Linear modeling and the coherence function

In this paper, a tutorial introduction to the magnitude squared (MS) coherence function, and its relation to linear modeling, is interleaved with new methods for characterizing bidirectional and unidirectional linear association between two time series. A new method for characterizing bidirectional linear association is based on the new concept of an MS coherence sequence which is the inverse z-transform of the MS coherence function. The MS coherence function is approximated as a single real-valued ratio of polynominals in z. The ratio is determined by a matrix equation whose entries are convolutions of correlation functions. Solutions for the corresponding MS coherence parameters employ an eigenvalue-eigenvector decomposition. The algorithm is compared to a traditional periodogram-based method using computer simulations. The computer simulations also demonstrate the effects of parameter overdetermination and overordering. The sensitivity of the matrix equations for the computer simulations is computed via the numerical linear algebra concept of condition number.