Questions tagged [multivariate-distribution]

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

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19 views

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Predict vector of random variables from historical data?

There's historical prices for gold, sp500, silver, iron. ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 $H(E | A,B,C,D)$. ...