Gaussian copula mixture models (GCMM) are the generalization of Gaussian Mixture models using the concept of copula. Its mathematical definition is given and the properties of likelihood function are studied in this paper. Based on these properties, extended Expectation Maximum algorithms are developed for estimating parameters for the mixture of copulas while marginal distributions corresponding to each component is estimated using separate nonparametric statistical methods. In the experiment, GCMM can achieve better goodness-of-fitting given the same number of clusters as GMM; furthermore, GCMM can utilize unsynchronized data on each dimension to achieve deeper mining of data.
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In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases
A novel positive dependence property is introduced, called positive measure inducing (PMI for short), being fulfilled by numerous copula classes, including Gaussian, Fr\'echet, Farlie-Gumbel-Morgenstern and Frank copulas; it is conjectured that even all positive quadrant dependent Archimedean copulas meet this property. From a geometric viewpoint, a PMI copula concentrates more mass near the main diagonal than in the opposite diagonal. A striking feature of PMI copulas is that they impose an ordering on a certain class of copula-induced measures of concordance, the latter originating in Edwards et al. (2004) and including Spearman's rho $\rho$ and Gini's gamma $\gamma$, leading to numerous new inequalities such as $3 \gamma \geq 2 \rho$. The measures of concordance within this class are estimated using (classical) empirical copulas and the intrinsic construction via empirical checkerboard copulas, and the estimators' asymptotic behaviour is determined. Building upon the presented inequalities, asymptotic tests are constructed having the potential of being used for detecting whether the underlying dependence structure of a given sample is PMI, which in turn can be used for excluding certain copula families from model building. The excellent performance of the tests is demonstrated in a simulation study and by means of a real-data example.
Restricted Boltzmann Machines are generative models that consist of a layer of hidden variables connected to another layer of visible units, and they are used to model the distribution over visible variables. In order to gain a higher representability power, many hidden units are commonly used, which, in combination with a large number of visible units, leads to a high number of trainable parameters. In this work we introduce the Structural Restricted Boltzmann Machine model, which taking advantage of the structure of the data in hand, constrains connections of hidden units to subsets of visible units in order to reduce significantly the number of trainable parameters, without compromising performance. As a possible area of application, we focus on image modelling. Based on the nature of the images, the structure of the connections is given in terms of spatial neighbourhoods over the pixels of the image that constitute the visible variables of the model. We conduct extensive experiments on various image domains. Image denoising is evaluated with corrupted images from the MNIST dataset. The generative power of our models is compared to vanilla RBMs, as well as their classification performance, which is assessed with five different image domains. Results show that our proposed model has a faster and more stable training, while also obtaining better results compared to an RBM with no constrained connections between its visible and hidden units.
In this paper, we consider the problem of finding perfectly balanced Boolean functions with high non-linearity values. Such functions have extensive applications in domains such as cryptography and error-correcting coding theory. We provide an approach for finding such functions by a local search method that exploits the structure of the underlying problem. Previous attempts in this vein typically focused on using the properties of the fitness landscape to guide the search. We opt for a different path in which we leverage the phenotype landscape (the mapping from genotypes to phenotypes) instead. In the context of the underlying problem, the phenotypes are represented by Walsh-Hadamard spectra of the candidate solutions (Boolean functions). We propose a novel selection criterion, under which the phenotypes are compared directly, and test whether its use increases the convergence speed (measured by the number of required spectra calculations) when compared to a competitive fitness function used in the literature. The results reveal promising convergence speed improvements for Boolean functions of sizes $N=6$ to $N=9$.
This paper presents a novel approach to Bayesian nonparametric spectral analysis of stationary multivariate time series. Starting with a parametric vector-autoregressive model, the parametric likelihood is nonparametrically adjusted in the frequency domain to account for potential deviations from parametric assumptions. We show mutual contiguity of the nonparametrically corrected likelihood, the multivariate Whittle likelihood approximation and the exact likelihood for Gaussian time series. A multivariate extension of the nonparametric Bernstein-Dirichlet process prior for univariate spectral densities to the space of Hermitian positive definite spectral density matrices is specified directly on the correction matrices. An infinite series representation of this prior is then used to develop a Markov chain Monte Carlo algorithm to sample from the posterior distribution. The code is made publicly available for ease of use and reproducibility. With this novel approach we provide a generalization of the multivariate Whittle-likelihood-based method of Meier et al. (2020) as well as an extension of the nonparametrically corrected likelihood for univariate stationary time series of Kirch et al. (2019) to the multivariate case. We demonstrate that the nonparametrically corrected likelihood combines the efficiencies of a parametric with the robustness of a nonparametric model. Its numerical accuracy is illustrated in a comprehensive simulation study. We illustrate its practical advantages by a spectral analysis of two environmental time series data sets: a bivariate time series of the Southern Oscillation Index and fish recruitment and time series of windspeed data at six locations in California.
