Recursive Kernel Trend [QuantAlgo]🟢 Overview
The Recursive Kernel Trend is a trend-following indicator built on a recursive residual estimator with adaptive rate scheduling. It applies one of six selectable filter structures to a residual-corrected recursion, modulates the update rate according to efficiency and volatility conditions, and confirms directional state through slope persistence. The result is a responsive yet controlled trend line that adapts its tracking behavior to market regime while filtering noise-driven fluctuations across every timeframe and instrument.
🟢 How It Works
The calculation begins with a residual between the selected price source and the current estimate. This residual drives a base recursive update whose rate is not fixed but scheduled on every bar:
resid = src - estimate
base = estimate + kern_rate * resid
The scheduled rate is produced by combining two adaptive weights. Efficiency weighting measures the ratio of net directional progress to total price path over a lookback window, raising the rate when movement is clean and lowering it during chop. Volatility weighting compares current ATR against a longer baseline and reduces the rate when volatility expands. The combined rate is then bounded by floor and ceiling limits and further scaled by an optional directional bias that applies different multipliers depending on whether price sits above or below the estimate:
eff_weight = eff_floor + (1.0 - eff_floor) * eff_ratio
vol_weight = math.min(math.max(1.0 / vol_ratio, 0.50), 1.75)
rate_sched = math.min(math.max(base_rate * eff_weight * vol_weight, rate_floor), rate_ceil)
kern_rate = rate_sched * bias
Six filter structures can be applied to the base update. Standard uses a single pass. Wilder halves the rate for smoother behavior. Double and Triple apply successive lag-compensated stages. Gaussian cascades four poles without compensation. Hull combines fast and slow passes then re-smooths the result. All structures receive the live scheduled rate so the adaptive weighting remains active.
A residual accumulator runs in parallel with the recursion. It retains a decaying memory of past residuals and applies a correction term that closes persistent offset during sustained trends. An optional ATR-based limiter can bound the accumulator to prevent overshoot after gaps or parabolic moves:
corr_acc := corr_acc * corr_decay + resid
estimate := kern_out + corr_weight * corr_acc
Directional state is derived from the slope of the finished estimate after a short smoothing window. A consecutive run of bars in the same slope direction must reach a confirmation threshold before the state is allowed to flip. This step prevents single-bar noise from reversing the trend color or firing alerts.
🟢 Signal Interpretation
▶ Bullish Trend (Long/Buy): When the smoothed slope of the estimate remains positive for the required number of confirmation bars, the indicator enters bullish state. The trend line and gradient layers switch to the bullish color. This condition identifies potential long or buy opportunities and remains active until an equal run of negative slope bars confirms a reversal.
▶ Bearish Trend (Short/Sell): When the smoothed slope remains negative for the required confirmation bars, the indicator enters bearish state. The visual elements switch to the bearish color. This condition identifies potential short or sell opportunities and holds until a confirmed positive run occurs.
🟢 Features
▶ Preconfigured Presets: Three parameter sets cover different trading approaches. Default targets swing trading on 1H to daily charts with balanced rate and confirmation. Fast Response raises the recursion rate and shortens confirmation for intraday charts where the indicator needs to adapt to shorter-duration moves. Smooth Trend lowers the rate and lengthens confirmation for position trading on daily and weekly timeframes, where the cost of a false flip is higher than the cost of a delayed one. Selecting a preset overrides the individual rate, efficiency, and state detection inputs.
▶ Built-in Alerts: Three alert conditions are provided. Bullish State Signal fires when the trend state flips from bearish to bullish. Bearish State Signal fires on the opposite transition. Any State Change combines both into a single notification.
▶ Visual Customization: Six color presets (Classic, Aqua, Cosmic, Cyber, Neon, Custom) coordinate the trend line and gradient layers. Optional bar coloring tints candles with the active state color at a configurable transparency.
*Tips: Layer the Recursive Kernel Trend with complementary analysis rather than treating it as a standalone trading tool. State flips hold most reliably when backed by participation, so combine each change with volume context, since a flip on expanding volume is far more likely to sustain than one on thin flow, and read the move against market structure, as a reversal that aligns with a clear swing high or low carries more significance than one in open space. Pairing this script with volume, open interest, CVD, market structure, and mean reversion indicators from our QuantAlgo toolkit can further validate a directional shift before entry. Indicator
Nadaraya-Watson Trend [QuantAlgo]🟢 Overview
The Nadaraya-Watson Trend indicator estimates a smooth, adaptive trend path by applying non-parametric kernel regression directly to price. For each bar it weights historical values inside a configurable lookback window with a chosen kernel function, normalizes those weights, and returns a single endpoint estimate that forms the plotted trend line. Bandwidth and kernel type control how aggressively recent bars dominate the estimate, optional residual bands express how far price is dispersed around that path, and slope based coloring with reversal markers make direction and turning points readable at a glance across any timeframe or instrument.
🟢 How It Works
The indicator is built around a one sided Nadaraya-Watson (NW) estimator: only the current bar and past bars enter the calculation, so the path behaves as a causal smoother rather than a centered, repainting fit. The pipeline has three stages: kernel weighting over the lookback window, normalized regression into a single trend value, and optional residual band construction from the same estimate.
First, effective bandwidth is formed from the configured bandwidth and multiplier. Each lag distance is then mapped to a kernel weight. Gaussian and Rational Quadratic keep infinite support with different decay shapes. Compact kernels (Epanechnikov, Triangular, Quartic, Cosine) only assign weight while the normalized lag stays inside the unit interval:
kernel_weight(float dist, float h, string ktype, float rq) =>
float w = 0.0
if h > 0.0
float u = dist / h
if ktype == 'Gaussian'
w := math.exp(-(dist * dist) / (2.0 * h * h))
else if ktype == 'Rational Quadratic'
w := math.pow(1.0 + (dist * dist) / (2.0 * rq * h * h), -rq)
else if math.abs(u) <= 1.0
if ktype == 'Epanechnikov'
w := 0.75 * (1.0 - u * u)
else if ktype == 'Triangular'
w := 1.0 - math.abs(u)
else if ktype == 'Quartic'
w := (15.0 / 16.0) * math.pow(1.0 - u * u, 2.0)
else if ktype == 'Cosine'
w := (math.pi / 4.0) * math.cos(math.pi * u / 2.0)
w
float h = bandwidth * h_mult
Next, the Nadaraya-Watson path is computed as the normalized weighted average of the selected source across the lookback window. Nearer bars dominate when bandwidth is low. Weight spreads more evenly when bandwidth is high, producing a smoother path:
float sum_w = 0.0
float sum_p = 0.0
for i = 0 to lookback
float w = kernel_weight(i, h, kernel_type, rel_weight)
sum_w += w
sum_p += src * w
float nw_trend = sum_w != 0.0 ? sum_p / sum_w : na
Finally, residual bands can be drawn from a kernel weighted mean absolute residual of the source versus the current NW estimate, scaled by the band multiplier. When price is tightly clustered around the path the envelope contracts. When price is dispersed the envelope expands, framing extension and compression relative to the same estimator that defines the trend:
float sum_abs = 0.0
float sum_res_w = 0.0
for i = 0 to lookback
float w = kernel_weight(i, h, kernel_type, rel_weight)
if w > 0.0 and not na(src ) and not na(nw_trend)
sum_abs += w * math.abs(src - nw_trend)
sum_res_w += w
float residual = sum_res_w != 0.0 ? sum_abs / sum_res_w : na
float upper = not na(nw_trend) and not na(residual) ? nw_trend + residual * band_mult : na
float lower = not na(nw_trend) and not na(residual) ? nw_trend - residual * band_mult : na
🟢 Signal Interpretation
▶ Bullish Path (Rising NW Line with Bullish Color): When the Nadaraya-Watson estimate is increasing bar to bar, the path and optional gradient fill plot in the bullish color, reading as an uptrend in the kernel smoothed series. Treat this as a long bias: strongest on the reversal marker with price holding above the path, or on pullbacks that respect the path while slope stays up. Bias weakens if price loses the path and the slope flattens or flips down.
▶ Bearish Path (Falling NW Line with Bearish Color): When the estimate is decreasing bar to bar, the path and fill plot in the bearish color, reading as a downtrend in the kernel smoothed series. Treat this as a short bias: strongest on the reversal marker with price holding below the path, or on bounces that fail at the path while slope stays down. Bias weakens if price reclaims the path and the slope flattens or flips up.
▶ Residual Bands (Optional Envelope Around the Path): With residual bands enabled, the upper and lower lines track a scaled kernel weighted residual around the NW path. Touches or closes beyond the outer band highlight price stretched away from the estimate. Returns toward the path after an extension often mark mean reversion relative to the kernel trend rather than a full regime change. Band width is derived from how widely the source has been scattered around the current NW estimate inside the lookback window
🟢 Features
▶ Preconfigured Presets: Three parameter sets tuned for different trading styles and timeframes. "Default" delivers balanced trend estimation for swing trading on 1H to daily charts, smoothing short lived noise while still responding to genuine directional turns. "Fast Response" is built for intraday work on 5 minute to 1H charts, keeping the path tighter to recent structure so turns register earlier at the cost of more frequent reversals in chop. "Smooth Trend" is aimed at position style reading on daily and weekly charts, forming a more stable baseline that flips only when the kernel path itself shifts with more conviction. Kernel type, residual bands, and visual options stay independently configurable under every preset.
▶ Kernel Library: Six kernel functions expand how the same endpoint Nadaraya-Watson framework assigns weight across the window. Gaussian is the classic smooth default with infinite support. Epanechnikov, Triangular, Quartic, and Cosine are compact kernels that fully exclude bars beyond the bandwidth scale. Rational Quadratic keeps infinite support with heavier tails, and its Relative Weighting input controls how much influence farther bars retain versus a Gaussian like decay. Switching kernels changes the shape of the single plotted path without adding a second model or external oscillator.
