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Questions tagged [multivariate-distribution]

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

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2answers
58 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 $K\times 1$...
0
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0answers
32 views

How Do I interpret this Q-Q plot?

I am new to statistics and I am having trouble interpreting the Q-Q plot above. Given the number of outliers in the plot, can I assume that the dataset being tested has a lot of bad data? Should I ...
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1answer
47 views

When does Bayesian classifier act as linear classifier?

I am reviewing my lectures in Machine Learning and my current topic is Bayesian Classifier. The context is the classification of two classes C1 and C2. My book (neural networks and learning machines ...
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0answers
8 views

How to calculate multiple correlation coefficients and test hypothesis?

I found the estimation of S and $\overline{x}$ of a data set to be approximately $\overline{x} = \begin{pmatrix} 185 & 151 & 183 & 149 \\ \end{pmatrix}'.$ and $S = \begin{pmatrix} 95 &...
0
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1answer
17 views

Correlation between two multivariate measures

I'm reading a paper, but I'm with a problem. The authors say: Let $\boldsymbol{X} = (X_1, \ldots, X_p)^T$ be a vector $m \times 1$ whose the estimative of variance is proportional to $\boldsymbol{\hat{...
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0answers
13 views

Frank copula with no dependence

From Wikipedia, the Frank copula is the function $C(u, v)$ such that: $$C(u, v) = \frac{1}{\theta} \log\!\left[ 1+\frac{(\exp(-\theta u)-1)(\exp(-\theta v)-1)}{\exp(-\theta)-1} \right]$$ for $\theta\...
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0answers
17 views

Sufficient conditions for multivariate MGF to be finite in neighborhood of zero

In the one-dimensional case, we have the following fact: (see Existence of the moment generating function and variance for proof) Proposition: The mgf $m(t)$ is finite in an open interval $(t_n,t_p)$ ...
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0answers
23 views

Estimating Pearson correlation for multivariate t distribution using Kendall rank correlation

I had a task to generate a certain amount of samples for $(X_{1},X_{2})$ with a bivariate $t$ distribution $t_{2}(\nu,\mu,\Sigma)$ and then estimate Pearson's correlation coefficient $\rho$ by using ...
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0answers
16 views

Projection of multivariate distribution to lower dimensional subspace

Say that $X \in \mathbb{R}^n$ is a vector of $n$ r.v.'s with pdf $p(x_1,\ldots,x_n)$. Let's consider now the linear map $Y = A X$ where $Y \in \mathbb{R}^m$ with $m < n$. I am seeking $p(y_1,\ldots,...
1
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1answer
59 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 $H(E | A,B,C,D)$. ...
0
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1answer
48 views

Multivariate distribution: calculate P(Y > b/2)

The joint probability function looks like this: The first step for calculating $P(Y > 2/b)$ is calculating $f_Y(y)$. Which I did like this: The problem here is that my x is still in my indicator,...
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1answer
96 views

Understanding this expression of the multivariate t-distribution

I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows $$ G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\ Z = N (...
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0answers
38 views

Multivariate $t$ ('mvt') - adjustment in emmeans in R

I am doing post-hoc comparisons of contrasts based on linear mixed models I built in R. I am using the emmeans package for the comparisons. One of the default ...
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0answers
98 views

multivariate normal distribution with mean vector 0 and covariance matrix Σ

I am newby in statistics and I have huge data with "p" variables and "n" samples. My data is a two dimensional matrix with "n" columns (each column is a sample) and "p" rows (each row is a variable). ...
0
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1answer
38 views

Unbiased estimator for Theta of a Normal Distribution

If $X_1,\ldots,X_n\sim \operatorname{iid} \operatorname N(\theta, \sigma^2)$, then verify that $\bar{X}_n$ is unbiased estimator for $\theta$ and that Cramer Rao bound is met? I am facing difficulty ...
0
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1answer
53 views

Finding joint probability distributions from marginal distributions

Question: I was solving test papers where I found this one. My doubt: I know to work with conditional probabilities and Jaccobian Transformation and part A and B can be done applying the above..But ...
0
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1answer
47 views

Mean and variance of probability density with multidimensional indicator function

I encountered the following question while studying machine learning: We are asked to calculate mean and covariance of a given probability density function $$p(x) = \frac{1}{16} \cdot 1_{0 \leq x_1 ...
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0answers
25 views

How to compare two multivariate datasets

I have two datasets consisting of 20 features, one set contains 50k records and the other 70k. I want to check if they are from the same population. Datasets contain discrete features and the ...
4
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2answers
52 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 ...
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0answers
86 views

Deriving predictive distribution

In Bayesian Regression, I am confused how to to get $f*$ and $\sigma*$, given $$y^∗ \mid \vec{y}\sim\mathcal{N}(f^∗ , σ^∗ )$$ $$ p(y^* \mid \vec{y}) = \int{p(y^* \mid \vec{w}) p(\vec{w} \mid \vec{y})...
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0answers
46 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{\...
5
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1answer
70 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 ...
0
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1answer
79 views

Variance of sum of random vectors - a proof

For nonrandom matrices $A(rXk)$,$ B(rXm)$, and $c(rX1)$, how does one show that $$\newcommand{\Var}{{\rm Var}}\newcommand{\Cov}{{\rm Cov}}\newcommand{\*}{{\times}} \Var(AX+BY+c)=A\Var(X)A′+ B \Var(...
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0answers
23 views

Multivariate Test for Mean Equivalence

I am looking for an equivalence test for multivariate means (arbitrary or normally distributed). Any suggestions or hints in the right direction are appreciated.
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0answers
27 views

Expectation of expressions involving sample covariance matrix and inverse of covariance matrix

Let $y_{ij}$, $i=1,2,\cdots,n_j$ be a random sample from $N_p(\mu_j,\Sigma_j)$, $j=1,2$. Let $$\overline{y}_j=\frac{1}{n_j}\sum_{i=1}^{n_j}y_{ij} \hspace{2mm} \mathrm{and}$$ $$S_j=\frac{1}{n_j-1}\sum_{...
2
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0answers
40 views

Check whether a random sample comes from an elliptical distribution?

