Multivariate Standard Normal Distribution Research Articles

Distinguishing newforms by the prime divisors of their Fourier coefficients

Given two non-CM newforms with integral Fourier coefficients, in this paper we study the number of distinct prime divisors of their Fourier coefficients in a probability way. Based on a multivariate version of the Erdős-Kac theorem, using the Galois representations attached to newforms and the effective Chebotarev density theorem, and assuming the generalized Riemann hypothesis, we show that the distribution of the number of distinct primes dividing the Fourier coefficients behaves like the standard multivariate normal distribution if these newforms are not twists of each other. As a consequence, we prove a multiplicity one result for modular forms under the generalized Riemann hypothesis.

High-dimensional scaling limits of piecewise deterministic sampling algorithms

Piecewise deterministic Markov processes are an important new tool in the design of Markov chain Monte Carlo algorithms. Two examples of fundamental importance are the bouncy particle sampler (BPS) and the zig–zag process (ZZ). In this paper scaling limits for both algorithms are determined. Here the dimensionality of the space tends towards infinity and the target distribution is the multivariate standard normal distribution. For several quantities of interest (angular momentum, first coordinate and negative log-density) the scaling limits show qualitatively very different and rich behaviour. Based on these scaling limits the performance of the two algorithms in high dimensions can be compared. Although for angular momentum both processes require only a computational effort of O(d) to obtain approximately independent samples, the computational effort for negative log-density and first coordinate differ: for these BPS requires O(d2) computational effort whereas ZZ requires O(d). Finally we provide a criterion for the choice of the refreshment rate of BPS.

Closed‐form expressions for maximum mean discrepancy with applications to Wasserstein auto‐encoders

The maximum mean discrepancy (MMD) has found numerous applications in statistics and machine learning, among which is its use as a penalty in the Wasserstein auto‐encoder (WAE). In this paper, we compute closed‐form expressions for estimating the Gaussian kernel‐based MMD between a given distribution and the standard multivariate normal distribution. This formula reveals a connection to the Baringhaus–Henze–Epps–Pulley (BHEP) statistic of the Henze–Zirkler test and provides further insights about the MMD. We introduce the standardized version of MMD as a penalty for the WAE training objective, allowing for a better interpretability of MMD values and more compatibility across different hyperparameter settings. Next, we propose code normalization–using batch normalization at the code layer–which has the benefits of making the kernel width selection easier, reducing the training effort, and preventing outliers in the aggregate code distribution. Our experiments on synthetic and real data show that the analytic formulation improves over the commonly used stochastic approximation of the MMD and demonstrate that code normalization provides significant benefits when training WAEs.

Gravitational-wave parameter estimation with autoregressive neural network flows

We introduce the use of autoregressive normalizing flows for rapid likelihood-free inference of binary black hole system parameters from gravitational-wave data with deep neural networks. A normalizing flow is an invertible mapping on a sample space that can be used to induce a transformation from a simple probability distribution to a more complex one: if the simple distribution can be rapidly sampled and its density evaluated, then so can the complex distribution. Our first application to gravitational waves uses an autoregressive flow, conditioned on detector strain data, to map a multivariate standard normal distribution into the posterior distribution over system parameters. We train the model on artificial strain data consisting of IMRPhenomPv2 waveforms drawn from a five-parameter $(m_1, m_2, \phi_0, t_c, d_L)$ prior and stationary Gaussian noise realizations with a fixed power spectral density. This gives performance comparable to current best deep-learning approaches to gravitational-wave parameter estimation. We then build a more powerful latent variable model by incorporating autoregressive flows within the variational autoencoder framework. This model has performance comparable to Markov chain Monte Carlo and, in particular, successfully models the multimodal $\phi_0$ posterior. Finally, we train the autoregressive latent variable model on an expanded parameter space, including also aligned spins $(\chi_{1z}, \chi_{2z})$ and binary inclination $\theta_{JN}$, and show that all parameters and degeneracies are well-recovered. In all cases, sampling is extremely fast, requiring less than two seconds to draw $10^4$ posterior samples.