▶ Residual Bands: Optional envelope around the NW path built from kernel weighted mean absolute residuals of the source versus the estimate, scaled by Band Multiplier. Enable when you want extension and compression context around the same trend line. Disable when you want only the path, gradient, and markers.
▶ Built-in Alerts: Five alert conditions support hands off monitoring. "Bullish Kernel Reversal" fires on the bar the path slope flips from down to up. "Bearish Kernel Reversal" fires on the bar the path slope flips from up to down. "Any Kernel Reversal" fires on either directional flip. "Source Cross Above Upper Band" and "Source Cross Below Lower Band" fire when the selected source crosses the residual envelope extremes. Alert messages include exchange, ticker, and timeframe for immediate context.
▶ Visual Customisation: Six color presets (Classic, Aqua, Cosmic, Cyber, Neon, and Custom) apply coordinated bullish and bearish colors to the path, gradient fill, residual bands, markers, optional bar coloring, and optional background coloring. Custom unlocks independent bullish and bearish color pickers. Gradient fill, residual bands, reversal markers, bar coloring, and background coloring can each be toggled so the chart stays as clean or as expressive as the workflow requires.
Indicator
Indicator
Iterative Locally Periodic EnvelopeThe Iterative Locally Periodic Envelope is a phase-conditioned kernel estimator with temporal locality and endogenous dispersion modeling, implemented as a Nadaraya–Watson estimator under a locally periodic kernel.
The locally periodic kernel defines similarity through cyclical phase alignment modulated by temporal proximity. Observations contribute to the estimator based on both their position within a repeating cycle structure and their recency, emphasizing structural recurrence with sensitivity to local regime conditions.
The indicator computes a latent equilibrium using a kernel-weighted mean and a dispersion measure using kernel-weighted variance under the same weighting structure. The resulting envelope reflects cycle-consistent deviation with temporal locality, rather than a conventional volatility band. All values are computed exclusively on closed historical bars using a bounded lookback window to ensure non-repainting behavior.
This indicator belongs to a broader class of iterative kernel-based envelopes that includes Gaussian, Rational Quadratic, and Periodic variants. All share a common Nadaraya–Watson estimation framework, differentiated by their kernel.
TRADING USES
The Iterative Locally Periodic Envelope is best interpreted as a cycle-aware structural estimator with adaptive temporal sensitivity, rather than a volatility-based band. The temporal locality component allows the estimator to adapt more readily to emerging regime shifts than the pure periodic variant.
Equilibrium Tracking
The latent equilibrium represents the phase-conditioned central tendency of price under locally periodic similarity weighting. Oscillations around this level reflect movement within a repeating structural cycle, with more recent phase-aligned observations contributing more strongly than temporally distant ones.
Cycle Regime Structure
The envelope emphasizes repeating structural behavior through phase recurrence weighting, modulated by temporal decay. Changes in symmetry, amplitude, or persistence of oscillation around the latent equilibrium may indicate transitions between cyclical regimes.
Mean Reversion Within Cycles
When a stable periodic structure is present, deviations from the latent equilibrium may revert toward phase-consistent levels. Mean-reversion behavior is conditioned on both cycle structure and temporal proximity.
Structural Extremes
Extreme deviations relative to the envelope correspond to phase-inconsistent states where cyclical structure becomes stretched or destabilized. Because the kernel incorporates temporal decay, these conditions are identified with greater sensitivity to recent price behavior.
State Estimation
The system defines a latent equilibrium as the inferred central cyclical state under joint phase and temporal weighting, with dispersion derived from kernel-weighted variance under identical constraints. This produces a structurally consistent representation of the market state that is sensitive to both cyclical position and local regime conditions.
LOCALLY PERIODIC ENVELOPE CONSTRUCTION
The envelope is constructed using kernel-weighted variance under the same locally periodic similarity measure used to estimate the latent equilibrium. The latent equilibrium defines the central state estimate and kernel-weighted variance defines dispersion under identical weighting, producing an endogenously determined envelope. The band width is fixed at ±1 kernel standard deviation with no multiplier, ensuring dispersion remains an intrinsic property of the locally periodic similarity structure rather than an externally imposed scaling parameter.
THEORY
The locally periodic kernel defines similarity in terms of cyclical phase recurrence modulated by temporal proximity. Observations contribute to the estimator based on alignment within a repeating cycle structure, with influence attenuated by temporal distance from the estimation point.
The estimator is formulated as a Nadaraya–Watson kernel regression under a locally periodic kernel, where weights are defined as:
k(i) = exp( -2 · sin²(πi / p) / L² ) · exp( -i² / 2L² )
Where:
p = period (cycle length)
L = lookback window (shared bandwidth parameter; effective smoothing scales with L²)
In this MacKay consistent formulation, the lookback window acts as a unified bandwidth parameter governing periodic phase selectivity and the Radial Basis Function (RBF) temporal decay envelope. The two components are coupled through L, producing a kernel that simultaneously emphasizes phase-aligned and temporally proximate observations.
As L increases, both the periodic and RBF components broaden, producing stronger smoothing across phase and time. As L decreases, phase selectivity and temporal locality both increase, making the estimator more sensitive to recent cycle-consistent observations.
This induces a similarity structure in which influence concentrates at phase-aligned intervals within a temporally bounded neighborhood. The resulting estimator defines a latent equilibrium governed by phase alignment and temporal proximity that can be interpreted as a locally stationary periodic extension of kernel regression on a circular phase manifold.
The key distinction from the pure periodic kernel is that phase-aligned observations at distant lags are progressively suppressed by the RBF decay term, allowing the estimator to adapt to structural drift while preserving cycle-aware weighting. During stable cyclical regimes the two estimators converge; during structural transitions the locally periodic variant adapts faster by downweighting older phase information.
CALIBRATION
As established in Gaussian Processes for Machine Learning (Rasmussen & Williams, 2006), the period should reflect the recurrence interval of the dominant cycle in the data, while the bandwidth parameter L controls how quickly similarity decays away from perfect phase alignment. For daily charts, common cycle anchors include the trading week (~5 bars), trading month (~21 bars), trading quarter (~63 bars), and trading year (~252 bars).
Length (Lookback / Bandwidth)
Controls structural depth of the estimator and acts as the unified bandwidth parameter for the periodic and RBF components; as L governs phase selectivity and temporal decay simultaneously, its effect is stronger than in the pure periodic variant. The default of 100 reflects the locally periodic kernel's temporal decay component; at longer lengths the RBF term weakens and behavior converges toward the pure periodic estimator.
- 50–100: high responsiveness, strong temporal locality, short-cycle sensitivity
- 150–250: balanced regime stability with moderate temporal decay
- 300+: broad structural smoothing, weak temporal decay, behavior converges toward pure periodic envelopes
Period (Cycle Length)
Defines the recurrence interval of the kernel and governs phase alignment and cyclical structure. Shorter periods increase phase resolution and cycle sensitivity, while longer periods emphasize broader structural recurrence. The period should reflect the dominant cycle present in the data, aligned with the anchor scales defined above.
Start At Bar
Offsets the kernel window backward from the most recent bars and excludes newer observations from the estimator. This ensures all calculations are based strictly on closed historical data and preserves non-repainting behavior.
MARKET USAGE
Stock, Forex, Crypto, Commodities, and Indices.
Performance is dependent on the presence of stable cyclical structure; in regimes lacking periodic coherence, the estimator converges toward a local smoother with reduced phase discrimination. Indicator
Iterative Periodic EnvelopeThe Iterative Periodic Envelope is a phase-conditioned kernel estimator with endogenous dispersion modeling, implemented as a Nadaraya–Watson estimator under a canonical periodic kernel.
The periodic kernel defines similarity through cyclical phase alignment rather than temporal proximity or multi-scale distance decay. Observations contribute to the estimator based on their position within a repeating cycle structure, emphasizing structural recurrence over linear time dependence.
The indicator computes a latent equilibrium using a kernel-weighted mean and a dispersion measure using kernel-weighted variance under the same weighting structure. The resulting envelope reflects cycle-consistent deviation, rather than a conventional volatility band. All values are computed exclusively on closed historical bars using a bounded lookback window, ensuring non-repainting behavior.
This indicator belongs to a broader class of iterative kernel-based envelopes that includes Gaussian and Rational Quadratic variants. All share a common Nadaraya–Watson estimation framework, differentiated by their kernel.
TRADING USES
The Iterative Periodic Envelope is best interpreted as a cycle-aware structural estimator rather than a volatility-based band.
Equilibrium Tracking
The latent equilibrium represents the phase-conditioned central tendency of price under periodic similarity weighting. Oscillations around this level reflect movement within a repeating structural cycle rather than directional drift.
Cycle Regime Structure
The envelope emphasizes repeating structural behavior through phase recurrence weighting. Changes in symmetry, amplitude, or persistence of oscillation around the latent equilibrium may indicate transitions between cyclical regimes.
Mean Reversion Within Cycles
When a stable periodic structure is present, deviations from the latent equilibrium may revert toward phase-consistent levels. This supports mean-reversion behavior that is conditioned on cycle structure rather than purely statistical dispersion.
Structural Extremes
Extreme deviations relative to the envelope correspond to phase-inconsistent states where cyclical structure becomes stretched or destabilized. These conditions often precede transitions such as cycle inversion, expansion, or compression.
State Estimation
The system defines a latent equilibrium as the inferred central cyclical state, with dispersion derived from kernel-weighted variance under identical periodic similarity constraints. This produces a structurally consistent representation of market state.
