Questions tagged [multivariate-distribution]

Probability distribution over vectors (as opposed to univariate distributions that are over numbers).

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28
votes
4answers
15k views

How to determine quantiles (isolines?) of a multivariate normal distribution

I'm interested in how one can calculate a quantile of a multivariate distribution. In the figures, I have drawn the 5% and 95% quantiles of a given univariate normal distribution (left). For the right ...
14
votes
4answers
2k views

Is this causation?

Consider the following joint distribution for the random variables $A$ and $B$: $$ \begin{array} {|r|r|}\hline & B=1 & B=2 \\ \hline A=1 & 49\% & 1\% \\ \hline A=2 & 49\% & 1\...
11
votes
2answers
4k views

Algorithms for computing multivariate Empirical distribution function (ECDF)?

One dimensional ECDF is fairly easy to compute. When it comes to two dimensions and up, however, online resources become sparse and hard to reach. Can anyone suggest, define and/or present efficient ...
8
votes
3answers
11k views

Generating Multivariate Uniform Distribution in R

Let $d$ a positive integer How to generate a sample of $n$ random variables with a multivariate uniform distribution on the cube $[a,b]^d$ in R? I don't know what to do. I know that I need a ...
8
votes
2answers
5k views

Multivariate Laplace distribution

I find it strange, but I can't find what multivariate Laplace distribution looks like. What is its pdf? I googled for a while but couldn't find a good description. I wasn't paying attention to ...
8
votes
2answers
741 views

Generate a random set of numbers with fixed sum and desired means and variances?

The Dirichlet distribution allows you to generate a sample of numbers $x_i$ with a prescribed sum, say $\sum_i x_i = 1$. Moreover, the parameters $\alpha$ allow some degree of control on the means of ...
7
votes
1answer
170 views

Probability distribution associated with nuclear norm?

The $\ell_1$ norm of model parameters is often added to loss functions because it induces sparsity in the solution of the overall cost function: $$ c(\theta) = -\log L(x|\theta) + \lambda ||\theta||_1$...
7
votes
0answers
146 views

Is multivariate Cauchy stable?

I am trying to prove (if possible), for given $A_{n\times n}$ and $B_{n\times n}$, there exists a $C_{n\times n}$ satisfying $$A\pmb{X}_1 + B\pmb{X}_2 \stackrel{D}{=} C\pmb{X},$$ where $X_1, ~X_2$, ...
6
votes
1answer
415 views

multivariate Student's t distribution: intuition for non-independence?

Consider a multivariate Student's t distribution, with parameters $\nu$ (d.f.), $\mu$ (location) and $\Sigma$ (shape). Does anyone have a good intuition for the individual components not being ...
5
votes
2answers
133 views

How to show that $\left|\left|E\left[\frac{x-X}{||x-X||}\right]\right|\right|\xrightarrow{||x||\to\infty}1$ for a multivariate distribution?

I am trying to prove that $\left|\left|E\left[\frac{x-X}{||x-X||}\right]\right|\right|\xrightarrow{||x||\to\infty}1$ for any multivariate distribution. My Progress: I have proved the result for ...
5
votes
2answers
266 views

Something like Mahalanobis distance when the copula is not Gaussian

Mahalanobis distance accounts for different variances of the marginal variables and correlations between the marginal variables. However, there is an implicit (maybe explicit) assumption that ...
5
votes
2answers
79 views

How to compare a new measurement to an existing multivariate distribution?

I have a dataset that describes the position and rotation of an object at different points in time using four dimensions. I want to use this sample of observations to get a sense of what positions and ...
5
votes
0answers
85 views

Copula entropy: calculation is borked?

I came across a pretty cool paper whose idea makes a lot of sense to me. Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-...
5
votes
0answers
524 views

Can we apply a constraint on the distribution of the layer output?

As far I understood, the hidden layer outputs can be anything based on the learning algorithm or optimization rules. I was wondering if it possible to some constraints on the layer output. For ...
4
votes
3answers
4k views

What's the difference between Multivariate Gaussian and Mixture of Gaussians?

What's the difference between Multivariate Gaussian and Mixture of Gaussians? If I have a Multivariate Gaussian and making all the data into ONE vector, is that a Mixture of Gaussians in 1 dimension?...
4
votes
1answer
493 views

What is the best way to sample points from an arbitrary 2D distribution?

I want to sample points $(x,y)$ randomly according to the Himmelblau function $$f(x,y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2\qquad -5\le x,y\le 5$$ which I treat as a multivariate probability density ...
4
votes
1answer
521 views

KL divergence for joint probability distributions?

