Dimensional Gaussian Distribution Research Articles

Ensemble-based unsupervised machine learning method for membership determination of open clusters using Mahalanobis distance

ABSTRACT We present an improved method for the determination of membership of an open cluster using ensemble-based unsupervised machine learning techniques. The working principle of this method relies on two stages: (i) choosing a suitable range of three astrometric parameters (π, μαcos δ, μδ) using k-nearest neighbour (kNN) algorithm on the data downloaded for the cluster within a smaller search radius; (ii) application of two component Gaussian mixture modelling (GMM) on the resulting one dimensional Gaussian distribution of Mahalanobis distance (MD) of stars using the range of parameters obtained from the earlier step, but with the data downloaded within a bigger search radius. MD is calculated from the mean of each of the parameters in three dimensions. Thus the use of MD reduces the input of the GMM from the 3D parameter space into a 1D parameter space for the cluster membership determination. The method has been tested on a few clusters including those which have overlaps in some/all the parameters using the data obtained from the Gaia DR3 data base. It is found that the approach can easily separate the cluster members from the field stars. The clean colour-magnitude diagrams and similar direction of proper motions of the member stars obtained for the clusters shows that this method is very efficient and robust in segregating the cluster members from the field stars.

Singularity, misspecification and the convergence rate of EM

A line of recent work has analyzed the behavior of the Expectation-Maximization (EM) algorithm in the well-specified setting, in which the population likelihood is locally strongly concave around its maximizing argument. Examples include suitably separated Gaussian mixture models and mixtures of linear regressions. We consider over-specified settings in which the number of fitted components is larger than the number of components in the true distribution. Such mis-specified settings can lead to singularity in the Fisher information matrix, and moreover, the maximum likelihood estimator based on $n$ i.i.d. samples in $d$ dimensions can have a nonstandard $\mathcal{O}((d/n)^{\frac{1}{4}})$ rate of convergence. Focusing on the simple setting of two-component mixtures fit to a $d$-dimensional Gaussian distribution, we study the behavior of the EM algorithm both when the mixture weights are different (unbalanced case), and are equal (balanced case). Our analysis reveals a sharp distinction between these two cases: in the former, the EM algorithm converges geometrically to a point at Euclidean distance of $\mathcal{O}((d/n)^{\frac{1}{2}})$ from the true parameter, whereas in the latter case, the convergence rate is exponentially slower, and the fixed point has a much lower $\mathcal{O}((d/n)^{\frac{1}{4}})$ accuracy. Analysis of this singular case requires the introduction of some novel techniques: in particular, we make use of a careful form of localization in the associated empirical process, and develop a recursive argument to progressively sharpen the statistical rate.

Simultaneous Test for Independence Among Subvectors of Several Moderately High Dimensional Multivariate Normal Distributions

Simultaneous Test for Independence Among Subvectors of Several Moderately High Dimensional Multivariate Normal Distributions

Determinant of sample correlation matrix with application

Let $\mathbf{x}_{1},\ldots ,\mathbf{x}_{n}$ be independent random vectors of a common $p$-dimensional normal distribution with population correlation matrix $\mathbf{R}_{n}$. The sample correlation matrix $\hat{\mathbf {R}}_{n}=(\hat{r}_{ij})_{p\times p}$ is generated from $\mathbf{x}_{1},\ldots ,\mathbf{x}_{n}$ such that $\hat{r}_{ij}$ is the Pearson correlation coefficient between the $i$th column and the $j$th column of the data matrix $(\mathbf{x}_{1},\ldots ,\mathbf{x}_{n})'$. The matrix $\hat{\mathbf {R}}_{n}$ is a popular object in multivariate analysis and it has many connections to other problems. We derive a central limit theorem (CLT) for the logarithm of the determinant of $\hat{\mathbf {R}}_{n}$ for a big class of $\mathbf{R}_{n}$. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. In particular, the CLT holds if $p/n$ has a nonzero limit and the smallest eigenvalue of $\mathbf{R}_{n}$ is larger than $1/2$. Besides, a formula of the moments of $\vert \hat{\mathbf {R}}_{n}\vert $ and a new method of showing weak convergence are introduced. We apply the CLT to a high-dimensional statistical test.