PERIODIC ENVELOPE CONSTRUCTION
The envelope is constructed using kernel-weighted variance under the same periodic similarity measure used to estimate the latent equilibrium. The latent equilibrium defines the central state estimate and kernel-weighted variance defines dispersion under identical weighting, producing an endogenously determined envelope. The band width is fixed at ±1 kernel standard deviation with no multiplier, ensuring dispersion remains an intrinsic property of the periodic similarity structure rather than an externally imposed scaling parameter.
THEORY
The periodic kernel defines similarity in terms of cyclical phase recurrence rather than linear temporal distance. Observations contribute to the estimator based on alignment within a repeating cycle structure.
The estimator is formulated as a Nadaraya–Watson kernel regression under a canonical periodic kernel, where weights are defined as:
k(i) = exp( -2 · sin²(πi / p) / L² )
Where:
p = period (cycle length)
L = lookback window (bandwidth parameter; effective smoothing scales with L²)
In this MacKay consistent formulation, the lookback window acts as a bandwidth control parameter, governing phase selectivity and structural smoothing. As L increases, the kernel becomes broader, producing stronger smoothing and reduced phase sensitivity. As L decreases, phase selectivity increases and the estimator becomes more locally sensitive to cyclical alignment.
This induces a cyclical similarity structure in which influence concentrates at recurring phase intervals. The resulting estimator defines a latent equilibrium governed by phase alignment rather than temporal proximity. This formulation can be interpreted as a periodic extension of kernel regression on a circular phase manifold.
CALIBRATION
Length (Lookback / Bandwidth)
Controls structural depth of the estimator and acts as the primary kernel bandwidth parameter.
- 50–100: high responsiveness, short-cycle sensitivity
- 150–250: balanced regime stability
- 300+: strong structural smoothing, reduced sensitivity to phase noise
Period (Cycle Length)
Defines the recurrence interval of the kernel and governs phase alignment and cyclical structure. Commonly aligns with dominant market rhythms such as intraday or macro-cycle structure.
- Lower values: faster cycle sensitivity
- Higher values: slower, broader structural cycles
Start At Bar
Offsets the kernel window backward from the most recent bars and excludes newer observations from the estimator. This ensures all calculations are based strictly on closed historical data and preserves non-repainting behavior.
MARKET USAGE
Stock, Forex, Crypto, Commodities, and Indices.
Performance is dependent on the presence of stable cyclical structure; in regimes lacking periodic coherence, the estimator converges toward a smoother, low-information state. 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
Iterative Rational Quadratic ChannelThe Iterative Rational Quadratic Channel is a kernel-based smoothing and state estimation framework that applies a Rational Quadratic kernel regression to price data, combined with a rolling standard deviation envelope to construct adaptive dynamic channel boundaries.
Unlike exponential kernel methods that prioritize recent data at the expense of historical context, the rational quadratic kernel introduces a heavy-tailed weighting structure that preserves multi-scale memory in price dynamics. This enables the channel to reflect not only short-term fluctuations, but also broader structural regime context.
The resulting channel is less reactive to micro-noise and more representative of persistent market structure, making it particularly effective for trend continuity analysis, regime modeling, and reducing sensitivity to false reversals.
Its primary utility is as a state estimation and regime-filtering tool for price behavior, rather than a pure high-frequency signal isolation tool.
TRADING USES
The Rational Quadratic Channel is best interpreted as a regime-aware structural filter rather than a purely reactive trading band.
Trend Continuity
The channel basis line (RQ smoothed price) provides a stable representation of underlying market direction. Sustained movement above or below the basis reflects trend persistence rather than short-lived fluctuations, making it useful for maintaining directional bias.
Regime Persistence
Due to the heavy-tailed memory of the rational quadratic kernel, historical price structure continues to influence current valuation. This produces smoother transitions between market phases and reduces sensitivity to short-term reversals, improving regime stability.
False Reversal Filtering
Compared to exponentially weighted kernels, the RQ channel reduces overreaction to transient volatility spikes. This helps filter out low-quality reversals driven by noise rather than structural change.
State Estimation
The channel functions as a continuous estimator of market state:
- The basis represents the inferred latent price state
- The envelope represents dynamic volatility dispersion around that state
This makes it well-suited for manual, semi-automated, and automated trading systems requiring a stable structural representation of price rather than raw responsiveness. Gradual shifts in the basis line and channel position can also serve as a framework for monitoring changes in trend direction and regime transitions over time.
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 rational quadratic kernel is a member of the scale-mixture family of Gaussian kernels and can be interpreted as a superposition of Gaussian processes operating at multiple length scales. This allows it to capture both local and global structure in time series data.
It is defined as:
k(i)=(1+i22αℓ2)−αk(i) = \left(1 + \frac{i^2}{2\alpha \ell^2}\right)^{-\alpha}k(i)=(1+2αℓ2i2)−α
Where:
---> α\alphaα controls tail heaviness (relativeWeight)
---> ℓ\ellℓ defines the characteristic scale (lookback)
Unlike Gaussian kernels, which enforce exponential decay and emphasize locality, the rational quadratic kernel follows a power-law decay. This allows older observations to retain influence over the estimator for longer periods, producing a smoothing effect that is inherently multi-scale and well-suited for modeling persistent structural behavior.
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 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, structural memory, and regime stability.
Length (Lookback)
Lower (50–100): More responsive, increased sensitivity to short-term structure
Medium (150–250): Balanced for swing trading and intermediate regimes
Higher (300+): Strong regime persistence, reduced sensitivity to noise
Relative Weight (Tail Sensitivity)
Controls how quickly historical influence decays:
Lower values (≈ 0.5 – 1.0):
- Behavior approaches Gaussian
- More responsive to recent price action
- Faster detection of trend changes
- Slightly more sensitive to noise
Higher values (≈ 2.0+):
- Stronger heavy-tail behavior
- Increased influence of older price data
- Smoother output and stronger regime anchoring
- Improved false reversal filtering
Start At Bar (Lag / Structural Anchoring)
Controls how much recent price data is excluded from the kernel calculation:
Lower values (0–10):
- Uses most recent data
- Faster reaction to price changes
- More sensitive to short-term volatility
Moderate values (10–30):
- Balanced responsiveness and stability
- Reduces noise without excessive lag
- Suitable for most trading environments
Higher values (30+):
- Strong structural anchoring
- Significantly reduced sensitivity to recent fluctuations
- Enhanced regime persistence
- Slower response to turning points
This parameter effectively introduces a controlled lag, allowing users to tune the tradeoff between responsiveness and regime stability.
MARKET USAGE
Stock, Forex, Crypto, Commodities, and Indices. Indicator
Exponential Nadaraya Watson kernel regression [Jamallo](2025)
Intro
Nadaraya-Watson (N-W) kernel regression is a non-parametric smoothing technique that estimates the underlying trend of a price series without assuming any fixed model shape (like a straight line or curve). Unlike a simple moving average which weights bars equally, or an EMA which applies a fixed exponential decay, N-W regression derives its curve by computing a weighted average of all prices within a lookback window — where the weights are determined by a kernel function.
The most common kernel used is the Gaussian kernel, which assigns weights in a bell-curve shape — prices closer to the center of the window receive higher weight, prices at the edges receive lower weight. Most N-W implementations use a pure symmetric Gaussian kernel, meaning every bar within the lookback window is weighted purely by its distance from the center, with no preference for recency.
This indicator uses a hybrid Exponential Nadaraya-Watson (ENW) kernel that combines two weighting forces simultaneously:
Gaussian spatial weight — bell-curve weighting centered on the window, same as standard N-W
Exponential time-decay weight — progressively heavier weighting on recent bars, similar to how an EMA behaves
Both weights are multiplied together for each bar, meaning a price bar must be both spatially central and recent to receive maximum influence. In practice this shifts the effective weight peak toward the recent end of the window, making the K Line more responsive to current price action than standard N-W.
Breakdown
K Line — ENW Kernel Regression
The central baseline of the indicator. A smooth adaptive curve derived from the hybrid ENW kernel applied to closing prices. Bandwidth is controlled via the Kernel Length and Alpha inputs — higher alpha increases the recency bias, lower alpha brings behavior closer to standard N-W.
Volatility Bands (Inner & Outer)
Rather than using ATR or standard deviation for band width, this indicator measures the absolute deviation of price from the K Line and smooths that deviation through the same ENW kernel. This means bands are fully adaptive — they expand and contract organically based on how far price has been straying from the regression curve, not a fixed statistical formula. Inner and outer bands are independently scaled via deviation multipliers.
A Line — Vervoort ATR Stop
A trailing stop built on the HLC4 price source with an ATR-based loss distance. Flips direction on a close beyond the stop level. Serves as the primary trend bias line — when above the K Line the fill turns bullish, when below it turns bearish. Cross signals (triangles) are plotted whenever the A Line crosses the K Line, marking potential trend shifts.
Signal Line — Vervoort ATR Stop (Secondary)
A second independent Vervoort trailing stop running on its own ATR period and multiplier settings. Typically configured looser than the A Line — wider multiplier, longer or equal period — so it acts as a slower confirmation layer. Useful for filtering noise on the A Line crosses: an A Line cross that also aligns with the Signal Line's bias carries more weight than one that doesn't.
-------
Nadaraya-Watson kernel regression concept — E. Nadaraya (1964), G.S. Watson (1964)
Vervoort ATR Stop — Sylvain Vervoort
Indicator
LOWESS Reversal & Continuation [UAlgo]LOWESS Reversal & Continuation is a trend sensitive signal indicator built around a local LOWESS style smoothing engine that adapts to both price structure and volatility. Its core objective is to separate meaningful directional shifts from routine market noise by combining smoothed trend estimation, ATR normalized slope analysis, acceleration filtering, and disciplined signal gating.
