Finding Patterns in Signals

Abstract

This chapter describes different aspects of finding patterns in signals. It starts out with the description of cross correlation, where one looks at the occurrence of a shorter feature in a longer signal. Two special cases are investigated in more detail. First, when the two signals have the same length one can ask how strong the linear relationship is between the two variables. This comparison of signals requires a normalization to eliminate trivial artifacts, and leads to the definition of the correlation coefficient. For multivariate data, the generaliztion from the correlation coefficient to the correlation matrix is explained. An intuitive interpretation of the correlation coefficient is obtained by looking how it is related to the best line fit to the two variables. In that case the square of the correlation coefficient, the coefficient of determination, quantifies which part of the signal change in one variable is explained by the corresponding change in the other variable. The second special case is obtained by comparing one signal to shifted versions of itself. This is called autocorrelation, and can be used to find hidden systematic patterns in signals. The final section of this chapter shows how the autocorrelation is used in time-series analysis (TSA), to obtain the maximum amount of information from a time series of data.