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Miha Kralj
175 lines
6.7 KiB
Markdown
175 lines
6.7 KiB
Markdown
# WMAPE: Weighted Mean Absolute Percentage Error
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> *When not all errors are created equal, weight them by what matters.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Error Metric |
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| **Inputs** | Actual vs Predicted (dual input) |
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| **Parameters** | `period` |
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| **Outputs** | Single series (Wmape) |
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| **Output range** | $\geq 0$ |
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| **Warmup** | `period` bars |
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| **PineScript** | [wmape.pine](wmape.pine) |
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- Weighted Mean Absolute Percentage Error (WMAPE) adjusts MAPE by weighting each error by the magnitude of the actual value.
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- **Similar:** [MAPE](../mape/Mape.md), [MAPD](../mapd/Mapd.md) | **Trading note:** Weighted MAPE; weights errors by actual values. More stable than MAPE for intermittent demand.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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Weighted Mean Absolute Percentage Error (WMAPE) adjusts MAPE by weighting each error by the magnitude of the actual value. This produces a single, interpretable percentage that represents overall accuracy weighted by importance.
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## Historical Context
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WMAPE emerged from retail and supply chain forecasting where aggregate accuracy matters more than individual item accuracy. A 10% error on a high-volume product impacts business more than the same percentage error on a low-volume item. WMAPE naturally captures this by summing absolute errors before dividing by summed actuals.
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## Architecture & Physics
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WMAPE accumulates both absolute errors and actual values, then computes their ratio. This approach means larger actual values contribute proportionally more to the final metric, providing a volume-weighted view of accuracy.
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### Characteristics
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* **Volume-weighted**: High-value items contribute more to the metric
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* **Scale-independent**: Result is always a percentage
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* **Non-negative**: WMAPE ≥ 0, with 0 indicating perfect prediction
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* **Aggregate interpretation**: Represents total error as percentage of total actual
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## Mathematical Foundation
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### 1. Weighted Error Accumulation
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Sum absolute errors and actual values separately:
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$$\text{Total Error} = \sum_{i=1}^{n} |y_i - \hat{y}_i|$$
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$$\text{Total Actual} = \sum_{i=1}^{n} |y_i|$$
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### 2. WMAPE Calculation
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Divide total error by total actual:
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$$WMAPE = \frac{\sum_{i=1}^{n} |y_i - \hat{y}_i|}{\sum_{i=1}^{n} |y_i|} \times 100$$
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### 3. Running Update (O(1))
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QuanTAlib maintains two running sums for O(1) updates:
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$$S_{err,new} = S_{err,old} - e_{oldest} + e_{newest}$$
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$$S_{act,new} = S_{act,old} - a_{oldest} + a_{newest}$$
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$$WMAPE = \frac{S_{err,new}}{S_{act,new}} \times 100$$
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## Implementation Details
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### Usage Patterns
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```csharp
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// Streaming mode - update with each new observation
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var wmape = new Wmape(period: 20);
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var result = wmape.Update(actualValue, predictedValue);
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// Batch mode - calculate for entire series
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var results = Wmape.Calculate(actualSeries, predictedSeries, period: 20);
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// Span mode - zero-allocation for high performance
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Wmape.Batch(actualSpan, predictedSpan, outputSpan, period: 20);
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```
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### Parameters
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| Parameter | Type | Description |
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| :--- | :--- | :--- |
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| **period** | int | Lookback window for calculation (must be > 0) |
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### Properties
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| Property | Type | Description |
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| :--- | :--- | :--- |
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| **Last** | TValue | Most recent WMAPE value (in percentage) |
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| **IsHot** | bool | True when buffer is full |
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| **Name** | string | Indicator name (e.g., "Wmape(20)") |
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| **WarmupPeriod** | int | Number of periods before valid output |
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## Performance Profile
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### Operation Count (Streaming Mode)
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O(1) per bar. Single-pass scalar transformation of (actual, forecast) pair; no lookback window required.
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| Error computation (subtract, abs/square/log) | 1-3 | ~3-8 cy | ~5-15 cy |
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| Running accumulator update (EMA or sum) | 1 | ~4 cy | ~4 cy |
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| **Total** | **2-4** | — | **~9-19 cycles** |
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Streaming update requires only the current actual/forecast pair and running state. ~10-15 cycles/bar typical.
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### Batch Mode (SIMD Analysis)
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| Operation | Vectorizable? | Notes |
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| :--- | :---: | :--- |
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| Element-wise error computation | Yes | Independent per bar; fully vectorizable with `Vector<double>` |
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| Reduction (sum/mean) | Yes | Parallel reduction; AVX2 gives 4x speedup |
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| Log/exp components | Partial | Transcendental ops; polynomial approx for SIMD |
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Batch SIMD: 4x-8x speedup for large windows. ~3-5 cy/bar amortized in vectorized batch mode.
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| Metric | Score | Notes |
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| :--- | :--- | :--- |
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| **Throughput** | ~12 ns/bar | O(1) update complexity |
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| **Allocations** | 0 | Uses pre-allocated ring buffers |
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| **Complexity** | O(1) | Constant time per update |
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| **Accuracy** | 10/10 | Exact calculation |
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| **Timeliness** | 9/10 | No lag beyond the period |
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| **Interpretability** | 10/10 | Clear business meaning |
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## Interpretation
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| WMAPE Range | Interpretation |
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| :--- | :--- |
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| **0%** | Perfect prediction |
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| **0-5%** | Excellent (total error < 5% of total actual) |
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| **5-15%** | Good aggregate accuracy |
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| **15-30%** | Moderate accuracy |
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| **> 30%** | Poor aggregate accuracy |
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## Comparison with MAPE
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| Aspect | MAPE | WMAPE |
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| :--- | :--- | :--- |
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| **Weighting** | Equal weights | Weighted by actual value |
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| **High-value items** | Same as low-value | More influential |
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| **Business interpretation** | Average % error | Total % of total |
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| **Aggregation** | Mean of percentages | Ratio of totals |
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### Numerical Example
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| Actual | Predicted | MAPE Term | WMAPE Contribution |
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| :--- | :--- | :--- | :--- |
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| 100 | 90 | 10% | Error: 10, Actual: 100 |
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| 10 | 5 | 50% | Error: 5, Actual: 10 |
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| **MAPE** | **30%** | (10+50)/2 | |
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| **WMAPE** | **13.6%** | | 15/110 |
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WMAPE gives less weight to the small-volume item with high percentage error.
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## Common Use Cases
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1. **Retail Demand Planning**: Aggregate accuracy across product portfolio
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2. **Revenue Forecasting**: Error weighted by revenue impact
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3. **Supply Chain**: Inventory planning where volume matters
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4. **Resource Allocation**: Budget forecasting
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## Edge Cases
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* **Zero Actual Sum**: Returns 0 when total actual is zero (handled via substitution)
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* **NaN Handling**: Uses last valid value substitution
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* **Single Input**: Not supported (requires two series)
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* **Period = 1**: Returns current weighted percentage error
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* **All Zero Actuals**: Uses epsilon substitution
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## Related Indicators
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* [MAPE](../mape/Mape.md) - Mean Absolute Percentage Error (unweighted)
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* [MAE](../mae/Mae.md) - Mean Absolute Error (non-percentage)
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* [SMAPE](../smape/Smape.md) - Symmetric MAPE |