Clustering of change patterns using Fourier coefficients

MOTIVATION To understand the behavior of genes, it is important to explore how the patterns of gene expression change over a time period because biologically related gene groups can share the same change patterns. Many clustering algorithms have been proposed to group observation data. However, because of the complexity of the underlying functions there have not been many studies on grouping data based on change patterns. In this study, the problem of finding similar change patterns is induced to clustering with the derivative Fourier coefficients. The sample Fourier coefficients not only provide information about the underlying functions, but also reduce the dimension. In addition, as their limiting distribution is a multivariate normal, a model-based clustering method incorporating statistical properties would be appropriate. RESULTS This work is aimed at discovering gene groups with similar change patterns that share similar biological properties. We developed a statistical model using derivative Fourier coefficients to identify similar change patterns of gene expression. We used a model-based method to cluster the Fourier series estimation of derivatives. The model-based method is advantageous over other methods in our proposed model because the sample Fourier coefficients asymptotically follow the multivariate normal distribution. Change patterns are automatically estimated with the Fourier representation in our model. Our model was tested in simulations and on real gene data sets. The simulation results showed that the model-based clustering method with the sample Fourier coefficients has a lower clustering error rate than K-means clustering. Even when the number of repeated time points was small, the same results were obtained. We also applied our model to cluster change patterns of yeast cell cycle microarray expression data with alpha-factor synchronization. It showed that, as the method clusters with the probability-neighboring data, the model-based clustering with our proposed model yielded biologically interpretable results. We expect that our proposed Fourier analysis with suitably chosen smoothing parameters could serve as a useful tool in classifying genes and interpreting possible biological change patterns. AVAILABILITY The R program is available upon the request.