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# Kaufman's Adaptive Z-Score (K-Score) Pro (v1.00)
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Statistical Adaptive Dispersion & Kinetic Elasticity Oscillator
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---
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## 1. Summary (Introduction)
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**K-Score Pro** is an advanced statistical momentum oscillator that measures price deviation from **Perry Kaufman's Adaptive Moving Average (KAMA)** normalized in units of standard deviation ($\sigma$).
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While traditional Z-Score indicators compute dispersion relative to rigid, lagging baselines (such as SMA or EMA), **K-Score replaces the static mean with a non-linear, regime-switching baseline**. This fundamentally transforms how statistical overbought and oversold states are interpreted:
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* **In Healthy Trends:** The KAMA baseline dynamically accelerates alongside price action. Consequently, K-Score remains within sustainable equilibrium bounds ($+0.5\sigma \dots +1.5\sigma$) rather than getting trapped in perpetual, false "overbought" readings.
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* **During Parabolic Blow-Offs:** If price accelerates faster than KAMA's maximum internal adaptation speed, the statistical "rubber band" stretches to extremes ($> +2.0\sigma \dots +2.5\sigma$), signaling a genuine kinetic climax.
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* **In Low-Efficiency Consolidations:** The KAMA baseline flattens completely into a horizontal line. Sudden price spikes outside the range produce instant $+2.0\sigma$ warnings, accurately identifying false breakouts and liquidity sweeps.
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```text
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┌────────────────────────────────────────────────────────────────────────┐
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│ K-SCORE DISPERSION MODEL │
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├────────────────────────────────────────────────────────────────────────┤
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│ K-Score = [ Price(t) - KAMA(t) ] / StandardDeviation(Price - KAMA, N) │
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│ Expressed in standardized Sigma Multiples (σ) │
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└────────────────────────────────────────────────────────────────────────┘
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```
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---
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## 2. Mathematical Foundations & Kinetic Dynamics
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```text
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+2.5σ (Extreme Exhaustion / Climax)
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───────────────────────────────────────────────────────────── DeepSkyBlue (Bull Climax)
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+1.5σ (Bullish Flow Threshold)
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- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - LightSkyBlue (Bull Flow)
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0.0σ (KAMA Equilibrium Mean)
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───────────────────────────────────────────────────────────── Gray (Noise / Equilibrium)
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-1.5σ (Bearish Flow Threshold)
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- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Coral (Bear Flow)
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-2.5σ (Extreme Exhaustion / Climax)
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───────────────────────────────────────────────────────────── OrangeRed (Bear Climax)
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```
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### 2.1. Mathematical Formulation
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#### 1. Adaptive Equilibrium Baseline ($\mu_t$)
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$$\mu_t = \text{KAMA}_t(\text{ER}, \text{Fast}, \text{Slow}, \text{Price})$$
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#### 2. Dispersion Variance Around KAMA
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Given a standard deviation lookback window $P_\sigma = \text{InpStDevPeriod}$:
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$$\sigma_t = \sqrt{\frac{1}{P_\sigma} \sum_{k=0}^{P_\sigma - 1} \left( P_{t-k} - \mu_t \right)^2}$$
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#### 3. Standardized K-Score Metric
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$$K\text{Score}_t = \begin{cases} \frac{P_t - \mu_t}{\sigma_t}, & \text{if } \sigma_t > 10^{-9} \\ 0.0, & \text{otherwise} \end{cases}$$
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---
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### 2.2. The Regime Duality Matrix (Why K-Score is Unique)
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| Market Environment | KAMA Baseline Behavior | K-Score Interpretation | Strategic Value |
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| :--- | :--- | :--- | :--- |
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| **Established Trend** | Accelerates with price ($\text{SC} \approx \alpha_{\text{fast}}$) | Stays moderate ($+0.5\sigma \dots +1.5\sigma$) | **Trend Retention:** Prevents premature exits during strong trend runs. |
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| **Parabolic Blow-off** | Price outruns fastest KAMA speed | Spikes to $> +2.0\sigma \dots +2.5\sigma$ | **Climax Warning:** Identifies unsustainable exhaustion spikes. |
