Abstract This paper proposes a fuzzy C-medoids-based clustering method with entropy regularization to solve the issue of grouping complex data as interval-valued time series. The dual nature of the data, that are both time-varying and interval-valued, needs to be considered and embedded into clustering techniques. In this work, a new dissimilarity measure, based on Dynamic Time Warping, is proposed. The performance of the new clustering procedure is evaluated through a simulation study and an application to financial time series.

Quite often real data exhibit non-normal features, such as asymmetry and heavy tails, and present a latent group structure. In this paper, we first propose the multivariate skew shifted exponential normal distribution that can account for these non-normal characteristics. Then, we use this distribution in a finite mixture modeling framework. An EM algorithm is illustrated for maximum-likelihood parameter estimation. We provide a simulation study that compares the fitting performance of our model with those of several alternative models. The comparison is also conducted on a real dataset concerning the log returns of four cryptocurrencies.

Abstract We consider the problem of inferring an unknown number of clusters in multinomial count data, by estimating finite mixtures of multinomial distributions with or without covariates. Both Maximum Likelihood (ML) as well as Bayesian estimation are taken into account. Under a Maximum Likelihood approach, we provide an Expectation–Maximization (EM) algorithm which exploits a careful initialization procedure combined with a ridge-stabilized implementation of the Newton-Raphson method in the M-step. Under a Bayesian setup, a stochastic gradient Markov chain Monte Carlo (MCMC) algorithm embedded within a prior parallel tempering scheme is devised. The number of clusters is selected according to the Integrated Completed Likelihood criterion in the ML approach and estimating the number of non-empty components in overfitting mixture models in the Bayesian case. Our method is illustrated in simulated data and applied to two real datasets. The proposed methods are implemented in a contributed package, available online.

This paper concerns the measurement and testing of equality of variability of functional data. We apply the distance standard deviation constructed based on distance correlation, which was recently introduced as a measure of spread. For functional data, the distance standard deviation seems to measure different kinds of variability, not only scale differences. Moreover, the distance standard deviation is just one real number, and for this reason, it is of more practical value than the covariance function, which is a more difficult object to interpret. For testing equality of variability in two groups, we propose a permutation method based on centered observations, which controls the type I error level much better than the standard permutation method. We also consider the applicability of other correlations to measure the variability of functional data. The finite sample properties of two-sample tests are investigated in extensive simulation studies. We also illustrate their use in five real data examples based on various data sets.

Symbolic data analysis (SDA) is an emerging area of statistics concerned with understanding and modelling data that takes distributional form (i.e. symbols), such as random lists, intervals and histograms. It was developed under the premise that the statistical unit of interest is the symbol, and that inference is required at this level. Here we consider a different perspective, which opens a new research direction in the field of SDA. We assume that, as with a standard statistical analysis, inference is required at the level of individual-level data. However, the individual-level data are unobserved, and are aggregated into observed symbols—group-based distributional-valued summaries—prior to the analysis. We introduce a novel general method for constructing likelihood functions for symbolic data based on a desired probability model for the underlying measurement-level data, while only observing the distributional summaries. This approach opens the door for new classes of symbol design and construction, in addition to developing SDA as a viable tool to enable and improve upon classical data analyses, particularly for very large and complex datasets. We illustrate this new direction for SDA research through several real and simulated data analyses, including a study of novel classes of multivariate symbol construction techniques.