QuanTAlib/lib/errors/tukey/TukeyBiweight.md

Tukey's Biweight: Robust Loss Function

"When outliers need to be silenced, not just quieted."

Tukey's Biweight (also called Bisquare) is a redescending M-estimator that completely ignores errors beyond a threshold. Unlike Huber loss which still penalizes large errors linearly, Tukey's biweight treats extreme outliers as if they don't exist.

Historical Context

Developed by John Tukey as part of his work on robust statistics in the 1970s, the biweight function was designed for situations where outliers are not just unusual but fundamentally different from the rest of the data. In such cases, including outliers at all (even with reduced influence) can corrupt the estimate.

Architecture & Physics

The biweight function is a smooth, bell-shaped curve that rises from 0, peaks at some finite error, and then descends back toward 0 for very large errors. Errors beyond the threshold c contribute nothing to the loss. This "redescending" property makes it extremely robust to gross outliers.

Properties

• Redescending: Large errors contribute zero loss (complete outlier rejection)
• Smooth: Continuously differentiable everywhere
• Bounded: Maximum loss is c²/6, regardless of error magnitude
• Tunable: Parameter c controls the outlier threshold

Mathematical Foundation

1. Tukey's Biweight Function

For each error, compute:

$$\rho(e) = \begin{cases} \frac{c^2}{6}\left[1 - \left(1 - \left(\frac{e}{c}\right)^2\right)^3\right] & \text{if } |e| \leq c \ \frac{c^2}{6} & \text{if } |e| > c \end{cases}$$

Where:

• e = y - \hat{y} = prediction error
• c = tuning constant (threshold)

2. Alternative Form

For |e| \leq c:

\rho(e) = \frac{c^2}{6}\left(1 - \left(1 - u^2\right)^3\right)

where u = e/c

3. Key Values

• At e = 0: \rho(0) = 0
• At e = c: \rho(c) = c^2/6 (maximum)
• For |e| > c: \rho(e) = c^2/6 (constant, flat)

4. Running Update (O(1))

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

S_{new} = S_{old} - \rho_{oldest} + \rho_{newest} TukeyBiweight = \frac{S_{new}}{n}

Implementation Details

Usage Patterns

// Streaming mode - with custom threshold
var tukey = new TukeyBiweight(period: 20, c: 4.685);
var result = tukey.Update(actualValue, predictedValue);

// Batch mode - calculate for entire series
var results = TukeyBiweight.Calculate(actualSeries, predictedSeries, period: 20, c: 4.685);

// Span mode - zero-allocation for high performance
TukeyBiweight.Batch(actualSpan, predictedSpan, outputSpan, period: 20, c: 4.685);

Parameters

Parameter	Type	Default	Description
period	int	-	Lookback window for averaging (must be > 0)
c	double	4.685	Tuning constant (must be > 0)

Properties

Property	Type	Description
Last	TValue	Most recent Tukey Biweight value
IsHot	bool	True when buffer is full
C	double	Current threshold parameter
Name	string	Indicator name (e.g., "TukeyBiweight(20,4.685)")
WarmupPeriod	int	Number of periods before valid output

Performance Profile

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
Robustness	10/10	Complete outlier rejection

Choosing c

c Value	Efficiency	Robustness	Use Case
4.685	95% at Gaussian	Moderate	Standard choice
6.0	98% at Gaussian	Lower	More outlier-tolerant
3.0	85% at Gaussian	Higher	More aggressive rejection
1.5	~70% at Gaussian	Very high	Extreme outlier rejection

The default c=4.685 achieves 95% efficiency for Gaussian data while providing good robustness.

Comparison with Other Robust Losses

Error Size	L2 (MSE)	Huber	Tukey
Small (< δ)	e²	e²/2	Growing
Medium (δ to c)	e²	δ|e| - δ²/2	Growing
Large (> c)	e² (huge)	δ|e| - δ²/2 (linear)	c²/6 (flat)
Very large	Explodes	Still grows	Constant

Key Insight

Tukey's biweight is the only loss function that completely stops penalizing errors beyond a threshold. A prediction error of 10 contributes the same as an error of 1000 if both exceed c.

Common Use Cases

1. Sensor Data: Reject faulty readings entirely
2. Financial Data: Ignore flash crashes or data errors
3. Image Processing: Robust edge detection
4. Scientific Measurement: Exclude instrument failures

Edge Cases

• Perfect Predictions: Returns exactly 0
• All Outliers: Returns c²/6 (maximum bounded loss)
• NaN Handling: Uses last valid value substitution
• Single Input: Not supported (requires two series)
• c = 0: Invalid (division issues)
• Errors exactly at c: Smooth transition (differentiable)

Related Indicators

• Huber - Huber Loss (linear, not redescending)
• MdAE - Median Absolute Error (robust via median)
• LogCosh - Log-Cosh Loss (smooth L1/L2 hybrid)