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1
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1answer
33 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
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0answers
40 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
20 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 $S$ ...
1
vote
1answer
37 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
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0answers
94 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 ...
3
votes
1answer
134 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
68 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
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0answers
54 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
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0answers
58 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
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0answers
29 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
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0answers
201 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
191 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
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1answer
124 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
2k 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
723 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
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5answers
599 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
113 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
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0answers
281 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
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1answer
778 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
442 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
291 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 ...