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| **Sideways Range** | Flattens into horizontal axis ($\text{SC} \approx \alpha_{\text{slow}}$) | Acts as a pure Gaussian range envelope | **Liquidity Trap Alert:** Identifies range-edge fades and fakeouts. |
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---
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### 2.3. Swapped Thermal 5-Zone Color Palette
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| State Index | Color | Classification | Sigma Level Trigger | Contextual Action |
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| :---: | :---: | :--- | :--- | :--- |
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| **0.0** | `clrGray` | **Noise / Equilibrium** | $\|\text{K-Score}\| \le 1.5\sigma$ | Normal distribution around KAMA; trend in balance. |
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| **1.0** | `clrLightSkyBlue` | **Bullish Flow** | $+1.5\sigma < \text{K-Score} \le +2.0\sigma$ | Healthy upward impulse expansion. |
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| **2.0** | `clrDeepSkyBlue` | **Bullish Climax** | $\text{K-Score} > +2.0\sigma$ | Parabolic blow-off top; tighten stops / scale out. |
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| **3.0** | `clrCoral` | **Bearish Flow** | $-2.0\sigma \le \text{K-Score} < -1.5\sigma$ | Healthy downward impulse expansion. |
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| **4.0** | `clrOrangeRed` | **Bearish Climax** | $\text{K-Score} < -2.0\sigma$ | Panic capitulation floor; cover shorts / prepare bounce. |
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---
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## 3. MQL5 Architecture & Engineering Standards
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```text
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┌────────────────────────────────────────────────────────┐
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│ KScore_Calculator.mqh │
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│ (Core Math Engine - Encapsulated CKamaCalculator) │
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└──────────────────────────┬─────────────────────────────┘
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│ Computes K-Score Values in O(1)
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▼
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┌────────────────────────────────────────────────────────┐
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│ KScore_Pro.mq5 │
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│ (Unified Wrapper: Native Timeframe & MTF Engine) │
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├──────────────────────────┬─────────────────────────────┤
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│ Buffer Layer (3) │ Centralized Framework │
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│ • BufferKScore (DATA) │ • DataSync_Tools.mqh │
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│ • BufferColors (INDEX) │ • MovingAverage_Engine │
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│ • BufferSignal (DATA) │ • Dynamic Sigma Levels │
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└──────────────────────────┴─────────────────────────────┘
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```
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1. **Embedded Engine Composition:** `CKScoreCalculator` encapsulates `CKamaCalculator` and `CHeikinAshi_Calculator` directly via composition, eliminating external memory leaks and pointer overhead.
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2. **2026 MTF Framework with DataSync Daemon:**
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* Multi-timeframe synchronization is handled asynchronously via `DataSync_Tools.mqh` through a 1-second `OnTimerUpdate` daemon.
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* **Forming LTF Block Flat-Force (Staircase Solution):** The mapping anchor dynamically resets to `first_bar_of_forming_htf`, ensuring all sub-bars of the active higher-timeframe candle update in real-time without visual distortion.
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---
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## 4. Parameters Reference
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### Timeframe Settings
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* `InpTimeframe` (*default: `PERIOD_CURRENT`*): Calculation timeframe. When set to `PERIOD_CURRENT`, it operates in native zero-lag mode. When set to a higher timeframe (e.g., `PERIOD_H1`, `PERIOD_D1`), it activates the synchronized MTF engine.
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### KAMA Core Settings
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* `InpErPeriod` (*default: `10`*): The lookback window ($N$) used for the KAMA Efficiency Ratio.
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* `InpFastEmaPeriod` (*default: `2`*): Fastest smoothing period ($F$) during high directional efficiency.
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* `InpSlowEmaPeriod` (*default: `30`*): Slowest smoothing period ($S$) during consolidation.
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* `InpStDevPeriod` (*default: `20`*): Lookback period ($P_\sigma$) for computing standard deviation variance around KAMA.
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* `InpSourcePrice` (*default: `PRICE_CLOSE_STD`*): Input price series (Standard OHLC or Synthetic Heikin Ashi).
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### Signal Line Settings
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* `InpShowSignal` (*default: `true`*): Toggle visibility of the Signal Moving Average line.
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* `InpSignalPeriod` (*default: `5`*): Lookback period for the signal line.
