Diagonally optimized spread: an optimized spread for quantifying local stationarity

In previous work, the spread has been presented as a means to quantify stationarity. This is done by estimating the support of the joint time-frequency correlation function known as the expected ambiguity function. Two fundamental issues concerning the spread are addressed here. The first is that the spread is not invariant under basis transformation. We address this problem by introducing the diagonally optimized spread, based on the proposition that the spread should be calculated using the covariance that is most nearly diagonal under basis transformation. The second issue is that in previous references to spread, the availability of covariance estimates have been assumed, which is an open problem non-stationary processes. A method to provide estimates of locally stationary processes was proposed by Mallat, Papanicolaou and Zhang. In their work they derive a method which calculates the basis which most nearly diagonalize the covariance matrix in the mean square sense. This method is ideally suited to our situation, and we extend it to include calculation of the diagonally optimized spread. The optimally diagonalized spread provides an improved indicator of non-stationarity and illustrates the connections between spread and the diagonizability of the covariance of a random process.