Multiscale Vehicular Expected Crashes Estimation with the Unnormalized Haar Wavelet Transform and Poisson’s Unbiased Risk Estimate

Abstract

In this paper, we present a multiscale approach to estimate the expected number of traffic crashes, (referred to here as expected crashes) from observed crash frequency data using the unnormalized Haar wavelet transform. The Haar transform decomposes crash count data into sums (scaling coefficients) and differences (wavelet coefficients) of Poisson distributed crash counts, which in turn are Poisson- and Skellam-distributed, respectively. This process compresses the expected crashes information into a few large wavelet detail coefficients; the random fluctuations of crash counts are relatively smaller and (evenly) spread over all wavelet detail coefficients. This allows us to effectively suppress the crashes’ random fluctuations by shrinking or thresholding the wavelet detail coefficients (i.e., setting their values below a specific threshold to zero). The appropriate amount of shrinking or thresholding is determined using Poisson’s unbiased risk estimate (PURE), which essentially minimizes the mean square error risk between the estimated expected crashes and the true unknown expected crashes. The approach can also be viewed from the point of view of nonparametric spatial clustering and smoothing, where suppression of wavelet coefficients results in smoothing of crash counts. However, in the proposed approach, the amount of smoothing is adaptive to the features at the different locations and scales. In areas where expected crashes are relatively uniform, the approach performs significant smoothing; in contrast, little smoothing is performed in areas where expected crashes vary significantly. We illustrate the method on simulated Poisson data as well as observed crash counts on an Interstate stretch in Virginia and compare it with the more traditional empirical Bayes approach. Results show good performance of the proposed approach.