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Physical activity (PA) is an important risk factor for many health outcomes. Wearable-devices such as accelerometers are increasingly used in biomedical studies to understand the associations between PA and health outcomes. Statistical analyses involving accelerometer data are challenging due to the following three characteristics: (i) high-dimensionality, (ii) temporal dependence, and (iii) measurement error. To address these challenges we treat accelerometer-based measures of physical activity as a single function-valued covariate prone to measurement error. Specifically, in order to determine the relationship between PA and a health outcome of interest, we propose a regression model with a functional covariate that accounts for measurement error. Using regression calibration, we develop a two-step estimation method for the model parameters and establish their consistency. A test is also proposed to test the significance of the estimated model parameters. Simulation studies are conducted to compare the proposed methods with existing alternative approaches under varying scenarios. Finally, the developed methods are used to assess the relationship between PA intensity and BMI obtained from the National Health and Nutrition Examination Survey data.

A phase-type distribution is the distribution of the time until absorption in a finite state-space time-homogeneous Markov jump process, with one absorbing state and the rest being transient. These distributions are mathematically tractable and conceptually attractive to model physical phenomena due to their interpretation in terms of a hidden Markov structure. Three recent extensions of regular phase-type distributions give rise to models which allow for heavy tails: discrete- or continuous-scaling; fractional-time semi-Markov extensions; and inhomogeneous time-change of the underlying Markov process. In this paper, we present a unifying theory for heavy-tailed phase-type distributions for which all three approaches are particular cases. Our main objective is to provide useful models for heavy-tailed phase-type distributions, but any other tail behavior is also captured by our specification. We provide relevant new examples and also show how existing approaches are naturally embedded. Subsequently, two multivariate extensions are presented, inspired by the univariate construction which can be considered as a matrix version of a frailty model. We provide fully explicit EM-algorithms for all models and illustrate them using synthetic and real-life data.

Pearson's chi-squared test is widely used to test the goodness of fit between categorical data and a given discrete distribution function. When the number of sets of the categorical data, say $k$, is a fixed integer, Pearson's chi-squared test statistic converges in distribution to a chi-squared distribution with $k-1$ degrees of freedom when the sample size $n$ goes to infinity. In real applications, the number $k$ often changes with $n$ and may be even much larger than $n$. By using the martingale techniques, we prove that Pearson's chi-squared test statistic converges to the normal under quite general conditions. We also propose a new test statistic which is more powerful than chi-squared test statistic based on our simulation study. A real application to lottery data is provided to illustrate our methodology.

We focus on controllable disentangled representation learning (C-Dis-RL), where users can control the partition of the disentangled latent space to factorize dataset attributes (concepts) for downstream tasks. Two general problems remain under-explored in current methods: (1) They lack comprehensive disentanglement constraints, especially missing the minimization of mutual information between different attributes across latent and observation domains. (2) They lack convexity constraints in disentangled latent space, which is important for meaningfully manipulating specific attributes for downstream tasks. To encourage both comprehensive C-Dis-RL and convexity simultaneously, we propose a simple yet efficient method: Controllable Interpolation Regularization (CIR), which creates a positive loop where the disentanglement and convexity can help each other. Specifically, we conduct controlled interpolation in latent space during training and 'reuse' the encoder to help form a 'perfect disentanglement' regularization. In that case, (a) disentanglement loss implicitly enlarges the potential 'understandable' distribution to encourage convexity; (b) convexity can in turn improve robust and precise disentanglement. CIR is a general module and we merge CIR with three different algorithms: ELEGANT, I2I-Dis, and GZS-Net to show the compatibility and effectiveness. Qualitative and quantitative experiments show improvement in C-Dis-RL and latent convexity by CIR. This further improves downstream tasks: controllable image synthesis, cross-modality image translation and zero-shot synthesis. More experiments demonstrate CIR can also improve other downstream tasks, such as new attribute value mining, data augmentation, and eliminating bias for fairness.

Compact data representations are one approach for improving generalization of learned functions. We explicitly illustrate the relationship between entropy and cardinality, both measures of compactness, including how gradient descent on the former reduces the latter. Whereas entropy is distribution sensitive, cardinality is not. We propose a third compactness measure that is a compromise between the two: expected cardinality, or the expected number of unique states in any finite number of draws, which is more meaningful than standard cardinality as it discounts states with negligible probability mass. We show that minimizing entropy also minimizes expected cardinality.

This article deals with the hypothesis test for the extremely heavy-tailed distributions with infinite mean or variance by using a truncated sample mean. We obtain three necessary and sufficient conditions under which the asymptotic distribution of the truncated test statistics converges to normal, neither normal nor stable or converges to $-\infty$ or the combination of stable distributions, respectively. The numerical simulation illustrates an application of the theoretical results above in the hypothesis testing.

Several non-linear functions and machine learning methods have been developed for flexible specification of the systematic utility in discrete choice models. However, they lack interpretability, do not ensure monotonicity conditions, and restrict substitution patterns. We address the first two challenges by modelling the systematic utility using the Choquet Integral (CI) function and the last one by embedding CI into the multinomial probit (MNP) choice probability kernel. We also extend the MNP-CI model to account for attribute cut-offs that enable a modeller to approximately mimic the semi-compensatory behaviour using the traditional choice experiment data. The MNP-CI model is estimated using a constrained maximum likelihood approach, and its statistical properties are validated through a comprehensive Monte Carlo study. The CI-based choice model is empirically advantageous as it captures interaction effects while maintaining monotonicity. It also provides information on the complementarity between pairs of attributes coupled with their importance ranking as a by-product of the estimation. These insights could potentially assist policymakers in making policies to improve the preference level for an alternative. These advantages of the MNP-CI model with attribute cut-offs are illustrated in an empirical application to understand New Yorkers' preferences towards mobility-on-demand services.

Recent work has argued that classification losses utilizing softmax cross-entropy are superior not only for fixed-set classification tasks, but also by outperforming losses developed specifically for open-set tasks including few-shot learning and retrieval. Softmax classifiers have been studied using different embedding geometries -- Euclidean, hyperbolic, and spherical -- and claims have been made about the superiority of one or another, but they have not been systematically compared with careful controls. We conduct an empirical investigation of embedding geometry on softmax losses for a variety of fixed-set classification and image retrieval tasks. An interesting property observed for the spherical losses lead us to propose a probabilistic classifier based on the von Mises-Fisher distribution, and we show that it is competitive with state-of-the-art methods while producing improved out-of-the-box calibration. We provide guidance regarding the trade-offs between losses and how to choose among them.

We propose two generic methods for improving semi-supervised learning (SSL). The first integrates weight perturbation (WP) into existing "consistency regularization" (CR) based methods. We implement WP by leveraging variational Bayesian inference (VBI). The second method proposes a novel consistency loss called "maximum uncertainty regularization" (MUR). While most consistency losses act on perturbations in the vicinity of each data point, MUR actively searches for "virtual" points situated beyond this region that cause the most uncertain class predictions. This allows MUR to impose smoothness on a wider area in the input-output manifold. Our experiments show clear improvements in classification errors of various CR based methods when they are combined with VBI or MUR or both.