On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey–Wilson, Al-Salam–Chihara and q-ultraspherical polynomials

We study properties of compactly supported, 4 parameter \newline $(\rho _{12},\rho _{23},\rho _{13},q)\in (-1,1)^{\times 4}$ family of continuous type 3 dimensional distributions, that have the property that for $q\rightarrow 1^{-}$ this family tends to some 3 dimensional Normal distribution. For $q=0$ we deal with 3 dimensional generalization of Kesten--McKay distribution. In a very special case when $\rho _{12}\rho _{13}\rho _{23}=q$ all one dimensional marginals are identical, semicircle distributions. We find both all marginal as well as all conditional distributions. Moreover, we find also families of polynomials that are orthogonalized by these one-dimensional margins and one-dimensional conditional distributions. Consequently, we find moments of both conditional and unconditional distributions of dimensions one and two. In particular, we show that all one-dimensional and two-dimensional conditional moments of, say, order $n$ and are polynomials of the same order $n$ in the conditioning random variables. Finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials. Among them are the famous Askey-Wilson, Al-Salam--Chihara polynomials considered in the complex, but conjugate, parameters, as well as q-Hermite and Rogers polynomials. It seems that this paper is one of the first papers that give a probabilistic interpretation of Rogers (continuous q-ultraspherical) polynomials.

An Efficient Non-Gaussian Sampling Method for High Sigma SRAM Yield Analysis

Yield 1 analysis of SRAM is a challenging issue, because the failure rates of SRAM cells are extremely small. In this article, an efficient non-Gaussian sampling method of cross entropy optimization is proposed for estimating the high sigma SRAM yield. Instead of sampling with the Gaussian distribution in existing methods, a non-Gaussian distribution, i.e., a joint one-dimensional generalized Pareto distribution and ( n -1)-dimensional Gaussian distribution, is taken as the function family of practical distribution, which is proved to be more suitable to fit the ideal distribution in the view of extreme failure event. To minimize the cross entropy between practical and ideal distributions, a sequential quadratic programing solver with multiple starting points strategy is applied for calculating the optimal parameters of practical distributions. Experimental results show that the proposed non-Gaussian sampling is a 2.2--4.1× speedup over the Gaussian sampling, on the whole, it is about a 1.6--2.3× speedup over state-of-the-art methods with low- and high-dimensional cases without loss of accuracy

A study of two high-dimensional likelihood ratio tests under alternative hypotheses

Let [Formula: see text] be a [Formula: see text]-dimensional normal distribution. Testing [Formula: see text] equal to a given matrix or [Formula: see text] equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension [Formula: see text] is fixed, it is known that the LRT statistics go to [Formula: see text]-distributions. When [Formula: see text] is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that [Formula: see text] and [Formula: see text] are proportional to each other. The condition [Formula: see text] suffices in our results.

Lyα emission-line reconstruction for high-zQSOs

We introduce an intrinsic Ly\alpha\ emission line profile reconstruction method for high-$z$ quasars (QSOs). This approach utilises a covariance matrix of emission line properties obtained from a large, moderate-$z$ ($2 \leq z \leq 2.5$), high signal to noise (S/N > 15) sample of BOSS QSOs. For each QSO, we complete a Monte Carlo Markov Chain fitting of the continuum and emission line properties and perform a visual quality assessment to construct a large database of robustly fit spectra. With this dataset, we construct a covariance matrix to describe the correlations between the high ionisation emission lines Ly\alpha, C IV, Si IV + O IV] and C III], and find it to be well approximated by an $N$-dimensional Gaussian distribution. This covariance matrix characterises the correlations between the line width, peak height and velocity offset from systemic while also allowing for the existence of broad and narrow line components for Ly\alpha\ and C IV. We illustrate how this covariance matrix allows us to statistically characterise the intrinsic Ly\alpha\ line solely from the observed spectrum redward of 1275\AA. This procedure can be used to reconstruct the intrinsic Ly\alpha\ line emission profile in cases where Ly\alpha\ may otherwise be obscured. Applying this reconstruction method to our sample of QSOs, we recovered the Ly\alpha\ line flux to within 15 per cent of the measured flux at 1205\AA\ (1220\AA) ~85 (90) per cent of the time.

