We use available measurements to estimate the unknown parameters (variance, smoothness parameter, and covariance length) of a covariance function by maximizing the joint Gaussian log-likelihood function. To overcome cubic complexity in the linear algebra, we approximate the discretized covariance function in the hierarchical (H-) matrix format. The H-matrix format has a log-linear computational cost and storage $O(kn \log n)$, where the rank $k$ is a small integer and $n$ is the number of locations. The H-matrix technique allows us to work with general covariance matrices in an efficient way, since H-matrices can approximate inhomogeneous covariance functions, with a fairly general mesh that is not necessarily axes-parallel, and neither the covariance matrix itself nor its inverse have to be sparse. We demonstrate our method with Monte Carlo simulations and an application to soil moisture data. The C, C++ codes and data are freely available.
The study of animal behavioural states inferred through hidden Markov models and similar state switching models has seen a significant increase in popularity in recent years. The ability to account for varying levels of behavioural scale has become possible through hierarchical hidden Markov models, but additional levels lead to higher complexity and increased correlation between model components. Maximum likelihood approaches to inference using the EM algorithm and direct optimisation of likelihoods are more frequently used, with Bayesian approaches being less favoured due to computational de…
Columbia University（哥伦比亚大学）；Princeton University（普林斯顿大学）
Implicit probabilistic models are a flexible class of models defined by a simulation process for data. They form the basis for theories which encompass our understanding of the physical world. Despite this fundamental nature, the use of implicit models remains limited due to challenges in specifying complex latent structure in them, and in performing inferences in such models with large data sets. In this paper, we first introduce hierarchical implicit models (HIMs). HIMs combine the idea of implicit densities with hierarchical Bayesian modeling, thereby defining models via simulators of data …
Positive-definite matrix-variate data is becoming popular in computer vision. The computer vision data descriptors in the form of Region Covariance Descriptors (RCD) are positive definite matrices, which extract the key features of the images. The RCDs are extensively used in image set classification. Some classification methods treating RCDs as Wishart distributed random matrices are being proposed. However, the majority of the current methods preclude the potential correlation among the RCDs caused by the so-called non-voxel information (e.g., subjects' ages and nose widths, etc). Modeling c…
Recent advances in Markov chain Monte Carlo (MCMC) extend the scope of Bayesian inference to models for which the likelihood function is intractable. Although these developments allow us to estimate model parameters, other basic problems such as estimating the marginal likelihood, a fundamental tool in Bayesian model selection, remain challenging. This is an important scientific limitation because testing psychological hypotheses with hierarchical models has proven difficult with current model selection methods. We propose an efficient method for estimating the marginal likelihood for models w…
Initialisation of the EM algorithm in model-based clustering is often crucial. Various starting points in the parameter space often lead to different local maxima of the likelihood function and, so to different clustering partitions. Among the several approaches available in the literature, model-based agglomerative hierarchical clustering is used to provide initial partitions in the popular MCLUST R package. This choice is computationally convenient and often yields good clustering partitions. However, in certain circumstances, poor initial partitions may cause the EM algorithm to converge to…
Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions. This led to the discovery of maximum likelihood duality between matrices of complementary ranks, a result proved subsequently by Draisma and Rodriguez.
The maximum likelihood estimates of an ARMA model can be obtained by the Kalman filter based on the state-space representation of the model. This paper presents an algorithm for computing gradient of the log-likelihood by an extending the Kalman filter without resorting to the numerical difference. Three examples of seasonal adjustment model and ARMA model are presented to exemplified the specification of structural matrices and initial matrices. An extension of the algorithm to compute the Hessian matrix is also shown.