Instead of relying on a conventional moving average crossover model, the script fits a locally weighted regression over a rolling window of recent bars. This produces a smoother and more context aware estimate of price direction, while still reacting fast enough to identify emerging reversals and pullback continuation opportunities. Because the estimate is local and weighted, recent bars have greater influence than distant bars, which helps the line remain relevant to current market conditions.
The indicator classifies signals into two practical categories. Reversal signals aim to identify transitions where directional pressure flips from negative to positive, or from positive to negative. Continuation signals aim to identify pullback resumption behavior inside an already established directional regime. This makes the script suitable for traders who want a single tool that can highlight both early trend change candidates and trend following re entry points.
To improve robustness, the script also supports optional robust weighting passes. These passes reduce the influence of outlier bars on the LOWESS fit, which is especially useful during abnormal spikes, illiquid conditions, or isolated volatility shocks. In addition, all slope readings are normalized by ATR, allowing the trend filter to scale more consistently across instruments, timeframes, and volatility regimes.
From a workflow perspective, the script is designed for clean chart usage. It colors the LOWESS line according to directional bias, draws an adaptive ATR based band around the curve, supports optional signal labels, limits on chart label clutter through an internal object manager, and provides alert conditions for all signal classes. The result is a visually compact but analytically rich framework for tracking trend reversals and continuation setups in real time.
🔹 Features
🔸 LOWESS Style Local Trend Estimation
The heart of the script is a locally weighted linear regression model applied over a rolling lookback window. Each bar inside the window receives a distance based weight, meaning bars closer to the current bar have a larger impact on the estimate. This produces a smooth trend line that is more adaptive than many standard moving average techniques and better suited for identifying subtle turning points.
🔸 Optional Robust Regression Passes
The indicator can apply additional robust weighting iterations after the initial fit. Residuals are measured relative to the first regression pass, and bars with unusually large residuals receive progressively lower influence in later passes. This reduces distortion from extreme candles and helps the LOWESS curve remain stable during irregular price events.
🔸 ATR Normalized Slope Filter
The script does not use raw slope in isolation. Instead, the LOWESS slope is divided by ATR, creating a volatility adjusted slope metric. This makes the directional threshold more portable across markets and timeframes. A slope that may be meaningful on a slow instrument can be very different on a high volatility asset, so ATR normalization creates a more balanced regime filter.
🔸 Reversal Signal Detection
Bullish and bearish reversal signals are triggered when the ATR normalized slope crosses the zero line and is confirmed by directional acceleration and price location relative to the LOWESS line. In other words, the script looks for a meaningful change in smoothed directional pressure, not simply a visual bend in the curve. This makes reversal signals more selective and better aligned with structural momentum shifts.
🔸 Continuation Signal Detection
Continuation logic is designed to capture trend resumption after a pullback. The script first requires an established directional regime, then checks whether price recently interacted with the LOWESS line, and finally waits for price to reclaim the trend direction with positive confirming acceleration. This helps distinguish genuine continuation behavior from random sideways oscillation around the curve.
🔸 Pullback Validation Window
A dedicated pullback lookback parameter ensures that continuation signals only occur when price has interacted with the LOWESS line within a recent number of bars. This prevents stale continuation triggers and keeps the setup focused on recent retracement behavior rather than distant historical interactions.
🔸 Close Confirmation Option
Signals can be gated so they only become valid after bar close. This is useful for traders who want to avoid intrabar flicker and premature triggers on live candles. When disabled, the script can respond more aggressively in real time, which may suit faster execution styles.
🔸 Signal Cooldown Logic
To reduce repetitive clustering, the indicator tracks the last occurrence of each signal type and imposes a cooldown period before another signal of the same class can be printed. Separate cooldown tracking is maintained for bullish reversals, bearish reversals, bullish continuations, and bearish continuations.
🔸 Adaptive ATR Band
An optional ATR based band can be plotted around the LOWESS line. This band provides a visual sense of dynamic range around the smoothed path and can help contextualize whether price is moving in a relatively stretched or balanced position around the trend estimate.
🔸 Directional Visual Coloring
The LOWESS curve changes color according to directional bias derived from the ATR normalized slope. This gives the user an immediate visual read on whether the smoothed trend pressure is currently positive, negative, or unavailable due to insufficient historical data.
🔸 Lightweight Label Management
When signal labels are enabled, the script uses an internal label book to store and manage plotted objects. Older labels are automatically deleted once the configured maximum is exceeded, helping keep the chart readable and preventing uncontrolled label buildup.
🔸 Full Alert Support
Alert conditions are included for all four event classes:
Bullish LOWESS Reversal
Bearish LOWESS Reversal
Bullish LOWESS Continuation
Bearish LOWESS Continuation
This allows the script to be used not only as a visual analysis tool, but also as an event driven signal framework for scanning and real time notification workflows.
🔹 Calculations
1) Rolling LOWESS Window Construction
For every bar, the script loads the most recent length values of the selected source into an internal rolling window. This window becomes the data sample used for the local regression fit.
method loadWindow(LowessEngine this, float seriesValue) =>
for i = 0 to this.length - 1
array.set(this.y, i, seriesValue )
Interpretation:
The regression is always fit on the latest rolling block of data.
The rightmost point in the window corresponds to the current estimation point.
This makes the smoothing local rather than global.
2) Distance Based LOWESS Weights
The script uses a tricube kernel to assign weights based on each point’s distance from the current bar inside the regression window. Bars nearer to the most recent observation receive a larger weight, while distant bars contribute less.
method buildDistanceWeights(LowessEngine this, int spanBars) =>
int x0 = this.length - 1
float bandwidth = math.max(spanBars, 1)
for i = 0 to this.length - 1
float u = math.abs(i - x0) / bandwidth
float w = u < 1 ? math.pow(1 - math.pow(u, 3), 3) : 0.0
array.set(this.baseW, i, w)
Interpretation of the conditions:
x0 is the current evaluation point inside the rolling window.
u is normalized distance from each historical point to the current point.
The tricube weight decays smoothly as distance increases.
Bars outside the effective span receive zero weight.
This is what gives the LOWESS fit its local character and helps it stay focused on recent structure.
3) Weighted Local Linear Regression
After weights are built, the script solves a weighted linear regression over the local window. The output is a local intercept and local slope. The final LOWESS estimate is the fitted value at the most recent point in the sample.
method solveWeightedLinear(LowessEngine this, array weights) =>
float s0 = 0.0
float s1 = 0.0
float s2 = 0.0
float t0 = 0.0
float t1 = 0.0
for i = 0 to this.length - 1
float w = array.get(weights, i)
float x = array.get(this.x, i)
float y = array.get(this.y, i)
s0 += w
s1 += w * x
s2 += w * x * x
t0 += w * y
t1 += w * x * y
float den = s0 * s2 - s1 * s1
if s0 <= 1e-10 or math.abs(den) <= 1e-10
this.slope := 0.0
this.intercept := array.get(this.y, this.length - 1)
else
this.slope := (s0 * t1 - s1 * t0) / den
this.intercept := (t0 - this.slope * s1) / s0
float x0 = this.length - 1
this.yhat := this.intercept + this.slope * x0
Interpretation:
this.slope measures the local directional gradient of the LOWESS fit.
this.yhat is the current LOWESS value plotted on the chart.
If the weighted regression becomes numerically unstable, the script falls back to a flat slope and uses the latest source value as intercept.
4) Robust Reweighting Passes
To reduce the impact of outliers, the script can run additional robust passes after the initial fit. It first calculates the absolute residual of each point relative to the fitted line, then computes a median based scale estimate, and finally applies a bisquare style robust weighting function.
method updateRobustWeights(LowessEngine this) =>
for i = 0 to this.length - 1
float xi = array.get(this.x, i)
float yi = array.get(this.y, i)
float fit = this.intercept + this.slope * xi
array.set(this.residualAbs, i, math.abs(yi - fit))
float med = this.residualAbs.median()
float scale = med * 6.0
if na(scale) or scale <= 1e-10
for i = 0 to this.length - 1
array.set(this.robustW, i, 1.0)
else
for i = 0 to this.length - 1
float u = array.get(this.residualAbs, i) / scale
float rw = u < 1 ? math.pow(1 - math.pow(u, 2), 2) : 0.0
array.set(this.robustW, i, rw)
Interpretation:
Large residual bars are treated as less trustworthy observations.
The median residual acts as a robust scale anchor.
Higher residuals receive smaller robust weights in subsequent fits.
This improves stability during abnormal spikes and irregular candles.
5) ATR Normalized Slope and Acceleration
Once the LOWESS fit is complete, the script converts raw slope into a volatility aware slope by dividing it by ATR. It also computes a first-difference style acceleration term to measure whether directional pressure is strengthening or weakening.
float slopeAtr = not na(slope) and atr > 0 ? slope / atr : na
float accelAtr = slopeAtr - nz(slopeAtr )
Interpretation:
slopeAtr expresses trend slope in ATR units.
accelAtr measures change in normalized slope from one bar to the next.
Positive acceleration supports bullish developments.
Negative acceleration supports bearish developments.
This combination helps the script distinguish a genuine regime shift from a weak or decaying slope condition.
6) Directional Regime Classification
The script uses a user defined ATR normalized threshold to determine whether the current smoothed state qualifies as a bullish or bearish directional regime.
bool upRegime = not na(slopeAtr) and slopeAtr > slopeThreshold
bool downRegime = not na(slopeAtr) and slopeAtr < -slopeThreshold
Interpretation:
A positive but very small slope is not automatically treated as a valid uptrend.
A negative but very small slope is not automatically treated as a valid downtrend.
The threshold acts as a noise filter that requires the trend estimate to have enough magnitude before continuation logic becomes eligible.
