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Miha Kralj
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- LTMA (Linear Trend Moving Average): Introduces a predictive moving average using dual cascaded EMAs for trend estimation. - MCNMA (McNicholl EMA): Implements a zero-lag TEMA using a cascaded EMA structure for enhanced responsiveness. - NLMA (Non-Lag Moving Average): Utilizes a damped cosine kernel to achieve reduced lag in moving averages. - NMA (Natural Moving Average): Adapts smoothing based on volatility profiles using a square-root kernel. - NYQMA (Nyquist Moving Average): Applies the Nyquist-Shannon theorem to prevent aliasing in cascaded moving averages. - RAIN (Rainbow Moving Average): Combines multiple SMA layers with weighted averages for multi-scale smoothing. - TRAMA (Trend Regularity Adaptive Moving Average): Adapts smoothing based on the frequency of new highs and lows in price data.
108 lines
4.2 KiB
Markdown
108 lines
4.2 KiB
Markdown
# CHOP: Choppiness Index
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The Choppiness Index is a non-directional regime indicator that measures whether the market is trending or trading sideways. It compares total price movement (sum of True Range) to net price movement (high-low channel width) using a logarithmic ratio, producing a bounded value where high readings indicate choppy/consolidating conditions and low readings indicate trending conditions. CHOP does not indicate direction — only whether directional strategies are likely to succeed. The logarithmic scaling normalizes the output to approximately 0-100 regardless of price level or volatility magnitude.
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## Historical Context
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Australian commodity trader E.W. Dreiss created the Choppiness Index to help traders avoid whipsaw losses by identifying market conditions unsuitable for trend-following strategies. The core insight is geometric: in a perfect trend, total bar-by-bar movement (sum of True Range) roughly equals the net distance traveled (channel width). In a choppy market, total movement greatly exceeds net progress — the market thrashes back and forth, accumulating True Range while the net channel stays narrow. The ratio between these two quantities, log-scaled to normalize across instruments and timeframes, produces a clean regime classifier. The conventional thresholds (38.2 and 61.8) are deliberately chosen as Fibonacci levels, though their efficacy is empirical rather than mathematical.
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## Architecture & Physics
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### 1. True Range Accumulation
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$$TR_t = \max(H_t - L_t,\; |H_t - C_{t-1}|,\; |L_t - C_{t-1}|)$$
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A rolling sum maintains $\sum_{i=1}^{N} TR_i$ over the lookback window.
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### 2. Price Channel Width
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The net price movement over the same window:
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$$\text{Channel} = \max(H_{t-N+1:t}) - \min(L_{t-N+1:t})$$
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### 3. Choppiness Index
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$$\text{CHOP} = 100 \times \frac{\log_{10}\!\left(\dfrac{\sum TR_N}{\text{Channel}}\right)}{\log_{10}(N)}$$
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The denominator $\log_{10}(N)$ normalizes the output so that the theoretical maximum approaches 100 (when $\sum TR = N \times \text{Channel}$, which occurs when every bar traverses the full channel).
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### 4. Complexity
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- **Time:** $O(N)$ per bar for min/max scanning of high/low buffers; rolling sum is $O(1)$
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- **Space:** $O(N)$ — three ring buffers (TR, highs, lows)
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- **Warmup:** $N$ bars
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## Mathematical Foundation
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### Parameters
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| Symbol | Parameter | Default | Constraint |
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|--------|-----------|---------|------------|
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| $N$ | period | 14 | $N \geq 2$ |
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### Pseudo-code
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```
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Initialize:
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trBuf = RingBuffer(period)
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highBuf = RingBuffer(period)
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lowBuf = RingBuffer(period)
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trSum = 0
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prevClose = NaN
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logPeriod = log10(period)
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On each bar (high, low, close, isNew):
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if !isNew: restore previous state
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// True Range
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if prevClose is valid:
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TR = max(high - low, |high - prevClose|, |low - prevClose|)
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else:
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TR = high - low
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// Rolling sum update
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if trBuf is full:
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trSum -= trBuf.Oldest
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trBuf.Add(TR)
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trSum += TR
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highBuf.Add(high)
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lowBuf.Add(low)
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// Channel width
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maxHigh = Max(highBuf)
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minLow = Min(lowBuf)
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channel = maxHigh - minLow
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// Choppiness Index
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if channel > 0 AND trSum > 0:
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CHOP = 100 × log10(trSum / channel) / logPeriod
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else:
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CHOP = 50 // neutral fallback
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prevClose = close
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output = Clamp(CHOP, 0, 100)
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```
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### Interpretation
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| CHOP Value | Market Regime | Strategy Implication |
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|------------|---------------|---------------------|
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| > 61.8 | High choppiness | Avoid trend-following; favor range strategies |
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| 38.2 - 61.8 | Ambiguous | Mixed conditions; reduced position sizing |
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| < 38.2 | Low choppiness | Market trending; favor momentum/breakout strategies |
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### Geometric Intuition
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- **Perfect trend (straight line):** $\sum TR \approx \text{Channel}$, so $\log_{10}(1) = 0$, CHOP $\to 0$
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- **Maximum chop (full traversal every bar):** $\sum TR \approx N \times \text{Channel}$, so $\log_{10}(N) / \log_{10}(N) = 1$, CHOP $\to 100$
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### Non-Directional Property
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CHOP is completely direction-agnostic. A strong uptrend and a strong downtrend produce identical low CHOP readings. Direction must be determined by a separate indicator (AMAT, ADX directional components, or simple price comparison).
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## Resources
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- Dreiss, E.W. — Choppiness Index (original development)
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- PineScript reference: `chop.pine` in indicator directory
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