pass adds measurable lag |
| `FILTER_ZEROLAG` | Regime detection, crossover systems that need the curve to track price tightly | Low-volume assets — Zero Lag amplifies high-frequency noise |
The three filters use the same underlying kernel with the same bandwidth, so switching between them does not require re-tuning. Default to `FILTER_SMOOTH` when in doubt — it is the best-behaved option across the widest range of assets and timeframes.
🟦 COMPATIBILITY
KernelLens targets Pine Script v6 and runs on every PulseWire chart — no exchange, asset class, or timeframe restriction.
- **Crypto** — Spot, futures, perpetual contracts
- **Forex** — All majors, minors, and exotics
- **Equities** — Stocks, ETFs, indices
- **Commodities** — Metals, energy, agriculture
- **Timeframes** — 1 minute through Monthly
The library is deterministic — given the same source and parameters, every bar of every symbol produces the same estimate. No calibration is needed across assets; the bandwidth parameter alone controls smoothness, and the kernel formulas are scale-free in the source dimension. Silverman's bandwidth helper automatically adapts to each asset's volatility.
🟦 TECHNICAL NOTES
- **Pine Script v6** — uses the modern type system, strict type checking, and the `switch` expression in the unified dispatcher
- **Non-repainting** — kernel outputs for any confirmed bar depend only on that bar's history; there is no look-ahead, no `request.security` with lookahead, and no dependency on the unconfirmed current bar unless `_phase = 0` is deliberately chosen
- **NA-safe iteration** — every bar lookup inside a kernel loop is guarded by `if not na(y)`, so chart history gaps and warm-up bars cannot poison the weighted sum
- **Division-by-zero protection** — every kernel's final division checks `den > 0.0` and returns `na` if the denominator collapses (which can only happen on truly empty windows)
- **Input validation** — every public function asserts its preconditions up front via `_assertFilter`, `_assertBandwidth`, `_assertPeriod`, `_assertAlpha`, and raises `runtime.error` with a descriptive message on misuse — no silent `na` fallbacks
- **Lazy filter evaluation** — the `"No Filter"` path never executes the second kernel pass; the `if`-branch check short-circuits, so single-pass mode is as cheap as a raw kernel call
- **Correct loop bounds** — `_depthInfinite`, `_depthCompact`, and `_depthPeriodic` compute the correct window size per kernel family, fixing the silent `_size = 1` bug that plagues every other published Pine kernel library
- **No persistent state** — the library is purely functional: no `var`, no arrays, no history buffers that grow over time; every export is a pure expression of `(inputs) → output`, so Pine's `max_*_count` limits cannot be exceeded and the library cannot leak memory
- **O(bandwidth) per bar per kernel call** — the loop depth is bounded by the constants in Section 0; there is no hidden quadratic behavior and the cost scales linearly with the user-chosen bandwidth
- **Unicode-safe comments** — the source uses academic notation (`σ`, `ℓ`, `α`, `ŷ`, `ℝ`) where it improves readability; all strings are plain ASCII for runtime compatibility
🟦 ACADEMIC REFERENCES
Every kernel and every formula in KernelLens is cited inline in the source. The combined bibliography:
- **Nadaraya, E. A. (1964).** On estimating regression. *Theory of Probability & Its Applications*, 9(1), 141–142.
- **Watson, G. S. (1964).** Smooth regression analysis. *Sankhyā: The Indian Journal of Statistics, Series A*, 26(4), 359–372.
- **Cleveland, W. S. (1979).** Robust locally weighted regression and smoothing scatterplots. *Journal of the American Statistical Association*, 74(368), 829–836. *(Tricube kernel, LOWESS.)*
- **Silverman, B. W. (1986).** *Density Estimation for Statistics and Data Analysis*. Chapman & Hall, London. *(Bandwidth rule of thumb.)*
- **Wand, M. P. & Jones, M. C. (1995).** *Kernel Smoothing*. Chapman & Hall. *(Unified treatment of all eight kernels.)*
- **MacKay, D. J. C. (1998).** Introduction to Gaussian Processes. *NIPS Tutorial*. *(Periodic and Rational Quadratic kernels.)*
- **Ehlers, J. F. (2000).** *Rocket Science for Traders*. John Wiley & Sons. *(Zero-lag smoothing trick.)*
- **Rasmussen, C. E. & Williams, C. K. I. (2006).** *Gaussian Processes for Machine Learning*. MIT Press. *(Locally Periodic and Rational Quadratic kernels.)*
🟦 VERSIONING & LICENSE
- **Version** — 1.0.0
- **Pine Script** — v6
- **License** — Mozilla Public License 2.0
- **Status** — Production-ready
KernelLens follows semantic versioning. Minor versions add new exports without breaking existing ones; patch versions fix bugs; major versions may change function signatures and will be announced in the changelog.
🟦 DISCLAIMER
KernelLens is a mathematical library for non-parametric regression on financial time series using the Nadaraya–Watson method. The library is provided solely for educational and research purposes and does not constitute financial, investment, or trading advice.
Kernel regression is a local smoothing technique. It estimates the mean of a source series in the neighborhood of the current bar based on historical data, but it does not predict future prices, does not generate trading signals on its own, and does not guarantee the profitability of any strategy built on top of its output.
