Abstract Integration against, and hence sampling from, high-dimensional probability distributions is of essential importance in many application areas and has been an active research area for decades. One approach that has drawn increasing attention in recent years has been the generation of samples from a target distribution $$\mathbb {P}_{\text {tar} }$$ P tar using transport maps: if $$\mathbb {P}_{\text {tar} } = T_\sharp \mathbb {P}_{\text {ref} }$$ P tar = T ♯ P ref is the pushforward of an easily-sampled probability distribution $$\mathbb {P}_{\text {ref} }$$ P ref under the transport map T , then the application of T to $$\mathbb {P}_{\text {ref} }$$ P ref -distributed samples yields $$\mathbb {P}_{\text {tar} }$$ P tar -distributed samples. This paper proposes the application of transport maps not just to random samples, but also to quasi-Monte Carlo points, higher-order nets, and sparse grids so that the transformed samples inherit the original convergence rates that are often better than $$N^{-1/2}$$ N - 1 / 2 , N being the number of samples/quadrature nodes. Our main result is the derivation of an explicit transport map for the case that $$\mathbb {P}_{\text {tar} }$$ P tar is a mixture of simple distributions, e.g. a Gaussian mixture, in which case application of the transport map T requires the solution of an explicit ODE with closed-form right-hand side. Mixture distributions are of particular applicability and interest since many methods proceed by first approximating $$\mathbb {P}_{\text {tar} }$$ P tar by a mixture and then sampling from that mixture (often using importance reweighting). Hence, this paper allows for the sampling step to provide a better convergence rate than $$N^{-1/2}$$ N - 1 / 2 for all such methods.

Abstract Outlier detection is an important data mining tool that becomes particularly challenging when dealing with nominal data. First and foremost, flagging observations as outlying requires a well-defined notion of nominal outlyingness. This paper presents a definition of nominal outlyingness and introduces a general framework for quantifying outlyingness of nominal data. The proposed framework makes use of ideas from the association rule mining literature and can be used for calculating scores that indicate how outlying a nominal observation is. Methods for determining the involved hyperparameter values are presented and the concepts of variable contributions and outlyingness depth are introduced, in an attempt to enhance interpretability of the results. The proposed framework is evaluated on both synthetic and publicly available data sets, demonstrating comparable performance to state-of-the-art frequent pattern mining algorithms and even outperforming them in certain cases. The ideas presented can serve as a tool for assessing the degree to which an observation differs from the rest of the data, under the assumption of sequences of nominal levels having been generated from a Multinomial distribution with varying event probabilities.