We derive bounds on the absolute values of the eigenvalues of special type of matrix rational functions using the following techniques/methods: (1) the Bauer-Fike theorem on an associated block matrix of the given matrix rational function, (2) by associating a real rational function, along with Rouch$\text{\'e}$ theorem for the matrix rational function and (3) by a numerical radius inequality for a block matrix for the matrix rational function. These bounds are compared when the coefficients are unitary matrices. A numerical example is given to illustrate the results obtained.
We consider linear random coefficient regression models, where the regressors are allowed to have a finite support. First, we investigate identifiability, and show that the means and the variances and covariances of the random coefficients are identified from the first two conditional moments of the response given the covariates if the support of the covariates, excluding the intercept, contains a Cartesian product with at least three points in each coordinate. We also discuss ientification of higher-order mixed moments, as well as partial identification in the presence of a binary regressor. Next we show the variable selection consistency of the adaptive LASSO for the variances and covariances of the random coefficients in finite and moderately high dimensions. This implies that the estimated covariance matrix will actually be positive semidefinite and hence a valid covariance matrix, in contrast to the estimate arising from a simple least squares fit. We illustrate the proposed method in a simulation study.
We consider blind ptychography, an imaging technique which aims to reconstruct an object of interest from a set of its diffraction patterns, each obtained by a local illumination. As the distribution of the light within the illuminated region, called the window, is unknown, it also has to be estimated as well. For the recovery, we consider gradient and stochastic gradient descent methods for the minimization of amplitude-base squared loss. In particular, this includes extended Ptychographic Iterative Engine as a special case of stochastic gradient descent. We show that all methods converge to a critical point at a sublinear rate with a proper choice of step sizes. We also discuss possibilities for larger step sizes.
Random graph models are playing an increasingly important role in science and industry, and finds their applications in a variety of fields ranging from social and traffic networks, to recommendation systems and molecular genetics. In this paper, we perform an in-depth analysis of the random Kronecker graph model proposed in \cite{leskovec2010kronecker}, when the number of graph vertices $N$ is large. Built upon recent advances in random matrix theory, we show, in the dense regime, that the random Kronecker graph adjacency matrix follows approximately a signal-plus-noise model, with a small-rank (of order at most $\log N$) signal matrix that is linear in the graph parameters and a random noise matrix having a quarter-circle-form singular value distribution. This observation allows us to propose a ``denoise-and-solve'' meta algorithm to approximately infer the graph parameters, with reduced computational complexity and (asymptotic) performance guarantee. Numerical experiments of graph inference and graph classification on both synthetic and realistic graphs are provided to support the advantageous performance of the proposed approach.
The Gaussian process latent variable model (GPLVM) is a popular probabilistic method used for nonlinear dimension reduction, matrix factorization, and state-space modeling. Inference for GPLVMs is computationally tractable only when the data likelihood is Gaussian. Moreover, inference for GPLVMs has typically been restricted to obtaining maximum a posteriori point estimates, which can lead to overfitting, or variational approximations, which mischaracterize the posterior uncertainty. Here, we present a method to perform Markov chain Monte Carlo (MCMC) inference for generalized Bayesian nonlinear latent variable modeling. The crucial insight necessary to generalize GPLVMs to arbitrary observation models is that we approximate the kernel function in the Gaussian process mappings with random Fourier features; this allows us to compute the gradient of the posterior in closed form with respect to the latent variables. We show that we can generalize GPLVMs to non-Gaussian observations, such as Poisson, negative binomial, and multinomial distributions, using our random feature latent variable model (RFLVM). Our generalized RFLVMs perform on par with state-of-the-art latent variable models on a wide range of applications, including motion capture, images, and text data for the purpose of estimating the latent structure and imputing the missing data of these complex data sets.
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