Questions tagged [multivariate-distribution]
Probability distribution over vectors (as opposed to univariate distributions that are over numbers).
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What are the best known techniques to verify that a GAN samples correctly from a given distribution?
I would like to know what are the best known techniques to check that a generative adversarial network (GAN) samples from the correct distribution.
Naively, I would say it all boils down to a ...
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13 views
Multivariate rare events
My data is something like this:
I have U urns, and I have taken a bag of $n$ objects from each urn. Each urn has $N$ objects, and I have sampled $n$ with replacement.
$n$ is comprised of coloured ...
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23 views
Algorithm to generate non normal (discrete and continuous) correlated variables using a correlation matrix
What I am trying to do here is, given a dataset (let's say n observations of N variables), and thus a correlation matrix M that result from said dataset, I would like to write a Python algorithm that ...
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21 views
Empirical conditional density of continuous variables
I have a dataframe, with data of several continuous variables. The variables are not independent.
My goal is to sample from the distribution that generated this data.
What's a relatively easy and ...
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1answer
50 views
Multivariate distributions
Let $x$ and $y$ be random variables with the following joint density function:
$f(x,y) = e^{-x}$ for $0<x< \infty$, and $0<y<1$
If $z= x+2y$, what is the joint density function of $x$ and ...
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1answer
11 views
How to choose what feature vector to plot in multivariate regression analysis?
I'm new to the field of machine learning and I have been having this doubt for a long time now. If we want to plot a scatter plot, we plot it as, x as a function of y where x is a 1-D array. But in ...
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1answer
20 views
Does the covariance of i.i.d. random vectors/multivariate random variables have any zero terms?
If we have i.i.d. random variables, $X$ and $Y$, then $\text{Cov}(X,Y)=0$.
But let's say we have i.i.d. random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ ...
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41 views
How to draw huge high-dimensional normally distributed vectors in a memory-efficient way
I would like to simulate a few hundred multivariate normally distributed vectors $H$ of dimension $10^6$ with covariance matrix $\Sigma$, preferrably in R. The individual entries of $\Sigma$ are ...
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1answer
33 views
What's the best function to test multivariate normality when sample size more than 3000?
Before MANOVA, I need to test multivariate normality.
Then I tried MVN::mvn() function in R, output as below:
...
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13 views
bivariate conditional joint sensor model
I am struggling to find $P\left( V_t | z \right)$ from $P\left( V_t | z , V_p \right)$. Here $z$ and $V_p$ are independent variables.
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28 views
Calculate fifth and sixth polynomials for Headrick (2002) method for non-normal multivariate distribution
I am trying to perform a 3-variable correlated multivariate Monte Carlo simulation. As the asset class returns are non-normal, I found the following function ...
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22 views
Likelihood of three dimensional data
I'm having a lot of trouble finding the likelihood and log-likelihood of $\tau$ that corresponds to the following equation:
$a_i = y(t_i, \tau) + \epsilon_i$
where $\epsilon_i \sim \mathcal{N}(0, \...
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1answer
16 views
Normal random matrix as linear transformation of standard normal random matrix?
Wikipedia has the following definition of a normal random vector:
A real random vector $\mathbf{X} = (X_1,\ldots,X_k)^{\mathrm T}$ is called a normal random vector if there exists a random $\ell$-...
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31 views
How to learn dependency of variables from data?
I have a data set $X$ that consist of $m$ vectors $\vec{x}$ of $n$ real valued components. Each vector component lies within a corresponding predefined interval of valid values, which is the same for ...
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51 views
Probability density of conditional multivariate distribution [duplicate]
We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning ${\boldsymbol Y}$ into
$${\boldsymbol Y}=\begin{bmatrix}{\boldsymbol y}_1 \\
{...
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2answers
43 views
How to prove a multivariate r.v. does not follow the nonparanormal distribution?
Background
You may find the definition of the non-paranormal distribution at the 2nd paragraph in p.2296 of this paper.
In short, $(X_1, \ldots, X_p)$ is non-paranormal if there exists a set of ...