An Extension of the Quadratic Error Function for Learning Imprecise Data in Multivariate Nonlinear Regression

Multivariate noises in the learning process are most of the time supposed to follow a standard multivariate normal distribution. This hypothesis does not often hold in many real-world situations. In this paper, we consider an approach based on multivariate skew-normal distribution. It allows for a multiple continuous variation from normality to nonnormality. We give an extension of the generalized least squares error function in a context of multivariate nonlinear regression to learn imprecise data. The simulation study and application case on real datasets conducted and based on multilayer perceptron neural networks (MLP) with bivariate continuous response and asymmetric revealed a significant gain in precision using the new quadratic error function for these types of data rather than using a classical generalized least squares error function having any covariance matrix.

Second Order Expansions for High-Dimension Low-Sample-Size Data Statistics in Random Setting

We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we get three different limit distributions: Normal, Student’s t-, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples.

A Geometry-Based Multiple Testing Correction for Contingency Tables by Truncated Normal Distribution

Inference procedure is a critical step of experimental researches to draw scientific conclusions especially in multiple testing. The false positive rate increases unless the unadjusted marginal p-values are corrected. Therefore, a multiple testing correction is necessary to adjust the p-values based on the number of tests to control type I error. We propose a multiple testing correction of MAX-test for a contingency table, where multiple χ2-tests are applied based on a truncated normal distribution (TND) estimation method by Botev. The table and tests are defined geometrically by contour hyperplanes in the degrees of freedom (df) dimensional space. A linear algebraic method called spherization transforms the shape of the space, defined by the contour hyperplanes of the distribution of tables sharing the same marginal counts. So, the stochastic distributions of these tables are transformed into a standard multivariate normal distribution in df-dimensional space. Geometrically, the p-value is defined by a convex polytope consisted of truncating hyperplanes of test’s contour lines in df-dimensional space. The TND approach of the Botev method was used to estimate the corrected p. Finally, the features of our approach were extracted using a real GWAS data.

Standardization of multivariate Gaussian mixture models and background adjustment of PET images in brain oncology

In brain oncology, it is routine to evaluate the progress or remission of the disease based on the differences between a pre-treatment and a post-treatment Positron Emission Tomography (PET) scan. Background adjustment is necessary to reduce confounding by tissue-dependent changes not related to the disease. When modeling the voxel intensities for the two scans as a bivariate Gaussian mixture, background adjustment translates into standardizing the mixture at each voxel, while tumor lesions present themselves as outliers to be detected. In this paper, we address the question of how to standardize the mixture to a standard multivariate normal distribution, so that the outliers (i.e., tumor lesions) can be detected using a statistical test. We show theoretically and numerically that the tail distribution of the standardized scores is favorably close to standard normal in a wide range of scenarios while being conservative at the tails, validating voxelwise hypothesis testing based on standardized scores. To address standardization in spatially heterogeneous image data, we propose a spatial and robust multivariate expectation-maximization (EM) algorithm, where prior class membership probabilities are provided by transformation of spatial probability template maps and the estimation of the class mean and covariances are robust to outliers. Simulations in both univariate and bivariate cases suggest that standardized scores with soft assignment have tail probabilities that are either very close to or more conservative than standard normal. The proposed methods are applied to a real data set from a PET phantom experiment, yet they are generic and can be used in other contexts.

A Comparison of the Separate and Concurrent Calibration Methods for the Full-Information Bifactor model.