Past performance of any model does not guarantee future results. Markets contain systemic risks that cannot be eliminated by any amount of mathematical rigor in the kernel itself. Responsibility for any trading decisions made using this library rests entirely with the user. Always apply sound capital management, conduct your own independent analysis, and never risk capital you are not prepared to lose.
The author assumes no liability for direct or indirect losses incurred through the use of KernelLens or any indicator built on top of it. Library
Nadaraya-Watson Dynamic Envelopes [identityKa]Welcome to another premium-grade, free-to-use trading tool developed by identityKa.
The Nadaraya-Watson Dynamic Envelopes is a highly advanced, visually smooth overlay indicator designed to capture dynamic support and resistance zones. Instead of relying on traditional, jagged moving averages, this indicator utilizes a Gaussian-inspired smoothing technique (via an optimized ALMA mathematical engine) to wrap the price action in a flowing, highly responsive channel.
Whether you are a day trader looking for mean-reversion setups or a swing trader identifying macro tops and bottoms, this indicator provides crystal-clear visual guidance without cluttering your chart.
🔥 Key Features:
Dynamic Smoothing Engine: Creates non-jagged, flowing upper (resistance) and lower (support) bands that adapt to market volatility.
Mean Reversion Signals: Prints distinct triangle shapes (Red for Bearish, Green for Bullish) when the price action sharply interacts with the outer envelopes, highlighting high-probability reversal zones.
Smart HUD Dashboard: Includes a clean, fully customizable on-chart dashboard that displays the current engine status, price position relative to the midline, and a real-time AI Suggestion (LONG, SHORT, or Dangerous) based on the algorithmic state.
100% Customizable: Every aspect of the indicator, from bandwidth smoothness and multipliers to theme colors (featuring the signature identityKa neon palette) and dashboard positioning, can be adjusted in the settings menu.
💡 How to Trade with It:
Mean Reversion: Look for bullish (green) triangle signals when the price touches or pierces the lower green envelope for potential long entries. Conversely, look for bearish (red) triangle signals at the upper red envelope for short entries.
Trend Continuation: In strong trending markets, the midline (white) often acts as dynamic support/resistance.
Pro Tip: This indicator works best when combined with a primary trend filter (such as our Alpha SuperTrend Signal). Avoid taking reversal signals against a very strong macroeconomic trend unless confirmed by other Price Action concepts.
⚙️ Settings & Alerts:
The script comes with built-in alert conditions for both Support Bounces (Bullish) and Resistance Rejections (Bearish), allowing you to automate your workflow. All settings are globally accessible in English.
Elevate your chart quality. Trade with precision. Trade with identityKa. Indicator
Adaptive Nadaraya-Watson (Non Repainting) [Metrify]To understand this implementation of the Nadaraya-Watson estimator, we have to look at the core equation governing non-parametric regression. This script aren't trying to average prices; we are trying to find the probability density of where price should be relative to its recent history.
1. The Kernel Physics (Bandwidth Modulation)
In standard kernel regression, you have a bandwidth parameter (h). This controls the "smoothness" of the curve. If h is too low, the curve jitters with every tick of noise. If h is too high, it acts like a sluggish SMA.
A static h fails because market volatility is dynamic. When the market explodes (high volatility), a tight bandwidth generates false signals. When the market sleeps, a wide bandwidth misses the micro-trends.
It try solving this by making h a function of the Asset's volatility ratio:
heff=h×max(0.5,min(SMA(ATR20,100)ATR20,2.0))
If the current ATR(20) is double the long-term average (100), the bandwidth doubles. This forces the estimator to "zoom out" during chaos, effectively ignoring noise that would otherwise look like a reversal.
vol_ratio = use_vol ? vol_raw / (vol_base == 0 ? 1 : vol_base) : 1.0
vol_mod = math.max(0.5, math.min(vol_ratio, 2.0))
h_eff = h_val * vol_mod
2. The Gaussian Loop (Endpoint Estimation)
Standard Nadaraya-Watson scripts repaint because they calculate the regression over a full window centered on the bar. To make this usable for live trading, we must calculate the Endpoint Estimate.
We iterate backward from the current bar (i=0) to the lookback limit. For every historical price Xi, we calculate a weight wi based on how far away it is in time (distance).
The weight is derived from the Gaussian Kernel function:
wi=exp(−2heff2i2)
Price data closer to the current bar (i=0) gets a weight near 1.0. Data further away (i=50) decays exponentially toward 0.
for i = 0 to lookback by 1
float dist = float(i)
float w = math.exp(-math.pow(dist, 2) / (2 * math.pow(h_eff, 2)))
num := num + w * src
den := den + w
3. Statistical Deviation (MAE vs. StDev)
Most Bollinger Band-style indicators use Standard Deviation (Root Mean Square). The problem with StDev is that it squares the errors, which heavily penalizes large outliers. In crypto or volatile forex pairs, one wick can blow out the bands for 20 bars.
This one use Mean Absolute Error (MAE) instead.
MAE=N1∑∣Price−y^∣
MAE is linear. It measures the average distance price strays from the kernel estimate without squaring the penalty. This creates "tighter" bands that adhere closer to price action during normal trend behavior but don't expand ridiculously during a flash crash.
