The Statistical Detection of Signals in Noisy Data

This chapter applies probability theory and statistics to data analysis and the statistical detection of signals. The characterisation of noise and the matched filter are discussed. Binary hypothesis testing starting from the one-dimensional case, then the two-dimensional case which bears most of the features of the multidimensional case, is discussed in detail. The composite hypothesis is described in a differential geometric framework. Then, the maximum likelihood method and the Rao-Fisher information matrix are discussed. Finally, various \(\chi ^2\) noise discriminators in the general framework of a generic \(\chi ^2\) are described. Here, the underlying mathematical structure of the \(\chi ^2\) , namely that of vector bundles, is made apparent.