The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
0answers
7 views

Hypothesis testing for matrices of RVs

I looking for the correct statistical test for comparing matrices of random variables - my IV being a categorical variable and my DVs being n by n matrices of random variables (which are themselves ...
1
vote
0answers
51 views

Write $\mathop {\mathbb E}[(X AX^h)]$ in function of $\mathop {\mathbb E}[(X X^h)]$?

Suppose $X$ is an $i \times j$ random matrix. In addition, $X$ has complex i.i.d. normal entries with $0$ as mean. We define $A$ (of dimension $j \times j$) as a deterministic matrix. Is it ...
4
votes
0answers
40 views

How to check if a distribution has undefined variance?

How can I determine if experimental data comes from a distribution where the variance is undefined (e.g. the Cauchy distribution)? I honestly have no idea how to attack this problem in a sensible way,...
0
votes
0answers
8 views

How to determine if a lower rank approximation exists?

In all the literature I read on low-rank approximations, I have yet to run across situations in which people first check to see if a lower rank exists. My understanding is that a matrix whose ...
0
votes
0answers
20 views

Order statistics of the diagonal terms of an inverse Wishart matrix

I have a question about the inverse Wishart matrix. In my understanding, consider $\mathbf H$ is a $n\times n$ matrix with each elements are complex Gaussian with zero mean unit variance. Then $\...
0
votes
0answers
17 views

Random complex orthogonal matrix

How can I uniformly extract a random complex orthogonal matrix $Ω∈O(3,ℂ)$? It is easily found in the literature the uniform measure for unitary and real matrices, but I couldn't find anything about ...
0
votes
1answer
46 views

Convergence of eigenvectors and eigenvalues of matrix that converges

For each random variable $X=x$, there is a symmetric positive definite matrix $M(x)$. Suppose there is a set of samples of random matrix $M_1,M_2,...,M_n$, where each $M_i$ is calculated based on the ...
1
vote
1answer
62 views

Smoothing Kernel Preventing Simulation of Semi-Circle Law of Random Matrices

In trying to understand the properties of random matrices in the book "Plane Answers to Complex Questions" by R.Christensen, I came across the Semi-Circle Law, and tried to reproduce it with ...
0
votes
0answers
102 views

Eigenvectors of a Wishart matrix

I have been trying to find a good source (or clarifications) to help me understand this point. I am very new to random matrix theory so any pointers will be appreciated. Here is what I think I have ...
2
votes
1answer
35 views

Spectral norm of a sparse Gaussian matrix

Suppose $G$ is an $m \times n$ matrix such that each entry of $G$ is a standard normal variable. We know that the spectral norm of $G$ scales as $\sqrt m + \sqrt n$. Now, given a set of indices $S$ ...
2
votes
0answers
95 views

Difference between recentred and scaled eigenvalues and the Tracy Widom distribution

I have been generating correlation matrices from independent normal data simulated using the MASS package. I do this k times and extract the eigenvalues of the matrices. I was interested in comparing ...
1
vote
0answers
201 views

(0,1) matrix probabilities with column and row sum constraints

Given an arbitrary length set of values $c=[1, 2, 3, 4, 5]$ and a max value $m=3$, construct a random (0,1) matrix with non-zero rows such that the column sums equal $c$ and the row sums do not ...
4
votes
1answer
229 views

Distribution of Trace of non-centered Wishart matrix

I am looking for the distribution of trace of the non-central Wishart matrix with different variations along different axes. Is there a general formula for such distribution? If not, is there a ...
2
votes
0answers
90 views

Largest eigenvalue of this random 2x2 matrix

Consider four random complex vectors $\mu_i$ of length $K$ whose entries are drawn from the complex normal distribution $\mathcal{CN}(\mathbf{0},\mathbb{1})$ centered in zero and of unit variance. ...
1
vote
0answers
64 views

Covariance of random vector multiplied with a random matrix

For a random vector $x$ multiplied by a non-random matrix $A$, $y=Ax$ the covariance matrix of $y$ is given by $\Sigma_y = E[Ax (Ax)^T] = E[Ax x^T A^T] = A E[x x^T ]A^T = A \Sigma_x A^T$, where $\...
2
votes
0answers
68 views

Why linear transformation can improve classification accuracy when the dimensionality of data is high?

