Questions tagged [multivariate-distribution]
Probability distribution over vectors (as opposed to univariate distributions that are over numbers).
102
questions
1
vote
1answer
52 views
How to perform joint inference on multivariate normal variables?
Suppose I have the following model:
$$\begin{aligned}
\text C &\sim \mathcal N \left(\mu, \delta^2\right) \\
\forall i: \text L_i | \text C = c &\sim \mathcal N \left(c, \lambda_i^2 \right) \\...
3
votes
1answer
44 views
Ornstein-Uhlenbeck process
Given a multivariate Ornstein-Uhlenbeck process that is a stochastic process, is it correct that each component of this process is a univariate Ornstein-Uhlenbeck process?
1
vote
1answer
33 views
Something like Mahalanobis distance when the copula is not Gaussian
Mahalanobis distance accounts for different variances of the marginal variables and correlations between the marginal variables. However, there is an implicit (maybe explicit) assumption that ...
2
votes
1answer
81 views
From beta distribution to Dirichlet: Estimation of the concentrantion parameters
Searching at least 3 hours about the connection between beta distribution and dirichlet. My problem is:
I have a collection of random variables $X_i \sim Beta(a_i, b_i)$. The parameters $a_i$ and $...
0
votes
0answers
36 views
How to generate multiple, non-independent samples from a multivariate normal distribution?
Suppose I have a multivariate normal (MVN) distribution:
$$\textbf{X} \sim MVN({\mu},\Sigma)$$
where $\Sigma \neq \sigma^2\textbf{I}$ i.e. the RVs within $\textbf{X}$ have some correlation structure....
2
votes
1answer
42 views
Decomposing a random variable into marginals and copula
I’m having trouble getting understanding how to actual construct a copula, from my understanding it captures the purely joint features of a joint distribution. I’ve been working with the following ...
0
votes
0answers
21 views
Quantifying information loss (KL divergence?) between a multivariate and a univariate discrete distribution
Let's say I have n discrete variables, n1, n2, ... n_n, each with a different scale, and another discrete variable ...
2
votes
0answers
82 views
Estimation of common variance in elliptical distribution
Suppose you observe a random $p$-vector $X$ which follows an elliptical
distribution with mean zero, covariance $\sigma^2 I$ and (unknown) distribution
function $g$. Given a single observation of $X$...
0
votes
1answer
40 views
How to prove that Normal Squared Distances follow a Chi-Square distribution?
Given a multivariate normal distribution
$f(x) = \frac{1}{\sqrt{(2 \pi)^n|\Sigma|}} \times \exp\left( -\frac{1}{2} (x-\mu)' \Sigma^{-1} (x-\mu)\right)$
how can I prove that $ (x-\mu)' \Sigma^{-1} (x-...
0
votes
1answer
71 views
Distribution of transformed multivariate log-normal
Let $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$ and $\mathbf{Y} = \text{exp}(\mathbf{X})$. If $Y_i$ is one of the components of $\mathbf{Y}$, what is the distribution of $\frac{\mathbf{Y}}{...
0
votes
1answer
128 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 ...
1
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0answers
23 views
Diffusion tensor as a covariance matrix
TLDR: In nuclear magnetic resonance (NMR), to study molecular diffusion we assume that molecules displace in 3D space according to a trivariate gaussian distribution. The variables are then the ...
4
votes
2answers
146 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
votes
0answers
39 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 ...
0
votes
1answer
248 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 ...
0
votes
0answers
20 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
votes
1answer
20 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{...
0
votes
0answers
14 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]$$
for $\theta\...
1
vote
0answers
25 views
Sufficient conditions for multivariate MGF to be finite in neighborhood of zero
In the one-dimensional case, we have the following fact: (see Existence of the moment generating function and variance for proof)
Proposition: The mgf $m(t)$ is finite in an open interval $(t_n,t_p)$ ...
1
vote
0answers
31 views
Estimating Pearson correlation for multivariate t distribution using Kendall rank correlation
I had a task to generate a certain amount of samples for $(X_{1},X_{2})$ with a bivariate $t$ distribution $t_{2}(\nu,\mu,\Sigma)$ and then estimate Pearson's correlation coefficient $\rho$ by using ...
