Optimizing the soft independent modeling of class analogy (SIMCA) using statistical prediction regions

The ultimate goal of a one-class classifier like the "rigorous" soft independent modeling of class analogy (SIMCA) is to predict with a certain confidence probability, the conformity of future objects with a given reference class. However, the SIMCA model, as currently implemented often suffers from an undercoverage problem, meaning that its observed sensitivity often falls far below the desired theoretical confidence probability, hence undermining its intended use as a predictive tool. To overcome the issue, the most reported strategy in the literature, involves incrementing the nominal confidence probability until the desired sensitivity is obtained in cross-validation. This article proposes a statistical prediction interval-based strategy as an alternative strategy to properly overcome this undercoverage issue. The strategy uses the concept of predictive distributions sensu stricto to construct statistical prediction regions for the metrics. Firstly, a procedure based on goodness-of-fit criteria is used to select the best-fitting family of probability models for each metric or its monotonic transformation, among several plausible candidate families of right-skewed probability distributions for positive random variables, including the gamma and the lognormal families. Secondly, assuming the best-fitting distribution, a generalized linear model is fitted to each metric data using the Bayesian method. This method enables to conveniently estimate uncertainties about the parameters of the selected distribution. Propagating these uncertainties to the best-fitting probability model of the metric enables to derive its so-called posterior predictive distribution, which is then used to set its critical limit. Overall, the evaluation of the proposed approach on a diversity of real datasets shows that it yields unbiased and more accurate sensitivities than existing methods which are not based on predictive densities. It can even yield better specificities than the strategy that attempts to improve sensitivities of existing methods by "optimizing" the type 1 error, especially in low sample sizes' contexts.