When calibrating items using multidimensional item response theory (MIRT) models, item response theory (IRT) calibration programs typically set the probability density of latent variables to a multivariate standard normal distribution to handle three types of indeterminacies: (a) the location of the origin, (b) the unit of measurement along each coordinate axis, and (c) the orientation of the coordinate axes. However, by doing so, item parameter estimates obtained from two independent calibration runs on nonequivalent groups are on two different coordinate systems. To handle this issue and place all the item parameter estimates on a common coordinate system, a process called linking is necessary. Although various linking methods have been introduced and studied for the full MIRT model, little research has been conducted on linking methods for the bifactor model. Thus, the purpose of this study was to provide detailed descriptions of two separate calibration methods and the concurrent calibration method for the bifactor model and to compare the three linking methods through simulation. In general, the concurrent calibration method provided more accurate linking results than the two separate calibration methods, demonstrating better recovery of the item parameters, item characteristic surfaces, and expected score distribution.

Embedding optimal transports in statistical manifolds

We consider Monge-Kantorovich optimal transport problems on ℝ d , d ≥ 1, with a convex cost function given by the cumulant generating function of a probability measure. Examples include theWasserstein-2 transport whose cost function is the square of the Euclidean distance and corresponds to the cumulant generating function of the multivariate standard normal distribution. The optimal coupling is usually described via an extended notion of convex/concave functions and their gradient maps. These extended notions are nonintuitive and do not satisfy useful inequalities such as Jensen’s inequality. Under mild regularity conditions, we show that all such extended gradient maps can be recovered as the usual supergradients of a nonnegative concave function on the space of probability distributions. This embedding provides a universal geometry for all such optimal transports and an unexpected connection with information geometry of exponential families of distributions.

Robust mixture multivariate linear regression by multivariate Laplace distribution

Assuming that the error terms follow a multivariate Laplace distribution, we propose a robust estimation procedure for mixture of multivariate linear regression models in this paper. Using the fact that the multivariate Laplace distribution is a scale mixture of the multivariate standard normal distribution, an efficient EM algorithm is designed to implement the proposed robust estimation procedure. The performance of the proposed algorithm is thoroughly evaluated by some simulation and comparison studies.

Adjusted adaptive Lasso for covariate model-building in nonlinear mixed-effect pharmacokinetic models.

One important aim in population pharmacokinetics (PK) and pharmacodynamics is identification and quantification of the relationships between the parameters and covariates. Lasso has been suggested as a technique for simultaneous estimation and covariate selection. In linear regression, it has been shown that Lasso possesses no oracle properties, which means it asymptotically performs as though the true underlying model was given in advance. Adaptive Lasso (ALasso) with appropriate initial weights is claimed to possess oracle properties; however, it can lead to poor predictive performance when there is multicollinearity between covariates. This simulation study implemented a new version of ALasso, called adjusted ALasso (AALasso), to take into account the ratio of the standard error of the maximum likelihood (ML) estimator to the ML coefficient as the initial weight in ALasso to deal with multicollinearity in non-linear mixed-effect models. The performance of AALasso was compared with that of ALasso and Lasso. PK data was simulated in four set-ups from a one-compartment bolus input model. Covariates were created by sampling from a multivariate standard normal distribution with no, low (0.2), moderate (0.5) or high (0.7) correlation. The true covariates influenced only clearance at different magnitudes. AALasso, ALasso and Lasso were compared in terms of mean absolute prediction error and error of the estimated covariate coefficient. The results show that AALasso performed better in small data sets, even in those in which a high correlation existed between covariates. This makes AALasso a promising method for covariate selection in nonlinear mixed-effect models.

Robust estimation of correlation coefficients among soil parameters under the multivariate normal framework

Based on limited amount of multivariate soil data Y, it is only possible to reliably estimate the marginal distributions and the correlations. A common practical approach of constructing the multivariate probability distribution of Y is to transform Y into standard normal data X and construct the multivariate standard normal distribution for X. This method is called the translation method. Its success depends on whether the Pearson product-moment correlations (δij) for X can be robustly estimated. This paper investigates the robustness for four methods of estimating δij. The emphasis is on the statistical uncertainty in the estimated δij when the amount of soil data is limited. It is found that the well known method that maps the Pearson correlations for Y to δij is the least robust, suffering the most significant statistical uncertainty. The causes for this non-robustness are investigated. The two methods that map the Spearman and Kendall rank correlations for Y to δij are quite robust. The method that converts Y to X and directly estimates δij is also robust as long as the conversion is based on properly chosen marginal distributions.