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* `InpSignalType` (*default: `EMA`*): Smoothing algorithm (`SMA`, `EMA`, `SMMA`, `LWMA`, `TMA`, `DEMA`, `TEMA`, `VWMA`).
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* `InpColorSignal` (*default: `clrFireBrick`*): Color applied to the signal line.
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### Indicator Levels (Sigma Units)
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* `InpLevelFlowHigh` (*default: `1.5`*): Bullish Flow warning boundary.
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* `InpLevelFlowLow` (*default: `-1.5`*): Bearish Flow warning boundary.
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* `InpLevelClimaxHigh` (*default: `2.0`*): Bullish Climax threshold (`DeepSkyBlue`).
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* `InpLevelClimaxLow` (*default: `-2.0`*): Bearish Climax threshold (`OrangeRed`).
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* `InpLevelExtremeHigh` (*default: `2.5`*): Extreme statistical over-extension level.
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* `InpLevelExtremeLow` (*default: `-2.5`*): Extreme statistical panic level.
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* `InpLevelColor` (*default: `clrSilver`*): Color of horizontal level lines.
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* `InpLevelStyle` (*default: `STYLE_DOT`*): Line style of horizontal level lines.
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---
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## 5. Quantitative Trading Strategies (Contextual Role)
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```text
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┌────────────────────────────────────────────────────────────────────────┐
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│ HOW TO USE K-SCORE IN A LIVE SYSTEM │
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├────────────────────────────────────────────────────────────────────────┤
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│ 1. Trend Filter: Use KAMA_Pro on Main Chart for Direction. │
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│ 2. Momentum Trigger: Use KAMA_Slope_Pro for Entry Acceleration. │
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│ 3. Elasticity Guard: Use K-Score_Pro to Prevent Buying Top / Selling │
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│ Bottom (Filter out trades when > +2.0σ / < -2.0σ).│
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│ 4. Climax Exit: Take profit when K-Score hits +2.5σ and crosses │
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│ back below Signal Line. │
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└────────────────────────────────────────────────────────────────────────┘
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```
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### 5.1. The Parabolic Climax Filter (Trade Invalidation)
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* **Rule:** Do **NOT** initiate new trend-following long positions if `K-Score > +2.0σ` (DeepSkyBlue), even if `KAMA_Slope_Pro` is strong green. The price is over-extended relative to its adaptive baseline and is prone to a sharp mean-reverting snapback.
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* **Rule:** Do **NOT** initiate new short breakout positions if `K-Score < -2.0σ` (OrangeRed).
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### 5.2. Mean-Reversion Exhaustion Exit
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* **Long Exit Trigger:** Price is in an uptrend, K-Score exceeds $+2.0\sigma$ or $+2.5\sigma$, and then crosses **below** the Signal MA line $\rightarrow$ Institutional signal to close long positions or lock in partial profits.
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* **Short Exit Trigger:** Price is in a downtrend, K-Score drops below $-2.0\sigma$ or $-2.5\sigma$, and then crosses **above** the Signal MA line $\rightarrow$ Institutional signal to cover short positions.
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### 5.3. Range Liquidity Sweep Trap (The Fakeout Reversal)
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* **Context:** `KAMA_Pro` is flat (Gray Slope).
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* **Setup:** A news spike thrusts price outside the range, driving K-Score to $+2.2\sigma$.
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* **Execution:** If the next candle fails to sustain momentum and closes back inside the range while K-Score drops below $+1.5\sigma$, fade the move back toward the KAMA centerline ($0.0\sigma$).
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---
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## 6. Indicator Buffer Map (For Developers & EA Integration)
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| Buffer Index | Name | Type | Description |
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| :---: | :---: | :---: | :--- |
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| **0** | `BufferKScore` | `INDICATOR_DATA` | Standardized K-Score Values in Sigma Multiples ($\sigma$) |
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| **1** | `BufferColors` | `INDICATOR_COLOR_INDEX` | Swapped Thermal 5-Zone Palette Index ($0.0 \dots 4.0$) |
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| **2** | `BufferSignal` | `INDICATOR_DATA` | Smoothed Signal Moving Average Plot |
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*All buffers strictly maintain non-series chronological order (`ArraySetAsSeries = false`), ensuring instant compatibility with Expert Advisors and scanner dashboards via `iCustom()`.*
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