I have a pair of joint probability distributions. I want to measure their similarity/dissimilarity. If they were single-dimensional probability distributions, then I could measure the Kullback–Leibler ...
4
votes
1answer
1k views

Are the marginals of the multivariate t distribution univariate Student t distributions?

Are the marginals of the Multivariate t distribution with $\nu$ degrees of freedom univariate Student t distributions with $\nu$ degrees of freedom?
4
votes
1answer
489 views

Multivariate normal distribution transformation

Suppose that $X $ has a multivariate normal distribution $X\sim MVN (\mu, \Sigma) $, How can I transform $X$ into $Z$ so that $Z\sim MVN(\mu, I) $ where $I$ is the identity matrix? For instance, ...
4
votes
1answer
232 views

Correlation Coefficient for Hypergeometric-type Distribution

Problem Statement: A box contains $N_1$ white balls, $N_2$ black balls, and $N_3$ red balls $(N_1+N_2+N_3=N).$ A random sample of $n$ balls is selected from the box (without replacement). Let $Y_1,Y_2,...
4
votes
2answers
295 views

Derive multivariate from bivariate normal distribution

Could anyone help me on the following. Let $K$ and $M$ be integers so that $K\geq3$ and $2\leq M < K$. Let $\boldsymbol{X}=(X_1, ..., X_K)^T$ be a random vector, $\boldsymbol{\mu}$ be a $K\times 1$...
4
votes
1answer
1k views

Multivariate conditional entropy

I would like to take data columns and compute the multivariate conditional entropy. For instance, suppose I have columns $A, B, C D, E$ and I want to compute the conditional entropy $H(E | A,B,C,D)$. ...
4
votes
1answer
147 views

Ornstein-Uhlenbeck process

Given a multivariate Ornstein-Uhlenbeck process that is a stochastic process, is it correct that each component of this process is a univariate Ornstein-Uhlenbeck process?
4
votes
1answer
796 views

Joint distribution of independent t-distributed random variables

The multivariate t distribution seems to be defined as a "ratio" of a vector of normal random variables and a single gamma (or chi-squared) random variable (independent from the vector of normal). ...
4
votes
1answer
170 views

A Multivariate Distribution for Linear Combinations of Independent Exponential Random Variables

Consider a random vector $\mathbf{X} \in \mathbb{R}^r$ whose components $X_j$ are independent exponential variables with different scale parameters $\beta_j$, $j=1,\dots,r$. Suppose I have a general $...
4
votes
2answers
143 views

Interpretation of multivariate conditional gaussian function form?

I've been reading over this Multivariate Gaussian conditional proof, trying to make sense of how the mean and variance of a gaussian conditional was derived. I've come to accept that unless I allocate ...
4
votes
0answers
69 views

Multivariate stable distribution

I know that if $\pmb{X}_1$ and $\pmb{X}_2$ are independent copies of a $n \times 1$ random vector $\pmb{X}$, then $\pmb{X}$ is said to be sum stable in $\mathbb{R}^n$ if $a\pmb{X}_1 + b\pmb{X}_2 \...
4
votes
0answers
111 views

Projection of a spherical t-distribution

I'm working with an $n$-dimensional spherical t-distribution for which the density is defined as $f(x) = \frac{\Gamma(\frac{n+\nu}{2})}{(\pi \sigma^2\nu)^{\frac n2}\Gamma(\frac \nu2)} \left( 1 + \...
3
votes
1answer
94 views

Can every multivariate distribution be expressed as a function of univariate distributions of the same random variables?

Can every multivariate distribution $p(X)$ of a multivariate random variable $X = [X_1, X_2, \dots, X_d]^{T} \in \mathbb{R}^d$, be defined as some function of univariate distributions on $X_i$? I ...
3
votes
2answers
299 views

What does i.i.d. mean for multivariate case?

When we say a random variable is i.i.d., it's often used to describe the dependency between the observations of that random variable, which I call the row dimension, indexed by time if it's a time ...
3
votes
3answers
2k views

Constraints on generating valid covariance matrices for multivariate normal distributions

I am interested in randomly generating multivariate normal distributions (MVND) as the underlying probability function to generate instances for a data stream. I understand that to do so requires two ...
3
votes
1answer
514 views

From beta distribution to Dirichlet: Estimation of the concentrantion parameters

Searching at least 3 hours about the connection between beta distribution and dirichlet. My problem is: I have a collection of random variables $X_i \sim Beta(a_i, b_i)$. The parameters $a_i$ and $...
3
votes
1answer
125 views

The mode of multivariate Gamma distribution

Let X, Y, Z be i.i.d. distributed Gamma random variables. What could the mode of the vector $(X, X+Y, X+Y+Z)$ be? Does the mode of a random vector equal the combination of the marginal modes?
3
votes
2answers
96 views

Can a random variable be uncorrelated with its product with a correlated random variable?

