Laplace Correction of Confusion Matrices to Produce Statistically Representative Confidence Intervals
During diagnostic algorithm development engine testing with implanted faults may be performed. The number of implanted faults is never large enough to truly capture the distribution in the confusion matrix. Misdiagnoses in particular are unlikely to be correctly represented. Misdiagnosis that could result in costly outcomes are frequently not captured in an implantation study, resulting in a deceptively reassuring zero value for the probability of it occurring. The Laplace correction can be applied to each element of the confusion matrix to improve the generated confidence interval. This also allows a confidence interval to be produced for zero value elements. Unfortunately, the choice of Laplace correction factor influences the size of the confidence interval, and without knowing the true distribution the best correction factor cannot be determined. The choice of correction factor depends on element probability, total sample size, number of faults and confidence level. The effect of the Laplace correction on the element probability is analytically examined to provide insight into the relative influence of the correction. This is followed by an examination of the influence of the element probability, total sample size, number of faults and confidence level on the required Laplace correction. This is achieved by sampling from known populations. A method of generating good confidence intervals on each element is proposed. This includes the production of a Laplace correction based on the sample size, number of faults and confidence level. This will allow consistent comparisons of Laplace corrected matrices rather than leaving the correction factor to each individual’s best engineering judgment.