7) Pullback Detection Relative to LOWESS
Continuation signals depend on recent interaction with the LOWESS line. The script checks how many bars have passed since price moved through the LOWESS curve in the opposite direction of the active regime.
int bullPbBars = int(nz(ta.barssince(low < lowess), 100000))
int bearPbBars = int(nz(ta.barssince(high > lowess), 100000))
bool bullPullbackRecent = bullPbBars <= pullbackLookback
bool bearPullbackRecent = bearPbBars <= pullbackLookback
Interpretation:
In a bullish regime, price must have recently dipped below the LOWESS line to qualify as a pullback.
In a bearish regime, price must have recently pushed above the LOWESS line to qualify as a pullback.
The pullbackLookback parameter controls how recent that interaction must be.
8) Reversal Signal Logic
Bullish and bearish reversal signals are built from zero line slope crossings, directional acceleration, and price confirmation relative to the LOWESS curve.
bool slopeCrossUp = ta.crossover(slopeAtr, 0)
bool slopeCrossDown = ta.crossunder(slopeAtr, 0)
bool bullRevRaw = enoughBars and slopeCrossUp and accelAtr > 0 and close > lowess
bool bearRevRaw = enoughBars and slopeCrossDown and accelAtr < 0 and close < lowess
Interpretation of the bullish reversal conditions:
slopeCrossUp means the normalized LOWESS slope has crossed from negative to positive.
accelAtr > 0 means the slope is improving, not merely touching zero.
close > lowess confirms that price is positioned above the smoothed trend estimate.
Interpretation of the bearish reversal conditions:
slopeCrossDown means the normalized LOWESS slope has crossed from positive to negative.
accelAtr < 0 confirms weakening trend pressure.
close < lowess confirms price is positioned below the LOWESS line.
This makes reversal signals more selective than a simple moving average crossover style event.
9) Continuation Signal Logic
Continuation signals are only allowed when a directional regime already exists, a recent pullback has occurred, price crosses back through the LOWESS line in trend direction, and acceleration confirms that the move is regaining strength.
bool priceCrossUp = ta.crossover(close, lowess)
bool priceCrossDown = ta.crossunder(close, lowess)
bool bullContRaw = enoughBars and upRegime and bullPullbackRecent and priceCrossUp and accelAtr > 0 and not bullRevRaw
bool bearContRaw = enoughBars and downRegime and bearPullbackRecent and priceCrossDown and accelAtr < 0 and not bearRevRaw
Interpretation of the bullish continuation conditions:
The LOWESS slope must already define an uptrend regime.
Price must have recently pulled back below the LOWESS line.
Price must cross back above the LOWESS line.
Acceleration must be positive.
A reversal signal takes priority, so continuation does not print if the same bar qualifies as a bullish reversal.
Interpretation of the bearish continuation conditions is the exact inverse.
10) Close Confirmation and Cooldown Control
The final signal is gated by an optional bar close confirmation and a per-signal cooldown filter.
bool gate = confirmClose ? barstate.isconfirmed : true
bool bullRev = gate and bullRevRaw and canBullRev(signalState, cooldownBars)
bool bearRev = gate and bearRevRaw and canBearRev(signalState, cooldownBars)
bool bullCont = gate and bullContRaw and canBullCont(signalState, cooldownBars)
bool bearCont = gate and bearContRaw and canBearCont(signalState, cooldownBars)
Interpretation:
When close confirmation is enabled, signals only become valid after the candle is closed.
Cooldown logic prevents repeated printing of the same signal class within a short number of bars.
This reduces visual clutter and avoids excessive re-triggering during choppy conditions.
11) Adaptive Band Construction
The script can draw an ATR-based envelope around the LOWESS line to provide volatility context.
float upperBand = showBand and not na(lowess) ? lowess + atr * bandAtrMult : na
float lowerBand = showBand and not na(lowess) ? lowess - atr * bandAtrMult : na
Interpretation:
The band expands and contracts with ATR.
This creates a dynamic visual frame around the LOWESS estimate.
It is not a signal by itself, but it helps contextualize the distance between current price and the smoothed trend path.
12) Visual Output and Alerts
The LOWESS line changes color according to slope direction, optional labels mark reversal and continuation events, and alert conditions are available for all four signal types.
alertcondition(bullRev, "Bullish LOWESS Reversal", "Bullish LOWESS reversal on {{ticker}}")
alertcondition(bearRev, "Bearish LOWESS Reversal", "Bearish LOWESS reversal on {{ticker}}")
alertcondition(bullCont, "Bullish LOWESS Continuation", "Bullish LOWESS continuation on {{ticker}}")
alertcondition(bearCont, "Bearish LOWESS Continuation", "Bearish LOWESS continuation on {{ticker}}")
In practical terms, this means the indicator can serve both as a visual discretionary analysis tool and as an alert driven framework for identifying smoothed trend reversals and pullback continuation opportunities with a volatility aware filter structure. Indicator
KDE Value Clouds [LuxAlgo]The KDE Value Clouds indicator is a quantitative tool that uses Kernel Density Estimation (KDE) to visualize the statistical distribution of price action, identifying high-density "Value Clouds" where the market has spent the most time.
🔶 USAGE
The indicator highlights areas of price " fair value " by calculating the probability density of price across a user-defined lookback period. Traders can use these density clusters to identify significant support and resistance levels that are often invisible to standard trend-following indicators.
🔹 Value Clouds
The " Value Clouds " appear directly on the price chart as gradient boxes. These clouds highlight regions where the density of price action exceeds the 50th percentile of the total distribution.
High Density (Bright Colors): Indicates a "Balance Area" where the market has reached a temporary equilibrium. These often act as magnets for price.
Low Density (Gaps): Indicates "Inefficiency" or fast moves where the market did not spend much time. These areas are often revisited or "filled" later.
🔹 KDE Profile & POC
On the right side of the chart, a smooth horizontal profile represents the continuous density function. The KDE POC (Point of Control) is the single price level with the highest calculated density within the lookback period, serving as the ultimate "anchor" for the current market regime.
🔹 How to use
Traders can look for price to "stall" or range within the bright Value Clouds, as these represent accepted price levels. When price moves into a "Gap" (a low-density area), it often moves quickly until it reaches the next cloud.
The KDE POC can be used as a primary support or resistance level; a breakout above a high-density cloud often signals a shift in market sentiment, while a rejection at the edge of a cloud suggests the market is still in a balanced state.
🔶 DETAILS
🔹 KDE vs. Volume Profile
A standard Volume Profile relies on "bins" (rectangles) to count volume at specific price steps. This can create "jagged" profiles that change drastically depending on the chosen row size.
The KDE Value Clouds approach is different because it uses a continuous probability function. Every price point in the lookback period contributes a small "bell curve" of influence to the total profile. This allows for a much smoother and more mathematically sound representation of where " Value " actually resides, regardless of arbitrary bin sizes.
The core of this indicator relies on two primary mathematical concepts:
Gaussian Kernel Estimation: Instead of simply counting occurrences, the script applies a Gaussian weight to every price point. This results in a "smooth" profile that captures the true shape of the price distribution.
Silverman’s Rule of Thumb: To prevent the clouds from being too noisy or too blurry, the indicator uses Silverman’s rule to calculate an optimal " Bandwidth ." This bandwidth adapts based on the standard deviation of the price data, ensuring the visualization stays relevant across different volatility regimes.
🔶 SETTINGS
🔹 Main Settings
Lookback Period: The number of bars used to calculate the price density. A higher lookback provides a "macro" view of value, while a lower lookback focuses on recent rotations.
Bandwidth Multiplier: Adjusts the "smoothness" of the KDE curve. Increasing this value will make the clouds broader and smoother; decreasing it will make them more granular.
Precision (Steps): Defines the vertical resolution of the density calculation. Higher values result in a more detailed profile.
🔹 Visualization
High/Low Density Colors: Customizes the gradient used for both the side profile and the on-chart clouds.
Profile Width (%): Controls how far the KDE profile extends horizontally across the right side of the chart.
Show Value Cloud on Chart: Toggles the visibility of the background "clouds" that highlight high-density price zones.
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
EDUVEST Lorentzian ClassificationEDUVEST Lorentzian Classification - Machine Learning Signal Detection
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█ ORIGINALITY
This indicator enhances the original Lorentzian Classification concept by jdehorty with EduVest's visual modifications and alert system integration. The core innovation is using Lorentzian distance instead of Euclidean distance for k-NN classification, providing more robust pattern recognition in financial markets.
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█ WHAT IT DOES
- Generates BUY/SELL signals using machine learning classification
- Displays kernel regression estimate for trend visualization
- Shows prediction values on each bar
- Provides trade statistics (Win Rate, W/L Ratio)
- Includes multiple filter options (Volatility, Regime, ADX, EMA, SMA)
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█ HOW IT WORKS
【Lorentzian Distance Calculation】
Unlike Euclidean distance, Lorentzian distance uses logarithmic transformation:
d = Σ log(1 + |xi - yi|)
This provides:
- Better handling of outliers
- More stable distance measurements
- Reduced sensitivity to extreme values
【Feature Engineering】
The classifier uses up to 5 configurable features:
- RSI (Relative Strength Index)
- WT (WaveTrend)
- CCI (Commodity Channel Index)
- ADX (Average Directional Index)
Each feature is normalized using the n_rsi, n_wt, n_cci, or n_adx functions.
