Non-diagonal Covariance Matrix Research Articles

Model independent estimation of the cosmography parameters using cosmic chronometers

Measurement of the universe expansion rate through the cosmic chronometers proves to be a novel approach to understanding the cosmic history. Although it provides a direct determination of the Hubble parameters at different redshifts, it suffers from underlying systematic uncertainties. In this work, we analyze the recent cosmic chronometers data with and without systematic uncertainties and investigate how they affect the results. We perform our analysis in both model dependent and independent methods to avoid any possible model bias. In the model dependent approach, we consider the \(\Lambda \)CDM, wCDM and CPL models. On the other hand, since the Gaussian process provides a unique tool to study data including a non-diagonal covariance matrix, our model independent analysis is based on the Gaussian process.

Bayesian Gaussian process factor analysis with copula for count data

Gaussian process factor analysis models aim to capture the latent trajectories that generated observed multivariate data by factorizing the observations. These methods assume conditional independence of observations, given the factor scores and loadings. For continuous data this restrictive assumption can be resolved by using a non-diagonal covariance matrix. However, this does not work for count data. We propose a new model which pairs a negative binomial Gaussian process factor analysis with a Gaussian copula to find latent trajectories and also accounts for the residual covariance not captured by these trajectories. We provide a fully Bayesian implementation of the model and use augmented likelihood for inference with Hamiltonian Monte Carlo. We compare the proposed method to other Gaussian process factor analysis models on 12 toy data sets, finding latent qualities of NBA teams, and forecasting disease counts. The results show that the proposed method is useful for latent structure extraction and out-of-sample prediction of multivariate counts.

On semiparametric modelling, estimation and inference for survival data subject to dependent censoring

Summary When modelling survival data, it is common to assume that the survival time $T$ is conditionally independent of the censoring time $C$ given a set of covariates. However, there are numerous situations in which this assumption is not realistic. The goal of this paper is therefore to develop a semiparametric normal transformation model which assumes that, after a proper nonparametric monotone transformation, the vector $(T, C)$ follows a linear model, and the vector of errors in this bivariate linear model follows a standard bivariate normal distribution with a possibly nondiagonal covariance matrix. We show that this semiparametric model is identifiable, and propose estimators of the nonparametric transformation, the regression coefficients and the correlation between the error terms. It is shown that the estimators of the model parameters and the transformation are consistent and asymptotically normal. We also assess the finite-sample performance of the proposed method by comparing it with an estimation method under a fully parametric model. Finally, our method is illustrated using data from the AIDS Clinical Trial Group 175 study.

Extended scalar sectors, effective operators and observed data

The available data on the 125 GeV scalar h is analysed to explore the room for new physics in the electroweak symmetry breaking sector. The first part of the study is model-independent, with h couplings to standard model particles scaled by quantities that are taken to be free parameters. At the same time, the additional loop contributions to h → γγ and h → Zγ, mediated by charged scalar contributions in the extended scalar sector, are treated in terms of gauge-invariant effective operators. Having justified this approach for cases where the concerned scalar masses are a little above the Z-boson mass, we fit the existing data to obtain marginalized 1σ and 2σ regions in the space of the coefficients of such effective operators, where the limit on the h → Zγ branching ratio is used as a constraint. The correlation between, say, the gluon fusion and vector-boson fusion channels, as reflected in a non-diagonal covariance matrix, is taken into account. After thus obtaining model-independent fits, the allowed values of the coefficients are translated into permissible regions of the parameter spaces of several specific models. In this spirit we constrain four different types of two Higgs doublet models, and also models with one or two Y = 2 scalar triplets, taking into account the correlatedness of the scale factors in h-interactions and the various couplings of charged Higgs states in each extended scenario.

