Questions tagged [multivariate-distribution]
Probability distribution over vectors (as opposed to univariate distributions that are over numbers).
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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.
...
4
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1answer
160 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
28 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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29 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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9 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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14 views
Predict vector of random variables from historical data?
There's historical prices for gold, sp500, silver, iron.
...
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1answer
57 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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34 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
77 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{\...
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1answer
78 views
How to perform joint inference on multivariate normal variables?
Suppose I have the following model:
$$\begin{aligned}
\text C &\sim \mathcal N \left(\mu, \delta^2\right) \\
\forall i: \text L_i | \text C = c &\sim \mathcal N \left(c, \lambda_i^2 \right) \\...
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1answer
62 views
Ornstein-Uhlenbeck process
Given a multivariate Ornstein-Uhlenbeck process that is a stochastic process, is it correct that each component of this process is a univariate Ornstein-Uhlenbeck process?
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1answer
52 views
Something like Mahalanobis distance when the copula is not Gaussian
Mahalanobis distance accounts for different variances of the marginal variables and correlations between the marginal variables. However, there is an implicit (maybe explicit) assumption that ...
2
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1answer
133 views
From beta distribution to Dirichlet: Estimation of the concentrantion parameters
Searching at least 3 hours about the connection between beta distribution and dirichlet. My problem is:
I have a collection of random variables $X_i \sim Beta(a_i, b_i)$. The parameters $a_i$ and $...
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0answers
43 views
How to generate multiple, non-independent samples from a multivariate normal distribution?
Suppose I have a multivariate normal (MVN) distribution:
$$\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
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1answer
46 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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24 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 ...
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0answers
85 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
54 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
75 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
376 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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33 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
192 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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41 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
360 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
25 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
17 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]$$
for $\theta\...
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0answers
38 views
Sufficient conditions for multivariate MGF to be finite in neighborhood of zero
In the one-dimensional case, we have the following fact: (see Existence of the moment generating function and variance for proof)
Proposition: The mgf $m(t)$ is finite in an open interval $(t_n,t_p)$ ...
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0answers
34 views
Estimating Pearson correlation for multivariate t distribution using Kendall rank correlation
I had a task to generate a certain amount of samples for $(X_{1},X_{2})$ with a bivariate $t$ distribution $t_{2}(\nu,\mu,\Sigma)$ and then estimate Pearson's correlation coefficient $\rho$ by using ...
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38 views
Projection of multivariate distribution to lower dimensional subspace
Say that $X \in \mathbb{R}^n$ is a vector of $n$ r.v.'s with pdf $p(x_1,\ldots,x_n)$. Let's consider now the linear map $Y = A X$ where $Y \in \mathbb{R}^m$ with $m < n$. I am seeking $p(y_1,\ldots,...
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1answer
306 views
Multivariate conditional entropy
I would like to take data columns and compute the multivariate conditional entropy. For instance, suppose I have columns $A, B, C D, E$ and I want to compute the conditional entropy $H(E | A,B,C,D)$. ...
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1answer
49 views
Multivariate distribution: calculate P(Y > b/2)
The joint probability function looks like this:
The first step for calculating $P(Y > 2/b)$ is calculating $f_Y(y)$.
Which I did like this:
The problem here is that my x is still in my indicator,...
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1answer
496 views
Understanding this expression of the multivariate t-distribution
I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows
$$
G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\
Z = N (...
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145 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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248 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). ...
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1answer
55 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 ...
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1answer
382 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 ...
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1answer
92 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
$$p(x) = \frac{1}{16} \cdot 1_{0 \leq x_1 ...
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52 views
How to compare two multivariate datasets
I have two datasets consisting of 20 features, one set contains 50k records and the other 70k. I want to check if they are from the same population.
Datasets contain discrete features and the ...
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2answers
61 views
How to compare a new measurement to an existing multivariate distribution?
I have a dataset that describes the position and rotation of an object at different points in time using four dimensions. I want to use this sample of observations to get a sense of what positions and ...
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0answers
52 views
Equality of two multivariate normal CDF's
Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
5
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1answer
152 views
multivariate Student's t distribution: intuition for non-independence?
Consider a multivariate Student's t distribution, with parameters $\nu$ (d.f.), $\mu$ (location) and $\Sigma$ (shape).
Does anyone have a good intuition for the individual components not being ...
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1answer
179 views
Variance of sum of random vectors - a proof
For nonrandom matrices $A(rXk)$,$ B(rXm)$, and $c(rX1)$, how does one show that
$$\newcommand{\Var}{{\rm Var}}\newcommand{\Cov}{{\rm Cov}}\newcommand{\*}{{\times}}
\Var(AX+BY+c)=A\Var(X)A′+ B \Var(...
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37 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
40 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 $$\overline{y}_j=\frac{1}{n_j}\sum_{i=1}^{n_j}y_{ij} \hspace{2mm} \mathrm{and}$$
$$S_j=\frac{1}{n_j-1}\sum_{...
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0answers
59 views
Check whether a random sample comes from an elliptical distribution?
How can I check whether it is a reasonable assumption to say that a multivariate sample $x_1,...,x_n$ comes from an elliptical distribution, such as a normal distribution or a t-distribution?
In the ...
4
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1answer
85 views
A Multivariate Distribution for Linear Combinations of Independent Exponential Random Variables
Consider a random vector $\mathbf{X} \in \mathbb{R}^r$ whose components $X_j$ are independent exponential variables with different scale parameters $\beta_j$, $j=1,\dots,r$. Suppose I have a general $...
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0answers
55 views
Mutual information between multivariate random variables? [duplicate]
I have read that mutual information only works with two random variables,
and that for 3 or more RVs there seems to be a variety of different measures (synergy, partial information decomposition, and ...
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1answer
136 views
Conditional distribution of multivariate Rayleigh distribution
The correlated Rayleigh envelopes using a set of zero-mean complex Gaussian RVs (Random Variables) is given by
$$G_{k}=\sigma_{k}(\sqrt{1-\lambda_k^2}X_k+\lambda_kX_0)+i\sigma_{k}(\sqrt{1-\lambda_k^2}...