【k-Nearest Neighbors Classification】
1. Calculate Lorentzian distance between current bar and historical bars
2. Find k nearest neighbors (default: 8)
3. Sum predictions from neighbors
4. Generate signal based on prediction sum (>0 = Long, <0 = Short)
【Kernel Regression】
Uses Rational Quadratic kernel for smooth trend estimation:
- Lookback Window: 8
- Relative Weighting: 8
- Regression Level: 25
【Filters】
- Volatility Filter: Filters signals during extreme volatility
- Regime Filter: Identifies market regime using threshold
- ADX Filter: Confirms trend strength
- EMA/SMA Filter: Trend direction confirmation
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█ HOW TO USE
【Recommended Settings】
- Timeframe: 15M, 1H, 4H, Daily
- Neighbors Count: 8 (default)
- Feature Count: 5 for comprehensive analysis
【Signal Interpretation】
- Green BUY label: Long entry signal
- Red SELL label: Short entry signal
- Bar colors: Green (bullish) / Red (bearish) prediction strength
【Trade Statistics Panel】
- Winrate: Historical win percentage
- Trades: Total (Wins|Losses)
- WL Ratio: Win/Loss ratio
- Early Signal Flips: Premature signal changes
【Filter Recommendations】
- Enable Volatility Filter for ranging markets
- Enable Regime Filter for trend confirmation
- Use EMA Filter (200) for higher timeframes
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█ CREDITS
Original Lorentzian Classification concept and MLExtensions library by jdehorty.
Enhanced with visual modifications and alert integration by EduVest.
License: Mozilla Public License 2.0 Indicator
Kernel Channel [BackQuant]Kernel Channel
A non-parametric, kernel-weighted trend channel that adapts to local structure, smooths noise without lagging like moving averages, and highlights volatility compressions, expansions, and directional bias through a flexible choice of kernels, band types, and squeeze logic.
What this is
This indicator builds a full trend channel using kernel regression rather than classical averaging. Instead of a simple moving average or exponential weighting, the midline is computed as a kernel-weighted expectation of past values. This allows it to adapt to local shape, give more weight to nearby bars, and reduce distortion from outliers.
You can think of it as a sliding local smoother where you define both the “window” of influence (Window Length) and the “locality strength” (Bandwidth). The result is a flexible midline with optional upper and lower bands derived from kernel-weighted ATR or kernel-weighted standard deviation, letting you visualize volatility in a structurally consistent way.
Three plotting modes help demonstrate this difference:
When the midline is shown alone, you get a smooth, adaptive baseline that behaves almost like a regression moving average, as shown in this view:
When full channels are enabled, you see how standard deviation reacts to local structure with dynamically widening and tightening bands, a mode illustrated here:
When ATR mode is chosen instead of StdDev, band width reflects breadth of movement rather than variance, creating a volatility-aware envelope like the example here:
Why kernels
Classical moving averages allocate fixed weights. Kernels let the user define weighting shape:
Epanechnikov — emphasizes bars near the current bar, fades fast, stable and smooth.
Triangular — linear decay, simple and responsive.
Laplacian — exponential decay from the current point, sharper reactivity.
Cosine — gentle periodic decay, balanced smoothness for trend filters.
Using these in combination with a bandwidth parameter gives fine control over smoothness vs responsiveness. Smaller bandwidths give sharper local sensitivity, larger bandwidths give smoother curvature.
How it works (core logic)
The indicator computes three building blocks:
1) Kernel-weighted midline
For every bar, a sliding window looks back Window Length bars. Each bar in this window receives a kernel weight depending on:
its index distance from the present
the chosen kernel shape
the bandwidth parameter (locality)
Weights form the denominator, weighted values form the numerator, and the resulting ratio is the kernel regression mean. This midline is the central trend.
2) Kernel-based width
You choose one of two band types:
Kernel ATR — ATR values are kernel-averaged, producing a smooth, volatility-based width that is not dependent on variance. Ideal for directional trend channels and regime separation.
Kernel StdDev — local variance around the midline is computed through kernel weighting. This produces a true statistical envelope that narrows in quiet periods and widens in noisy areas.
Width is scaled using Band Multiplier , controlling how far the envelope extends.
3) Upper and lower channels
Provided midline and width exist, the channel edges are:
Upper = midline + bandMult × width
Lower = midline − bandMult × width
These create smooth structures around price that adapt continuously.
Plotting modes
The indicator supports multiple visual styles depending on what you want to emphasize.
When only the midline is displayed, you get a pure kernel trend: a smooth regression-like curve that reacts to local structure while filtering noise, demonstrated here: This provides a clean read on direction and slope.
With full channels enabled, the behavior of the bands becomes visible. Standard deviation mode creates elastic boundaries that tighten during compressions and widen during turbulence, which you can see in the band-focused demonstration: This helps identify expansion events, volatility clusters, and breakouts.
ATR mode shifts interpretation from statistical variance to raw movement amplitude. This makes channels less sensitive to outliers and more consistent across trend phases, as shown in this ATR variation example: This mode is particularly useful for breakout systems and bar-range regimes.
Regime detection and bar coloring
The slope of the midline defines directional bias:
Up-slope → green
Down-slope → red
Flat → gray
A secondary regime filter compares close to the channel:
Trend Up Strong — close above upper band and midline rising.
Trend Down Strong — close below lower band and midline falling.
Trend Up Weak — close between midline and upper band with rising slope.
Trend Down Weak — close between lower band and midline with falling slope.
Compression mode — squeeze conditions.
Bar coloring is optional and can be toggled for cleaner charts.
Squeeze logic
The indicator includes non-standard squeeze detection based on relative width , defined as:
width / |midline|
This gives a dimensionless measure of how “tight” or “loose” the channel is, normalized for trend level.
A rolling window evaluates the percentile rank of current width relative to past behavior. If the width is in the lowest X% of its last N observations, the script flags a squeeze environment. This highlights compression regions that may precede breakouts or regime shifts.
Deviation highlighting
When using Kernel StdDev mode, you may enable deviation flags that highlight bars where price moves outside the channel:
Above upper band → bullish momentum overextension
Below lower band → bearish momentum overextension
This is turned off in ATR mode because ATR widths do not represent distributional variance.
Alerts included
Kernel Channel Long — midline turns up.
Kernel Channel Short — midline turns down.
Price Crossed Midline — crossover or crossunder of the midline.
Price Above Upper — early momentum expansion.
Price Below Lower — downward volatility expansion.
These help automate regime changes and breakout detection.
How to use it
Trend identification
The midline acts as a bias filter. Rising midline means trend strength upward, falling midline means downward behavior. The channel width contextualizes confidence.
Breakout anticipation
Kernel StdDev compressions highlight areas where price is coiling. Breakouts often follow narrow relative width. ATR mode provides structural expansion cues that are smooth and robust.
Mean reversion
StdDev mode is suitable for fade setups. Moves to outer bands during low volatility often revert to the midline.
Continuation logic
If price breaks above the upper band while midline is rising, the indicator flags strong directional expansion. Same logic for breakdowns on the lower band.
Volatility characterization
Kernel ATR maps raw bar movements and is excellent for identifying regime shifts in markets where variance is unstable.
Tuning guidance
For smoother long-term trend tracking
Larger window (150–300).
Moderate bandwidth (1.0–2.0).
Epanechnikov or Cosine kernel.
ATR mode for stable envelopes.
For swing trading / short-term structure
Window length around 50–100.
Bandwidth 0.6–1.2.
Triangular for speed, Laplacian for sharper reactions.
StdDev bands for precise volatility compression.
For breakout systems
Smaller bandwidth for sharp local detection.
ATR mode for stable envelopes.
Enable squeeze highlighting for identifying setups early.
For mean-reversion systems
Use StdDev bands.
Moderate window length.
Highlight deviations to locate overextended bars.
Settings overview
Kernel Settings
Source
Window Length
Bandwidth
Kernel Type (Epanechnikov, Triangular, Laplacian, Cosine)
Channel Width
Band Type (Kernel ATR or Kernel StdDev)
Band Multiplier
Visuals
Show Bands
Color Bars By Regime
Highlight Squeeze Periods
Highlight Deviation
Lookback and Percentile settings
Colors for uptrend, downtrend, squeeze, flat
Trading applications
Trend filtering — trade only in direction of the midline slope.
Breakout confirmation — expansion outside the bands while slope agrees.
Squeeze timing — compression periods often precede the next directional leg.
Volatility-aware stops — ATR mode makes channel edges suitable for adaptive stop placement.
Structural swing mapping — StdDev bands help locate midline pullbacks vs distributional extremes.
Bias rotation — bar coloring highlights when regime shifts occur.
Notes
The Kernel Channel is not a signal generator by itself, but a structural map. It helps classify trend direction, volatility environment, distribution shape, and compression cycles. Combine it with your entry and exit framework, risk parameters, and higher-timeframe confirmation.
It is designed to behave consistently across markets, to avoid the bluntness of classical averages, and to reveal subtle curvature in price that traditional channels miss. Adjust kernel type, bandwidth, and band source to match the noise profile of your instrument, then use squeeze logic and deviation highlighting to guide timing.
Indicator
Multiple Symbol Trend Screener [Pineify]Multiple Symbol Trend Screener Pineify – Ultimate Multi-Indicator Scanner for PulseWire
Empower your trading with deep market insights across multiple symbols using this feature-rich Pine Script screener. The Multiple Symbol Trend Screener Pineify enables traders to monitor and compare trends, reversals, and consolidations in real-time across the biggest equity symbols on PulseWire, through a synergistic blend of popular technical indicators.
Key Features
Monitor up to 15 symbols and their trends simultaneously
Integrates 7 professional-grade indicators: MA Distance, Aroon, Parabolic SAR (PSAR), ADX, Supertrend, Keltner Channel, and BBTrend
Color-coded table display for instant visual assessment
Customizable lookback periods, indicator types, and calculation methods
SEO optimized for multi-symbol trend detection, screener, and advanced PulseWire indicator
How It Works
This indicator leverages PulseWire’s Pine Script v6 and request.security() to process multiple symbols across selected timeframes. Data populates a dynamic table, updating each cell based on the calculated value of every underlying indicator. MA Distance highlights deviation from moving averages; Aroon flags emerging trend strength; PSAR marks potential trend reversals; ADX assesses trend momentum; Supertrend detects bullish/bearish phases; Keltner Channel and BBTrend offer volatility and power insights.
