QuanTAlib/lib/errors/theilu/TheilU.md

Theil's U: Theil's U Statistic

The forecast that matters is the one that beats a naive guess.

Property	Value
Category	Error Metric
Inputs	Actual vs Predicted (dual input)
Parameters	period
Outputs	Single series (TheilU)
Output range	\geq 0
Warmup	period bars
PineScript	theilu.pine

• Theil's U Statistic measures forecast accuracy relative to a naive no-change forecast.
• Similar: MASE, Rsquared | Trading note: Theil's U statistic; <1 = forecast beats naïve, >1 = worse than naïve random walk.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

Theil's U Statistic measures forecast accuracy relative to a naive no-change forecast. A value below 1 indicates the model outperforms simply predicting that tomorrow equals today; above 1 means you'd be better off not forecasting at all.

Historical Context

Developed by Dutch econometrician Henri Theil in the 1960s, Theil's U was designed to evaluate economic forecasts against the simplest possible benchmark: the assumption of no change. This was revolutionary because many sophisticated models fail to beat this naive approach, especially in financial markets.

Architecture & Physics

Theil's U computes two parallel error metrics: one for the forecast and one for a naive prediction. The ratio reveals whether the forecasting effort adds value. A forecast might have low absolute error but still be worse than doing nothing.

Properties

• Relative benchmark: Compares against naive no-change forecast
• Scale-independent: Ratio is unitless
• Interpretable threshold: U = 1 is the break-even point
• Range: 0 to ∞, with 0 being perfect and > 1 being worse than naive

Mathematical Foundation

1. Forecast Error

Calculate squared errors for the actual forecast:

FPE = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2

Where:

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

2. Naive Error

Calculate squared errors for naive prediction (previous actual):

NPE = \sum_{i=1}^{n} (y_i - y_{i-1})^2

3. Theil's U Calculation

Take the ratio of forecast to naive:

U = \sqrt{\frac{FPE}{NPE}} = \sqrt{\frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i - y_{i-1})^2}}

4. Running Update (O(1))

QuanTAlib maintains running sums of both squared error terms:

S_{f,new} = S_{f,old} - e_{f,oldest}^2 + e_{f,newest}^2 S_{n,new} = S_{n,old} - e_{n,oldest}^2 + e_{n,newest}^2 U = \sqrt{\frac{S_{f,new}}{S_{n,new}}}

Implementation Details

Usage Patterns

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

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

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

Parameters

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

Properties

Property	Type	Description
Last	TValue	Most recent Theil's U value
IsHot	bool	True when buffer is full
Name	string	Indicator name (e.g., "TheilU(20)")
WarmupPeriod	int	Number of periods before valid output

Performance Profile

Operation Count (Streaming Mode)

Theil's U statistic: U = sqrt(MSE_forecast) / sqrt(MSE_naive). Requires two running mean-squared-error accumulators.

Operation	Count	Cost (cycles)	Subtotal
Forecast MSE update (e^2 + EMA)	2	~5 cy	~10 cy
Naive MSE update (naive_e^2 + EMA)	2	~5 cy	~10 cy
U = sqrt(MSE_f) / sqrt(MSE_n)	2	~15 cy	~30 cy
Total	~6	—	~50 cycles

O(1) per bar. Two parallel EMA accumulators + ratio with sqrt. ~50 cycles/bar.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
Squared error accumulation	Yes	Element-wise squares + reduction
sqrt ratio	No	Single scalar at end

Batch MSE accumulation vectorizable; final ratio is scalar. ~8 cy/bar for squared-error accumulation.

Metric	Score	Notes
Throughput	~15 ns/bar	O(1) update complexity
Allocations	0	Uses pre-allocated ring buffers
Complexity	O(1)	Constant time per update
Accuracy	10/10	Exact calculation
Timeliness	9/10	No lag beyond the period
Interpretability	10/10	Clear benchmark comparison

Interpretation

Theil's U	Interpretation
0	Perfect prediction
< 0.5	Excellent (error < 50% of naive)
0.5 - 0.8	Good forecasting skill
0.8 - 1.0	Marginal improvement over naive
= 1.0	Equal to naive forecast
> 1.0	Worse than naive (model adds noise)

Why Use Theil's U?

Scenario	Low MAE but High U	High MAE but Low U
Meaning	Series is easy to predict	Model adds value despite errors
Example	Stable prices, any model works	Volatile prices, model captures moves
Recommendation	Use simpler model	Keep using the model

Common Use Cases

1. Economic Forecasting: Evaluate macro predictions against random walk
2. Financial Markets: Test trading signals against buy-and-hold
3. Model Selection: Choose models that beat naive benchmarks
4. Forecast Validation: Ensure forecasting effort is worthwhile

Edge Cases

• Zero Naive Error: Returns infinity when series is perfectly flat (naive is perfect)
• NaN Handling: Uses last valid value substitution
• Single Input: Not supported (requires two series)
• Period = 1: Returns 0 (insufficient data for naive comparison)
• First Value: Needs at least 2 values for naive benchmark

Related Indicators

• RMSE - Root Mean Squared Error (absolute, not relative)
• MASE - Mean Absolute Scaled Error (similar concept)
• R-Squared - Coefficient of Determination