Pine Script
float error = math.abs(src - y_hat)
float mae = ta.sma(error, lookback)
We project two sets of bands:
Inner Band (Balanced): The "Noise Zone". Price inside here is considered random walk.
Outer Band (Precision): The "Exhaustion Zone". Price reaching here is statistically unlikely (2.8x MAE).
Input & Visual Summary
Kernel Physics:
h_val: The base smoothness. Lower (e.g., 6) = faster, noisier. Higher (e.g., 10) = slower, smoother.
use_vol: Keep this TRUE. It prevents the bands from being too tight during news events.
Envelope Statistics:
mult_in / mult_out: These are your risk settings. 1.5/2.8 is a standard deviation-like setting suited for MAE.
Indicator
Indicator
Kernel Regression Envelope with SMI OscillatorThis script combines the predictive capabilities of the **Nadaraya-Watson estimator**, implemented by the esteemed jdehorty (credit to him for his excellent work on the `KernelFunctions` library and the original Nadaraya-Watson Envelope indicator), with the confirmation strength of the **Stochastic Momentum Index (SMI)** to create a dynamic trend reversal strategy. The core idea is to identify potential overbought and oversold conditions using the Nadaraya-Watson Envelope and then confirm these signals with the SMI before entering a trade.
**Understanding the Nadaraya-Watson Envelope:**
The Nadaraya-Watson estimator is a non-parametric regression technique that essentially calculates a weighted average of past price data to estimate the current underlying trend. Unlike simple moving averages that give equal weight to all past data within a defined period, the Nadaraya-Watson estimator uses a **kernel function** (in this case, the Rational Quadratic Kernel) to assign weights. The key parameters influencing this estimation are:
* **Lookback Window (h):** This determines how many historical bars are considered for the estimation. A larger window results in a smoother estimation, while a smaller window makes it more reactive to recent price changes.
* **Relative Weighting (alpha):** This parameter controls the influence of different time frames in the estimation. Lower values emphasize longer-term price action, while higher values make the estimator more sensitive to shorter-term movements.
* **Start Regression at Bar (x\_0):** This allows you to exclude the potentially volatile initial bars of a chart from the calculation, leading to a more stable estimation.
The script calculates the Nadaraya-Watson estimation for the closing price (`yhat_close`), as well as the highs (`yhat_high`) and lows (`yhat_low`). The `yhat_close` is then used as the central trend line.
**Dynamic Envelope Bands with ATR:**
To identify potential entry and exit points around the Nadaraya-Watson estimation, the script uses **Average True Range (ATR)** to create dynamic envelope bands. ATR measures the volatility of the price. By multiplying the ATR by different factors (`nearFactor` and `farFactor`), we create multiple bands:
* **Near Bands:** These are closer to the Nadaraya-Watson estimation and are intended to identify potential immediate overbought or oversold zones.
* **Far Bands:** These are further away and can act as potential take-profit or stop-loss levels, representing more extreme price extensions.
The script calculates both near and far upper and lower bands, as well as an average between the near and far bands. This provides a nuanced view of potential support and resistance levels around the estimated trend.
**Confirming Reversals with the Stochastic Momentum Index (SMI):**
While the Nadaraya-Watson Envelope identifies potential overextended conditions, the **Stochastic Momentum Index (SMI)** is used to confirm a potential trend reversal. The SMI, unlike a traditional stochastic oscillator, oscillates around a zero line. It measures the location of the current closing price relative to the median of the high/low range over a specified period.
The script calculates the SMI on a **higher timeframe** (defined by the "Timeframe" input) to gain a broader perspective on the market momentum. This helps to filter out potential whipsaws and false signals that might occur on the current chart's timeframe. The SMI calculation involves:
* **%K Length:** The lookback period for calculating the highest high and lowest low.
* **%D Length:** The period for smoothing the relative range.
* **EMA Length:** The period for smoothing the SMI itself.
The script uses a double EMA for smoothing within the SMI calculation for added smoothness.
**How the Indicators Work Together in the Strategy:**
The strategy enters a long position when:
1. The closing price crosses below the **near lower band** of the Nadaraya-Watson Envelope, suggesting a potential oversold condition.
2. The SMI crosses above its EMA, indicating positive momentum.
3. The SMI value is below -50, further supporting the oversold idea on the higher timeframe.
Conversely, the strategy enters a short position when:
1. The closing price crosses above the **near upper band** of the Nadaraya-Watson Envelope, suggesting a potential overbought condition.
2. The SMI crosses below its EMA, indicating negative momentum.
3. The SMI value is above 50, further supporting the overbought idea on the higher timeframe.
Trades are closed when the price crosses the **far band** in the opposite direction of the trade. A stop-loss is also implemented based on a fixed value.
**In essence:** The Nadaraya-Watson Envelope identifies areas where the price might be deviating significantly from its estimated trend. The SMI, calculated on a higher timeframe, then acts as a confirmation signal, suggesting that the momentum is shifting in the direction of a potential reversal. The ATR-based bands provide dynamic entry and exit points based on the current volatility.