Set up your preferred symbols and timeframes
Each indicator runs its calculation per symbol using its parameter group
All results are displayed in a table for a comprehensive dashboard view
Trading Ideas and Insights
Traders can use this screener for cross-market comparison, directional bias, entry/exit filtering, and comprehensive trend evaluation. The screener is excellent for swing trading, day trading, and portfolio tracking. It enables confirmation across multiple frameworks — for example, spotting momentum with ADX before confirming direction with Supertrend and PSAR.
Identify correlated movements or divergences across selected assets
Spot synchronized trend changes for basket trading ideas
Filter symbols by volatility, strength, or trend status for precise trade selection
How Multiple Indicators Work Together
The screener’s edge lies in its intelligent correlation of popular indicators. MA Distance measures the proximity to chosen moving averages, ideal for spotting overbought/oversold conditions. Aroon reveals the strength of new price trends, PSAR indicates reversal signals, and ADX quantifies the momentum of these trends. Supertrend provides a directional phase, while Keltner Channel & BBTrend analyze volatility shifts and band compressions. This amalgamation allows for a robust, multi-dimensional market snapshot, capturing details missed by single-indicator tools.
By displaying all key metrics side-by-side, the screener enables holistic decision-making, revealing confluence zones and contradiction areas across multiple tickers and timeframes.
Unique Aspects
Original implementation combining seven independent trend and momentum indicators for each symbol
Rich customization for symbols, timeframes, and all indicator parameters
Intuitive color-coding for quick reading of bullish/bearish/neutral signals
Comprehensive dashboard for instant actionable insights
How to Use
Load the indicator onto your PulseWire chart
Go to the script’s settings and input your preferred symbols and relevant timeframes
Set your desired parameters for each indicator group: Moving Average type, Aroon length, PSAR values, ADX smoothing, etc.
Observe the results in the top-right table, then use it to filter candidates and validate trade setups
The screener is suitable for all timeframes and asset classes available on PulseWire. Make sure your chart’s timeframe matches the one used in the scanner for optimal accuracy.
Customization
Choose up to 15 symbols to monitor in a single dashboard
Customize lookback periods, indicator types, colors, and display settings
Configure alerting options and thresholds for advanced trade automation
Conclusion
The Multiple Symbol Trend Screener Pineify sets a new standard for multi-asset screening on PulseWire. By elegantly merging seven proven technical indicators, the screener delivers powerful trend detection, reversal analysis, and volatility monitoring — all in one dashboard. Take your trading to new heights with in-depth, customizable market surveillance.
Indicator
Indicator
Half Causal EstimatorOverview
The Half Causal Estimator is a specialized filtering method that provides responsive averages of market variables (volume, true range, or price change) with significantly reduced time delay compared to traditional moving averages. It employs a hybrid approach that leverages both historical data and time-of-day patterns to create a timely representation of market activity while maintaining smooth output.
Core Concept
Traditional moving averages suffer from time lag, which can delay signals and reduce their effectiveness for real-time decision making. The Half Causal Estimator addresses this limitation by using a non-causal filtering method that incorporates recent historical data (the causal component) alongside expected future behavior based on time-of-day patterns (the non-causal component).
This dual approach allows the filter to respond more quickly to changing market conditions while maintaining smoothness. The name "Half Causal" refers to this hybrid methodology—half of the data window comes from actual historical observations, while the other half is derived from time-of-day patterns observed over multiple days. By incorporating these "future" values from past patterns, the estimator can reduce the inherent lag present in traditional moving averages.
How It Works
The indicator operates through several coordinated steps. First, it stores and organizes market data by specific times of day (minutes/hours). Then it builds a profile of typical behavior for each time period. For calculations, it creates a filtering window where half consists of recent actual data and half consists of expected future values based on historical time-of-day patterns. Finally, it applies a kernel-based smoothing function to weight the values in this composite window.
This approach is particularly effective because market variables like volume, true range, and price changes tend to follow recognizable intraday patterns (they are positive values without DC components). By leveraging these patterns, the indicator doesn't try to predict future values in the traditional sense, but rather incorporates the average historical behavior at those future times into the current estimate.
The benefit of using this "average future data" approach is that it counteracts the lag inherent in traditional moving averages. In a standard moving average, recent price action is underweighted because older data points hold equal influence. By incorporating time-of-day averages for future periods, the Half Causal Estimator essentially shifts the center of the filter window closer to the current bar, resulting in more timely outputs while maintaining smoothing benefits.
Understanding Kernel Smoothing
At the heart of the Half Causal Estimator is kernel smoothing, a statistical technique that creates weighted averages where points closer to the center receive higher weights. This approach offers several advantages over simple moving averages. Unlike simple moving averages that weight all points equally, kernel smoothing applies a mathematically defined weight distribution. The weighting function helps minimize the impact of outliers and random fluctuations. Additionally, by adjusting the kernel width parameter, users can fine-tune the balance between responsiveness and smoothness.
The indicator supports three kernel types. The Gaussian kernel uses a bell-shaped distribution that weights central points heavily while still considering distant points. The Epanechnikov kernel employs a parabolic function that provides efficient noise reduction with a finite support range. The Triangular kernel applies a linear weighting that decreases uniformly from center to edges. These kernel functions provide the mathematical foundation for how the filter processes the combined window of past and "future" data points.
Applicable Data Sources
The indicator can be applied to three different data sources: volume (the trading volume of the security), true range (expressed as a percentage, measuring volatility), and change (the absolute percentage change from one closing price to the next).
Each of these variables shares the characteristic of being consistently positive and exhibiting cyclical intraday patterns, making them ideal candidates for this filtering approach.
Practical Applications
The Half Causal Estimator excels in scenarios where timely information is crucial. It helps in identifying volume climaxes or diminishing volume trends earlier than conventional indicators. It can detect changes in volatility patterns with reduced lag. The indicator is also useful for recognizing shifts in price momentum before they become obvious in price action, and providing smoother data for algorithmic trading systems that require reduced noise without sacrificing timeliness.
When volatility or volume spikes occur, conventional moving averages typically lag behind, potentially causing missed opportunities or delayed responses. The Half Causal Estimator produces signals that align more closely with actual market turns.
Technical Implementation
The implementation of the Half Causal Estimator involves several technical components working together. Data collection and organization is the first step—the indicator maintains a data structure that organizes market data by specific times of day. This creates a historical record of how volume, true range, or price change typically behaves at each minute/hour of the trading day.
For each calculation, the indicator constructs a composite window consisting of recent actual data points from the current session (the causal half) and historical averages for upcoming time periods from previous sessions (the non-causal half). The selected kernel function is then applied to this composite window, creating a weighted average where points closer to the center receive higher weights according to the mathematical properties of the chosen kernel. Finally, the kernel weights are normalized to ensure the output maintains proper scaling regardless of the kernel type or width parameter.
This framework enables the indicator to leverage the predictable time-of-day components in market data without trying to predict specific future values. Instead, it uses average historical patterns to reduce lag while maintaining the statistical benefits of smoothing techniques.
Configuration Options
The indicator provides several customization options. The data period setting determines the number of days of observations to store (0 uses all available data). Filter length controls the number of historical data points for the filter (total window size is length × 2 - 1). Filter width adjusts the width of the kernel function. Users can also select between Gaussian, Epanechnikov, and Triangular kernel functions, and customize visual settings such as colors and line width.
These parameters allow for fine-tuning the balance between responsiveness and smoothness based on individual trading preferences and the specific characteristics of the traded instrument.
Limitations
The indicator requires minute-based intraday timeframes, securities with volume data (when using volume as the source), and sufficient historical data to establish time-of-day patterns.
Conclusion
The Half Causal Estimator represents an innovative approach to technical analysis that addresses one of the fundamental limitations of traditional indicators: time lag. By incorporating time-of-day patterns into its calculations, it provides a more timely representation of market variables while maintaining the noise-reduction benefits of smoothing. This makes it a valuable tool for traders who need to make decisions based on real-time information about volume, volatility, or price changes. Indicator
Machine Learning SupertrendThe Machine Learning Supertrend is an advanced trend-following indicator that enhances the traditional Supertrend with Gaussian Process Regression (GPR) and kernel-based learning. Unlike conventional methods that rely purely on historical ATR values, this indicator integrates machine learning techniques to dynamically estimate volatility and forecast future price movements, resulting in a more adaptive and robust trend detection system.
At the core of this indicator lies Gaussian Process Regression (GPR), which utilizes a Radial Basis Function (RBF) kernel to model price distributions and anticipate future trends. Instead of simply looking at past price action, it constructs a kernel matrix, enabling a probabilistic approach to price forecasting. This allows the indicator to not only detect current trends but also project potential trend reversals with greater accuracy.
By applying machine learning to ATR estimation, the ML Supertrend dynamically adjusts its thresholds based on predicted values rather than a fixed multiplier. This makes the trend signals more responsive to market conditions, reducing false signals and minimizing whipsaws often seen with traditional Supertrend indicators. The upper and lower bands are no longer static but evolve based on the underlying price structure, improving the reliability of trend shifts.
When the price crosses these adaptive levels, the indicator detects a trend change and plots it accordingly. Green signifies a bullish trend, while red indicates a bearish one. Alerts can also be triggered when the trend shifts, allowing traders to react quickly to potential reversals.
What makes this approach powerful is its ability to adapt to different market conditions. Traditional ATR-based methods use fixed parameters that might not always be optimal, whereas this ML-driven Supertrend continuously refines its estimations based on real-time data. The result is a more intelligent, less lagging, and highly adaptive trend-following tool.