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9 views
Modelling probability distribution of multivariate dataset
my background is not in stats so apologies if this is an obvious question.
I have a dataset with 60,000 observations, and 128 features. Each feature follows a hurdle distribution with either a gamma, ...
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34 views
Distribution of quadratic forms in mixed model
I have a question related to the distribution or asymptotic distribution of quadratic forms that arise in the linear mixed model. Suppose,
$$Y=X\beta + H\delta + \epsilon$$ where $\epsilon \sim N_n(0,...
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1answer
29 views
Conditional Distribution Multivariate Normal Distribution [closed]
Let $X_1, X_2, X_3$, be jointly distributed according to a multivariate normal distribution.
$[X_1, X_2, X_3]^T\sim N(\mu=[0,0,0]^T , \Sigma = [[5,0,0],[0,2,1], 0,1,3]])$
$U = X_1 + 2X_2$ and $V = ...
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11 views
Calculating the credible interval of each variable in a multivariate distribution
I have a k-dimensional Dirichlet posterior with parameters $\alpha_1,...,\alpha_k$ and variables $\theta_1,...,\theta_k$. This posterior comes from a Dirichlet prior and a multinomial likelihood.
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1answer
197 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, ...
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1answer
23 views
DeGroot P.155 integration problem for multivariate distributions
I am stuck with the integral for equation 3.7.4 and do not see how it was done. Could someone provide me with some hints or resources to read around?
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1answer
48 views
Confusion over multinomial and multivariate- hypergeometric distributions
You, your parents, your sister, go to visit grandma for her birthday. Grandma made a cake for the party. If she puts $20$ raisins in the cake at random in the cake, and she divides the cake into $5$ ...
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35 views
Simulations using correlated random numbers from Multivariate Normal and fat-tailed distributions
This question makes use of the LaplacesDemon package in R, but it is not a coding question, so I believe this is the most appropriate forum.
First, the unsurprising results. I generate k correlated ...
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10 views
How to check if set of 50-d vectors comes from the multivariable normal distribution in Python?
I have a data set that consists of 18000 rows and 50 columns. Each row represents an observation and each observation is a vector with 50 components.
Is there any way in Python for me to check if ...
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17 views
Predict vector of random variables from historical data?
There's historical prices for gold, sp500, silver, iron.
...
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1answer
75 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 ...
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36 views
What information in general is necessary to fully specify a multivariate distribution?
Given some multivariate probability distribution, we can fully describe it with its density or mass function -- we can associate each point in the space with either a probability density or mass, ...
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10 views
Tail Dependency of Multivariate T-distribution
In my time series class, my prof said if $\mathbf{Y}$ has a multivariate t-distribution, then $Y_i$ and $Y_j$ are dependent because of the tail dependence. Can someone give an intuitive and/or ...
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0answers
85 views
Sum of Log Chi-Squared Asymptotic Distribution
I'd like to find the asymptotic distribution of
$$\sqrt{n}\left(\log|\mathbf{S}| - \log|\boldsymbol{\Sigma}|\right), ~~~~~n \rightarrow \infty$$
where $\mathbf{S} \sim W_j\left(n, \frac{\boldsymbol{\...
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2answers
492 views
What is the conditional expectation of the exponential functional?
Consider the function $g(W)=-e^{-W}$, where $W$ is some random variable s.t.$W=X+YZ$. Furthermore, it holds that all the random variables $X,Y,Z$ follow the normal distribution with the following ...
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1answer
85 views
How to perform joint inference on multivariate normal variables?
Suppose I have the following model:
$$\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) \\...
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1answer
79 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?
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1answer
69 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 ...
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1answer
192 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 $...
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0answers
49 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....
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1answer
49 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 ...
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87 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 $X$...
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1answer
62 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-...
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1answer
80 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}}{...
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1answer
696 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 ...
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44 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 ...
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2answers
216 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 $K\times 1$...
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1answer
414 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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1answer
26 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
20 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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43 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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39 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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50 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,...
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1answer
442 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 $H(E | A,B,C,D)$. ...