Optimisation of the Population Monte Carlo algorithm: Application to constraining isocurvature models with cosmic microwave background data

AbstractWe optimise the parameters of the Population Monte Carlo algorithm using numerical simulations. The optimisation is based on an efficiency statistic related to the number of samples evaluated prior to convergence, and is applied to a D ‐dimensional Gaussian distribution to derive optimal scaling laws for the algorithm parameters. More complex distributions such as the banana and bimodal distributions are also studied. We apply these results to a cosmological parameter estimation problem that uses CMB anisotropy data from the WMAP nine‐year release to constrain a six parameter adiabatic model and a fifteen parameter admixture model, consisting of correlated adiabatic and isocurvature perturbations. In the case of the adiabatic model and the admixture model we find that the number of sample points increase by factors of 3 and 20, respectively, relative to the optimal Gaussian case. This is due to degeneracies in the underlying parameter space. The WMAP nine‐year data constrain the admixture model to have an isocurvature fraction of 36.3 ± 2.8 %. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Snow-melt flood frequency analysis by means of copula based 2D probability distributions for the Narew River in Poland

Abstract Study region Narew River in Northeastern Poland. Study focus Three methods for frequency analysis of snowmelt floods were compared. Two dimensional (2D) normal distribution and copula-based 2D probability distributions were applied to statistically describe floods with two parameters (flood peak Q max,f and flood volume V f ). Two copula functions from different classes – the elliptical Gaussian copula and Archimedean 1-parameter Gumbel–Hougaard copula – were evaluated based on measurements. New hydrological insights for the region The results indicated that the 2D normal probability distribution model gives a better probabilistic description of snowmelt floods characterized by the 2-dimensional random variable ( Q max,f , V f ) compared to the elliptical Gaussian copula and Archimedean 1-parameter Gumbel–Hougaard copula models, in particular from the view point of probability of exceedance as well as complexity and time of computation. Nevertheless, the copula approach offers a new perspective in estimating the 2D probability distribution for multidimensional random variables. Results showed that the 2D model for snowmelt floods built using the Gumbel–Hougaard copula is much better than the model built using the Gaussian copula.

Gradient Scan Gibbs Sampler: An Efficient Algorithm for High-Dimensional Gaussian Distributions

This paper deals with Gibbs samplers that include high dimensional conditional Gaussian distributions. It proposes an efficient algorithm that avoids the high dimensional Gaussian sampling and relies on a random excursion along a small set of directions. The algorithm is proved to converge, i.e. the drawn samples are asymptotically distributed according to the target distribution. Our main motivation is in inverse problems related to general linear observation models and their solution in a hierarchical Bayesian framework implemented through sampling algorithms. It finds direct applications in semi-blind/unsupervised methods as well as in some non-Gaussian methods. The paper provides an illustration focused on the unsupervised estimation for super-resolution methods.

Adaptive Bayesian inference in the Gaussian sequence model using exponential-variance priors

We revisit the problem of estimating the mean of an infinite dimensional normal distribution in a Bayesian paradigm. Of particular interest is obtaining adaptive estimation procedures so that the posterior distribution attains optimal rate of convergence without the knowledge of the true smoothness of the underlying parameter of interest. Belitser & Ghosal (2003) studied a class of power-variance priors and obtained adaptive posterior convergence rates assuming that the underlying smoothness lies inside a countable set on which the prior is specified. In this article, we propose a different class of exponential-variance priors, which leads to optimal rate of posterior convergence (up to a logarithmic factor) adaptively over all the smoothness levels in the positive real line. Our proposal draws a close parallel with signal estimation in a white noise model using rescaled Gaussian process prior with squared exponential covariance kernel.

Vehicle Moving State Recognition Method Based on Anomaly Detection and Its Applications on Dynamic In-car Navigation Systems

In order to efiectively determine whether a vehicle is turning or not, we propose a method to map arbitrary consecutive GPS heading information to two dimensional feature space. Through analyzing the distribution of the data, we decide to use two dimensional Gaussian distribution to build the anomaly detection model. After that, the feature space can be divided into two categories (one indicates the vehicle is traveling on a straight road and the other indicates it is doing a turning maneuver) by setting the threshold of the model. Finally, we make use of the labeled data and F-measure method to choose the threshold value. The experimental results show that the model built in this way has good generalization. Based on the above research achievement, a vehicle moving state recognition learning system for Dynamic In-Car Navigation Systems is designed and implemented, and this system is applied to improving the map-matching algorithm. The improved algorithm is tested on a complex urban road network and the experimental results show that the new algorithm can improve the performance of the junction match.