Estimating observation error covariance matrix of seismic data from a perspective of image denoising

Estimating observation error covariance matrix properly is a key step towards successful seismic history matching. Typically, observation errors of seismic data are spatially correlated; therefore, the observation error covariance matrix is non-diagonal. Estimating such a non-diagonal covariance matrix is the focus of the current study. We decompose the estimation into two steps: (1) estimate observation errors and (2) construct covariance matrix based on the estimated observation errors. Our focus is on step (1), whereas at step (2) we use a procedure similar to that in Aanonsen et al. 2003. In Aanonsen et al. 2003, step (1) is carried out using a local moving average algorithm. By treating seismic data as an image, this algorithm can be interpreted as a discrete convolution between an image and a rectangular window function. Following the perspective of image processing, we consider three types of image denoising methods, namely, local moving average with different window functions (as an extension of the method in Aanonsen et al. 2003), non-local means denoising and wavelet denoising. The performance of these three algorithms is compared using both synthetic and field seismic data. It is found that, in our investigated cases, the wavelet denoising method leads to the best performance in most of the time.

Evolving Type-2 Fuzzy Classifier

Evolving fuzzy classifiers (EFCs) have achieved immense success in dealing with nonstationary data streams because of their flexible characteristics. Nonetheless, most real-world data streams feature highly uncertain characteristics, which cannot be handled by the type-1 EFC. A novel interval type-2 fuzzy classifier, namely evolving type-2 classifier (eT2Class), is proposed in this paper, which constructs an evolving working principle in the framework of interval type-2 fuzzy system. The eT2Class commences its learning process from scratch with an empty or initially trained rule base, and its fuzzy rules can be automatically grown, pruned, recalled, and merged on the fly referring to summarization power and generalization power of data streams. In addition, the eT2Class is driven by a generalized interval type-2 fuzzy rule, where the premise part is composed of the multivariate Gaussian function with an uncertain nondiagonal covariance matrix, while employing a subset of the nonlinear Chebyshev polynomial as the rule consequents. The efficacy of the eT2Class has been rigorously assessed by numerous real-world and artificial study cases, benchmarked against state-of-the-art classifiers, and validated through various statistical tests. Our numerical results demonstrate that the eT2Class produces more reliable classification rates, while retaining more compact and parsimonious rule base than state-of-the-art EFCs recently published in the literature.

A dimensionally reduced finite mixture model for multilevel data

Recently, different mixture models have been proposed for multilevel data, generally requiring the local independence assumption. In this work, this assumption is relaxed by allowing each mixture component at the lower level of the hierarchical structure to be modeled according to a multivariate Gaussian distribution with a non-diagonal covariance matrix. For high-dimensional problems, this solution can lead to highly parameterized models. In this proposal, the trade-off between model parsimony and flexibility is governed by assuming a latent factor generative model.

The influence of the correlations between an asteroid’s orbital parameters on the estimation of the probability of planetary collision by the Monte Carlo method

The probability of an asteroid colliding with a planet can be estimated by the Monte Carlo method, in particular, through the statistical simulation of the possible initial conditions for the motion of an asteroid based on the probability density distribution set by the respective covariance matrix to be further projected with the orbital model onto the supposed time point of the collision. Hence, the collision probability is calculated as the ratio between the number of projected (virtual) asteroids striking the planet and their total number. The main problem is that different elements of the initial conditions (orbit or state vector) are correlated and, therefore, cannot be simulated independently. These correlations are reflected in the nondiagonal covariance matrix of the solution. The matrix is diagonalized by an orthogonal transformation. In the uncertainty domain constructed from the diagonal matrix elements, the initial values for each of the six orbital elements are simulated independently from the other elements, but with the accounting for their normal distribution. The program for calculating the normal distribution is based on the central limit theorem. Each sample of the initial values for the six orbital elements is transferred to the initial reference frame using an inverse transformation. Then, numerical integration is used to track the asteroid’s motion along the respective orbit to predict a possible impact event. Asteroids 99942 Apophis and 2007 WD5 are used as examples to show that disregarding the correlations when diagonalizing the covariance matrix to set the initial conditions may seriously distort the collision probability estimates. The paper gives the probabilities of the collisions of Apophis with the Earth and asteroid 2007 WD5 with Mars calculated by the author from observation sets showing nonzero collision probabilities. The author’s estimates are compared to those calculated by NASA.

