Regularized Within-Class Precision Matrix Based PLDA in Text-Dependent Speaker Verification

In the field of speaker verification, probabilistic linear discriminant analysis (PLDA) is the dominant method for back-end scoring. To estimate the PLDA model, the between-class covariance and within-class precision matrices must be estimated from samples. However, the empirical covariance/precision estimated from samples has estimation errors due to the limited number of samples available. In this paper, we propose a method to improve the conventional PLDA by estimating the PLDA model using the regularized within-class precision matrix. We use graphical least absolute shrinking and selection operator (GLASSO) for the regularization. The GLASSO regularization decreases the estimation errors in the empirical precision matrix by making the precision matrix sparse, which corresponds to the reflection of the conditional independence structure. The experimental results on text-dependent speaker verification reveal that the proposed method reduce the relative equal error rate by up to 23% compared with the conventional PLDA.

Assessment of Tscherning-Rapp covariance in Earth gravity modeling using gravity gradient and GPS/leveling observations

<p>Determination of Earth’s gravity field in a high accuracy needs different complementary data and also methods to combine these data in an optimized procedure. Newly invented resources such as GPS, GRACE, and GOCE provide various data with different distribution which makes it possible to reach this aim. Least Squares Collocation (LSC) is one of the methods that help to mix different data types via covariance function which correlates the different involved parameters within the procedure. One way to construct such covariance functions is involving two steps within the remove-compute-restore (RCR) procedure: first, calculation of an empirical covariance function from observations which the gravitational effects of global gravity field (Long-wavelength) and topography/bathymetry have been subtracted from it and then fitting the Tscherning–Rapp analytical covariance model to the empirical one. According to the corresponding studies, the accuracy of LSC is directly related to the ability to localize the covariance function which itself depends on the data distribution. In this study, we have analyzed the data distribution and geometrically fitting factors, on GPS/Leveling and GOCE gradient data by considering the various case studies with different data distributions. To make the assessment of the covariance determination possible, the residual observations were divided into two datasets namely, observations and control points. The observations point served as input data within the LSC procedure using the Tscherning – Rapp covariance model and the control points used to evaluate the accuracy of the LSC in gravity gradient, gravity anomaly, and geoid predicting and then the covariance estimation. The results of this study show that the Tscherning-Rapp (1974) covariance has different outcomes over different quantities. For example, it models accurate enough the empirical covariance of gradient gravity but requires more analysis for gravity anomalies and GPS/Leveling quantities to reach the optimized results in terms of STD of difference between the computed and control points.</p>

Calendar Spread Exchange Options Pricing with Gaussian Random Fields

Most of the models leading to an analytical expression for option prices are based on the assumption that underlying asset returns evolve according to a Brownian motion with drift. For some asset classes like commodities, a Brownian model does not fit empirical covariance and autocorrelation structures. This failure to replicate the covariance introduces a bias in the valuation of calendar spread exchange options. As the payoff of these options depends on two asset values at different times, particular care must be taken for the modeling of covariance and autocorrelation. This article proposes a simple alternative model for asset prices with sub-exponential, exponential and hyper-exponential autocovariance structures. In the proposed approach, price processes are seen as conditional Gaussian fields indexed by the time. In general, this process is not a semi-martingale, and therefore, we cannot rely on stochastic differential calculus to evaluate options. However, option prices are still calculable by the technique of the change of numeraire. A numerical illustration confirms the important influence of the covariance structure in the valuation of calendar spread exchange options for Brent against WTI crude oil and for gold against silver.

Ensemble-based estimates of eigenvector error for empirical covariance matrices

Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $\{\lambda _i\}$ and eigenvectors $\{\boldsymbol{u}_i\}$ of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $n$ to obtain empirical eigenvalues $\{\tilde{\lambda }_i\}$ and eigenvectors $\{\tilde{\boldsymbol{u}}_i\}$, and therefore understanding the error so introduced is of central importance. We analyse eigenvector error $\|\boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2$ while leveraging the assumption that the true covariance matrix having size $p$ is drawn from a matrix ensemble with known spectral properties—particularly, we assume the distribution of population eigenvalues weakly converges as $p\to \infty $ to a spectral density $\rho (\lambda )$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $p$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $\lambda $ and approximate the distribution of the expected square error $r= \mathbb{E}\left [\| \boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2\right ]$ across the matrix ensemble for all $\boldsymbol{u}_i$ associated with $\lambda _i=\lambda $. We find, for example, that for sufficiently large matrix size $p$ and sample size $n> p$, the probability density of $r$ scales as $1/nr^2$. This power-law scaling implies that the eigenvector error is extremely heterogeneous—even if $r$ is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments.