QuanTAlib/lib/errors/mae/Mae.md

MAE: Mean Absolute Error

Property	Value
Category	Error Metric
Inputs	Source (close)
Parameters	period
Outputs	Single series (MAE)
Output range	\geq 0
Warmup	1 bar

TL;DR

• Mean Absolute Error (MAE) measures the average magnitude of errors in a set of predictions, without considering their direction.
• Parameterized by period.
• Output range: \geq 0.
• Requires 1 bar of warmup before first valid output (IsHot = true).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

"When you need to know how wrong you are on average, without the drama of squared errors."

Mean Absolute Error (MAE) measures the average magnitude of errors in a set of predictions, without considering their direction. It represents the average of the absolute differences between actual and predicted values.

Historical Context

MAE is one of the oldest and most intuitive error metrics in statistics. Its simplicity and interpretability have made it a staple in regression analysis, forecasting, and model evaluation since the early days of statistical analysis.

Architecture & Physics

MAE treats all errors equally, making it more robust to outliers compared to squared-error metrics like MSE. The absolute value operation removes directionality, focusing purely on error magnitude.

Properties

• Non-negative: MAE ≥ 0, with 0 indicating perfect prediction
• Same units: Unlike MSE, MAE is in the same units as the original data
• Linear sensitivity: Each unit of error contributes equally to the final metric
• Robust: Less sensitive to outliers than squared-error metrics

Mathematical Foundation

1. Absolute Error

For each observation, calculate the absolute difference between actual and predicted values:

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

Where:

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

2. Mean Calculation

Average the absolute errors over the period:

MAE = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{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} MAE = \frac{S_{new}}{n}

Implementation Details

Usage Patterns

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

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

// Span mode - zero-allocation for high performance
Mae.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 MAE value
IsHot	bool	True when buffer is full
Name	string	Indicator name (e.g., "Mae(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	~10 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

MAE Range	Interpretation
0	Perfect prediction
Low	Predictions are close to actual values
High	Large average prediction error

Comparison with Other Metrics

Metric	Outlier Sensitivity	Units	Interpretation
MAE	Low	Same as data	Average absolute error
MSE	High	Squared units	Penalizes large errors more
RMSE	High	Same as data	MSE in original units
MAPE	Varies	Percentage	Relative error

Common Use Cases

1. Forecast Evaluation: Measure prediction accuracy over time
2. Model Comparison: Compare different prediction models
3. Trading Strategy: Track signal accuracy
4. Risk Assessment: Monitor prediction reliability

Edge Cases

• Identical Values: Returns 0 when actual equals predicted
• NaN Handling: Uses last valid value substitution
• Single Input: Not supported (requires two series)
• Period = 1: Returns current absolute error

Related Indicators

• MSE - Mean Squared Error
• RMSE - Root Mean Squared Error
• MAPE - Mean Absolute Percentage Error