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2
votes
0answers
32 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
37 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
51 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 ...
0
votes
0answers
18 views

Smallest Spectral Norm / Deviation Inequality

Consider $A_{m \times n}$ be an i.i.d. random matrix with finite first to fourth moments. There is a good number of asymptotic and non-asymptotic results regarding the spectral norm of $A$, $\|A\|_2$, ...
0
votes
0answers
7 views

Modeling data using matrices

I have multiple square matrices represent the relations between members of a club. Is there any statistical method to represent these metrices as one matrix.
1
vote
0answers
21 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, ...
0
votes
0answers
126 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
73 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 ...
5
votes
1answer
110 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 ...
3
votes
1answer
1k 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, ...
7
votes
2answers
520 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
431 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
93 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? ...
4
votes
0answers
243 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 ...
18
votes
1answer
684 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
351 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
266 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 ...