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Exploration of the search space of Gaussian graphical models for paired data. (arXiv:2303.05561v1 [stat.ML]) arxiv.org/abs/2303.05561

Exploration of the search space of Gaussian graphical models for paired data

We consider the problem of learning a Gaussian graphical model in the case where the observations come from two dependent groups sharing the same variables. We focus on a family of coloured Gaussian graphical models specifically suited for the paired data problem. Commonly, graphical models are ordered by the submodel relationship so that the search space is a lattice, called the model inclusion lattice. We introduce a novel order between models, named the twin order. We show that, embedded with this order, the model space is a lattice that, unlike the model inclusion lattice, is distributive. Furthermore, we provide the relevant rules for the computation of the neighbours of a model. The latter are more efficient than the same operations in the model inclusion lattice, and are then exploited to achieve a more efficient exploration of the search space. These results can be applied to improve the efficiency of both greedy and Bayesian model search procedures. Here we implement a stepwise backward elimination procedure and evaluate its performance by means of simulations. Finally, the procedure is applied to learn a brain network from fMRI data where the two groups correspond to the left and right hemispheres, respectively.

arxiv.org

Variance-aware robust reinforcement learning with linear function approximation with heavy-tailed rewards. (arXiv:2303.05606v1 [cs.LG]) arxiv.org/abs/2303.05606

Variance-aware robust reinforcement learning with linear function approximation with heavy-tailed rewards

This paper presents two algorithms, AdaOFUL and VARA, for online sequential decision-making in the presence of heavy-tailed rewards with only finite variances. For linear stochastic bandits, we address the issue of heavy-tailed rewards by modifying the adaptive Huber regression and proposing AdaOFUL. AdaOFUL achieves a state-of-the-art regret bound of $\widetilde{\mathcal{O}}\big(d\big(\sum_{t=1}^T ν_{t}^2\big)^{1/2}+d\big)$ as if the rewards were uniformly bounded, where $ν_{t}^2$ is the observed conditional variance of the reward at round $t$, $d$ is the feature dimension, and $\widetilde{\mathcal{O}}(\cdot)$ hides logarithmic dependence. Building upon AdaOFUL, we propose VARA for linear MDPs, which achieves a tighter variance-aware regret bound of $\widetilde{\mathcal{O}}(d\sqrt{H\mathcal{G}^*K})$. Here, $H$ is the length of episodes, $K$ is the number of episodes, and $\mathcal{G}^*$ is a smaller instance-dependent quantity that can be bounded by other instance-dependent quantities when additional structural conditions on the MDP are satisfied. Our regret bound is superior to the current state-of-the-art bounds in three ways: (1) it depends on a tighter instance-dependent quantity and has optimal dependence on $d$ and $H$, (2) we can obtain further instance-dependent bounds of $\mathcal{G}^*$ under additional structural conditions on the MDP, and (3) our regret bound is valid even when rewards have only finite variances, achieving a level of generality unmatched by previous works. Overall, our modified adaptive Huber regression algorithm may serve as a useful building block in the design of algorithms for online problems with heavy-tailed rewards.

arxiv.org

On the Unlikelihood of D-Separation. (arXiv:2303.05628v1 [cs.LG]) arxiv.org/abs/2303.05628

On the Unlikelihood of D-Separation

Causal discovery aims to recover a causal graph from data generated by it; constraint based methods do so by searching for a d-separating conditioning set of nodes in the graph via an oracle. In this paper, we provide analytic evidence that on large graphs, d-separation is a rare phenomenon, even when guaranteed to exist, unless the graph is extremely sparse. We then provide an analytic average case analysis of the PC Algorithm for causal discovery, as well as a variant of the SGS Algorithm we call UniformSGS. We consider a set $V=\{v_1,\ldots,v_n\}$ of nodes, and generate a random DAG $G=(V,E)$ where $(v_a, v_b) \in E$ with i.i.d. probability $p_1$ if $a<b$ and $0$ if $a > b$. We provide upper bounds on the probability that a subset of $V-\{x,y\}$ d-separates $x$ and $y$, conditional on $x$ and $y$ being d-separable; our upper bounds decay exponentially fast to $0$ as $|V| \rightarrow \infty$. For the PC Algorithm, while it is known that its worst-case guarantees fail on non-sparse graphs, we show that the same is true for the average case, and that the sparsity requirement is quite demanding: for good performance, the density must go to $0$ as $|V| \rightarrow \infty$ even in the average case. For UniformSGS, while it is known that the running time is exponential for existing edges, we show that in the average case, that is the expected running time for most non-existing edges as well.