**How to Use the Script:**
1. **Apply the script to your chart.**
2. **Adjust the "Kernel Settings":**
* **Lookback Window (h):** Experiment with different values to find the smoothness that best suits the asset and timeframe you are trading. Lower values make the envelope more reactive, while higher values make it smoother.
* **Relative Weighting (alpha):** Adjust to control the influence of different timeframes on the Nadaraya-Watson estimation.
* **Start Regression at Bar (x\_0):** Increase this value if you want to exclude the initial, potentially volatile, bars from the calculation.
* **Stoploss:** Set your desired stop-loss value.
3. **Adjust the "SMI" settings:**
* **%K Length, %D Length, EMA Length:** These parameters control the sensitivity and smoothness of the SMI. Experiment to find settings that work well for your trading style.
* **Timeframe:** Select the higher timeframe you want to use for SMI confirmation.
4. **Adjust the "ATR Length" and "Near/Far ATR Factor":** These settings control the width and sensitivity of the envelope bands. Smaller ATR lengths make the bands more reactive to recent volatility.
5. **Customize the "Color Settings"** to your preference.
6. **Observe the plots:**
* The **Nadaraya-Watson Estimation (yhat)** line represents the estimated underlying trend.
* The **near and far upper and lower bands** visualize potential overbought and oversold zones based on the ATR.
* The **fill areas** highlight the regions between the near and far bands.
7. **Look for entry signals:** A long entry is considered when the price touches or crosses below the lower near band and the SMI confirms upward momentum. A short entry is considered when the price touches or crosses above the upper near band and the SMI confirms downward momentum.
8. **Manage your trades:** The script provides exit signals when the price crosses the far band. The fixed stop-loss will also close trades if the price moves against your position.
**Justification for Combining Nadaraya-Watson Envelope and SMI:**
The combination of the Nadaraya-Watson Envelope and the SMI provides a more robust approach to identifying potential trend reversals compared to using either indicator in isolation. The Nadaraya-Watson Envelope excels at identifying potential areas where the price is overextended relative to its recent history. However, relying solely on the envelope can lead to false signals, especially in choppy or volatile markets. By incorporating the SMI as a confirmation tool, we add a momentum filter that helps to validate the potential reversals signaled by the envelope. The higher timeframe SMI further helps to filter out noise and focus on more significant shifts in momentum. The ATR-based bands add a dynamic element to the entry and exit points, adapting to the current market volatility. This mashup aims to leverage the strengths of each indicator to create a more reliable trading strategy.
Strategy
Holt-Winters Forecast BandsDescription:
The Holt-Winters Adaptive Bands indicator combines seasonal trend forecasting with adaptive volatility bands. It uses the Holt-Winters triple exponential smoothing model to project future price trends, while Nadaraya-Watson smoothed bands highlight dynamic support and resistance zones.
This indicator is ideal for traders seeking to predict future price movements and visualize potential market turning points. By focusing on broader seasonal and trend data, it provides insight into both short- and long-term market directions. It’s particularly effective for swing trading and medium-to-long-term trend analysis on timeframes like daily and 4-hour charts, although it can be adjusted for other timeframes.
Key Features:
Holt-Winters Forecast Line: The core of this indicator is the Holt-Winters model, which uses three components — level, trend, and seasonality — to project future prices. This model is widely used for time-series forecasting, and in this script, it provides a dynamic forecast line that predicts where price might move based on historical patterns.
Adaptive Volatility Bands: The shaded areas around the forecast line are based on Nadaraya-Watson smoothing of historical price data. These bands provide a visual representation of potential support and resistance levels, adapting to recent volatility in the market. The bands' fill colors (red for upper and green for lower) allow traders to identify potential reversal zones without cluttering the chart.
Dynamic Confidence Levels: The indicator adapts its forecast based on market volatility, using inputs such as average true range (ATR) and price deviations. This means that in high-volatility conditions, the bands may widen to account for increased price movements, helping traders gauge the current market environment.
How to Use:
Forecasting: Use the forecast line to gain insight into potential future price direction. This line provides a directional bias, helping traders anticipate whether the price may continue along a trend or reverse.
Support and Resistance Zones: The shaded bands act as dynamic support and resistance zones. When price enters the upper (red) band, it may be in an overbought area, while the lower (green) band may indicate oversold conditions. These bands adjust with volatility, so they reflect the current market conditions rather than fixed levels.
Timeframe Recommendations:
This indicator performs best on daily and 4-hour charts due to its reliance on trend and seasonality. It can be used on lower timeframes, but accuracy may vary due to increased price noise.
For traders looking to capture swing trades, the daily and 4-hour timeframes provide a balance of trend stability and signal reliability.
Adjustable Settings:
Alpha, Beta, and Gamma: These settings control the level, trend, and seasonality components of the forecast. Alpha is generally the most sensitive setting for adjusting responsiveness to recent price movements, while Beta and Gamma help fine-tune the trend and seasonal adjustments.
Band Smoothing and Deviation: These settings control the lookback period and width of the volatility bands, allowing users to customize how closely the bands follow price action.
Parameters:
Prediction Length: Sets the length of the forecast, determining how far into the future the prediction line extends.
Season Length: Defines the seasonality cycle. A setting of 14 is typical for bi-weekly cycles, but this can be adjusted based on observed market cycles.
Alpha, Beta, Gamma: These parameters adjust the Holt-Winters model's sensitivity to recent prices, trends, and seasonal patterns.
Band Smoothing: Determines the smoothing applied to the bands, making them either more reactive or smoother.
Ideal Use Cases:
Swing Trading and Trend Following: The Holt-Winters model is particularly suited for capturing larger market trends. Use the forecast line to determine trend direction and the bands to gauge support/resistance levels for potential entries or exits.
Identifying Reversal Zones: The adaptive bands act as dynamic overbought and oversold zones, giving traders potential reversal areas when price reaches these levels.
Important Notes:
No Buy/Sell Signals: This indicator does not produce direct buy or sell signals. It’s intended for visual trend analysis and support/resistance identification, leaving trade decisions to the user.
Not for High-Frequency Trading: Due to the nature of the Holt-Winters model, this indicator is optimized for higher timeframes like the daily and 4-hour charts. It may not be suitable for high-frequency or scalping strategies on very short timeframes.
Adjust for Volatility: If using the indicator on lower timeframes or more volatile assets, consider adjusting the band smoothing and prediction length settings for better responsiveness. Indicator
Kernel Regression RibbonKernel Regression Ribbon is a flexible, visually pleasing trend identification tool. Plotting 8 different kernel regressions of different types and parameters allows the user to see where levels of support and resistance are being tested, retested and broken.
What’s Kernel Regression?
A statistical method for estimating the best fitting curve for a dataset, in this case, a time/price chart.
How’s Kernel Regression different from a Moving Average?
A Moving Average is basically a simple form of Kernel Regression, in that it uses a fixed (Retangular) Kernel function. In an MA, all data points are weighted equally over its length. However, a Kernel function reacts more to data points that are closer to the current point. This means it will adapt more quickly to changes in data than an MA. Due to this adaptability, Kernel functions often form part of Machine Learning.
Using this indicator:
Explore the default Regular mode first to get a feel for the inputs, which are more numerous than for MAs. Try out different settings, filters and intervals to get the best out of each kernel. Not all parameters are available for each KR. There are info tips to explain this in the menu, but I’ve also included handy, optional labels on the chart for each KR as a more accessible guide.
Once you know your way round the Regular mode, check out the Presets and start changing the parameters of each kernel to your liking in the “User KR1, KR2, … “ mode. Each kernel type has its strong and weak points. Blending different kernels is where this indicator comes into its own. Give your charts a funky shine!
This indicator does NOT repaint.
This script acknowledges, and hopefully showcases, the great work of @veryfid Kernel Regression Toolkit.
Indicator
Indicator
Nadaraya-Watson Envelope Strategy (Non-Repainting) Log ScaleIn the diverse world of trading strategies, the Nadaraya-Watson Envelope Strategy offers a different approach. Grounded in mathematical analysis, this strategy utilizes the Nadaraya-Watson kernel regression, a method traditionally employed for interpreting complex data patterns.
At the core of this strategy lies the concept of 'envelopes', which are essentially dynamic volatility bands formed around the price based on a custom Average True Range (ATR). These envelopes help provide guidance on potential market entry and exit points. The strategy suggests considering a buy when the price crosses the lower envelope and a sell when it crosses the upper envelope.
One distinctive characteristic of the Nadaraya-Watson Envelope Strategy is its use of a logarithmic scale, as opposed to a linear scale. The logarithmic scale can be advantageous when dealing with larger timeframes and assets with wide-ranging price movements.
The strategy is implemented using Pine Script v5, and includes several adjustable parameters such as the lookback window, relative weighting, and the regression start point, providing a level of flexibility.
However, it's important to maintain a balanced view. While the use of mathematical models like the Nadaraya-Watson kernel regression may provide insightful data analysis, no strategy can guarantee success. Thorough backtesting, understanding the mathematical principles involved, and sound risk management are always essential when applying any trading strategy.
The Nadaraya-Watson Envelope Strategy thus offers another tool for traders to consider. As with all strategies, its effectiveness will largely depend on the trader's understanding, application, and the specific market conditions. Strategy
Adaptive Price Channel (log scale)The field of technical analysis is consistently expanding, with numerous indicators used for market forecasting. Amongst them, a novel indicator dubbed the Adaptive Price Channel (log scale), inspired by the renowned Nadaraya-Watson Envelope (LuxAlgo) from LuxAlgo, is gaining traction for its distinctive features and versatility. Unlike its predecessor, the Adaptive Price Channel (log scale) is applicable on a logarithmic scale, thereby allowing it to be utilized on both smaller and larger timeframes.
1. Key Features
The Adaptive Price Channel (log scale) is founded on the trading view Pinescript language, version 5, with its primary aim to maximize the versatility and scalability of trading indicators. It allows traders to adapt it according to their preferred timeframe, thereby making it applicable for a wide range of trading strategies.
Its bandwidth can be adjusted through the input parameters, offering traders the flexibility to manipulate the indicator according to their strategic requirements. Furthermore, it provides an option for repainting smoothing. This option enables users to control the repainting effect in which the historical output of the indicator may change over time. When disabled, the indicator provides the endpoints of the calculations, ensuring consistency in historical values.
