User soroosh - MathOverflow

Two-Factor Factor Analysis Model

2012-11-20T04:31:00Z

Hi everyone,

I want to build the following factor analysis model and I've samples observations of data (i.e., $m(s)$), labeled with factor $s$. The observations are affected by both $y(s)$ and also an external factor (i.e., $x$). But, I don't know how to fit the model parameters and also estimate the factors ($x$ and $y$), for a given observation.

Thanks for your help.

$m(s) = \mu + Vy(s) + Ux$

$m(s)_{~n\times 1}$

$\mu_{~n\times 1}$

$V_{~n\times p}$

$y(s)_{~k\times 1}$

$U_{~n\times q}$

$x_{~q\times 1}$

$x,y \sim ~ N(0,I)$

$p,q < n$

Deriving Conditional Probability Distribution

2012-04-29T04:04:55Z

Given the following joint normal PDF of $X \in \mathcal{R}^K$ and $Y,Z \in \mathcal{R}$

$p(\begin{bmatrix} X \\ Y \\ Z \end{bmatrix}) = \mathcal{N}(\begin{bmatrix} \mu_X \\ \mu_Y \\ \mu_Z \end{bmatrix},\begin{bmatrix} \Sigma_{XX} \ \Sigma_{XY} \ \Sigma_{XZ} \\ \Sigma_{YX} \ \Sigma_{YY} \ \Sigma_{YZ} \\ \Sigma_{ZX} \ \Sigma_{ZY} \ \Sigma_{ZZ} \end{bmatrix})$

Can we derive the closed form expression for the following PDF?

$P(X|A)$ (or equivalently $P(X|A^2)$)

where, $A = \sqrt{Y^2+Z^2}$

Derivative in Matrix Calculus

2012-03-01T19:38:29Z

Hi everyone, Given the two full rank matrices $X$ and $A$,

$X_{n\times n},~~(rank(X) = n)$

$A_{m\times n},~~(rank(A) = m \le n)$

Can I get a closed form expression for the following derivative? Thanks in advance.

$\frac{\partial det(X-XA'(AXA')^{-1}AX)}{\partial A}=?$

Robust entropy-like measure for analyzing uncertainity

2011-09-14T20:39:21Z

I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which results into the following equation for MVG distributions:

H(s) = ln(sqrt((2πe)^k*det(cov)))

H(s) = 0.5*[k*ln(2πe)+sum(log(eigs))]

Where, Sigma(Σ) is the covariance matrix. Thus, it's just a constant term plus sum of logarithm of eigenvalues of covariance matrix. But, when I've a small eigenvalue among eigenvalues, the small value affects the whole thing dramatically. In other words, this measure is not robust to small eigenvalues. On the other hand if I use sum of eigenvalues itself (instead of logarithmic scales), I won't face this issue. I was wondering if there is any other measures of uncertainty which may result in to sum of eigs instead of sum of log(eigs)?

closed form expression for Rényi entropy for multivariate Gaussian distributions

2011-09-15T16:30:21Z

Is there any closed form expression for Rényi entropy of a set variables with multivariate Gaussian distribution?

Comment by Soroosh

2012-04-29T04:12:38Z

You are right. My question was not clear enough. So, I've posted the correct and precise problem (the following link). I'll remove this post. :) mathoverflow.net/questions/95486/…

Comment by Soroosh

2012-04-16T02:38:57Z

I assume the product measure implies their independence. However, I'm interested in the dependency between the variables. The variable with Rayleigh distribution (i.e., $R$) is actually the magnitude of a complex number with i.i.d imaginary and real parts. $R$ is the variable of interest in my problem. Now, given a set of normally distributed variables (i.e., $X$), I want to estimate $R$. Therefore, I want to build the joint distribution of $R$ and $X$.

Comment by Soroosh

2012-03-01T23:22:19Z

Thanks for your detailed and helpful response.

Comment by Soroosh

2011-09-14T23:14:23Z

Is this "total variance" measure used as a measure of uncertainty in literature?