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Nonnegative matrix factorization can be used to automatically detect topics within a corpus in an unsupervised fashion. The technique amounts to an approximation of a nonnegative matrix as the product of two nonnegative matrices of lower rank. In certain applications it is desirable to extract topics and use them to predict quantitative outcomes. In this paper, we show Nonnegative Matrix Factorization can be combined with regression on a continuous response variable by minimizing a penalty function that adds a weighted regression error to a matrix factorization error. We show theoretically that as the weighting increases, the regression error in training decreases weakly. We test our method on synthetic data and real data coming from Rate My Professors reviews to predict an instructor's rating from the text in their reviews. In practice, when used as a dimensionality reduction method (when the number of topics chosen in the model is fewer than the true number of topics), the method performs better than doing regression after topics are identified—both during training and testing—and it retrains interpretability. [ABSTRACT FROM AUTHOR]

We present an unsupervised method to detect anomalous time series among a collection of time series. To do so, we extend traditional Kernel Density Estimation for estimating probability distributions in Euclidean space to Hilbert spaces. The estimated probability densities we derive can be obtained formally through treating each series as a point in a Hilbert space, placing a kernel at those points, and summing the kernels (a "point approach"), or through using Kernel Density Estimation to approximate the distributions of Fourier mode coefficients to infer a probability density (a "Fourier approach"). We refer to these approaches as Functional Kernel Density Estimation for Anomaly Detection as they both yield functionals that can score a time series for how anomalous it is. Both methods naturally handle missing data and apply to a variety of settings, performing well when compared with an outlyingness score derived from a boxplot method for functional data, with a Principal Component Analysis approach for functional data, and with the Functional Isolation Forest method. We illustrate the use of the proposed methods with aviation safety report data from the International Air Transport Association (IATA). [ABSTRACT FROM AUTHOR]