QuanTAlib/lib/errors/mpe/Mpe.md

MPE: Mean Percentage Error

"MAPE tells you how wrong you are; MPE tells you which direction you're wrong in."

Mean Percentage Error measures the average percentage difference between actual and predicted values while preserving the sign. Unlike MAPE, which takes absolute values, MPE reveals systematic bias in predictions—whether a model consistently over-predicts or under-predicts.

Architecture & Physics

MPE computes the signed percentage error for each data point and averages over a rolling window:

\text{MPE} = \frac{100}{n} \sum_{i=1}^{n} \frac{(\text{actual}_i - \text{predicted}_i)}{\text{actual}_i}

The sign preservation makes MPE invaluable for bias detection:

• Positive MPE: Model systematically under-predicts (actual > predicted)
• Negative MPE: Model systematically over-predicts (actual < predicted)
• MPE near zero: No systematic bias (though individual errors may be large)

Bias Detection

Consider a weather forecasting model:

• If MPE = +15%, the model consistently predicts temperatures 15% lower than actual
• If MPE = -10%, the model consistently predicts temperatures 10% higher than actual
• If MPE ≈ 0% but MAPE = 20%, errors cancel out (no bias) but magnitude is still significant

Mathematical Foundation

1. Point-wise Percentage Error

For each observation:

e_i = 100 \times \frac{\text{actual}_i - \text{predicted}_i}{\text{actual}_i}

2. Rolling Average

Over a period n:

\text{MPE}_t = \frac{1}{n} \sum_{i=t-n+1}^{t} e_i

3. Relationship to MAPE

\text{MAPE} = \frac{100}{n} \sum |e_i / 100| \text{MPE} = \frac{100}{n} \sum (e_i / 100)

When errors are consistently in one direction: |\text{MPE}| \approx \text{MAPE} When errors alternate: |\text{MPE}| < \text{MAPE}

Performance Profile

Metric	Score	Notes
Throughput	15 ns/bar	O(1) via running sum
Allocations	0	Zero-allocation hot path
Complexity	O(1)	Constant per update
Bias Detection	10/10	Primary strength
Magnitude Info	3/10	Errors can cancel
Scale Independence	9/10	Percentage-based
Outlier Sensitivity	5/10	Linear in error magnitude

Usage

// Streaming mode - bias detection in real-time
var mpe = new Mpe(20);

// Actual values consistently higher than predictions
mpe.Update(actual: 105.0, predicted: 100.0); // +5%
mpe.Update(actual: 110.0, predicted: 100.0); // +10%
// MPE will be positive, indicating under-prediction bias

double currentBias = mpe.Last.Value;
if (currentBias > 5.0)
    Console.WriteLine("Model is under-predicting by {0:F1}%", currentBias);
else if (currentBias < -5.0)
    Console.WriteLine("Model is over-predicting by {0:F1}%", Math.Abs(currentBias));
else
    Console.WriteLine("Model shows no significant bias");

// Batch mode - analyze historical predictions
var actual = new TSeries { 100, 105, 98, 102, 101 };
var predicted = new TSeries { 95, 100, 95, 100, 100 };
var results = Mpe.Calculate(actual, predicted, period: 3);

// Span mode - zero-allocation bulk processing
Span<double> output = stackalloc double[1000];
Mpe.Batch(actualSpan, predictedSpan, output, period: 20);

Interpretation Guide

MPE Value	Interpretation	Action
> +10%	Severe under-prediction	Add positive bias correction
+5% to +10%	Moderate under-prediction	Consider model recalibration
-5% to +5%	Acceptable bias range	Monitor for drift
-10% to -5%	Moderate over-prediction	Consider model recalibration
< -10%	Severe over-prediction	Add negative bias correction

Comparison with Related Metrics

Metric	Formula	Preserves Sign	Use Case
MPE	100 × (A-P)/A	✓	Bias detection
MAPE	100 × |A-P|/A	✗	Magnitude only
ME	A - P	✓	Absolute bias
MAE	|A - P|	✗	Absolute magnitude

Common Pitfalls

1. Zero Actuals

MPE is undefined when actual = 0. The implementation uses epsilon fallback:

double divisor = Math.Abs(actual) < 1e-10 ? 1e-10 : actual;

2. Cancellation Effect

Errors of opposite signs cancel out. A model alternating between +50% and -50% errors would show MPE ≈ 0%, masking severe inaccuracy.

Solution: Use MPE alongside MAPE:

• Low MAPE + Low |MPE|: Good model
• Low MAPE + High |MPE|: Unlikely (mathematically constrained)
• High MAPE + Low |MPE|: High variance, no bias
• High MAPE + High |MPE|: High variance with bias

3. Asymmetric Bounds

Unlike MAPE (bounded at 0% to ∞), MPE can range from -∞ to +100%:

• Maximum positive: actual = 100, predicted = 0 → MPE = +100%
• No upper bound on negative: actual = 100, predicted = 1000 → MPE = -900%

See Also

• MAPE - Unsigned percentage error for magnitude
• ME - Signed absolute error for absolute bias
• MAE - Unsigned absolute error for magnitude