Distribution of a Quadratic Form in Normal Vectors

Series representations of distributions of quadratic form in the normal vectors and generalised variance

Hotelling's T 2 is a generalization of the squared univariate t and used for testing hypotheses about a mean vector μ . Specifically, T 2 is a measure of the distance between a sample mean vector and a hypothesized population vector in the metric of the sample covariance matrix. T 2 can also be used for testing the equality of two independent multivariate normal mean vectors assuming that the populations have a common covariance matrix, for testing whether p means measured over time are equal, or the equality of response vectors measured over time. Rejection of a test based on T 2 does not indicate which of the p responses may have contributed to that decision.

Chapter 6 - Quadratic forms and Eigenvalue problems

The estimation of covariance matrices of gene expressions has many applications in cancer systems biology. Many gene expression studies, however, are hampered by low sample size and it has therefore become popular to increase sample size by collecting gene expression data across studies. Motivated by the traditional meta-analysis using random effects models, we present a hierarchical random covariance model and use it for the meta-analysis of gene correlation networks across 11 large-scale gene expression studies of diffuse large B-cell lymphoma (DLBCL). We suggest to use a maximum likelihood estimator for the underlying common covariance matrix and introduce an EM algorithm for estimation. By simulation experiments comparing the estimated covariance matrices by cophenetic correlation and Kullback-Leibler divergence the suggested estimator showed to perform better or not worse than a simple pooled estimator. In a posthoc analysis of the estimated common covariance matrix for the DLBCL data we were able to identify novel biologically meaningful gene correlation networks with eigengenes of prognostic value. In conclusion, the method seems to provide a generally applicable framework for meta-analysis, when multiple features are measured and believed to share a common covariance matrix obscured by study dependent noise.

Quadratic Forms and Graphs

In this paper the researchers are presenting an upper bound for the distribution function of quadratic forms in normal vector with mean zero and positive definite covariance matrix. They also will show that the new upper bound is more precise than the one introduced by Okamoto [4] and the one introduced by Siddiqui [5]. Theoretical Error bounds for both, the new and Okamoto upper bounds are derived in this paper. For larger number of terms in any given positive definite quadratic form, a rough and easier upper bound is suggested.

Let $x: 1 \times n$ be a row vector of random variables such that each $x_i (i = 1,2, \cdots, n)$ is independently normally distributed with mean $\mu_i$ and variance one. Many authors have studied the distribution of what we shall call a univariate quadratic form $xAx'$, where $A$ is a positive definite $n \times n$ matrix. Three types of representation of the distribution have been developed, namely (i) power series about the origin, (ii) mixtures of chi-squares (we refer to these as Ruben-type representations), and (iii) series of Laguerre polynomials. Though all three types of expansion yield correct convergent representations, it has been found that the Laguerre series representation is computationally the most convenient and effective throughout the range of interesting values of the argument. Let $X: p \times n$ be a matrix whose column vectors are independently and identically distributed in multivariate normal distributions having zero mean vector and variance covariance matrix $\Sigma$. If $L$ is a positive definite $n \times n$ matrix, we refer to the $p \times p$ matrix $S = XLX'$ as a positive definite quadratic form with matrix argument. Khatri [11] has given a representation of the density function of the distribution of $S$, somewhat similar to the Ruben-type expansion for the univariate case. In this paper we express the density function of $S$ in terms of Laguerre polynomials with matrix argument. Our results can be easily extended to quadratic forms in a matrix argument when the common multivariate distribution of the column vectors is complex. In Section 2, we give definitions and notations. Section 3 gives some results on integration over orthogonal groups, and in Section 4 we derive the main results.

13 - Matrices and related results

Approximations to the expectation of a ratio of quadratic forms in normal variables have been proposed by Ullah and Srivastava (1994) and Lieberman (1994), the former suggests a small disturbance approximation and the latter a Laplace approximation. Another approximation based on Nagars (1959) method is derived. Each proposal relies upon taking the expectation of an approximation to the ratio. This paper considers the expectation of the particular ratio where x is an n-dimensional normal random vector with non-zero mean and A is a symmetric matrix. The exact formula for E(r) is given in Smith (1993) and involves confluent hypergeometric functions. Asymptotic expansions for these functions are applied enabling a large n approximation and a small disturbance approximation to E(r) to be developed. The accuracies of the approximations are studied in a variety of settings. It is demonstrated that approximations based on large n asymptotics, such as the Nagar, are more reliable than other methods

The program FIT performs Krishnaiah’s finite intersection test procedure on the mean vectors from k multivariate populations. The test procedure is valid under the following assumptions: a) the k populations are distributed as multivariate normal, b) the covariance matrices of the k populations are equal. We can perform twosided or one-sided tests. The common covariance matrix, ∑ = (σij) may be unknown or known. When ∑ is unknown, the test statistics are distributed as multivariate F or multivariate t for the two-sided test or the one-sided test respectively. In the case when ∑ is known, then the test statistics are distributed as multivariate chi-square or multivariate normal for the two-sided test or the one-sided test respectively. The program FIT computes suitable bounds on the required percentage points of these distributions.

SUMMARY This paper is concerned with the characterization of the variance covariance matrix for a normal family in which certain quadratic forms are independent and follow cχ 2 distributions. It is also proved that the necessary and sufficient condition for any quadratic or linear form to be distributed independently of a sum of a finite number of non-negative definite quadratic forms is that it is distributed independently of each one of them.

Some characterizations of the multivariate t distribution

Many useful statistics equal the ratio of a possibly noncentral chi-square to a quadratic form in Gaussian variables with all positive weights. Expressing the density and distribution function as positively weighted sums of corresponding F functions has many advantages. The mixture forms have analytic value when embedded within a more complex problem. The mixture forms also have computational value. The expansions work well with quadratic forms having few components and small degrees of freedom. A more general algorithm from earlier literature can take longer or fail to converge in the same setting. Many approximations have been suggested for the problem. A positively weighted noncentral quadratic form can always have two moments matched to a noncentral chi-square. For a single quadratic form, the noncentral form performs neither uniformly more or less accurately than older approximations. The approach also gives a noncentral F approximation for any ratio of a positively weighted noncentral form to a positively weighted central quadratic form. The method provides better accuracy for noncentral ratios than approximations based on a single chi-square. The accuracy suffices for many practical applications, such as power analysis, even with few degrees of freedom. Naturally the approximation proves much faster and simpler to compute than any exact method. Embedding the approximation in analytic expressions provides simple forms which correctly guarantee only positive values have nonzero probabilities, and also automatically reduce to partially or fully exact results when either quadratic form has only one term.

A robust clustering method for detection of abnormal situations in a process with multiple steady-state operation modes

Uncorrelated Local Maximum Margin Criterion: An Efficient Dimensionality Reduction Method for Text Classification