arxiv.org

Towards better traffic volume estimation: Tackling both underdetermined and non-equilibrium problems via a correlation adaptive graph convolution network. (arXiv:2303.05660v1 [stat.ML]) arxiv.org/abs/2303.05660

Towards better traffic volume estimation: Tackling both underdetermined and non-equilibrium problems via a correlation adaptive graph convolution network

Traffic volume is an indispensable ingredient to provide fine-grained information for traffic management and control. However, due to limited deployment of traffic sensors, obtaining full-scale volume information is far from easy. Existing works on this topic primarily focus on improving the overall estimation accuracy of a particular method and ignore the underlying challenges of volume estimation, thereby having inferior performances on some critical tasks. This paper studies two key problems with regard to traffic volume estimation: (1) underdetermined traffic flows caused by undetected movements, and (2) non-equilibrium traffic flows arise from congestion propagation. Here we demonstrate a graph-based deep learning method that can offer a data-driven, model-free and correlation adaptive approach to tackle the above issues and perform accurate network-wide traffic volume estimation. Particularly, in order to quantify the dynamic and nonlinear relationships between traffic speed and volume for the estimation of underdetermined flows, a speed patternadaptive adjacent matrix based on graph attention is developed and integrated into the graph convolution process, to capture non-local correlations between sensors. To measure the impacts of non-equilibrium flows, a temporal masked and clipped attention combined with a gated temporal convolution layer is customized to capture time-asynchronous correlations between upstream and downstream sensors. We then evaluate our model on a real-world highway traffic volume dataset and compare it with several benchmark models. It is demonstrated that the proposed model achieves high estimation accuracy even under 20% sensor coverage rate and outperforms other baselines significantly, especially on underdetermined and non-equilibrium flow locations. Furthermore, comprehensive quantitative model analysis are also carried out to justify the model designs.

arxiv.org

Prevalence and major risk factors of non-communicable diseases: A Hospital-based Cross-Sectional Study in Dhaka, Bangladesh. (arXiv:2303.04808v1 [q-bio.QM]) arxiv.org/abs/2303.04808

Prevalence and major risk factors of non-communicable diseases: A Hospital-based Cross-Sectional Study in Dhaka, Bangladesh

Objective: The study aimed to determine the prevalence of several non-communicable diseases (NCD) and analyze risk factors among adult patients seeking nutritional guidance in Dhaka, Bangladesh. Result: Our study observed the relationships between gender, age groups, obesity, and NCDs (DM, CKD, IBS, CVD, CRD, thyroid). The most frequently reported NCD was cardiovascular issues (CVD), which was present in 83.56% of all participants. CVD was more common in male participants. Consequently, male participants had a higher blood pressure distribution than females. Diabetes mellitus (DM), on the other hand, did not have a gender-based inclination. Both CVD and DM had an age-based progression. Our study showed that chronic respiratory illness was more frequent in middle-aged participants than in younger or elderly individuals. Based on the data, every one in five hospitalized patients was obese. We analyzed the co-morbidities and found that 31.5% of the population has only one NCD, 30.1% has two NCDs, and 38.3% has more than two NCDs. Besides, 86.25% of all diabetic patients had cardiovascular issues. All thyroid patients in our study had CVD. Using a t-test, we found a relationship between CKD and thyroid (p-value 0.061). Males under 35 years have a statistically significant relationship between thyroid and chronic respiratory diseases (p-value 0.018). We also found an association between DM and CKD among patients over 65 (p-value 0.038). Moreover, there has been a statistically significant relationship between CKD and Thyroid (P < 0.05) for those below 35 and 35-65. We used a two-way ANOVA test to find the statistically significant interaction of heart issues and chronic respiratory illness, in combination with diabetes. The combination of DM and RTI also affected CKD in male patients over 65 years old.