Moreover, the Adaptive Price Channel (log scale) allows for color customization, thereby improving visibility and user-friendliness. The colors of the indicator's upward and downward directions can be changed according to the user's preference.
2. Working Mechanism
The Adaptive Price Channel (log scale) uses the logarithm of the source, which is typically the closing price of a trading instrument. It leverages a Gaussian function that exponentially decreases the further the price moves away from the mean, accounting for both positive and negative values. The bandwidth of the Gaussian function can be adjusted to adapt to different market conditions.
Additionally, the Adaptive Price Channel (log scale) features an array of 500 lines for each bar, which helps in defining the boundaries or envelope for price movements. The calculations are executed using the Nadaraya-Watson estimator, which uses kernel regression for non-parametric analysis.
The calculated values for the upper and lower bounds of the envelope are then converted back from the logarithmic scale using the exponential function. This calculation process continues for each bar until the last bar in the data set.
To ensure optimal performance, the Adaptive Price Channel (log scale) uses dynamic repainting. If the repainting mode is enabled, it adjusts the smoothing of the indicator for the entire historical data, making the results more accurate.
3. Visualization and Alerts
The Adaptive Price Channel (log scale) offers an array of visual aids, including labels and plots. The upper and lower bounds of the envelope are plotted, and the indicator triggers labels at points where the closing price crosses these boundaries. These labels serve as alerts for potential trading opportunities.
4. Conclusion
The Adaptive Price Channel (log scale) is an innovative and adaptable trading indicator, drawing inspiration from its predecessor but introducing unique features to increase its versatility. By providing a repainting option, it ensures consistent historical values, thereby enhancing the reliability of the indicator. Furthermore, the capability to operate on a logarithmic scale broadens its usability for different timeframes. The Adaptive Price Channel (log scale) is a powerful tool for any trader, facilitating a better understanding of market dynamics, and enabling more informed decision-making. Indicator
Machine Learning: Lorentzian Classification█ OVERVIEW
A Lorentzian Distance Classifier (LDC) is a Machine Learning classification algorithm capable of categorizing historical data from a multi-dimensional feature space. This indicator demonstrates how Lorentzian Classification can also be used to predict the direction of future price movements when used as the distance metric for a novel implementation of an Approximate Nearest Neighbors (ANN) algorithm.
█ BACKGROUND
In physics, Lorentzian space is perhaps best known for its role in describing the curvature of space-time in Einstein's theory of General Relativity (2). Interestingly, however, this abstract concept from theoretical physics also has tangible real-world applications in trading.
Recently, it was hypothesized that Lorentzian space was also well-suited for analyzing time-series data (4), (5). This hypothesis has been supported by several empirical studies that demonstrate that Lorentzian distance is more robust to outliers and noise than the more commonly used Euclidean distance (1), (3), (6). Furthermore, Lorentzian distance was also shown to outperform dozens of other highly regarded distance metrics, including Manhattan distance, Bhattacharyya similarity, and Cosine similarity (1), (3). Outside of Dynamic Time Warping based approaches, which are unfortunately too computationally intensive for PineScript at this time, the Lorentzian Distance metric consistently scores the highest mean accuracy over a wide variety of time series data sets (1).
Euclidean distance is commonly used as the default distance metric for NN-based search algorithms, but it may not always be the best choice when dealing with financial market data. This is because financial market data can be significantly impacted by proximity to major world events such as FOMC Meetings and Black Swan events. This event-based distortion of market data can be framed as similar to the gravitational warping caused by a massive object on the space-time continuum. For financial markets, the analogous continuum that experiences warping can be referred to as "price-time".
Below is a side-by-side comparison of how neighborhoods of similar historical points appear in three-dimensional Euclidean Space and Lorentzian Space:
This figure demonstrates how Lorentzian space can better accommodate the warping of price-time since the Lorentzian distance function compresses the Euclidean neighborhood in such a way that the new neighborhood distribution in Lorentzian space tends to cluster around each of the major feature axes in addition to the origin itself. This means that, even though some nearest neighbors will be the same regardless of the distance metric used, Lorentzian space will also allow for the consideration of historical points that would otherwise never be considered with a Euclidean distance metric.
Intuitively, the advantage inherent in the Lorentzian distance metric makes sense. For example, it is logical that the price action that occurs in the hours after Chairman Powell finishes delivering a speech would resemble at least some of the previous times when he finished delivering a speech. This may be true regardless of other factors, such as whether or not the market was overbought or oversold at the time or if the macro conditions were more bullish or bearish overall. These historical reference points are extremely valuable for predictive models, yet the Euclidean distance metric would miss these neighbors entirely, often in favor of irrelevant data points from the day before the event. By using Lorentzian distance as a metric, the ML model is instead able to consider the warping of price-time caused by the event and, ultimately, transcend the temporal bias imposed on it by the time series.
For more information on the implementation details of the Approximate Nearest Neighbors (ANN) algorithm used in this indicator, please refer to the detailed comments in the source code.
█ HOW TO USE
Below is an explanatory breakdown of the different parts of this indicator as it appears in the interface:
Below is an explanation of the different settings for this indicator:
General Settings:
Source - This has a default value of "hlc3" and is used to control the input data source.