This indicator is particularly useful for traders looking to enhance trend-following strategies with AI-driven insights. It reduces noise, improves signal reliability, and even offers a degree of trend forecasting, making it ideal for those who want a more advanced and dynamic alternative to standard Supertrend indicators.
This indicator is provided for educational and informational purposes only. It does not constitute financial advice, and past performance is not indicative of future results. Trading involves risk, and users should conduct their own research and use proper risk management before making investment decisions. Indicator
Trend Regression Kernel [IkkeOmar]Kernel by @jdehorty huge shoutout to him! This is only an idea for how I use it when trading
All credit for the kernel goes to him, I did not make the kernel! I don't know how to make it more clear.
I use this to assist with top-down analysis.
timeframe I want to trade : timeframe to analyse with white noise and kernel:
1m : 1H
5m : 2H
15m : 4H
1H : 1D
In the chart you see that I have the 1H open, I use the white noise at a "lower setting length" (55 in this case), I change the source of to be the kernel on the higher timeframe. When a new trend is detected by the White noise I wait for price to retest the kernel before building a position. Another case described below:
Here i use the adaptive MCVF (I have made this free for everyone on PulseWire) to buy when price is below the kernel while the trend for the white noise is bullish .
Notice that the Kernel is set on the 4H timeframe! The source of the white noise is the kernel!
Here is an example in a bearish trend:
Notice, I am on the 5m chart, kernel uses the 2H chart and the source of the white noise is the kernel.
I use the adaptive MCVF to help me get entries AFTER the first touch of the kernel.
Mandatory code explanation, with respect to the house rules:
Input settings:
Input Settings:
The script provides various input parameters to customize the indicator:
src: The source of price data, defaulted to closing prices.
h, r, x_0: Parameters for Kernel 1.
h2, r2, x_2: Parameters for Kernel 2.
Kernel Regression Functions:
Two functions kernel_regression1 and kernel_regression2 are defined to perform kernel regression calculations.
These functions estimate the trend using the Nadaraya-Watson kernel non-parametric regression method.
They take the source data (_src), the size of the data series (_size), and the lookback window (_h) as inputs.
They iterate over the data series and calculate the weighted sum of the values based on the specified kernel parameters.
The result is divided by the cumulative weight to obtain the estimated value.
Estimations:
The kernel_regression1 and kernel_regression2 functions are called with the respective parameters to estimate trends (yhat1 and yhat2).
Buy and Sell Signals:
Buy and sell signals are generated based on crossover and crossunder conditions between the two trend estimates (yhat1 and yhat2).
buySignal is true when yhat1 crosses above yhat2.
SellSignal is true when yhat1 crosses below yhat2.
Plotting:
The average of the two trend estimates (yhat1 and yhat2) is calculated and plotted.
The color of the plot is determined based on whether yhat1 is greater than yhat2, less than yhat2, or equal to yhat2.
Buy and sell signals are plotted using triangle shapes below and above bars, respectively.
Alerts:
Alert conditions are set based on buy and sell signals. Alerts are triggered when a crossover (long signal) or crossunder (short signal) occurs.
The alerts include information about the signal type, symbol, and price.
It's important to mention that the buy and sell signals from the indicator is very discretionary, I rarely use them, and if I do it's if they are in confluence with a correction i am biased towards or if it has confluence with some of my other systems.
The adaptive MCVF and White noise is free for everyone on PulseWire, linked below:)
Huge shoutout to @jdehorty, original kernel below:
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Kernels©2024, GoemonYae; copied from @jdehorty's "KernelFunctions" on 2024-03-09 to ensure future dependency compatibility. Will also add more functions to this script.
Library "KernelFunctions"
This library provides non-repainting kernel functions for Nadaraya-Watson estimator implementations. This allows for easy substition/comparison of different kernel functions for one another in indicators. Furthermore, kernels can easily be combined with other kernels to create newer, more customized kernels.
rationalQuadratic(_src, _lookback, _relativeWeight, startAtBar)
Rational Quadratic Kernel - An infinite sum of Gaussian Kernels of different length scales.
Parameters:
_src (float) : The source series.
_lookback (simple int) : The number of bars used for the estimation. This is a sliding value that represents the most recent historical bars.
_relativeWeight (simple float) : Relative weighting of time frames. Smaller values resut in a more stretched out curve and larger values will result in a more wiggly curve. As this value approaches zero, the longer time frames will exert more influence on the estimation. As this value approaches infinity, the behavior of the Rational Quadratic Kernel will become identical to the Gaussian kernel.
startAtBar (simple int)
Returns: yhat The estimated values according to the Rational Quadratic Kernel.
gaussian(_src, _lookback, startAtBar)
Gaussian Kernel - A weighted average of the source series. The weights are determined by the Radial Basis Function (RBF).
Parameters:
_src (float) : The source series.
_lookback (simple int) : The number of bars used for the estimation. This is a sliding value that represents the most recent historical bars.
startAtBar (simple int)
Returns: yhat The estimated values according to the Gaussian Kernel.
periodic(_src, _lookback, _period, startAtBar)
Periodic Kernel - The periodic kernel (derived by David Mackay) allows one to model functions which repeat themselves exactly.
Parameters:
_src (float) : The source series.
_lookback (simple int) : The number of bars used for the estimation. This is a sliding value that represents the most recent historical bars.
_period (simple int) : The distance between repititions of the function.
startAtBar (simple int)
Returns: yhat The estimated values according to the Periodic Kernel.
locallyPeriodic(_src, _lookback, _period, startAtBar)
Locally Periodic Kernel - The locally periodic kernel is a periodic function that slowly varies with time. It is the product of the Periodic Kernel and the Gaussian Kernel.
Parameters:
_src (float) : The source series.
_lookback (simple int) : The number of bars used for the estimation. This is a sliding value that represents the most recent historical bars.
_period (simple int) : The distance between repititions of the function.
startAtBar (simple int)
Returns: yhat The estimated values according to the Locally Periodic Kernel. Library
Bandwidth Volatility - Silverman Rule of thumb EstimatorOverview
This indicator calculates volatility using the Rule of Thumb bandwidth estimator and incorporating the standard deviations of returns to get historical volatility. There are two options: one for the original rule of thumb bandwidth estimator, and another for the modified rule of thumb estimator. This indicator comes with the bandwidth , which is shown with the color gradient columns, which are colored by a percentile of the bandwidth, and the moving average of the bandwidth, which is the dark shaded area.
The rule of thumb bandwidth estimator is a simple and quick method for estimating the bandwidth parameter in kernel density estimation (KSE) or kernel regression. It provides a rough approximation of the bandwidth without requiring extensive computation resources or fine-tuning. One common rule of thumb estimator is Silverman rule, which is given by
h = 1.06*σ*n^(-1/5)
where
h is the bandwidth
σ is the standard deviation of the data
n is the number of data points
This rule of thumb is based on assuming a Gaussian kernel and aims to strike a balance between over-smoothing and under-smoothing the data. It is simple to implement and usually provides reasonable bandwidth estimates for a wide range of datasets. However , it is important to note that this rule of thumb may not always have optimal results, especially for non-Gaussian or multimodal distributions. In such cases, a modified bandwidth selection, such as cross-validation or even applying a log transformation (if the data is right-skewed), may be preferable.
How it works:
This indicator computes the bandwidth volatility using returns, which are used in the standard deviation calculation. It then estimates the bandwidth based on either the Silverman rule of thumb or a modified version considering the interquartile range. The percentile ranks of the bandwidth estimate are then used to visualize the volatility levels, identify high and low volatility periods, and show them with colors.
Modified Rule of thumb Bandwidth:
The modified rule of thumb bandwidth formula combines elements of standard deviations and interquartile ranges, scaled by a multiplier of 0.9 and inversely with a number of periods. This modification aims to provide a more robust and adaptable bandwidth estimation method, particularly suitable for financial time series data with potentially skewed or heavy-tailed data.
Formula for Modified Rule of Thumb Bandwidth:
h = 0.9 * min(σ, (IQR/1.34))*n^(-1/5)
This modification introduces the use of the IQR divided by 1.34 as an alternative to the standard deviation. It aims to improve the estimation, mainly when the underlying distribution deviates from a perfect Gaussian distribution.
Analysis
Rule of thumb Bandwidth: Provides a broader perspective on volatility trends, smoothing out short-term fluctuations and focusing more on the overall shape of the density function.
Historical Volatility: Offers a more granular view of volatility, capturing day-to-day or intra-period fluctuations in asset prices and returns.
Modelling Requirements
Rule of thumb Bandwidth: Provides a broader perspective on volatility trends, smoothing out short-term fluctuations and focusing more on the overall shape of the density function.
Historical Volatility: Offers a more granular view of volatility, capturing day-to-day or intra-period fluctuations in asset prices and returns.
Pros of Bandwidth as a volatility measure
Robust to Data Distribution: Bandwidth volatility, especially when estimated using robust methods like Silverman's rule of thumb or its modifications, can be less sensitive to outliers and non-normal distributions compared to some other measures of volatility
Flexibility: It can be applied to a wide range of data types and can adapt to different underlying data distributions, making it versatile for various analytical tasks.
How can traders use this indicator?
In finance, volatility is thought to be a mean-reverting process. So when volatility is at an extreme low, it is expected that a volatility expansion happens, which comes with bigger movements in price, and when volatility is at an extreme high, it is expected for volatility to eventually decrease, leading to smaller price moves, and many traders view this as an area to take profit in.
In the context of this indicator, low volatility is thought of as having the green color, which indicates a low percentile value, and also being below the moving average. High volatility is thought of as having the yellow color and possibly being above the moving average, showing that you can eventually expect volatility to decrease.
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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.
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