arxiv.org

Agnostic PAC Learning of k-juntas Using L2-Polynomial Regression. (arXiv:2303.04859v1 [cs.LG]) arxiv.org/abs/2303.04859

Agnostic PAC Learning of k-juntas Using L2-Polynomial Regression

Many conventional learning algorithms rely on loss functions other than the natural 0-1 loss for computational efficiency and theoretical tractability. Among them are approaches based on absolute loss (L1 regression) and square loss (L2 regression). The first is proved to be an \textit{agnostic} PAC learner for various important concept classes such as \textit{juntas}, and \textit{half-spaces}. On the other hand, the second is preferable because of its computational efficiency, which is linear in the sample size. However, PAC learnability is still unknown as guarantees have been proved only under distributional restrictions. The question of whether L2 regression is an agnostic PAC learner for 0-1 loss has been open since 1993 and yet has to be answered. This paper resolves this problem for the junta class on the Boolean cube -- proving agnostic PAC learning of k-juntas using L2 polynomial regression. Moreover, we present a new PAC learning algorithm based on the Boolean Fourier expansion with lower computational complexity. Fourier-based algorithms, such as Linial et al. (1993), have been used under distributional restrictions, such as uniform distribution. We show that with an appropriate change, one can apply those algorithms in agnostic settings without any distributional assumption. We prove our results by connecting the PAC learning with 0-1 loss to the minimum mean square estimation (MMSE) problem. We derive an elegant upper bound on the 0-1 loss in terms of the MMSE error and show that the sign of the MMSE is a PAC learner for any concept class containing it.

arxiv.org

The shape of the relative frailty variance induced by discrete random effect distributions in univariate and multivariate survival models. (arXiv:2303.04915v1 [stat.ME]) arxiv.org/abs/2303.04915

The shape of the relative frailty variance induced by discrete random effect distributions in univariate and multivariate survival models

In statistical models for the analysis of time-to-event data, individual heterogeneity is usually accounted for by means of one or more random effects, also known as frailties. In the vast majority of the literature, the random effect is assumed to follow a continuous probability distribution. However, in some areas of application, a discrete frailty distribution may be more appropriate. We investigate and compare various existing families of discrete univariate and shared frailty models by taking as our focus the variance of the relative frailty distribution in survivors. The relative frailty variance (RFV) among survivors provides a readily interpretable measure of how the heterogeneity of a population, as represented by a frailty model, evolves over time. We explore the shape of the RFV for the purpose of model selection and review available discrete random effect distributions in this context. We find non-monotone trajectories of the RFV for discrete univariate and shared frailty models, which is a rare property. Furthermore, we proof that for discrete time-invariant univariate and shared frailty models with (without) an atom at zero, the limit of the RFV approaches infinity (zero), if the support of the discrete distribution can be arranged in ascending order. Through the one-to-one relationship of the RFV with the cross-ratio function in shared frailty models, which we generalize to the higher-variate case, our results also apply to patterns of association within a cluster. Extensions and contrasts to discrete time-varying frailty models and contrasts to correlated discrete frailty models are discussed.

arxiv.org

Predicting Elite NBA Lineups Using Individual Player Order Statistics. (arXiv:2303.04963v1 [stat.AP]) arxiv.org/abs/2303.04963

Predicting Elite NBA Lineups Using Individual Player Order Statistics

NBA team managers and owners try to acquire high-performing players. An important consideration in these decisions is how well the new players will perform in combination with their teammates. Our objective is to identify elite five-person lineups, which we define as those having a positive plus-minus per minute (PMM). Using individual player order statistics, our model can identify an elite lineup even if the five players in the lineup have never played together, which can inform player acquisition decisions, salary negotiations, and real-time coaching decisions. We combine seven classification tools into a unanimous consent classifier (all-or-nothing classifier, or ANC) in which a lineup is predicted to be elite only if all seven classifiers predict it to be elite. In this way, we achieve high positive predictive value (i.e., precision), the likelihood that a lineup classified as elite will indeed have a positive PMM. We train and test the model on individual player and lineup data from the 2017-18 season and use the model to predict the performance of lineups drawn from all 30 NBA teams' 2018-19 regular season rosters. Although the ANC is conservative and misses some high-performing lineups, it achieves high precision and recommends positionally balanced lineups.

arxiv.org
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