On the Properties of Estimates of Monotonic Mean Vectors for Multivariate Normal Distributions

Problems concerning estimation of parameters and determination the statistic, when it is known a priori that some of these parameters are subject to certain order restrictions, are of considerable interest. In the present paper, we consider the estimators of the monotonic mean vectors for two dimensional normal distributions and compare those with the unrestricted maximum likelihood estimators under two different cases. One case is that covariance matrices are known, the other one is that covariance matrices are completely unknown and unequal. We show that when the covariance matrices are known, under the squared error loss function which is similar to the mahalanobis distance, the obtained multivariate isotonic regression estimators, motivated by estimators given in Robertson et al. (1988), which are the estimators given by Sasabuchi et al. (1983) and Sasabuchi et al. (1992), have the smaller risk than the unrestricted maximum likelihood estimators uniformly, but when the covariance matrices are unknown and unequal, the estimators have the smaller risk than the unrestricted maximum likelihood estimators only over some special sets which are defined on the covariance matrices. To illustrate the results two numerical examples are presented.

Joint mutual information-based input variable selection for multivariate time series modeling

For modeling of multivariate time series, input variable selection is a key problem. This paper presents the estimation of joint mutual information and its application in input variable selection problems. Mutual information is a commonly used measure for variable selection. To improve the performance of input variable selection, we propose a novel high-dimensional mutual information estimator based on copula entropy, which is estimated by the truncated k-nearest neighbor method. Simulations on high dimensional Gaussian distributions substantiate the effectiveness of the proposed mutual information estimator. A relationship between the joint mutual information and the copula entropy is derived, which is used for joint mutual information estimation. Then the proposed estimator is applied to input variable selection for multivariate time series modeling based on the criterion of max dependency and max–min dependency. A stop criterion is proposed to terminate the selection process automatically. Simulation results show that the input variable selection method works well on both synthetic and real life dataset.

Convergence Analysis of the Variance in Gaussian Belief Propagation

It is known that Gaussian belief propagation (BP) is a low-complexity algorithm for (approximately) computing the marginal distribution of a high dimensional Gaussian distribution. However, in loopy factor graph, it is important to determine whether Gaussian BP converges. In general, the convergence conditions for Gaussian BP variances and means are not necessarily the same, and this paper focuses on the convergence condition of Gaussian BP variances. In particular, by describing the message-passing process of Gaussian BP as a set of updating functions, the necessary and sufficient convergence conditions of Gaussian BP variances are derived under both synchronous and asynchronous schedulings, with the converged variances proved to be independent of the initialization as long as it is chosen from the proposed set. The necessary and sufficient convergence condition is further expressed in the form of a semi-definite programming (SDP) optimization problem, thus can be verified more efficiently compared to the existing convergence condition based on computation tree. The relationship between the proposed convergence condition and the existing one based on computation tree is also established analytically. Numerical examples are presented to corroborate the established theories.

Efficient Gaussian Sampling for Solving Large-Scale Inverse Problems Using MCMC

The resolution of many large-scale inverse problems using MCMC methods requires a step of drawing samples from a high dimensional Gaussian distribution. While direct Gaussian sampling techniques, such as those based on Cholesky factorization, induce an excessive numerical complexity and memory requirement, sequential coordinate sampling methods present a low rate of convergence. Based on the reversible jump Markov chain framework, this paper proposes an efficient Gaussian sampling algorithm having a reduced computation cost and memory usage, while maintaining the theoretical convergence of the sampler. The main feature of the algorithm is to perform an approximate resolution of a linear system with a truncation level adjusted using a self-tuning adaptive scheme allowing to achieve the minimal computation cost per effective sample. The connection between this algorithm and some existing strategies is given and its performance is illustrated on a linear inverse problem of image resolution enhancement.