Neighbors Count - This has a default value of 8, a minimum value of 1, a maximum value of 100, and a step of 1. It is used to control the number of neighbors to consider.
Max Bars Back - This has a default value of 2000.
Feature Count - This has a default value of 5, a minimum value of 2, and a maximum value of 5. It controls the number of features to use for ML predictions.
Color Compression - This has a default value of 1, a minimum value of 1, and a maximum value of 10. It is used to control the compression factor for adjusting the intensity of the color scale.
Show Exits - This has a default value of false. It controls whether to show the exit threshold on the chart.
Use Dynamic Exits - This has a default value of false. It is used to control whether to attempt to let profits ride by dynamically adjusting the exit threshold based on kernel regression.
Feature Engineering Settings:
Note: The Feature Engineering section is for fine-tuning the features used for ML predictions. The default values are optimized for the 4H to 12H timeframes for most charts, but they should also work reasonably well for other timeframes. By default, the model can support features that accept two parameters (Parameter A and Parameter B, respectively). Even though there are only 4 features provided by default, the same feature with different settings counts as two separate features. If the feature only accepts one parameter, then the second parameter will default to EMA-based smoothing with a default value of 1. These features represent the most effective combination I have encountered in my testing, but additional features may be added as additional options in the future.
Feature 1 - This has a default value of "RSI" and options are: "RSI", "WT", "CCI", "ADX".
Feature 2 - This has a default value of "WT" and options are: "RSI", "WT", "CCI", "ADX".
Feature 3 - This has a default value of "CCI" and options are: "RSI", "WT", "CCI", "ADX".
Feature 4 - This has a default value of "ADX" and options are: "RSI", "WT", "CCI", "ADX".
Feature 5 - This has a default value of "RSI" and options are: "RSI", "WT", "CCI", "ADX".
Filters Settings:
Use Volatility Filter - This has a default value of true. It is used to control whether to use the volatility filter.
Use Regime Filter - This has a default value of true. It is used to control whether to use the trend detection filter.
Use ADX Filter - This has a default value of false. It is used to control whether to use the ADX filter.
Regime Threshold - This has a default value of -0.1, a minimum value of -10, a maximum value of 10, and a step of 0.1. It is used to control the Regime Detection filter for detecting Trending/Ranging markets.
ADX Threshold - This has a default value of 20, a minimum value of 0, a maximum value of 100, and a step of 1. It is used to control the threshold for detecting Trending/Ranging markets.
Kernel Regression Settings:
Trade with Kernel - This has a default value of true. It is used to control whether to trade with the kernel.
Show Kernel Estimate - This has a default value of true. It is used to control whether to show the kernel estimate.
Lookback Window - This has a default value of 8 and a minimum value of 3. It is used to control the number of bars used for the estimation. Recommended range: 3-50
Relative Weighting - This has a default value of 8 and a step size of 0.25. It is used to control the relative weighting of time frames. Recommended range: 0.25-25
Start Regression at Bar - This has a default value of 25. It is used to control the bar index on which to start regression. Recommended range: 0-25
Display Settings:
Show Bar Colors - This has a default value of true. It is used to control whether to show the bar colors.
Show Bar Prediction Values - This has a default value of true. It controls whether to show the ML model's evaluation of each bar as an integer.
Use ATR Offset - This has a default value of false. It controls whether to use the ATR offset instead of the bar prediction offset.
Bar Prediction Offset - This has a default value of 0 and a minimum value of 0. It is used to control the offset of the bar predictions as a percentage from the bar high or close.
Backtesting Settings:
Show Backtest Results - This has a default value of true. It is used to control whether to display the win rate of the given configuration.
█ WORKS CITED
(1) R. Giusti and G. E. A. P. A. Batista, "An Empirical Comparison of Dissimilarity Measures for Time Series Classification," 2013 Brazilian Conference on Intelligent Systems, Oct. 2013, DOI: 10.1109/bracis.2013.22.
(2) Y. Kerimbekov, H. Ş. Bilge, and H. H. Uğurlu, "The use of Lorentzian distance metric in classification problems," Pattern Recognition Letters, vol. 84, 170–176, Dec. 2016, DOI: 10.1016/j.patrec.2016.09.006.
(3) A. Bagnall, A. Bostrom, J. Large, and J. Lines, "The Great Time Series Classification Bake Off: An Experimental Evaluation of Recently Proposed Algorithms." ResearchGate, Feb. 04, 2016.
(4) H. Ş. Bilge, Yerzhan Kerimbekov, and Hasan Hüseyin Uğurlu, "A new classification method by using Lorentzian distance metric," ResearchGate, Sep. 02, 2015.
(5) Y. Kerimbekov and H. Şakir Bilge, "Lorentzian Distance Classifier for Multiple Features," Proceedings of the 6th International Conference on Pattern Recognition Applications and Methods, 2017, DOI: 10.5220/0006197004930501.
(6) V. Surya Prasath et al., "Effects of Distance Measure Choice on KNN Classifier Performance - A Review." .
█ ACKNOWLEDGEMENTS
@veryfid - For many invaluable insights, discussions, and advice that helped to shape this project.
