Questions tagged [multivariate-normal-distribution]

The multivariate normal distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. (Also called, multivariate Gaussian)

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Marginal Likelihood Computation for Bayesian Linear Model

Given a simple Bayesian linear model with $N$ observations $y = X\beta + \varepsilon \quad \quad \varepsilon \sim \mathcal{N}(0, \Sigma)$ with known error variance-covariance matrix $\Sigma$ and ...
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Distribution of exponent of multivariate normal distribution

My slides say that the exponent of a multivariate normal distribution, $(\mathbf{X} - \boldsymbol{\mu})^\text{T} \boldsymbol{\Sigma}^{-1} (\mathbf{X} - \boldsymbol{\mu})$, follows a chi squared ...
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The principal submatrix of projection matrix with Gaussian design

I've come across a phenomenon from a simulation that I'm very curious about. But I don't know how to start my analysis. So, I am asking for some guidance. Thanks! Denote by $\mathbf{H}$ the principal ...
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Covariance matrix of multivariate normal when negative values are made zero

Let $x$ be $n$ dimensionally multivariate normally distributed with mean $\mu$ and covariance matrix $\Sigma$. Now let $y$ be random variables defined by \begin{equation} y_i= \begin{cases} 0, ...
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What is the expected value of $X_i/\|X\|^2$ when $X \sim \mathcal{N}(\mu, \sigma^2I)$

Let $X$ be an N-dimensional normal random vector with non-zero mean $\mu$ and diagonal covariance matrix $\sigma^2I$. I would like to understand if it is possible to derive the expected value of the ...
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Wrapped normal distribution variance from angle between two multivariate normal distributions

Suppose there are two 2-d multivariate normal distributions , as in the image below: There is no correlation between the x and y components in the distributions, and the variance for each dimension ...
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How to bound the probability of multivariate gaussian vector norm?

Let's say $v \in \mathbb{R}^n \sim \mathcal{N}(0, \sigma I)$. That is, $v$ is a gaussian random vector, whose entries are distributed $\mathcal{N}(0, \sigma)$ i.i.d. From the book "C. Giraud. ...
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Sum of dependent multivariate gaussians

Note: I have already seen this Wikipedia article, and similar questions on this website: 1. Given two dependent multivariate Gaussian random variables, is the sum also a multivariate Gaussian? $X \sim ...
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Let $\textbf{y}=AX\textbf{w}+\textbf{z}$ where $\textbf{w}$ and $\textbf{z}$ are i.i.d. vectors, why does $p(\textbf{z})=p(\textbf{y}|X,\textbf{w})$?

I am a student at the undergraduate level and I have been reading a paper in the information theory/compressed sensing literature which derives a likelihood function that I do not understand. In this ...
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linear transform of a random variable follows multivariate normal, what is the distribution before the transform?

$x$ is a $n\times 1$ random vector,$A$ is a $m\times n$ matrix. Given that $x$'s linear transform $z = Ax$ follows a multivariate normal distribution: $$ z = Ax \sim N(\mu_z,\Sigma_z) $$ The question ...
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Calculating weighted covariance matrix of a weighted finite mixture of multivariate normal distributions

I am trying to calculate the weighted covariance matrix for a finite mixture of multivariate normal distributions. I read this post here and this one here, but the first post is focused on uniformly ...
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Parametrizing and Sampling Multivariate Garch Parameters Metropolis-Hastings MCMC

My question is how to sample multivariate GARCH parameters from a proposal distribution (multivariate normal) for a Metropolis-Hastings algorithm. Considering the different dimensions of the parameter ...
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How to calculate a cross-product like R^2?

$\lbrace Y_1, Y_2, \boldsymbol{X} \rbrace$ are jointly normally distributed (it is not essential to assume normality, I think). Let $\Sigma_{X}$ be the variance-covariance matrix of $\boldsymbol{X}.$ ...
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Probabilistic Machine Learning: Product of gaussian pdfs of samples is equal to gaussian pdf of sample mean

I'm currently reading the book Probabilistic Machine Learning: An Introduction by Kevin P. Murphy and I'm stuck on the derivation of a formula in section 3.3.4 (Example: inferring an unknown vector) ...
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Joint normality of a vector derived from joint normal vectors

Suppose that we have two random vectors following joint normal distributions: $$X=[x_1,x_2]'\sim N(0,\Sigma_X)\quad \textrm{and}\quad Y=[y_1,y_2,y_3]'\sim N(0,\Sigma_Y).$$ In this setup, I am ...
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Joint normality of RV made up of jointly normal RVs

I'm reading a paper where (simplifying): $Y_{i} = \beta_\theta \theta_i + \beta_x X_i + \epsilon_i $, $\epsilon_i \sim N(0, \sigma_\epsilon^2) \perp (\theta_i, X_i) $, and $\beta_\theta$ and $\beta_x$ ...
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Joint Normality of a Random Vector with elements from a Joint Normal Distribution