A Robust Multivariate EWMA Control Chart for Detecting Sparse Mean Shifts

In multivariate statistical process control (MSPC) applications, process mean shifts sometimes occur in only a few components. To solve this MSPC problem, many control charts were proposed in the literature. Most of these charts assumed that the multivariate quality characteristics are normally distributed. Among them, the control chart proposed by Zou and Qiu (2009), incorporating the least absolute shrinkage and selection operator (LASSO) method into the EWMA scheme, has the best overall performance. In this paper, we extend the classical multivariate LASSO control chart to a robust version that has an affine-invariance property and is distribution free under the family of elliptical direction distributions, indicating that the in-control run-length distribution is the same for any continuous distribution in this family and the control limit can be acquired from the multivariate standard normal distribution. Our simulation results show that the proposed method is very efficient in detecting various sparse shifts under heavy-tailed and skewed multivariate distributions. In addition, it is easy to implement with an iterative algorithm and the least angle regression (LARS) algorithm. White-wine data illustrates that the proposed control chart performs quite well in applications.

The internal branch lengths of the Kingman coalescent

In the Kingman coalescent tree the length of order $r$ is defined as the sum of the lengths of all branches that support $r$ leaves. For $r=1$ these branches are external, while for $r\ge2$ they are internal and carry a subtree with $r$ leaves. In this paper we prove that for any $s\in\mathbb{N}$ the vector of rescaled lengths of orders $1\le r\le s$ converges to the multivariate standard normal distribution as the number of leaves of the Kingman coalescent tends to infinity. To this end we use a coupling argument which shows that for any $r\ge2$ the (internal) length of order $r$ behaves asymptotically in the same way as the length of order 1 (i.e., the external length).

Generalized Linear Factor Models: A New Local EM Estimation Algorithm

In this article, a general approach to latent variable models based on an underlying generalized linear model (GLM) with factor analysis observation process is introduced. We call these models Generalized Linear Factor Models (GLFM). The observations are produced from a general model framework that involves observed and latent variables that are assumed to be distributed in the exponential family. More specifically, we concentrate on situations where the observed variables are both discretely measured (e.g., binomial, Poisson) and continuously distributed (e.g., gamma). The common latent factors are assumed to be independent with a standard multivariate normal distribution. Practical details of training such models with a new local expectation-maximization (EM) algorithm, which can be considered as a generalized EM-type algorithm, are also discussed. In conjunction with an approximated version of the Fisher score algorithm (FSA), we show how to calculate maximum likelihood estimates of the model parameters, and to yield inferences about the unobservable path of the common factors. The methodology is illustrated by an extensive Monte Carlo simulation study and the results show promising performance.

Comparison of hierarchical cluster analysis methods by cophenetic correlation

This study proposes the best clustering method(s) for different distance measures under two different conditions using the cophenetic correlation coefficient. In the first one, the data has multivariate standard normal distribution without outliers for and the second one is with outliers (5%) for . The proposed method is applied to simulated multivariate normal data via MATLAB software. According the results of simulation the Average (especially for ) and Centroid (especially for and ) methods are recommended at both conditions. This study hopes to contribute to literature for making better decisions on selection of appropriate cluster methods by using subgroup sizes, variable numbers, subgroup means and variances.

