Efficient computation of the discrete autocorrelation wavelet inner product matrix

Discrete autocorrelation (a.c.) wavelets have recently been applied in the statistical analysis of locally stationary time series for local spectral modelling and estimation. This article proposes a fast recursive construction of the inner product matrix of discrete a.c. wavelets which is required by the statistical analysis. The recursion connects neighbouring elements on diagonals of the inner product matrix using a two-scale property of the a.c. wavelets. The recursive method is an (log (N)³) operation which compares favourably with the (N log N) operations required by the brute force approach. We conclude by describing an efficient construction of the inner product matrix in the (separable) two-dimensional case.