Questions tagged [multivariate-distribution]
Probability distribution over vectors (as opposed to univariate distributions that are over numbers).
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Derive the posterior multivariate normal distribution
I have a question when I was reading the book Latent Variable Models and Factor Analysis: A Unified Approach by Bartholomew, Knott and Moustaki. Here it is:
Suppose that $\mathbf{x}=(x_1, x_2, ..., ...
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Degrees of freedom in a likelihood ratio test - multivariate normal vs univariate normal and Archimedean copula
Hopefully the title is self explanatory! To be more specific, I have three datasets. First, I fit them to a multivariate normal distribution, and calculate the log-likelihood. Then, I fit normal ...
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Maximum correlation matching between 2 sets of points is creating clumps
I am describing here an issue with mapping 2 sets of points that has been bugging me for some time. Any input would be greatly appreciated!
The Task:
I have 2 sets of points living in a multivariate ...
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Why does the multivariate data generated by a copula in R not exhibit the prespecified correlation?
I am using the package copula in R to generate a bivariate sample. The marginal distributions are binomial with p=0.5 and ...
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Covariance of multivariate negative binomial with random effects
I am fitting a negative binomial-2 regression model where there is a multivariate normal random effects term. I would like to find an equation for the covariance of two outcomes. In "the ...
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Correlation and $z$-Transformation for Vectors With Correlation Structure
Consider a set of $p$ pairs $(x_1,y_1),...,(x_p,y_p)$, the sample correlation coefficient is
$$r=\frac{\sum_{i=1}^p (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^p (x_i-\bar{x})^2} \sqrt{\sum_{i=1}^p (...
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Modelling the joint pmf of 2 correlated variables as p(x)*pmf(E(y|x))
Let x,y be 2 correlated counts. We want to model the joint pmf p(x,y). We know that p(x,y) = p(x)p(y|x) = p(y)(x|y). However, what happens when we don't know y|x, but we can estimate E(y|x)? Can't we ...
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How may I find the distribution of a transformation of multivariate random variables?
Forgive me if this question has already been asked on here, but I could only find posts if the multivariate random variables were multivariate Gaussian.
Suppose we have two multivariate random ...
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Hotelling's $T^2$ chart for subgroups with unequal size
I have been reading about Hotelling's $T^2$ control charts and I'm unsure on how to deal with the case where the mean observations come from unequal-sized subgroups.
Consider $m$ observations $\mathbf{...
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Optimizing parameters for a non-standard probability density function
We have a non-standard multivariate probability density function, P(x | q), where x is a vector, and q are the parameters of the density. I get events ...
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PMF of the Independent Multivariate Bernoulli Distribution
I was reading this paper on the Multivariate Bernoulli Distribution, which provides the general form of the PMF in equation 3.1. The paper refers to this as the probability distribution function, but ...
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Sampling from multivariate probability distribution
What's the best way to sample from multivariate probability density functions that are proportional to $\exp(-\|x\|_2)$ or $\|x\|_2^p \exp(-\|x\|_2)$ for some positive integer $p$ with $x \in \mathbb{...
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Multivariate Normal Distribution. How do we apply this to dataset?
I am having a hard time understanding the concept of a multivariate normal distribution.
From my understanding, it assumes each group is normally distributed, making one joint normal distribution with ...
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Parametric copulas with marginals that are regressions
In Dependence Modeling with Copulas (Harry Joe) I'm struggling to interpret the meaning of a statement. In Chaper 5.1, it is stated:
Parametric inference for copulas
For dependence modeling with ...
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Multivariate sample covariance
I have a set of $X_1,...,X_n$ samples with covariance $\Sigma_1,...,\Sigma_n$.
The multivariate sample mean is then $$ \left(\sum_{i=1}^n \Sigma_i^{-1} \right)^{-1} \left(\sum_{i=1}^n \Sigma_i^{-1} ...
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Product of Two t-distribution Formulas
Does the product of two t-distribution formulas with same degrees of freedom simplify?
$T_v(x; \mu_1, \Sigma_1)T_v(x; \mu_2, \Sigma_2) =\ ?...$
In the normal case it simplifies to:
$\mathcal{N}(x; \...