Composite likelihood inference has gained much popularity thanks to its computational manageability and its theoretical properties. Unfortunately, performing composite likelihood ratio tests is inconvenient because of their awkward asymptotic distribution. There are many proposals for adjusting composite likelihood ratio tests in order to recover an asymptotic chi square distribution, but they all depend on the sensitivity and variability matrices. The same is true for Wald-type and score-type counterparts. In realistic applications sensitivity and variability matrices usually need to be estim…
Physics-based covariance models provide a systematic way to construct covariance models that are consistent with the underlying physical laws in Gaussian process analysis. The unknown parameters in the covariance models can be estimated using maximum likelihood estimation, but direct construction of the covariance matrix and classical strategies of computing with it requires $n$ physical model runs, $n^2$ storage complexity, and $n^3$ computational complexity. To address such challenges, we propose to approximate the discretized covariance function using hierarchical matrices. By utilizing ran…
Finding a set of nested partitions of a dataset is useful to uncover relevant structure at different scales, and is often dealt with a data-dependent methodology. In this paper, we introduce a general two-step methodology for model-based hierarchical clustering. Considering the integrated classification likelihood criterion as an objective function, this work applies to every discrete latent variable models (DLVMs) where this quantity is tractable. The first step of the methodology involves maximizing the criterion with respect to the partition. Addressing the known problem of sub-optimal loca…
HLIBCov: Parallel Hierarchical Matrix Approximation of Large Covariance Matrices and Likelihoods with Applications in Parameter Identification
RWTH Aachen（亚琛工业大学）；Max-Planck-Institut für Mathematik in den Naturwissenschaften（马克斯·普朗克自然科学数学研究所）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
We present a fast direct algorithm for computing symmetric factorizations, i.e. $A = WW^T$, of symmetric positive-definite hierarchical matrices with weak-admissibility conditions. The computational cost for the symmetric factorization scales as $\mathcal{O}(n \log^2 n)$ for hierarchically off-diagonal low-rank matrices. Once this factorization is obtained, the cost for inversion, application, and determinant computation scales as $\mathcal{O}(n \log n)$. In particular, this allows for the near optimal generation of correlated random variates in the case where $A$ is a covariance matrix. This …
Hierarchical models allow for heterogeneous behaviours in a population while simultaneously borrowing estimation strength across all subpopulations. Unfortunately, existing likelihood-based methods for fitting hierarchical models have high computational demands, and these demands have limited their adoption in large-scale prediction and inference problems. This paper proposes a moment-based procedure for estimating the parameters of a hierarchical model which has its roots in a method originally introduced by Cochran in 1937. The method trades statistical efficiency for computational efficienc…
Statistical analysis of massive datasets very often implies expensive linear algebra operations with large dense matrices. Typical tasks are an estimation of unknown parameters of the underlying statistical model and prediction of missing values. We developed the H-MLE procedure, which solves these tasks. The unknown parameters can be estimated by maximizing the joint Gaussian log-likelihood function, which depends on a covariance matrix. To decrease high computational cost, we approximate the covariance matrix in the hierarchical (H-) matrix format. The H-matrix technique allows us to work wi…
We propose a penalized likelihood method to fit the linear discriminant analysis model when the predictor is matrix valued. We simultaneously estimate the means and the precision matrix, which we assume has a Kronecker product decomposition. Our penalties encourage pairs of response category mean matrices to have equal entries and also encourage zeros in the precision matrix. To compute our estimators, we use a blockwise coordinate descent algorithm. To update the optimization variables corresponding to response category mean matrices, we use an alternating minimization algorithm that takes ad…
In this paper we consider the problem of computing the likelihood of the profile of a discrete distribution, i.e., the probability of observing the multiset of element frequencies, and computing a profile maximum likelihood (PML) distribution, i.e., a distribution with the maximum profile likelihood. For each problem we provide polynomial time algorithms that given $n$ i.i.d.\ samples from a discrete distribution, achieve an approximation factor of $\exp\left(-O(\sqrt{n} \log n) \right)$, improving upon the previous best-known bound achievable in polynomial time of $\exp(-O(n^{2/3} \log n))$ (…
We introduce Gaussian orthogonal latent factor processes for modeling and predicting large correlated data. To handle the computational challenge, we first decompose the likelihood function of the Gaussian random field with a multi-dimensional input domain into a product of densities at the orthogonal components with lower-dimensional inputs. The continuous-time Kalman filter is implemented to compute the likelihood function efficiently without making approximations. We also show that the posterior distribution of the factor processes is independent, as a consequence of prior independence of f…