Suppose that $X=[X_1,X_2,X_3,X_4]\sim N(0, \Sigma)$ (i.e. $X$ follows a joint normal distribution). Define $Y=X_1+X_2$ and $Z=X_3+X_4$. Here, as far as I know, each of $Y$ and $Z$ is normally ...
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The distribution function for for the sum of variables which follow a multivariate normal? [duplicate]

Suppose we have two random variables that follow a multivariate normal distribution, $[x,y]\sim MVN([a,b], \begin{bmatrix} \rho_1 & \rho_3 \\ \rho_3 & \rho_2 \end{bmatrix})$ Then what is the ...
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Generating data in the desired correlation structure from the multivariate normal distribution in R

I want to derive 200 variables from the multivariate normal distribution. I will divide these 200 variables into 3 blocks. The 1st block will consist of 40 variables and will be low-moderately ...
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What's the conditional variance of the chain X -> Y -> Z?

If I have a cascade of 3 random variables, represented as a Bayesian Graph: $X\rightarrow Y \rightarrow Z$, is there a simple formula for $\sigma_{X|Z}$? Further, assume all the variables are normal, ...
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How to derive solution to loss function of GLasso for precision matrix

I am trying to find the parameter $\hat\omega = min_{\omega}\Big(-log|\omega| + tr(S\omega) + \sum_{i,j}\lambda|\omega_{ij}|\Big)$ This is to regularize the precision matrix $\omega$ for the GLasso. ...
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How can I compute rectangular confidence regions for parameters using R?

Simultaneous confidence regions for multivariate parameters (say, a confidence region for multivariate mean, or for regression parameters) usually find an elliptical region when the parameters' ...
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Correlation of multivariate distributions without "slope"

Wikipedia has this image showing different correlations: It says: The correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (...
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Non-positive definite matrix problem for desired correlation structure

I want to derive a correlation matrix such that block1 is 0.1 within itself, block2 is 0.1 within itself and 0.7 with block1, and the remaining variables are 0.01 within itself and with other blocks ...
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Is it possible to derive the joint probability distribution of squared OLS residuals under the classical linear regression assumptions?

Consider the linear regression model, $$ \boldsymbol{y}=\boldsymbol{X\beta}+\boldsymbol{\epsilon}, $$ where $\boldsymbol{y}$ is an $n$-vector of responses, $\boldsymbol{X}$ is an $n\times p$ matrix of ...
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Generating multivariate random variable with normal and exponential marginals

I have a collection of data points of the form $[U, V, X, Y]$, where $U$ ~ $N(\mu_1, \sigma_1)$; $V$ ~ $N(\mu_2, \sigma_2)$; $X$ ~ $exp(\lambda_1)$; and $Y$ ~ $exp(\lambda_2)$, and I am looking to ...
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Statistical analysis (comparison) of time course experiments

I have data sets that represent multiple measurements over time. The data sets come from biological experiments in which we measure something in N individual cells, with around 250 measurements over a ...
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Sum of exponential of MVN dimensions

Consider drawing N random variables $x_{i}$ for $1\leq i \leq N$ from a multivariate normal with $\mu = (\mu_1, .. \mu_i, .. \mu_N)$ and $ \Sigma_{N \times N} = [\sigma_{ij}]$ (equivalently, N ...
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Correlation matrix from pairwise correlations with specified structure

I need to simulate multivariate normal samples with a pre-specified correlation structure. The structure is such that the bigger the (GPS) distance between two points, the smaller the correlation (...
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Density of Multivariate Normal Distribution (MVN): Dimension = 1 or k?

I have a very naive question about the density function of the MVN distribution. According to the wiki page, the density function formula sometimes has a constant k ...
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How to split the data into multiple (normal) distributions or clusters?

I am trying to train a machine learning model to predict yield on fields, based on multiple parameters, such as elevation, humidity, and nitrogen content. Observing the historical harvest data, I ...
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Multivariate Gaussian probability mass inside a sphere

Assume I have some d-dimensional multivariate gaussian $X\sim\mathcal{N}\left(\mu,\Sigma\right)$ and some sphere $C=\left\{ x:\left\Vert x-z\right\Vert_2\le r\right\}\subseteq\mathbb{R}^{d}$. I was ...
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Linear transformation between two multivariate normal distribution

Suppose I have two multivariate normal distribution $N_1$, $N_2$. If I know the mean and cov of $N_1$, $N_2$:$\mu_1$, $\Sigma_1$, $\mu_2$, $\Sigma_2$. Can I find a linear transformation which makes ...
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How to generate a Multivariate Normal Distribution out of means and a correlation coefficient in R?