How can I check whether it is a reasonable assumption to say that a multivariate sample $x_1,...,x_n$ comes from an elliptical distribution, such as a normal distribution or a t-distribution? In the ...
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0answers
30 views

How to use Copulas to Combine Multivariate Conditional Probability with Univariate Conditional Probability?

This is sure to be an odd one, but here goes. I'm trying to estimate P(X|Y, Z) by the distributions of P(X|Y) and P(X|Z). I've thus far been trying to using copulas to achieve that aim, but I'm not ...
4
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1answer
64 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 $...
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0answers
46 views

Mutual information between multivariate random variables? [duplicate]

I have read that mutual information only works with two random variables, and that for 3 or more RVs there seems to be a variety of different measures (synergy, partial information decomposition, and ...
1
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1answer
77 views

Conditional distribution of multivariate Rayleigh distribution

The correlated Rayleigh envelopes using a set of zero-mean complex Gaussian RVs (Random Variables) is given by $$G_{k}=\sigma_{k}(\sqrt{1-\lambda_k^2}X_k+\lambda_kX_0)+i\sigma_{k}(\sqrt{1-\lambda_k^2}...
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0answers
102 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 $X_1, ~X_2$, ...
1
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3answers
96 views

Hessian of Log of Matrix-t distribution

I am trying to calculate the hessian of the log of the matrix-t distribution. I know that the log of the matrix-t distribution can be written: $$\log T_{N\times P}(X| \nu, M, \Sigma, \Omega) \propto -\...
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0answers
20 views

Bayesian updating of MVN with joint signal

Suppose t and l are independent variables that are normally distributed with known means and variances. Suppose I get a signal s = f(t, l), where I know f(.) which is deterministic. How do I define ...
1
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1answer
72 views

If $\mathbf{x} \sim N(\mathbf{0,I})$ and $\mathbf{y} = \mathbf{Ax}$, what does $\mathbf{A}^T \mathbf{A}$ represent?

If $\mathbf{x} \sim N(\mathbf{0,I})$ then $\mathbf{AA}^T$ is the covariance matrix of $\mathbf{y} = \mathbf{Ax}$, but what does $\mathbf{A}^T \mathbf{A}$ represent? In some places I have seen ...
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0answers
18 views

Multivariate goodness-of-fit diagnostics

In a multivariate setting, we can assess the goodness-of-fit of a $p$-dimensional multivariate distribution to a set of $p$-dimensional multivariate data. Using, for example, the squared Mahalanobis ...
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0answers
80 views

Joint distribution of sample means from multivariate normal with unknown covariance matrix

The distribution of a (standardised) sample mean from a univariate normal distribution with unknown variance is given by the Student's t-distribution (Student's t-distribution). What about a joint ...
2
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1answer
209 views

MGF of the multivariate hypergeometric distribution

Does the multivariate hypergeometric distribution, for sampling without replacement from multiple objects, have a known form for the moment generating function?
4
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0answers
54 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 \...
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0answers
65 views

How does one choose a random isotropic direction and then have the vector have norm 1? [duplicate]

I want to choose a random vector in high dimensions such that it all directions have the same uniform chance (i.e. isotropic in all directions). My current idea is the following algorithm: sample v ...
0
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0answers
22 views

Mutual information for independent subsystems with non-Gaussian distributions

I seem to have found contradiction when computing mutual information for a multivariate Cauchy distribution. Below, I list a few things that I think are true. But I expect that at least one of them ...
4
votes
1answer
245 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). ...
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0answers
200 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 ...
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0answers
92 views

Transforming a multivariate binary time series to be stationary

I have a multivariate (multi-response) dataset with, for example, 10 different binary responses. I'm interested in an AR(p) model, determining how the responses at previous time steps relate to the ...
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0answers
28 views

Categorical probability distribution that captures “some” permutation invariance / mirror symmetry

I'm fitting something similar to a naive Bayes model to a data set where each data point has six features, $A_1$, $B_1$, $C_1$, $A_2$, $B_2$ and $C_2$. $A_1$ and $A_2$ can both take values in {$a_{1}$,...
2
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1answer
203 views

Correlation matrix for multivariate Cauchy distribution

I have found an equation for the entropy of a $p$-variate Cauchy distribution here [page 70]: $H(X,R) = \frac{1}{2}\log(\det(R))+f(p)\,,$ where $X=(X_1,X_2,\dots,X_p)$ is vector of random variables ...
1
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1answer
107 views

Why do I need multivariate normality tests?

I am new to time series analysis and would like to test a multivariate time series (12 components) for normality. I found several straightforward normality tests and some multivariate normality tests. ...
0
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1answer
94 views

finding Pr(X+Y > 500) given the following joint probability density function

I am reviewing for a probability and statistics class. I am stuck on a problem despite repeated attempts. (THIS IS NOT HOMEWORK!) The questions is: Consider an electronic system with two components. ...
0
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1answer
56 views

Show that $(\mathbf{x}, \boldsymbol\Theta\mathbf{x}+\boldsymbol\eta)$ is jointly normal

This is from Theodoridis' Machine Learning, exercise 3.16. Suppose $\mathbf{x}$ is a vector of jointly normal random variables with covariance matrix $\boldsymbol\Sigma_x$. Let $$\mathbf{y} = \...
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0answers
49 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 ...
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3answers
1k 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?...