Questions tagged [multivariate-normal]

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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Efficient sampling from a multivariate Gaussian Mixture distribution for a given CDF level

I have a multivariate Gaussian Mixture (GM) distribution. I am wondering if there is any more efficient way of drawing samples (i.e., identify the iso-surface) from a multivariate Gaussian Mixture ...
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What is the distribution of single variable in multivariate normal given the other variables are fixed? [duplicate]

Say we have a multivariate normal: $$\boldsymbol{X} \sim \cal{N}(\boldsymbol{\mu}, \Sigma)$$ then the distribution of each variable $x_i$ is a Gaussian normal, $p(x_i|\mu_i,\Sigma_{ii}) = \cal{N}(\...
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PYMC: estimate parameters of two correlated processes [closed]

Let's assume we have two instantaneously correlated processes: $S_1, S_2$ both driven by Geometric Brownian Motion such that \begin{equation} \left( {\begin{array}{*{20}{c}} {d{S_1}(t)}\\ {d{S_2}(t)} \...
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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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Calculate conditional probability using sample from 3variate normal distribution

I am working in R and have a matrix (200X3) named X where I have simulated data from a 3variate normal distribution given its parameters (variance-covariance matrix & and mu). My task is to ...
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Mutual information between subsets of variables in the multivariate normal distribution

Let $\vec{X}$ be a random vector following a multi-variate normal distribution $P(\vec X)$ with covariance matrix $\Sigma$ and zero means (for simplicity). Consider a partition of $\vec X$ into two ...
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Covariance/Correlation of Multivariate Normal

Suppose $\mathbf{Y} = (Y_1, \dots, Y_n)'$, where $\mathbf{Y} \sim N(\boldsymbol{\mu}, \boldsymbol{\Sigma})$. I'm interested in the details of how one goes about finding $$\text{Corr}(Y_i, E[Y_i|\...
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Self-Study: If $x \sim \mbox N_p(\mu; V)$ and A is a matrix then how to show that $E[(x-\mu)(x-\mu)'A(x-\mu) = 0$?

If we assume the random vector $x$ to be normally distributed with $N_p(\mu; V)$ then $E[(x-\mu)(x-\mu)'(x-\mu)] = 0_p$. If I am not mistaken, this can be shown using the moment generating function of ...
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64 views

Does R use Tukey or Tukey-Kramer test corrected for unequal sample size and does it use the multivariate t distribution?

Does R use the classic Tukey HSD test for balanced data or corrects for unbalanced ones with the Tukey-Kramer approach? I mean the stats::TukeyHSD(). I found, that the Dunnet procedure uses ...
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Can I have high variance and correlation at the same time?

I am a newbie in stats who is self-studying bivariate normal. I am trying to understand how variance and correlation are interpreted graphically. But I am puzzled by the next question: if I have two ...
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90 views

Find $\delta$ such that sparse covariance matrix is positive definite

Good day, I was looking through some papers to help with my project assignment that wants me to implements 2 lasso approaches. I am having trouble simulating the samples from a MVN distribution. $\...
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Multivariate Gaussian FItting

When trying to approximate a distribution of random vectors D by using multivariate gaussian what properties must we ensure that D has ie; what distributions can be estimated by Multivariate gaussian ...
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39 views

How can I find $\rho$, given $P(4 < Y < 16|X=5)=0.9544$?

Let $X$ and $Y$ have a bivariate normal distribution with $\mu_X=5, \mu_Y=10, \sigma^2_X=1, \sigma^2_Y=25, \rho >0$. If $P(4 < Y < 16|X=5)=0.9544$, I would like to find $\rho$. I know that ...
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Linear combination of two bivariate densities

Suppose $\binom{X}{Y}$ follows a bivariate density $f=0.5f_{1}+0.5f_{2}$ where f$_{1}$ and $f_{2}$ are densities of $N(\mu_1, \Sigma)$ and $N(\mu_2, \Sigma)$. $\mu_1 = \binom{1}{1} , \mu_2= \binom{-1}...
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Showing t-distribution from multivariate standard normals

I came across a paper that assumes the following has a t-distribution: Let $W = \frac{\mathbf{a}'\mathbf{X}}{\sqrt{\mathbf{X}'\mathbf{X}}}$ and $\mathbf{a}' \in \mathbb{R}^n$ with $\mathbf{a}'\...
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1answer
51 views

Bivariate normal distribution from independent random variables

Let $X_1$ and $X_2$ random variables such that $X_1+X_2$ and $X_1-X_2$ have independent standard normal distributions. Show that $x=(X_1, X_2)$ has a bivariate normal distribution. My work: Since $...
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Expected value for f(x) [closed]

I am reading an article and trying to extend their case to a multivariate case. I have the function $f_{i} (x)=\frac{1}{|Σ|}f_{0}((x-μ_{i})'Σ^{-1}(x-μ_{i}))$, where $f_{0}(.)$ is a base density ...
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How to create a variance-covariance matrix for forecasted fantasy basketball scores?