A Characteristics Grouping Algorithm in DHMM Speech Recognition

The paper introduced a speech feature grouping algorithm for the speech recognition system based traditional Markov in accordance with the large computation of the traditional hidden Markov model and the Viterbi algorithm as well as the Gaussian mixture distribution probability. For the speech characteristic parameters, clustering was executed by K-Means algorithm on the basis of the first and second segmentation, and then obtained the grouped characteristic parameters and the parameters to be grouped, and the speech samples can be divided into different characteristic group according to these two parameters. On this basis, a grouping training algorithm was proposed by using the redundant, which improved the accuracy of grouping the speech characteristic by clustering algorithm. Compared with the traditional HMM method, the amount of calculation can be reduced more than 60% in the case of ensuring the speech recognition rate. In the traditional HMM speech recognition system, and for the isolated word, the classic Viterbi algorithm perfectly solved the decoding problem with iterative approach in the mathematics. But in the speech recognition systems, for the large vocabulary in the practical application of acoustic decoding, the amount of computation that the algorithm required is considerable. Assuming the vocabulary that a speech recognition system can recognize is 1000, then to establish a model for each word, and assuming they have the same number of states N, in order to facilitate the estimation and calculation, these models must be connected to a large model. So, the state number of the large model is 1000 times larger than that of original model. Because the orders of computation that the Viterbi algorithm required is N2T (T is the number of input speech frame), the computation amount of Viterbi algorithm of large models increased four orders than that of the original phoneme model. Moreover, according to the experiment, it was found that the computation was mainly used for calculating the multi- dimensional (mixture) Gaussian distribution probability of the obtained observation vector for each state of every frame. Therefore, the real-time requirement for the small mobile device with large vocabulary HMM speech recognition system is difficult to meet because of the limit computing capability. Contrary to the large amount of computation for the Viterbi algorithm and the Gaussian mixture distribution, the paper presented a grouping model of speech characteristics, before the matching calculation of the input speech done, the grouping judgment was executed firstly and obtained the group that the input speech belonged to, then to calculate the HMM parameters of input speech matching with that of all the speech in the same group. Then combined with the improved HMM speech recognition system, the efficiency can be improved.

SU-E-T-232: Proton Source Modeling for Geant4 Monte Carlo Simulations.

To investigate the effect of initial proton beam source placement, distribution and angle on the proton dose distribution in a therapeutic nozzle using Geant4. We performed Geant4 Monte Carlo simulations of a passively scattered proton treatment nozzle. Accurate geometry including all elements in the treatment room was used. Protons were generated just inside the vacuum pipe using one of two models. First, a standard two dimensional Gaussian distribution of proton starting position was used with a small random angle added to the initial direction. The size of the Gaussian distribution and the random angle were set to match measured beam spot size and angular spread at the exit window. Second, a point source of protons further back in the vacuum pipe with a small random angle was used. The distance of the point source to exit window and the random angle were set to match the spot size and angular deviation used for the Gaussian distribution. Depth dose curves and orthogonal beam profiles were examined to determine changes between the two models. Orthogonal beam profiles for large apertures showed changes of up to 6.5% between the two models with the point source showing much better agreement with measured data. Depth dose curves and orthogonal profiles for small apertures were unaffected. For large apertures, the average difference compared to measured data was of 1.9% and 0.7% and the max difference was 5.0% and 1.6% for Gaussian and point sources, respectively. The point source more realistically models the proton distribution in the vacuum pipe by correlating the proton position with the direction. For certain scattering setups and large apertures point source modeling is necessary to accurate match measured data with Monte Carlo simulations.

On the computation of some properties of testing homogeneity of multivariate normal mean vectors against an order Restriction

In the present article we are interested in presenting some properties of testing homogeneity of multivariate normal mean vectors against an order restriction for two cases, the covariance matrices are known, and the case that they have an unknown scale factor. This problem of testing with these two different cases was considered by Sasabuchi et al. (1983) and Kulatunga and Sasabuchi (1984). They only derived the test statistic and studied its null distribution. In this article, we obtain the critical values for the proposed test statistic by Kulatunga and Sasabuchi (1984) for the first case, at different significance levels for some of the two and three dimensional normal distributions. The power and p-value are computed using Monte Carlo simulation. We consider the case that covariance matrices have an unknown scale factor. In this case the specific conditions are given which under those the estimator of the unknown scale factor does not exist. Also we derive the unique test statistic. Some properties of this test, for instance, the critical values and power are computed by simulation study.