QuanTAlib/lib/errors/mrae/Mrae.md

MRAE: Mean Relative Absolute Error

When you need to understand your error in the context of what you're predicting.

Property	Value
Category	Error Metric
Inputs	Actual, Predicted (dual series)
Parameters	period
Outputs	Single series (MRAE)
Output range	\geq 0
Warmup	period bars
PineScript	mrae.pine

• Mean Relative Absolute Error (MRAE) measures the average magnitude of errors relative to the actual values.
• Similar: RAE, MASE | Trading note: Mean Relative Absolute Error; ratio of errors to benchmark errors. Scale-free comparison metric.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

Mean Relative Absolute Error (MRAE) measures the average magnitude of errors relative to the actual values. This normalization makes the metric scale-independent and easier to interpret across different datasets.

Historical Context

MRAE emerged as an alternative to MAPE for situations where relative error measurement is important but where the issues with percentage-based metrics (like undefined values when actuals are zero) need to be handled differently. It provides a bounded, interpretable measure of prediction accuracy.

Architecture & Physics

MRAE divides each absolute error by the actual value, providing context for the error magnitude. The error of 5 means something different when predicting 10 versus predicting 1000, and MRAE captures this distinction.

Properties

• Scale-independent: Comparable across different data magnitudes
• Non-negative: MRAE ≥ 0, with 0 indicating perfect prediction
• Interpretable: A value of 0.1 means 10% average relative error
• Denominator sensitivity: Undefined when actual values are zero (handled via substitution)

Mathematical Foundation

1. Relative Absolute Error

For each observation, calculate the relative error:

e_i = \frac{|y_i - \hat{y}_i|}{|y_i|}

Where:

• y_i = actual value
• \hat{y}_i = predicted value

2. Mean Calculation

Average the relative errors over the period:

MRAE = \frac{1}{n} \sum_{i=1}^{n} \frac{|y_i - \hat{y}_i|}{|y_i|}

3. Running Update (O(1))

QuanTAlib uses a ring buffer with running sum for O(1) updates:

S_{new} = S_{old} - e_{oldest} + e_{newest} MRAE = \frac{S_{new}}{n}

Implementation Details

Usage Patterns

// Streaming mode - update with each new observation
var mrae = new Mrae(period: 20);
var result = mrae.Update(actualValue, predictedValue);

// Batch mode - calculate for entire series
var results = Mrae.Calculate(actualSeries, predictedSeries, period: 20);

// Span mode - zero-allocation for high performance
Mrae.Batch(actualSpan, predictedSpan, outputSpan, period: 20);

Parameters

Parameter	Type	Description
period	int	Lookback window for averaging (must be > 0)

Properties

Property	Type	Description
Last	TValue	Most recent MRAE value
IsHot	bool	True when buffer is full
Name	string	Indicator name (e.g., "Mrae(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	~15 ns/bar	O(1) update complexity
Allocations	0	Uses pre-allocated ring buffer
Complexity	O(1)	Constant time per update
Accuracy	10/10	Exact calculation
Timeliness	9/10	No lag beyond the period
Smoothness	7/10	Moderate smoothing

Interpretation

MRAE Range	Interpretation
0	Perfect prediction
0 - 0.1	Excellent (< 10% average relative error)
0.1 - 0.3	Good (10-30% average relative error)
> 0.3	Poor (> 30% average relative error)

Comparison with Other Metrics

Metric	Scale-Independent	Zero-Safe	Symmetry
MRAE	Yes	No (uses substitution)	No
MAPE	Yes	No	No
MAE	No	Yes	Yes
SMAPE	Yes	Partially	Yes

Common Use Cases

1. Financial Forecasting: Compare prediction accuracy across different asset prices
2. Demand Forecasting: Normalize errors across products with varying sales volumes
3. Model Comparison: Compare models on datasets with different scales
4. Time Series Analysis: Track relative prediction quality over time

Edge Cases

• Zero Actual Values: Substitutes with small epsilon (1e-10) to avoid division by zero
• NaN Handling: Uses last valid value substitution
• Single Input: Not supported (requires two series)
• Period = 1: Returns current relative absolute error

Related Indicators

• MAE - Mean Absolute Error (non-relative)
• MAPE - Mean Absolute Percentage Error
• SMAPE - Symmetric Mean Absolute Percentage Error