0
votes
0answers
30 views
Projection of multivariate distribution to lower dimensional subspace
Say that $X \in \mathbb{R}^n$ is a vector of $n$ r.v.'s with pdf $p(x_1,\ldots,x_n)$. Let's consider now the linear map $Y = A X$ where $Y \in \mathbb{R}^m$ with $m < n$. I am seeking $p(y_1,\ldots,...
1
vote
1answer
213 views
Multivariate conditional entropy
I would like to take data columns and compute the multivariate conditional entropy. For instance, suppose I have columns $A, B, C D, E$ and I want to compute the conditional entropy $H(E | A,B,C,D)$. ...
0
votes
1answer
49 views
Multivariate distribution: calculate P(Y > b/2)
The joint probability function looks like this:
The first step for calculating $P(Y > 2/b)$ is calculating $f_Y(y)$.
Which I did like this:
The problem here is that my x is still in my indicator,...
0
votes
1answer
337 views
Understanding this expression of the multivariate t-distribution
I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows
$$
G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\
Z = N (...
1
vote
0answers
95 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 ...
0
votes
0answers
145 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
votes
1answer
53 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
votes
1answer
230 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
votes
1answer
83 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
$$p(x) = \frac{1}{16} \cdot 1_{0 \leq x_1 ...
1
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0answers
41 views
How to compare two multivariate datasets
I have two datasets consisting of 20 features, one set contains 50k records and the other 70k. I want to check if they are from the same population.
Datasets contain discrete features and the ...
5
votes
2answers
57 views
How to compare a new measurement to an existing multivariate distribution?
I have a dataset that describes the position and rotation of an object at different points in time using four dimensions. I want to use this sample of observations to get a sense of what positions and ...
3
votes
0answers
50 views
Equality of two multivariate normal CDF's
Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
5
votes
1answer
110 views
multivariate Student's t distribution: intuition for non-independence?
Consider a multivariate Student's t distribution, with parameters $\nu$ (d.f.), $\mu$ (location) and $\Sigma$ (shape).
Does anyone have a good intuition for the individual components not being ...
0
votes
1answer
135 views
Variance of sum of random vectors - a proof
For nonrandom matrices $A(rXk)$,$ B(rXm)$, and $c(rX1)$, how does one show that
$$\newcommand{\Var}{{\rm Var}}\newcommand{\Cov}{{\rm Cov}}\newcommand{\*}{{\times}}
\Var(AX+BY+c)=A\Var(X)A′+ B \Var(...
1
vote
0answers
31 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.
1
vote
0answers
37 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 $$\overline{y}_j=\frac{1}{n_j}\sum_{i=1}^{n_j}y_{ij} \hspace{2mm} \mathrm{and}$$
$$S_j=\frac{1}{n_j-1}\sum_{...
2
votes
0answers
52 views
Check whether a random sample comes from an elliptical distribution?
How can I check whether it is a reasonable assumption to say that a multivariate sample $x_1,...,x_n$ comes from an elliptical distribution, such as a normal distribution or a t-distribution?
In the ...
4
votes
1answer
72 views
A Multivariate Distribution for Linear Combinations of Independent Exponential Random Variables
Consider a random vector $\mathbf{X} \in \mathbb{R}^r$ whose components $X_j$ are independent exponential variables with different scale parameters $\beta_j$, $j=1,\dots,r$. Suppose I have a general $...
1
vote
0answers
51 views
Mutual information between multivariate random variables? [duplicate]
I have read that mutual information only works with two random variables,
and that for 3 or more RVs there seems to be a variety of different measures (synergy, partial information decomposition, and ...
1
vote
1answer
117 views
Conditional distribution of multivariate Rayleigh distribution
The correlated Rayleigh envelopes using a set of zero-mean complex Gaussian RVs (Random Variables) is given by
$$G_{k}=\sigma_{k}(\sqrt{1-\lambda_k^2}X_k+\lambda_kX_0)+i\sigma_{k}(\sqrt{1-\lambda_k^2}...
7
votes
0answers
107 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
vote
3answers
110 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 -\...
1
vote
1answer
74 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 ...
2
votes
1answer
451 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
votes
0answers
59 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 \...
0
votes
0answers
97 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 ...
4
votes
1answer
334 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).
...
2
votes
0answers
316 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 ...
1
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0answers
146 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 ...
0
votes
0answers
34 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 {$a_{1}$,...
