A nonlinear least-squares technique for analyzing probability of detection data that uses all the data without grouping
William Julius Richter · The Journal of the Acoustical Society of America · 1991
The purpose of a probability of detection (PD) experiment is to quantify the ability of a system to detect signals embedded in noise. The first step in the classical technique for analyzing data from such an experiment is to least-squares fit the data to a Gaussian probability integral using Gaussian probability paper or probits such that the Gaussian probability integral plots as a straight line. Because of the ease with which data can be fitted to a straight line, the classical technique does not require a modern digital computer. The classical technique has been effective over the years in many applications, however its shortcomings are (1) groups of data points must exist at several signal to noise ratios (SNRs) with about the same number of data points at each SNR and (2) groups with SNRs where detections occur 100 or 0% cannot be used. Data must be regrouped or thrown out to meet these criteria. The nonlinear least-squares (NLLS) technique described in this paper avoids these problems by least squares fitting the data to a Gaussian probability integral directly. This is now possible with modern digital computers. NLLS allows each data point to be used independently as a sample probability of detection of 1 or 0. This avoids the need for data to be grouped at all and permits using all the data. Results are compared with the classical technique, and favorable agreement is attained where all the conditions of the classical technique apply. Examples where all these conditions are not present are also provided, and the advantage of the new technique is demonstrated. The analysis is performed with a program written in basic and run on a home computer. In applying the new technique to the examples, it was noted that errors in quantifying the ability of a system to detect signals can be large. Further analysis showed that the size of the errors depended on the location of the SNRs, and an iterative method is demonstrated for selecting SNRs such that reasonable accuracy is obtained with a small number (50) of data samples.