I have three basketball players who have played in games together and I want to find a Variance-Covariance matrix that will be as accurate as possible for their fantasy points in an upcoming game. My ...
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1answer
40 views

Formula for marginal density of multivariate normal (Bayesian)? [closed]

I would like to ask if anyone knows the formula for the marginal density of a multivariate normal. I could not find it anywhere. Say, $x | \mu \sim \mathcal{N}(\mu, \Sigma)$ and $\pi(\mu) \sim \...
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How to decompose covariance matrice, multiplied by constant, to sample from multivariate normal? [closed]

I need to sample from multivariate normal distribution with mean vector $\mu$ and covariance matrix $\Sigma_1$. For that I want to use the decomposition of $\Sigma_1$ into $UΛ{U}^T$ and samle as $\...
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Multivariate Gaussian distribution explanation needed [duplicate]

I am pretty new in statistics. I Googled the multivariate Gaussian distribution, but still have no idea how to solve this. I tried to make $\mu_{x} \rightarrow a\mu_{x} \ and\, \mu_{y} \rightarrow ...
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Testing equality of marginal means of a multivariate normal random vector given estimated means and covariances

$x=(x_1,\dots,x_p)^\top$ is a random vector the elements of which are estimators that are known to have an asymptotic joint normal distribution $N(\mu,\Sigma)$. I have the estimated mean vector $\bar{...
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variables that are normaly distributed but their joint distribution is not multivariate normal with ρ = 0.5 [duplicate]

can you give me an example or explain me how to find one. It can be with copulas
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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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How is the mahalanobis distance like the euclidean distance? [duplicate]

Let's say $\vec{x}$ is an $n$ dimensional observation, $\vec{\mu}$ the $n$ dimensional mean of the sample that $\vec{x}$ is from and $\Sigma$ the $n \times n$ covariance matrix of that sample. Then ...
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Suggest a model for this dataset

I have a time series data set (the Old Faithful geyser data available here: http://www.gatsby.ucl.ac.uk/teaching/courses/ml1-2012/geyser.txt). Plotting the eruption duration on the x axis and the ...
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55 views

Sum of Normal Random Variables | Different Dimensions

I would like to generate one random numer $single\sim N(0,1)$ and create the vector that contains only this one number: $one = [single, single, ..., single]$ . Later, I would like to combine with ...
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Multivariate bayesian inference: learnig about the mean of a variable by observing another variable

I want to derive a Bayesian learning procedure where I don't only learn from my own signal, but also from other signals which are correlated to mine. I thought it could simply work with Bayesian ...
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Shape of parameters marginal posterior in hierarchical Bayes model

Consider a generic hierarchical Bayes model with data $y_i\sim p(y|\theta_i)$, dependent of parameters $\theta_i\sim p(\theta|\phi)$ and hyperparameters $\phi\sim p(\phi)$. Furthermore, assume that $\...
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Degenerate multivariate normal in Maximum Likelihood Estimator (Akaike's Info Criterion, BIC, LR Test usage)

Let's suppose that the considered set of random variable has a covariance matrix which is psd. Therefore the Gaussian pdf must be written in its degenerate form, where the determninat of the ...
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Fitting a Gaussian to two Gaussians [duplicate]

Let's say I have a dataset set of 2D points. I break the dataset to 2 subsets, equal in number. Then I fit a bivariate (2D) Gaussian to each subset. So I have two bivariate gaussians, each with its ...
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Problem with two correlated random normals

Imagine you have a two-dimensional multivariate normal random variable with $\mu = [0, 0]$ and $\Sigma\ = \begin{bmatrix}1 & r\\r & 1\end{bmatrix}$. (Conceptually, you have two random normal ...
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Delta method for vector valued functions

Suppose I have an estimator $B\in\mathbb{R}^m$ converging to $\beta$, such that $$ \sqrt{n}(B-\beta)\rightarrow\mathcal{N}(0,\Sigma). $$ I am interested in a quantity $\mathbf{h}(B):\ \mathbb{R}^m\...
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Conditional distribution of a function of a random vector given conditional distribution of random vector

