11. Correntropy for Random Processes: Properties and Applications in Signal Processing

The previous chapter defined cross-correntropy for the case of a pair of scalar random variables, and presented applications in statistical inference. This chapter extends the definition of correntropy for the case of random (or stochastic) processes, which are index sets of random variables. In statistical signal processing the index set is time; we are interested in random variables that are a function of time and the goal is to quantify their statistical dependencies (although the index set can also be defined over inputs or channels of multivariate random variables). The autocorrelation function, which measures the statistical dependency between random variables at two different times, is conventionally utilized for this goal. Hence, we generalize the definition of autocorrelation to an autocorrentropy function. The name correntropywas coined to reflect the fact that the function “looks like” correlation but the sum over the lags (or over dimensions of the multivariate random variable) is the information potential (i.e., the argument of Renyi’s quadratic entropy). The definition of cross-correntropy for random variables carries over to time series with a minor but important change in the domain of the variables that now are an index set of lags. When it is clear from the context, we simplify the terminology and refer to the different functions (autocorrentropy, or crosscorrentropy) simply as correntropy function, but keep the word “function” to distinguish them from Chapter 10 quantities.