@capissimo - For open sourcing his interesting ideas regarding various KNN implementations in PineScript, several of which helped inspire my original undertaking of this project.
@RikkiTavi - For many invaluable physics-related conversations and for his helping me develop a mechanism for visualizing various distance algorithms in 3D using JavaScript
@jlaurel - For invaluable literature recommendations that helped me to understand the underlying subject matter of this project.
@annutara - For help in beta-testing this indicator and for sharing many helpful ideas and insights early on in its development.
@jasontaylor7 - For helping to beta-test this indicator and for many helpful conversations that helped to shape my backtesting workflow
@meddymarkusvanhala - For helping to beta-test this indicator
@dlbnext - For incredibly detailed backtesting testing of this indicator and for sharing numerous ideas on how the user experience could be improved. Indicator
Nadaraya-Watson: Envelope (Non-Repainting)Due to popular request, this is an envelope implementation of my non-repainting Nadaraya-Watson indicator using the Rational Quadratic Kernel. For more information on this implementation, please refer to the original indicator located here:
What is an Envelope?
In technical analysis, an "envelope" typically refers to a pair of upper and lower bounds that surrounds price action to help characterize extreme overbought and oversold conditions. Envelopes are often derived from a simple moving average (SMA) and are placed at a predefined distance above and below the SMA from which they were generated. However, envelopes do not necessarily need to be derived from a moving average; they can be derived from any estimator, including a kernel estimator such as Nadaraya-Watson.
How to use this indicator?
Overall, this indicator offers a high degree of flexibility, and the location of the envelope's bands can be adjusted by (1) tweaking the parameters for the Rational Quadratic Kernel and (2) adjusting the lookback window for the custom ATR calculation. In a trending market, it is often helpful to use the Nadaraya-Watson estimate line as a floating SR and/or reversal zone. In a ranging market, it is often more convenient to use the two Upper Bands and two Lower Bands as reversal zones.
How are the Upper and Lower bounds calculated?
In this indicator, the Rational Quadratic (RQ) Kernel estimates the price value at each bar in a user-defined lookback window. From this estimation, the upper and lower bounds of the envelope are calculated based on a custom ATR calculated from the kernel estimations for the high, low, and close series, respectively. These calculations are then scaled against a user-defined multiplier, which can be used to further customize the Upper and Lower bounds for a given chart.
How to use Kernel Estimations like this for other indicators?
Kernel Functions are highly underrated, and when calibrated correctly, they have the potential to provide more value than any mundane moving average. For those interested in using non-repainting Kernel Estimations for technical analysis, I have written a Kernel Functions library that makes it easy to access various well-known kernel functions quickly. The Rational Quadratic Kernel is used in this implementation, but one can conveniently swap out other kernels from the library by modifying only a single line of code. For more details and usage examples, please refer to the Kernel Functions library located here:
Indicator
Nadaraya-Watson non repainting [LPWN]// ENGLISH
The problem of the wonderfuls Nadaraya-Watson indicators is that they repainting, @jdehorty made an aproximation of the Nadaraya-Watson Estimator using raational Quadratic Kernel so i used this indicator as inspiration i just added the Upper and lower band using ATR with this we get an aproximation of Nadaraya-Watson Envelope without repainting
Settings:
Bandwidth. This is the number of bars that the indicator will use as a lookback window.
Relative Weighting Parameter. The alpha parameter for the Rational Quadratic Kernel function. This is a hyperparameter that controls the smoothness of the curve. A lower value of alpha will result in a smoother, more stretched-out curve, while a lower value will result in a more wiggly curve with a tighter fit to the data. As this parameter approaches 0, the longer time frames will exert more influence on the estimation, and as it approaches infinity, the curve will become identical to the one produced by the Gaussian Kernel.
Color Smoothing. Toggles the mechanism for coloring the estimation plot between rate of change and cross over modes.
ATR Period. Period to calculate the ATR (upper and lower bands)
Multiplier. Separation of the bands
// SPANISH
El problema de los maravillosos indicadores de Nadaraya-Watson es que repintan, @jdehorty hizo una aproximación delNadaraya-Watson Estimator usando un Kernel cuadrático racional, así que usé este indicador como inspiración y solo agregamos la banda superior e inferior usando ATR con esto obtenemos una aproximación de Nadaraya-Watson Envelope sin volver a pintar
Configuración:
Banda ancha. Este es el número de barras que el indicador utilizará como ventana retrospectiva.
Parámetro de ponderación relativa. El parámetro alfa para la función Rational Quadratic Kernel. Este es un hiperparámetro que controla la suavidad de la curva. Un valor más bajo de alfa dará como resultado una curva más suave y estirada, mientras que un valor más bajo dará como resultado una curva más ondulada con un ajuste más ajustado a los datos. A medida que este parámetro se acerque a 0, los marcos de tiempo más largos ejercerán más influencia en la estimación y, a medida que se acerque al infinito, la curva será idéntica a la que produce el Gaussian Kernel.
Suavizado de color. Alterna el mecanismo para colorear el gráfico de estimación entre la tasa de cambio y los modos cruzados.
Período ATR. Periodo para calcular el ATR (bandas superior e inferior)
Multiplicador. Separación de las bandas Indicator
