On Entropy of Probability Integral Transformed Time Series

The goal of this paper is to investigate the changes of entropy estimates when the amplitude distribution of the time series is equalized using the probability integral transformation. The data we analyzed were with known properties—pseudo-random signals with known distributions, mutually coupled using statistical or deterministic methods that include generators of statistically dependent distributions, linear and non-linear transforms, and deterministic chaos. The signal pairs were coupled using a correlation coefficient ranging from zero to one. The dependence of the signal samples is achieved by moving average filter and non-linear equations. The applied coupling methods are checked using statistical tests for correlation. The changes in signal regularity are checked by a multifractal spectrum. The probability integral transformation is then applied to cardiovascular time series—systolic blood pressure and pulse interval—acquired from the laboratory animals and represented the results of entropy estimations. We derived an expression for the reference value of entropy in the probability integral transformed signals. We also experimentally evaluated the reliability of entropy estimates concerning the matching probabilities.

Density Forecast Evaluations via a Simulation-Based Dynamic Probability Integral Transformation*

This paper presents simulation-based density forecast evaluation methods using particle filters. The simulation-based dynamic probability integral transformation or log-likelihood evaluation method is combined with the existing density forecast evaluation methods. This methodology is applicable to various density forecast models, such as log stochastic volatility models and affine jump diffusion (AJD) models, for which the probability integral transform or likelihood computation is difficult due to the presence of latent stochastic volatilities. This methodology is applied to the daily S&P 500 stock index. The empirical results show that the AJD models with jumps perform the best for out-of-sample evaluations.