The means are 109 and 115, the correlation coefficient is 0.797153. I know about the mvrnorm() function, but it requires a "a positive-definite symmetric matrix specifying the covariance matrix ...
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Discarding highly correlated variables

I have got a dataset of $N$ observations with $k$ predictors $x_1, ..., x_k$ and a vector $y$ of binary responses, where $y \in \{0,1\}$. I assume that given a class my data comes from a $k-$...
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Overlap coefficient for two multidimensional normal distributions

For two PDFs $f_1(x)$ and $f_2(x)$ the overlap coefficient (OVL) measures the similarity between two distributions through the overlapping area of their distribution functions and is given by the ...
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Joint distribution of sample correlations of variables taken from a multivariate normal distribution

Let us assume multivariate normal vector $(X_1, \cdots, X_n)$ with mean vector $\mu$ and variance-covariance matrix $\Sigma$. A sample correlation will not exactly equal its population parameter, but ...
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Can we rule out $\frac{1}{2^n} + \frac{1}{2 \pi (n-1)} \left( \sum_{\substack{i,j \in \{ 1, \cdots, n \} \\ i < j}} \sin^{-1} (\rho_{i,j}) \right)$?

I am curious about orthant probabilities for the multivariate normal distribution for any finite dimension $n$. While Wikipedia currently doesn't seem mention these quantities the Wolfram Mathworld ...
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calculate correlation coefficient with singular values of covariance matrix

Given a normal distribution where the covariance matrix $\Sigma$ has known singular values $s_1$, $s_2$, ... $s_m$, what are the Pearson's correlation coefficient values?
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Expected Value of $x_1 \exp(a_1 x_1 + a_2 x_2 ... a_n x_n)$ when X is multivariate $N(0, \Sigma)$ [closed]

Which is the $E(x_1 \exp(a_1 x_1 + a_2 x_2 + \dotsm + a_n x_n)$ when X is an n-random vector distributed multivariate normal (0, $\Sigma$).
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Analogous result to Isserlis' theorem for mixed absolute product-moments of multivariate normal distribution

Suppose that $(X_1, \cdots, X_n)$ have a joint normal distribution. If $n = 2m + 1$, then $\mathbb{E} \left[ \prod_{j=1}^n X_j \right] = 0$. This can be argued from the symmetry of the multivariate ...
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Iterative generalized ridge regression

I am looking for some references. Assume I have a series of observable input/output pairs $(y_t, X_t)$ for which I assume the following relations to hold: $$\beta_t\text{ are i.i.d. }\sim N(\bar{\beta}...
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How to apply the conditional expectation of the multivariate normal distribution to fill gaps in data?

I have a data matrix $X \in \mathbb{R}^{m \times 4}$, where $m$ is any number of rows, whose data follow a multivariate normal (MVN) distribution. Suppose that for a given row $i$, the data for the ...
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Metric to measure how "standard Gaussian" a set of samples is?

Assume that I have a set of $N\in\mathbb{R}^{D}$ samples from some otherwise unknown multivariate distribution $p$. I seek a metric which might tell me how "close" $p$ is to a standard ...
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completing the square in multivariate gaussian

I have a question while I studying PRML - Gaussian Distribution. When completing the square in multivariate Gaussian, in above equation, I'm wondering why
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Appendix A on Variational Gaussian Process State Space Model

On Frigola et al in the Supplementary material A, equation (19) is: $\prod_{t=1}^{T}p(\mathbf{f}_t|\mathbf{f}_{1:t-1},\mathbf{x}_{0:t-1},\mathbf{u})=\mathcal{N}(\mathbf{f}_{1:T}|\mathbf{K}_{0:T-1,\...
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How can population variance be estimated from a bivariate sample?

Let's assume a bivariate population with a correlation $\rho$ and a common $\sigma$ so that $\Sigma = \sigma^2 \begin{pmatrix}1 & \rho \\ \rho & 1\end{pmatrix}$. I would like to know the ...
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What is the distribution of the $k^{th}$ highest value of a multivariate normal distribution

Let X be an N-dimensional multivariate normal, $X \sim N(\mu,\Sigma)$ where $\mu$ is Nx1 and $\Sigma$ is NxN. If we take a draw of $X$ from this distribution and then sort $X$ from largest to smallest,...
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Understanding Plackett's singular MVN correlation matrix

I'm trying to follow the paper "A Reduction Formula for Normal Multivariate Integrals" (Plackett, Robin L., 1954) which proposes reduction formulae for calculating the cumulative ...
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Expected distance under Gaussian noise

Summary I'm working on a tracking problem, where I'm trying to estimate the position of an object that moves in on plane. In my simulator, at each sampling step I generate a measurement that is given ...

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