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Distribution of covariance parameter resulting from sum of known- and unknown-covariance noise processes
Let's say I have a set $X$ of $N$ random $p$-dimensional vectors generated by $\mathbf{x}_i = \boldsymbol{\mu} + \Psi_i^{1/2} \boldsymbol{\xi}_i + \Sigma^{1/2} \boldsymbol{\zeta}_i$, where $\...
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How do you count degrees of freedom in multivariate distributions?
When dealing with a multivariate distribution (e.g., a multivariate t-distribution), how does dimensionality play into degrees of freedom? Let's say you have N measurements of a d-dimensional vector, ...
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Generate multivariate distributions of lognormal and normal distribution in python
I need to generate random numbers from 3 correlated distributions. First two of them are lognormal and the final one is normal, i.e. for X, ...
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multicomp package and emmeans package produce different adjust pvalues for Dunnett procedure [closed]
For Dunnett adjustment, multicomp package and emmeans package in R give different results. Anyone knows why? Thanks. Please see ...
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Estimate multivariate distribution with several variables on real data (continuous and categoricals) and sample from it
I have a complex dataset, collected through a survey, with both continuous (such as Age, Body mass index, etc..) and categorical variables (i.e. Gender, Education, etc..). I want to estimate their ...
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Difference between random vector and joint distribution
My definition of a random vector is a vector $(x_{1},...,x_{n})$ that maps from a sample space to ${R}^{n}$. An example of this (in my understanding) would be a random process such as drawing one card ...
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Multivariate and integral representations explanation
I would to understand more on the multivariate and integral representations. I am fascinated by the possibility of a multivariate distribution like this one:
$$\mathbb{P}(X_1 \le Z + a, X_2 \le Z + a, ...
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Calculation of multivariate probability mass function
How to calculate the following multivariate probability mass function:
$P(X_1-X = n, X_2-X = n, ..., X_{N-1}-X = n)$
Where $n$ and $N$ are positive integers, and $X_i$ and $X$ are iid random variables ...
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Quantile function of multivariate distributions from empirical samples
Let's say I have a k-dimensional multivariate normal distribution $MVN(0,\Sigma)$. Denote random vector $X \sim MVN(0,\Sigma)$ as $X = (x_1, x_2, \dots, x_k)$. It is trivial to find $P(x_1 \leq c_1, ...
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Approximating a bivariate distribution with another distribution, which method to use?
Let $X \sim F(;\theta)$ and $Y \sim G(;\eta)$ be two independent continuous random variables. The greek symbols represent the parameters of those distributions. I can easily sample from these ...
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Find the conditional PDF of a multivariate normal distribution given a constraint [duplicate]
Problem to solve
We have a vector of random variables $\textbf{X}=(X_1,X_2)$ issued from a bivariate normal distribution. In particular, $\mu = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$, $\Sigma = \begin{...
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Joint testing of proportions, analogous to the Hotelling $T^2$ test?
Hotelling's $T^2$ test examine if two multivariate Gaussian distributions with the same covariance matrix have the same mean vector. Assuming the two distributions to have the same covariance matrix ...
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Standard deviation of a function of random variables
I was a bit confused recently with why two jointly distributed variables will exhibit a smaller scatter as the correlation between the random variables increases (assuming only positive values for the ...
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Probability distribution of the product of three dependent continuous random variables
Given the joint probability distribution for three dependent continuous random variables, I want to find a formula to compute the probability distribution for the product of these 3 random variables. ...
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Kullback–Leibler divergence between two multivariate t distributions with different degrees of freedom?
I want to calculate the Kullback–Leibler divergence
between two multivariate $t$ distributions with different degrees of freedom (say $\nu_1$ and $\nu_2$), but same location and scale matrix, for ...
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Generate nonnegative variates with mean 1 and specified variance-covariance
Problem
In several applications in surveys, it would be helpful to be able to generate a set of $R$ $n$-dimensional variates with the following properties:
Has mean vector $1$
Has a specified ...
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Statistic for significance test comparing transition matrix to null
I have several pitch sequences (mini songs), that look something like this when plotted as a pitch profile (piano roll):
I've coded each sequence in terms of its constituent interval sizes, and ...
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Measure the "actual" number of dimentions in a multivariate distribution
Consider a 3D multivariate normal distribution $x\sim N(0,\Sigma)$ where
$$\Sigma=\begin{bmatrix}1 &1 &0 \\ 1&1&0 \\ 0 &0& 1 \end{bmatrix}$$
Since $x_1$ and $x_2$ are fully ...
