QuanTAlib/lib/errors/me/Me.md

ME: Mean Error (Mean Bias Error)

"Sometimes you need to know not just how wrong you are, but which direction you're wrong in."

Mean Error (ME), also known as Mean Bias Error, measures the average error between actual and predicted values while preserving the sign. Unlike MAE, ME reveals systematic bias in predictions: whether a model consistently over-predicts or under-predicts.

Historical Context

ME is one of the fundamental error metrics in statistics and forecasting. While MAE and MSE focus on error magnitude, ME fills the critical role of detecting directional bias. A model could have low MAE but significant ME, indicating consistent over or under-prediction that cancels out when measuring magnitude alone.

Architecture & Physics

ME preserves the sign of errors, allowing positive and negative errors to cancel each other. This makes it ideal for detecting systematic bias but unsuitable for measuring prediction accuracy alone.

Properties

• Can be negative: ME can be positive, negative, or zero
• Positive ME: Model under-predicts (actual > predicted on average)
• Negative ME: Model over-predicts (actual < predicted on average)
• Zero ME: No systematic bias (but not necessarily accurate)
• Same units: ME is in the same units as the original data
• Cancellation: Errors can cancel out, hiding large individual errors

Mathematical Foundation

1. Error Calculation

For each observation, calculate the signed 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 errors over the period:

ME = \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} ME = \frac{S_{new}}{n}

Implementation Details

Usage Patterns

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

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

// Span mode - zero-allocation for high performance
Me.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 ME value
IsHot	bool	True when buffer is full
Name	string	Indicator name (e.g., "Me(20)")
WarmupPeriod	int	Number of periods before valid output

Performance Profile

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

ME Value	Interpretation
ME > 0	Systematic under-prediction (actual > predicted)
ME = 0	No systematic bias
ME < 0	Systematic over-prediction (actual < predicted)

Comparison with Other Metrics

Metric	Shows Bias	Units	Use Case
ME	Yes	Same as data	Detect systematic bias
MAE	No	Same as data	Average error magnitude
MSE	No	Squared units	Penalize large errors
MPE	Yes	Percentage	Relative bias

Common Use Cases

1. Bias Detection: Identify if a model consistently over or under-predicts
2. Model Calibration: Use ME to adjust model outputs
3. Forecast Evaluation: Distinguish between random errors and systematic bias
4. Trading Signals: Detect directional bias in price predictions

Warning: Cancellation Problem

ME can be misleading when errors cancel out:

var me = new Me(4);
me.Update(110, 100); // Error: +10
me.Update(90, 100);  // Error: -10
me.Update(110, 100); // Error: +10
me.Update(90, 100);  // Error: -10
// ME = 0, but individual errors are large!

Always use ME alongside MAE or MSE to get a complete picture.

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 signed error
• Balanced Errors: Can return 0 even with large individual errors

Related Indicators

• MAE - Mean Absolute Error (magnitude only)
• MSE - Mean Squared Error
• MPE - Mean Percentage Error (relative bias)