I have a random variable $X.$ I want to find a random variable $Y$ such that $Y$ is correlated with $X,$ but $Y$ is not correlated with the product of $X$ and $Y.$ Is it always possible?
3
votes
1answer
140 views

Mean of Generalization of the Dirichlet Distribution

I know that if $X_{1},X_{2},...X_{n}$ are independent $\mathrm{Gamma}(\alpha_{i},\theta)$ - distributed variables (notice they all have the same scale parameter $\theta$) and $Y_{i}=\frac{X_{i}}{\sum_{...
3
votes
1answer
203 views

Generating pairs of random variables with given covariance and gamma marginals

I have shape parameters $k_X, k_Y$ and scale parameters $\theta_X, \theta_Y$, as well as a covariance $\sigma_{XY}$. How do I generate random variables $(X,Y)$ such that the marginals are gamma ...
3
votes
1answer
2k views

Marginalizing a high-dimensional multivariate Gaussian distribution

I have an 11-dimensional multivariate Gaussian, with a covariance matrix with non-zero values in every element. My goal is to marginalize this down to 4 dimensions, but I'm having some computational ...
3
votes
2answers
835 views

Cholesky Decomposition And Multivariate Distributions

I'm studying multivariate distributions in general and I keep coming across an expression like: $X = \mu + A Y$ where $ \mu $ and $A $ are constant vectors of dimensions d x 1 and d x k respectively....
3
votes
0answers
56 views

If normally distributed random vector $X$ has a PD covariance matrix, then any conditional distribution induced by $X$ has a PD covariance matrix?

Suppose I have a random vector $X$ whose distribution is joint normal. I know that the covariance matrix of $X$ is positive definite. I wonder if I partition $X$ in any way: e.g., $X=(X_1, X_2)$, then ...
3
votes
0answers
97 views

Mutual information relationship to copula entropy is borked?

I have posted a related Question based on a paper, Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-54. In the paper, they ...
3
votes
0answers
59 views

Equality of two multivariate normal CDF's

Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
3
votes
0answers
91 views

Why is conditional independence more important than marginal independence?

Graphical models are based on the idea of representing certain types of conditional independences in a (joint) distribution via a graph, and are an active research area. As argued (correctly I believe ...
3
votes
0answers
84 views

Swiss Cheese Distributions

I am curious about a normal distribution with no probability mass in certain regions, sort of like the complement of the truncated normal. In particular, it will have zero mass in a circular region. ...
3
votes
0answers
581 views

GLM in Matrix Notation

I would like to verify my thoughts here concerning matrix notation of generalized linear models (i.e. generalized general linear models). A classical generalized linear model is given by $$ Y_i = h(\...
3
votes
0answers
76 views

What graphical model formalism should I use to model state-space model?

I have a state-space model of a greenhouse control model that I'd like to transform into a probabilistic graphical model (to make it easier for non-technical managers to understand relationships among ...
3
votes
1answer
113 views

Probability distribution transformation of variables question

Problem: Hi there, I'm stuck trying to derive an equation stated in a research paper relating to Bayesian statistics in Cosmology (the paper is: http://mnras.oxfordjournals.org/content/398/4/2049....
2
votes
2answers
2k views

How does Gibbs sampling produce values for a variable using the univariate conditional probability?

I have a question about Gibbs sampling for generating samples. The Gibbs sampling algorithm is often stated. $x^0 = (x_1^0, x_2^0, \ldots, x_n^0)$ //initialize random values for $t=1$ in $T$ //...
2
votes
1answer
64 views

Inner Product for Geometric Interpretation of Multivariate Random Vectors

I was looking into the geometric interpretation of random variables as random vectors in a vector space. The textbook I'm referring to defined $\operatorname {Cov}(X,Y)$ as the inner product for any ...
2
votes
2answers
61 views

Why is there covariance instead of simply variance in multidimensional normal distributions?

Maybe it is just because I'm only experimenting with 2 dimensional normal distributions, but multi-dimensional normal distributions for me seem like just multiple one dimensional normal distributions. ...
2
votes
2answers
68 views

How to prove a multivariate r.v. does not follow the nonparanormal distribution?

Background You may find the definition of the non-paranormal distribution at the 2nd paragraph in p.2296 of this paper. In short, $(X_1, \ldots, X_p)$ is non-paranormal if there exists a set of ...