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We present a non-parametric Bayesian latent variable model capable of learning dependency structures across dimensions in a multivariate setting. Our approach is based on flexible Gaussian process priors for the generative mappings and interchangeable Dirichlet process priors to learn the structure. The introduction of the Dirichlet process as a specific structural prior allows our model to circumvent issues associated with previous Gaussian process latent variable models. Inference is performed by deriving an efficient variational bound on the marginal log-likelihood on the model.
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In this paper, we propose the multivariate quantile Bayesian structural time series (MQBSTS) model for the joint quantile time series forecast, which is the first such model for correlated multivariate time series to the authors best knowledge. The MQBSTS model also enables quantile based feature selection in its regression component where each time series has its own pool of contemporaneous external time series predictors, which is the first time that a fully data-driven quantile feature selection technique applicable to time series data to the authors best knowledge. Different from most machine learning algorithms, the MQBSTS model has very few hyper-parameters to tune, requires small datasets to train, converges fast, and is executable on ordinary personal computers. Extensive examinations on simulated data and empirical data confirmed that the MQBSTS model has superior performance in feature selection, parameter estimation, and forecast.
The identification of relevant features, i.e., the driving variables that determine a process or the property of a system, is an essential part of the analysis of data sets whose entries are described by a large number of variables. The preferred measure for quantifying the relevance of nonlinear statistical dependencies is mutual information, which requires as input probability distributions. Probability distributions cannot be reliably sampled and estimated from limited data, especially for real-valued data samples such as lengths or energies. Here, we introduce total cumulative mutual information (TCMI), a measure of the relevance of mutual dependencies based on cumulative probability distributions. TCMI can be estimated directly from sample data and is a non-parametric, robust and deterministic measure that facilitates comparisons and rankings between feature sets with different cardinality. The ranking induced by TCMI allows for feature selection, i.e., the identification of the set of relevant features that are statistical related to the process or the property of a system, while taking into account the number of data samples as well as the cardinality of the feature subsets. We evaluate the performance of our measure with simulated data, compare its performance with similar multivariate dependence measures, and demonstrate the effectiveness of our feature selection method on a set of standard data sets and a typical scenario in materials science.
We develop a novel hybrid method for Bayesian network structure learning called partitioned hybrid greedy search (pHGS), composed of three distinct yet compatible new algorithms: Partitioned PC (pPC) accelerates skeleton learning via a divide-and-conquer strategy, $p$-value adjacency thresholding (PATH) effectively accomplishes parameter tuning with a single execution, and hybrid greedy initialization (HGI) maximally utilizes constraint-based information to obtain a high-scoring and well-performing initial graph for greedy search. We establish structure learning consistency of our algorithms in the large-sample limit, and empirically validate our methods individually and collectively through extensive numerical comparisons. The combined merits of pPC and PATH achieve significant computational reductions compared to the PC algorithm without sacrificing the accuracy of estimated structures, and our generally applicable HGI strategy reliably improves the estimation structural accuracy of popular hybrid algorithms with negligible additional computational expense. Our empirical results demonstrate the superior empirical performance of pHGS against many state-of-the-art structure learning algorithms.
We propose a class of kernel-based two-sample tests, which aim to determine whether two sets of samples are drawn from the same distribution. Our tests are constructed from kernels parameterized by deep neural nets, trained to maximize test power. These tests adapt to variations in distribution smoothness and shape over space, and are especially suited to high dimensions and complex data. By contrast, the simpler kernels used in prior kernel testing work are spatially homogeneous, and adaptive only in lengthscale. We explain how this scheme includes popular classifier-based two-sample tests as a special case, but improves on them in general. We provide the first proof of consistency for the proposed adaptation method, which applies both to kernels on deep features and to simpler radial basis kernels or multiple kernel learning. In experiments, we establish the superior performance of our deep kernels in hypothesis testing on benchmark and real-world data. The code of our deep-kernel-based two sample tests is available at https://github.com/fengliu90/DK-for-TST.
This paper deals with inference and prediction for multiple correlated time series, where one has also the choice of using a candidate pool of contemporaneous predictors for each target series. Starting with a structural model for the time-series, Bayesian tools are used for model fitting, prediction, and feature selection, thus extending some recent work along these lines for the univariate case. The Bayesian paradigm in this multivariate setting helps the model avoid overfitting as well as capture correlations among the multiple time series with the various state components. The model provides needed flexibility to choose a different set of components and available predictors for each target series. The cyclical component in the model can handle large variations in the short term, which may be caused by external shocks. We run extensive simulations to investigate properties such as estimation accuracy and performance in forecasting. We then run an empirical study with one-step-ahead prediction on the max log return of a portfolio of stocks that involve four leading financial institutions. Both the simulation studies and the extensive empirical study confirm that this multivariate model outperforms three other benchmark models, viz. a model that treats each target series as independent, the autoregressive integrated moving average model with regression (ARIMAX), and the multivariate ARIMAX (MARIMAX) model.
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