Let $X$ be an $m\times n$ ($m$: number of records, and $n$: number of attributes) dataset. When the number of attributes $n$ is large and the dataset $X$ is noisy, classification gets more ...
1
vote
0answers
33 views

Statistical distribution of a scaled complex Wishart matrix

Given that $\mathbf{x}\in\mathbb{C}^{N\times 1}$, where $\mathbf{x}\sim\mathcal{CN}(0,\mathbf{R})$ then $\mathbf{x}\mathbf{x}^\mathrm{H}$ is complex Wishart, i.e, $\mathbf{x}\mathbf{x}^\mathrm{H}\sim\...
1
vote
0answers
261 views

Generate correlated multivariate normal samples with copula

I've seen examples of constructing multivariate distribution with univariate marginals coupled together via a normal copula (see Mvdc function from copula package ...
1
vote
0answers
284 views

Generating and verifying uniformly distrubuted orthogonal matrices

I would like to generate a uniformly distributed $n \times n$ orthogonal matrix. There seems to be several such methods; see this question and the oft-cited paper by Stewart. Using a QR decompostion ...
6
votes
1answer
140 views

Distribution of eigenvalues given one is known

I'm familiar with using insights from Random Matrix Theory to determine the number of principal components from the PCA of a covariance/correlation matrix to use to form factors. If the eigenvalue ...
6
votes
1answer
5k views

Expected value and variance of trace function

For random variables $X \in \mathbb{R}^h$, and a positive semi-definite matrix $A$: Is there a simplified expression for the expected value, $\mathop {\mathbb E}[Tr(X^TAX)]$ and variance, $Var[Tr(X^...
7
votes
2answers
1k views

Generating random variables satisfying constraints

I need to generate a list of random variables $\bf{x}$ subject to constraints that can be expressed in the form $\bf{E}x=b$ where $\bf{E}$ is an $m \times n $ matrix if $\bf{x}$ has $n$ entries. In ...
6
votes
5answers
767 views

Generating random matrices with sum and maximality constraints

I'd like to generate a random square matrix such that the rows are normalized to one and the diagonal elements are the maximum of their column. If there an efficient way to sample these matrices ...
3
votes
1answer
118 views

Generation of orthogonal matrices “close” to identity

Suppose I want to generate a $n \times n$ orthogonal matrix $H$ (that is, $H^T H=I$) but with the property that $1-e < (tr H)/n < 1+e$ for some pre-specified tolerance $e$. How can I do this? ...
5
votes
0answers
316 views

Generating random matrices with specific equality constraints

Suppose I want to generate a nonnegative $n \times n$ matrix $\mathbf A$ for an odd $n$ (say, $n=5$ for a good enough example), such that the individual elements are drawn from a uniform ...
20
votes
1answer
864 views

Random matrices with constraints on row and column length

I need to generate random non-square matrices with $R$ rows and $C$ columns, elements randomly distributed with mean = 0, and constrained such that the length (L2 norm) of each row is $1$ and the ...
2
votes
1answer
539 views

Second moment of draws from a multivariate normal covariance matrix

Suppose I have an $N\times N$ covariance matrix that describes a multivariate normal joint distribution. Now take 100,000 draws of the covariance matrix. I measure the variance of values for each ...
3
votes
1answer
331 views

PCA with covariance matrix calculated using random matrix theory in R

I would like to perform a PCA and use the covariance matrix obtained by the random matrix theory. Is there an implementation of this in R? I am currently using the standard prcomp function from ...