Apache Commons Mathematics Library
Commons Math is a library of lightweight, self-contained mathematics and statistics components addressing the most common problems not available in the Java programming language or Commons Lang. …

Model Explanation System (MES)
We propose a general model explanation system (MES) for “explaining” the output of black box classifiers. In this introduction we use the motivating example of a classifier trained to detect fraud in a credit card transaction history. The key aspect is that we provide explanations applicable to a single prediction, rather than provide an interpretable set of parameters. The labels in the provided examples are usually negative. Hence, we focus on explaining positive predictions (alerts). In many classification applications, but especially in fraud detection, there is an expectation of false positives. Alerts are given to a human analyst before any further action is taken. Analysts often insist on understanding “why” there was an alert, since an opaque alert makes it difficult for them to proceed. Analogous scenarios occur in computer vision , credit risk , spam detection , etc. Furthermore, the MES framework is useful for model criticism. In the world of generative models, practitioners often generate synthetic data from a trained model to get an idea of “what the model is doing”. Our MES framework augments such tools. As an added benefit, MES is applicable to completely non-probabilistic black boxes that only provide hard labels. In Section 3 we use MES to visualize the decisions of a face recognition system. …

QR Decomposition
In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm. If A has n linearly independent columns, then the first n columns of Q form an orthonormal basis for the column space of A. More generally, the first k columns of Q form an orthonormal basis for the span of the first k columns of A for any 1 ≤ k ≤ n. The fact that any column k of A only depends on the first k columns of Q is responsible for the triangular form of R. …