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Miha Kralj
136 lines
5.4 KiB
Markdown
136 lines
5.4 KiB
Markdown
# MASE: Mean Absolute Scaled Error
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> *A good forecast is one that's better than guessing. MASE tells you exactly how much better.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Error Metric |
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| **Inputs** | Actual, Predicted (dual series) |
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| **Parameters** | `period` |
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| **Outputs** | Single series (Mase) |
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| **Output range** | $\geq 0$ |
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| **Warmup** | `period + 1` bars |
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| **PineScript** | [mase.pine](mase.pine) |
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- Mean Absolute Scaled Error (MASE) normalizes forecast errors by the average error of a naive "random walk" forecast (using the previous value as th...
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- **Similar:** [MAE](../mae/Mae.md), [RAE](../rae/Rae.md) | **Trading note:** Mean Absolute Scaled Error; compares forecast to naïve random-walk baseline. <1 = better than naïve.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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Mean Absolute Scaled Error (MASE) normalizes forecast errors by the average error of a naive "random walk" forecast (using the previous value as the prediction). This makes MASE scale-independent and interpretable across different time series.
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## Architecture & Physics
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MASE computes a ratio: the mean absolute error of your predictions divided by the mean absolute error of a naive forecast. The naive forecast simply predicts that tomorrow's value equals today's value.
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### Interpretation Guide
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| MASE Value | Interpretation |
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| ---------- | -------------- |
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| **MASE < 1** | Forecast is better than naive (good) |
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| **MASE = 1** | Forecast equals naive performance |
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| **MASE > 1** | Forecast is worse than naive (bad) |
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| **MASE = 0** | Perfect forecast |
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The naive baseline captures the inherent "forecastability" of the series. A highly volatile series has a larger naive error, making a given absolute error less significant.
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## Mathematical Foundation
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### 1. Absolute Error
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$$e_t = |y_t - \hat{y}_t|$$
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### 2. Naive Forecast Scale
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$$\text{Scale} = \frac{1}{n-1} \sum_{i=2}^{n} |y_i - y_{i-1}|$$
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The scale represents the average absolute change from one period to the next.
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### 3. Mean Absolute Scaled Error
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$$\text{MASE} = \frac{\frac{1}{n} \sum_{t=1}^{n} |y_t - \hat{y}_t|}{\frac{1}{n-1} \sum_{i=2}^{n} |y_i - y_{i-1}|}$$
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Or more simply:
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$$\text{MASE} = \frac{\text{MAE}}{\text{Scale}}$$
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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** | ~35 ns/bar | Dual running sums for error and scale |
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| **Allocations** | 0 | Zero-allocation implementation |
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| **Complexity** | O(1) | Constant time per update |
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| **Accuracy** | 9/10 | Handles edge cases well |
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| **Timeliness** | 7/10 | Rolling window introduces lag |
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| **Robustness** | 10/10 | Works with zero/negative values |
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## Common Pitfalls
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### Flat Series Problem
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When the actual series is constant (no change between values), the scale becomes zero. The implementation handles this by returning the raw MAE when scale is near zero.
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### Initial Warmup
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The scale calculation requires at least two values (to compute differences). During warmup, MASE defaults to MAE / 1.0.
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### Different from Other Scaled Metrics
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Unlike MAPE which scales by actual values, MASE scales by the difficulty of the forecasting problem itself.
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## Usage
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```csharp
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// Create MASE calculator with period 14
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var mase = new Mase(14);
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// Stream values
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var result = mase.Update(actual, predicted);
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Console.WriteLine($"MASE: {result.Value:F4}");
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// MASE < 1 = better than naive, MASE > 1 = worse than naive
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// Batch calculation
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var maseSeries = Mase.Calculate(actualSeries, predictedSeries, 14);
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// Zero-allocation span version
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Mase.Batch(actualSpan, predictedSpan, outputSpan, 14);
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```
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## Comparison with Other Error Metrics
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| Metric | Scale-Independent | Handles Zero | Symmetric | Interpretable |
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| ------ | ----------------- | ------------ | --------- | ------------- |
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| **MASE** | ✅ | ✅ | ✅ | ✅ (vs naive) |
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| **MAPE** | ✅ | ❌ | ❌ | ✅ (% error) |
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| **SMAPE** | ✅ | ⚠️ | ✅ | ⚠️ (bounded %) |
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| **MAE** | ❌ | ✅ | ✅ | ❌ (raw units) |
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| **RMSE** | ❌ | ✅ | ✅ | ❌ (raw units) |
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MASE is particularly valuable when:
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* Comparing forecasts across different series
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* Evaluating against a natural baseline (naive forecast)
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* Working with data that includes zeros
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* Needing symmetric treatment of over/under predictions |