A probabilistic approach to calculating the reliability of over-year storage reservoirs with persistent gaussian inflow

► A method to calculate the reliability of reservoirs with persistent Gaussian inflow. ► The method is based on the probability distribution of the critical period. ► Calculations are based only on the autocorrelation function of the Gaussian inflow. ► The method is useful in planning stage prior to using a detailed simulation model. The reliability of over-year storage reservoirs has been traditionally estimated by using Monte Carlo simulation (behavior analysis), numerical solution of the stochastic reservoir equation (difference or integral equations), or by using approximations and empirical relationships. The case of persistent inflow, which is the case for most natural streams, adds complexity to the problem. Apart from simulation, which is the most flexible approach, other methods have had partial success in this case, and are mainly limited to the case of Markovian autocorrelation. This paper suggests an alternative method for calculating the reliability of over-year storage reservoirs with persistent gaussian inflow using basic probability principles and the numerical integration of the standard multi-variate normal distribution. It is shown that the classical probabilities of the reservoir being empty and those of spilling can be directly calculated based on the distribution of the critical drawdown and filling periods. The results obtained by the application of the proposed method compare very well with those obtained by Monte Carlo simulation and those reported in previous studies. The proposed method thus offers another alternative that should be useful in handling complex problems.

A UNIFIED APPROACH FOR SHEWHART-TYPE PHASE I CONTROL CHARTS FOR THE MEAN

The false alarm probability (FAP) is the metric typically used to design and evaluate the performance of Phase I control charts. It is shown that in situations where the exact or the asymptotic joint p.d.f. of the standardized charting statistics follows a singular standard multivariate normal distribution with a common negative correlation, the FAP of some Shewhart-type Phase I charts for the mean can be expressed as a multiple integral of the joint p.d.f. Hence the required charting constants can be calculated (and the control chart can be implemented) by evaluating this integral. A table with the charting constants is provided for some popular choices of the nominal FAP (denoted FAP0) and the number of Phase I samples, m. The proposed methodology is useful to unify some existing Phase I charts and is illustrated with two charts from the literature: the p chart for the fraction nonconforming and the [Formula: see text] chart for the mean. A summary and some conclusions are provided.

Power, selection bias and predictive performance of the Population Pharmacokinetic Covariate Model.

Identification and quantification of covariate relations is often an important part of population pharmacokinetic/pharmacodynamic (PK/PD) modelling. The covariate model is regularly built in a stepwise manner. With such methods, selection bias may be a problem if only statistically significant covariates are accepted into the model. Competition between multiple covariates may further increase selection bias, especially when there is a moderate to high correlation between the covariates. This can also result in a loss of power to find the true covariates. The aim of this simulation study was to investigate the effect on power, selection bias and predictive performance of the covariate model, when altering study design and system-related quantities. Data sets with 20-1000 subjects were investigated. Five covariates were created by sampling from a multivariate standard normal distribution. The true covariate was set up to have no, low, moderate and high correlation to the other four covariates, respectively. Data sets, in which each individual had two or three PK observations, were simulated using a one-compartment i.v. bolus model. The true covariate influenced clearance according to one of several magnitudes. Different magnitudes of residual error and inter-individual variability in the structural model parameters were also introduced to the simulation model. A total of 7400 replicate data sets were simulated independently for each combination of the above conditions. Models with one of the five simulated covariates influencing clearance and the model without any covariate were fitted to the data. The probability of selecting (according to a pre-specified P-value) the different covariates, along with the estimated covariate coefficient, was recorded. The results show that selection bias is very high for small data sets (< or = 50 subjects) simulated with a weak covariate effect. If selected under these circumstances, the covariate coefficient is on average estimated to be more than twice its true value, making the covariate model useless for predictive purposes. Surprisingly, even though competition from false covariates caused substantial loss in the power of selecting the true covariate, the already high selection bias increased only marginally. This means that the bias due to competition is negligible if statistical significance is also required for covariate selection. Bias and predictive performance are direct functions of power, only indirectly affected by study design and system-related quantities. Mainly because of selection bias, low-powered covariates can be expected to harm the predictive performance when selected. For the same reason these low-powered covariates may falsely appear to be clinically relevant when selected. If the aim of an analysis is predictive modelling, we do not recommend stepwise selection or significance testing of covariates to be performed on small or moderately sized data sets (<50-100 subjects).