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Co-authored-by: Claude Opus 4.5 <noreply@anthropic.com> Co-authored-by: aider (openrouter/anthropic/claude-sonnet-4) <aider@aider.chat> Co-authored-by: Warp <agent@warp.dev>
128 lines
3.8 KiB
Markdown
128 lines
3.8 KiB
Markdown
# ZLEMA: Zero-Lag Exponential Moving Average
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## EMA with lag compensation via a zero-lag signal
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> "ZLEMA does not erase lag. It predicts just enough to act early, then pays the price in overshoot."
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ZLEMA takes a standard EMA and feeds it a **zero-lag signal**: current price minus a lagged price. This produces a smoother that responds faster than EMA without going fully raw. It is not magic. It shifts some lag into controlled overshoot.
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## Historical Context
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ZLEMA is a widely used variation on EMA intended to reduce delay without abandoning exponential smoothing. It appears in multiple technical analysis toolkits and is often described as a "predictive EMA." The prediction is simple: extrapolate the current price by subtracting a lagged value.
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## Architecture & Physics
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### Pipeline
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1. **Lag estimate**
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$$\text{lag} = \max(1, \text{round}((N-1)/2))$$
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2. **Zero-lag signal**
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$$s_t = 2 \cdot x_t - x_{t-\text{lag}}$$
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3. **EMA smoothing**
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$$\text{ZLEMA}_t = \text{EMA}(s_t, \alpha)$$
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### Warmup compensation
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ZLEMA uses EMA bias compensation during warmup:
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$$y_t^{*} = \frac{y_t}{1 - (1 - \alpha)^t}$$
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This avoids the early-stage bias toward zero and makes the first values usable.
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## Math Foundation
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**EMA update:**
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$$y_t = y_{t-1} + \alpha (s_t - y_{t-1})$$
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**Zero-lag signal:**
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$$s_t = 2 \cdot x_t - x_{t-\text{lag}}$$
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**Alpha from period:**
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$$\alpha = \frac{2}{N + 1}$$
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## Performance Profile
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### Operation Count (Streaming Mode, Scalar)
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**Hot path (after warmup, compensation complete):**
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| FMA | 2 | 4 | 8 |
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| MUL | 1 | 3 | 3 |
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| **Total** | **3** | | **~11 cycles** |
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The hot path consists of:
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1. Zero-lag signal: `FMA(2.0, val, -lagged)` 1 FMA
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2. EMA core: `FMA(zlemaRaw, beta, alpha * signal)` 1 FMA + 1 MUL
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**Warmup path (with bias compensation):**
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| FMA | 2 | 4 | 8 |
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| MUL | 2 | 3 | 6 |
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| DIV | 1 | 15 | 15 |
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| CMP | 2 | 1 | 2 |
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| **Total** | **7** | | **~31 cycles** |
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Additional warmup operations:
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- Decay tracking: `e *= beta` 1 MUL
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- Bias compensation: `zlemaRaw / (1 - e)` 1 DIV
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- Hot/compensated checks 2 CMP
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### Batch Mode (SIMD Analysis)
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ZLEMA is an IIR filter with lag buffer dependency not directly vectorizable across bars. However, within-bar operations use FMA intrinsics.
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| Optimization | Benefit |
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| :--- | :--- |
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| FMA instructions | ~11 cycles vs ~14 scalar |
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| stackalloc buffer | Zero heap allocation for lag d256 |
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 8/10 | Matches PineScript reference |
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| **Timeliness** | 8/10 | Faster response than EMA |
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| **Overshoot** | 6/10 | Predictive signal causes overshoot on reversals |
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| **Smoothness** | 7/10 | Between EMA and raw price |
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## Validation
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ZLEMA is validated against a PineScript reference implementation.
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| Library | Status | Tolerance | Notes |
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|:---|:---|:---|:---|
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| **TA-Lib** | N/A | - | No ZLEMA in TA-Lib |
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| **Skender** | N/A | - | No ZLEMA in Skender |
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| **Tulip** | Partial | - | Tulip has `zlema` but not used here |
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| **Ooples** | N/A | - | No ZLEMA in Ooples |
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| **PineScript** | ? Passed | 1e-10 | Matches `lib/trends_IIR/zlema/zlema.pine` |
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## Common Pitfalls
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1. **Overshoot on turns**
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The zero-lag signal is a forward estimate. It can overshoot when price reverses sharply. This is expected behavior.
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2. **Period semantics**
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ZLEMA uses EMA alpha; the lag term is derived from period but not equivalent to a window length. Do not compare ZLEMA period directly to SMA window length.
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3. **Warmup discipline**
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Use `IsHot` / `WarmupPeriod` before acting on signals. Early values are bias-corrected but still unstable.
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4. **Non-finite data**
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NaN or Infinity is replaced with the last valid value. Before the first valid sample, output is `NaN`. |