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