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Variance estimators for the Lempel-Ziv entropy rate estimator.
José M Amigó1, Matthew B Kennel
1Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain.
Chaos (Woodbury, N.Y.)
|January 4, 2007
Summary
We developed a simple way to estimate the error for the Lempel-Ziv entropy rate, a key measure for symbolic time series. This method provides a reliable "error bar" for the entropy rate estimate.
Area of Science:
- Information Theory
- Time Series Analysis
- Statistical Inference
Background:
- The Lempel-Ziv algorithm (1976) offers a parameter-free method for estimating entropy rate in symbolic time series.
- Quantifying the uncertainty of this entropy rate estimate is crucial for reliable analysis.
Purpose of the Study:
- To derive an analytical variance estimate for the Lempel-Ziv entropy rate estimator.
- To provide a computable method for generating "error bars" for entropy rate estimations.
- To compare the derived variance estimate with a time-series-based bootstrap method.
Main Methods:
- Derivation of an analytical variance estimate for the Lempel-Ziv entropy rate.
- Computation of the variance estimate directly from observed symbolic time series data.
- Comparison with a time-series bootstrap procedure for variance estimation.
Main Results:
- An easily computable analytical variance estimate for the Lempel-Ziv entropy rate was successfully derived.
- The derived method requires minimal additional computational effort beyond the entropy rate calculation.
- The analytical estimate provides a justified quantification of expected fluctuations in the entropy rate estimate.
Conclusions:
- The derived analytical variance estimate offers a practical and efficient way to quantify uncertainty in Lempel-Ziv entropy rate estimations.
- This method provides a reliable "error bar" for symbolic time series analysis.
- The findings support the use of this analytical approach for assessing the reliability of entropy rate estimates.
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