Efficient computation of the inner product matrix of discrete autocorrelation wavelets

Discrete autocorrelation wavelets have recently been applied in the statistical analysis of locally stationary time series. This form of time series analysis also requires the construction of the inner product matrix of such wavelets. To date, both discrete autocorrelation wavelets and the inner product matrix have been constructed via a brute force technique which proves to be computationally expensive. We propose a recursive approach for the construction of discrete autocorrelation wavelets. This is an O(2J) operation, which compares favourably with the O(22J ) operations required by the brute-force approach. An alternative method of constructing the inner product matrix is also proposed. This recursive construction relates neighbouring elements of the inner product matrix, utilising the recursive structure which connects discrete autocorrelation wavelets at varying scales. We conclude by suggesting a construction for the inner product matrix of (separable) two-dimensional autocorrelation wavelets, together with an outline of how this matrix may be constructed efficiently.