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Test for multivariate stationarity
Given a bivariate time series $(X_{1t}, X_{2t})$ with $X_{it}\sim N(0,1)$, I want to know if there is a way of testing if the bivariate distribution does not change over time. This is, if $F_{12}(x_1,...
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Distribution of iid variates with fixed sum
I have $n$ bins, and a pool containing $N$ balls, $N > n$. I do a consecutive $N$ times: I take one of the balls from the pool and put it into a bin randomly selected with uniform probability $1/n$....
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Creating multivariate distribution using marginal: can I use a copula?
I am modeling a multivariate distribution, where I already have the distributions of the marginals. Let's call the marginals, $f_{i}(x)$ for every $i$ marginal distribution, where $i=1,2,3$. We ...
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Contour lines for multivariate Laplace distribution
In the p-variate normal distribution ($\mathbf{N_p(\mu,\Sigma)}$), the solid ellipsoid of x values satisfying
$(\mathbf{x-\mu)'\Sigma^{-1}(\mathbf{x}-\mu)}\leq \chi^2_p(\alpha)$
has probability $1-\...
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When to specify multivariate versus univariate priors on parameters?
Suppose a linear regression model:
$$y \sim Normal(\beta X, \sigma)$$
For our purposes, assume $y$ is a univariate outcome and $X$ is a design matrix containing an intercept and one additional ...
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Joint Uniform Distribution Probability Problem
Let $X \sim U(0,1) $ and $Y \sim U(0,x) $. Calculate $$ \Pr(X >0.5 | Y= 0.25)$$
Is this a trick question ? Since $\Pr(Y = 0.25) = 0$, right ?
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Mistake in Multivariable Transformation
The answer is $\frac{19}{24}$ but I don't know where is my mistake. The question is this.
Let $$f_{X,Y}(x,y) = \begin{cases} x+y, \quad 0<x<1, 0<y<1 \\ 0, \quad \quad \quad \text{elsewhere}...
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Sampling from multivariate Bernoulli
Suppose you have a vector p drawn from a multivariate Beta distribution (not a Dirichlet), such as the one described here ( How to construct a multivariate Beta distribution? ) with a Gaussian copula.
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Why is it difficult to sample from a multivariate distribution? [closed]
The Monte Carlo Markov Chain method requires sampling from a multivariate distribution. This is because the Markov Chain process requires dependent draws. See 1:55 at https://www.youtube.com/watch?v=...
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How to fit a copula when zeros abound?
I am modelling a joint distribution for two random variables: $F(x,y)$. I observe $n$ data points $(x^{}_{i},y^{}_{i})^{N}_{i=1}$. I would like to model $F$ as the product of its marginals and a ...
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Fitting a copula vs. directly fitting a multivariate distribution
I understand that the joint density of two random variables $f(x,y)$ can be decomposed as the product of its marginals and a copula: $f(x,y) = g(x)k(y) \times c(G(x),K(y))$. Alternatively this may be ...
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Probability of getting k different colored balls from an urn with K different colors of balls, each color has the same number of balls
Let's say I have an urn with $n$ balls, with $K$ different colors of balls, where each color has the same number of balls: $\frac{n}{K}$. Given I reach into the urn and grab a ball (without ...
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What is the distribution of a random linear combination of gamma random variables?
Let $U\sim \mathcal U(0, 1)$ be a random variable uniformly distributed over the interval $[0, 1]$. Let $X_1, X_2\sim \Gamma(a, b)$ be two iid random variables with a Gamma distribution. Now it is ...
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generalization of univariate pdf under constraints
I'm looking for generalizations of a univariate probability distribution function. The pdf is $$ \varphi(x)=(2\sqrt{s}K_1(2\sqrt{s}))^{-1}e^{\frac{s}{\log x}}. $$
for $K_1$ a modified Bessel function, ...
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Multivariate Probability Distribution with Linear Conditional Expectation
I want to know what probability distribution has the linearity property of the conditional expectation.
To be specific, suppose that we have three random variables named $v_1,\;v_2,\;v_3$.
Then, if $[...
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Random variate of a singular Wishart distribution with non-integral degrees of freedom
Let's say that I have a random variable $X$ that follows a singular Wishart distribution with $\nu$ degrees of freedom and a shape matrix $\Sigma$ that is $p\times p$. We can write "symbolically&...
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