Penalized mixtures of factor analyzers with application to clustering high-dimensional microarray data

Model-based clustering has been widely used, e.g. in microarray data analysis. Since for high-dimensional data variable selection is necessary, several penalized model-based clustering methods have been proposed tørealize simultaneous variable selection and clustering. However, the existing methods all assume that the variables are independent with the use of diagonal covariance matrices. To model non-independence of variables (e.g. correlated gene expressions) while alleviating the problem with the large number of unknown parameters associated with a general non-diagonal covariance matrix, we generalize the mixture of factor analyzers to that with penalization, which, among others, can effectively realize variable selection. We use simulated data and real microarray data to illustrate the utility and advantages of the proposed method over several existing ones.

Statistical image reconstruction from correlated data with applications to PET

Most statistical reconstruction methods for emission tomography are designed for data modeled as conditionally independent Poisson variates. In reality, due to scanner detectors, electronics and data processing, correlations are introduced into the data resulting in dependent variates. In general, these correlations are ignored because they are difficult to measure and lead to computationally challenging statistical reconstruction algorithms. This work addresses the second concern, seeking to simplify the reconstruction of correlated data and provide a more precise image estimate than the conventional independent methods. In general, correlated variates have a large non-diagonal covariance matrix that is computationally challenging to use as a weighting term in a reconstruction algorithm. This work proposes two methods to simplify the use of a non-diagonal covariance matrix as the weighting term by (a) limiting the number of dimensions in which the correlations are modeled and (b) adopting flexible, yet computationally tractable, models for correlation structure. We apply and test these methods with simple simulated PET data and data processed with the Fourier rebinning algorithm which include the one-dimensional correlations in the axial direction and the two-dimensional correlations in the transaxial directions. The methods are incorporated into a penalized weighted least-squares 2D reconstruction and compared with a conventional maximum a posteriori approach.

Model inversion by parameter fit using NN emulating the forward model — Evaluation of indirect measurements

The usage of inverse models to derive parameters of interest from measurements is widespread in science and technology. The operational usage of many inverse models became feasible just by emulation of the inverse model via a neural net (NN). This paper shows how NNs can be used to improve inversion accuracy by minimizing the sum of error squares. The procedure is very fast as it takes advantage of the Jacobian which is a byproduct of the NN calculation. An example from remote sensing is shown. It is also possible to take into account a non-diagonal covariance matrix of the measurement to derive the covariance matrix of the retrieved parameters.

Confidence intervals for linear discrete inverse problems with a non-negativity constraint

We consider the problem of constructing confidence intervals for linear functionals βtx of a non-negative vector x given discrete indirect noisy observations y = Kx + ε. The intervals are defined by the range of the functional over constrained projected contours of a quadratic data misfit function. Rust and O'Leary (1994 J. Comput. Graph. Stat.3 67) derived similar intervals but we show that some conditions are required for the intervals to have the correct coverage and provide valid intervals for the cases when the conditions are not met. The problem is reduced to two simple cases: for β > 0 a closed formula for the intervals can be used on reduced one-dimensional data without the need to solve an optimization problem. For β ≯ 0 the optimization is reduced to a two-dimensional problem with K = I but a non-diagonal covariance matrix.

A nonlinear preconditioned quasi-Newton method without inversion of a first-guess covariance matrix in variational analyses

Aquasi-Newton method developed for adopting a non-diagonal first-guess covariance matrix in nonlinearvariational analyses (i.e. variational analyses adopting a non-quadratic cost function) is presented.As an example, we also show the effect of the nonlinear relationship between temperature and seasurfaceheight in three-dimensional variational ocean analyses.

Analyzing Simulation Experiments with Common Random Numbers

To analyze simulation runs which use the same random numbers, the blocking concept of experimental design is not needed. Instead, this paper applies a linear regression model with a nondiagonal covariance matrix. This covariance matrix does not need to have a specific pattern such as constant covariances. A simple example yields surprising results. The paper proposes a new framework for the error analysis. This framework consists of three factors (namely, common random numbers, replication, model validity), each with three levels.