Ca Confidence Intervals

The BCa confidence intervals for fixed values of SNR were obtained by a bootstrapping method in which images are the sampling units. Suppose that there are B bootstrap samples. We took B = 2000. Each bootstrap sample was generated by sampling randomly (with replacement) 450 times from our set of 450 (subjective score, SNR) pairs. Huge computational savings can be realized in that, for the same set of images being sampled, one bootstrap sample can be used simultaneously for different SNRs. For a fixed SNR, E is the fitted expected subjective score based on the model as computed for the original data, and E*(b) is the value computed for the bth bootstrap sample. The 100(1 — 2a)% BCa confidence interval will be of the form 

where E*(a) is the 100ath percentile of the bootstrap distribution of E; a1 and a2 are defined by

O is the standard normal cumulative distribution function, and z(^ is the 100;6th percentile (so, for example, z(0 95) = 1.645). The "bias correction'' z0 and "acceleration constant'' a remain to be defined. E*(b) is the value of E for the bth bootstrap sample, and E is the value computed for the original data. Then where O—1 is the Gaussian quantile function (so, for example, O—1(0.95) = 1.645). Suppose that there are n images in all, and let E(j) be the computed value of E when the ith image is deleted (so the computation is done on n — 1 images). Let

Then

= En=i[E(.) — E(i)]3 fl^6{En=1[E(.) — E(,-)]2}3/2'

There are two main differences between the BCa confidence intervals described here and the Scheffe confidence intervals described in the previous chapter. The Scheffe method produces a simultaneous confidence interval, that is, one that provides upper and lower limits for the entire curve at once. The BCa method supplies pointwise intervals, valid for specific points along the x-axis. It is not currently known how to extend the BCa method to simultaneous intervals. The second difference is that the Scheffe intervals are always symmetric about the curve, regardless of whether there are any constraints on the range of the variables. Sensitivity and PVP, for example, have a maximum value of 1. The expected value of sensitivity at a particular bit rate may be very close to that upper limit, and when obtaining a Scheffe confidence interval for that curve, the confidence interval may exceed 1, since it is necessarily symmetric about the curve. In that case, the upper confidence curve must be thresholded at 1. The BCa method has the advantage of providing intervals that are not necessarily symmetric, but respect the fact that the values of the response variable lie within a small constrained range.