Generalized Prediction Intervals for Arbitrary Distributed High-Dimensional Data

This paper generalizes the traditional statistical concept of prediction intervals for arbitrary probability density functions in high-dimensional feature spaces by introducing significance level distributions, which provides interval-independent probabilities for continuous random variables. The advantage of the transformation of a probability density function into a significance level distribution is that it enables one-class classification or outlier detection in a direct manner.

💡 Research Summary

The paper tackles the long‑standing problem of constructing prediction intervals for data that live in high‑dimensional feature spaces and follow arbitrary probability density functions (PDFs). Traditional prediction intervals rely on the cumulative distribution function (CDF) of a univariate or low‑dimensional variable and define an interval that contains a prescribed proportion (e.g., 95 %) of the probability mass. When the dimensionality grows, two fundamental issues arise: (1) the probability mass becomes extremely sparse, making interval boundaries hard to estimate, and (2) the shape of equal‑density surfaces (level sets) can be highly irregular, so a simple rectangular or ellipsoidal interval is no longer appropriate.

To overcome these obstacles, the authors introduce the concept of a significance level distribution. The key insight is to treat the density value itself as a random variable. For a random vector X with density f(x), they define Y = f(X). The cumulative distribution of Y, denoted G(y) = P