A Preliminary Test for Nonlinear Structure

We present a preliminary test for nonlinear structure in large data sets. This procedure consists of transforming the data to remove the correlations, then discretizing the data and finally, studying the cell counts in the resulting contingency table. Formal tests can be performed using the usual chi-squared test statistic. A simple Poisson approximation is useful for informal examination of the distribution of the cell counts. We derive the limiting joint distribution of the cell counts and the limiting distribution of the chi-squared statistic. We also present simulation results to check the accuracy of the limiting distribution of the chi-squared statistic, and then present several examples using both simulated and real data sets. Our procedure can detect a variety of different types of structure. In our examples we examine: data with clustering, digitized speech data, and data consisting of the output from a faulty random number generator. The chi-squared statistic can be used in a..