Let $\mathbf{X}=(X_1,...,X_n)^T$ be a multivariate normal distribution. Now we have $\mathbf{Y}=(Y_1,...,Y_n)^T$ defined by $Y_i = e^{X_i}$. Let $\mathbf{Y^1}, \mathbf{Y^2}$ be partitions of $\mathbf{...
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Computing probability density function at a point, given the covariance matrix and mean

(Edited for clarity.) Say I have the variance-covariance matrix $\mathbf{V}$ and mean $\mathbf{\mu}$ of a multivariate normal distribution. Given a sample, $\mathbf{s}$, can I compute/estimate the ...
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Conjugate prior for DPGMMs using Gibbs sampling

I am using Gibbs sampling to infer DPGMMs. The prior for multivariate Gaussians is Normal-inverse Wishart. But it turns out that the covariances are not estimated accurately. Here is codes and results....
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Conditional probability density function (PDF) of bivariate normal distribution

Let $X$ and $Y$ have bivariate normal PDF with correlation coefficient $\rho$, i.e.,: $f(x,y)=\frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}}\exp{(-\frac{1}{2(1-\rho^2)}[\frac{(x-\mu_X)^2}{\sigma_X^2}+\...
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Explanatory variable independent of the response, yet has non-zero beta

My intuition was that if an explanatory variable is independent of the response then in a multiple regression it should have a $\beta$ of zero. Consider however the following very simple example: the ...
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How to characterize the effect of $(\textrm{Diag}(\Sigma^{-1}))^{-1}$ badly approximating $\textrm{Diag}(\Sigma)$

I have an almost singular covariance matrix $\Sigma\in\mathbb{R}^{n\times n}$ that has a few large eigenvalues, followed by many many comparatively very small ev's. If I were to try to approximate ...
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1answer
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Compute the mean and std of a random variable from its correlation with known random variables?

Context I'm running statistical simulations on IQ distribution among people and I would like to sample the IQ of a child given : The general distribution of IQ following : $$ \mathcal{N}(100, 15.5) ...
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Multivariate normal error with autocorrelation in second dimension

I am trying to forecast my model, but am unsure how to so in terms of error distribution (using mvrnorm). The model itself essentially estimates numbers over time and state (a matrix time x state). ...
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84 views

Deriving the conditional distribution of a multivariate normal, for inequalities

This question is slightly related to Deriving the conditional distributions of a multivariate normal distribution. In that question, the following situation was given. If $Y$ follows a multivariate ...
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Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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How to calculate Bhattacharya distance for singular multivariate normal distributions?

I am applying Bhattacharya distance to multivariate normal distributions $$D_{B}={1 \over 8}({\boldsymbol \mu }_{1}-{\boldsymbol \mu }_{2})^{T}{\boldsymbol \Sigma }^{{-1}}({\boldsymbol \mu }_{1}-{...
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2answers
106 views

Integrating with a multivariate Gaussian

I need to figure out the steps to solve the following integral, where $Q(\mathbf{w})$ is a multivariate Gaussian with mean $\overline{\mathbf{w}}$ and covariance $\mathbf{C}$: \begin{align}\int Q(\...
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1answer
48 views

Estimate parameters of a normal knowing the density function

I have a matrix with 3 columns. The first column contains values of a variable $x_1 \in [-1,1]$. The second column contains values of a variable $x_2 \in [-1,1]$. The third column contains a variable $...
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2answers
154 views

Distribution of $XY$ when $(X,Y) \sim BVN(0,0,1,1,\rho)$

The question is pretty much in the title, I need to find an approximate distribution of $XY$ when $(X,Y)$ follow a Bivariate Normal Distribution where $X$ and $Y$ are each $N(0,1)$ distributed and $...
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Simplification of bivariate normal $\phi_2(x,y,\rho)$ at $y=y_F$ (i.e. fixing one of the axes)

Suppose we start off with the traditional standard bivariate normal distribution: $$\phi_2(x,y|\rho,\mu_x=0,\mu_y=0,\sigma_x=1,\sigma_y=1)=\frac{1}{2\pi\sqrt{1-\rho^2}}\exp \left(-\frac{x^2-2\rho x y ...
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58 views

Using Hotelling's T-statistic to find an elliptic confidence set

The problem: We have samples of sizes ${n_1} = 25,{n_2} = 15,{n_3} = 30$ drawn independently from $N\left( {{\mu _i},{\sigma ^2}} \right),i = 1,2,3$ (normal distributions with same variance). We have $...
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20 views

Phase marginal for a multivariate complex Gaussian density

Suppose $z$ is a random variable taking values in $\mathbb{C}^n$ and admitting the complex Gaussian density $p(z;W) \propto \exp{(-\frac{1}{2}z^*Wz)}$, where $W$ is Hermitian. Let $r$ be the vector of ...