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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Approximation for Multivariate Gaussian

I am reading a paper, where they say they approximate a 2D multivariate Gaussian distribution by its second moment. The corresponding formula is the following: $\displaystyle d(u) = \frac{1}{1 + (u - \...
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Multivariate normal distribution vs. sampling multiple times from univariate normal distribution

This may come across as a stupid question, but I am confused as to the difference between a multivariate normal distribution and sampling multiple times from a single univariate distribution. Lets say ...
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How to build a probability distribution from locations with accuracies?

I have a set of $n$ GPS locations $l_i$ with latitude, longitude in degrees and accuracy in meters, corresponding to $3 σ$, i.e. the probability ≈ 0.997) $(lat_i, lng_i, acc_i)$ or $(lat_i, lng_i, σ_i ...
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How does the reparameterisation trick work for multivariate Gaussians?

I understand that for sampling from a univariate Gaussian, we can use $x = g(\epsilon) = \mu + \epsilon \sigma $ and then differentiate this transformation with respect to $\mu, \sigma$. How does ...
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Fisher information of $\rho$ in a symmetric normal $N_p(\mathbf 0,\Sigma)$ distribution

Suppose $\boldsymbol X=(X_1,\ldots,X_p)'\sim N_p(\mathbf0,\Sigma)$ where $\Sigma=(1-\rho)I_p+\rho\mathbf1\mathbf1'$ is positive definite. The objective is to obtain the asymptotic variance of the MLE ...
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Distribution of the minimum of the components of a multivariate normal random variable [duplicate]

Let $\mathbf{X} = (X_1, \dots, X_p)^\mathsf{T}$ be a $p$-dimensional random variable following a multivariate normal distribution with mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{...
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Cholesky decomposition lower triangular in Gaussian process sampling

I am trying to intuitively understand the Cholesky decomposition in gaussian process function sampling. I understand it as as the square root of the covariance matrix being the multivariate ...
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Does lack of data affect covariance matrix estimate?

I am building some experiments using the multivariate normal probability density function to estimate the likelihood of a given sample to come from a distribution. For that, the PDF is built using as ...
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Combining two covariance matrices — Multiplying two multi-variate Gaussian PDFs

I want to multiply two Normal probability density functions, $$ {\displaystyle f_{\mathbf {X} }(x_{1},\ldots ,x_{k})={\frac {\exp \left(-{\frac {1}{2}}({\mathbf {x} }-{\boldsymbol {\mu }})^{\mathrm {T}...
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Sampling from multivariate normal using a pseudo-inverse [duplicate]

Say I have an improper multivariate normal distribution with singular precision matrix $Q$. I can compute the pseudo-inverse of $Q$ and use this as a covariance matrix, i.e. $\Sigma = Q ^{+}$. I seem ...
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Mean and variance for multivariate truncated normal

Does anyone have a reference for mean and variance of a multivariate normal truncated along a single axis? I.e. $\mathbb{E}[X | x_i > 0]$ and $Var[X | x_i > 0]$, where $X= [x_1,..,x_n] \sim \...
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Fast multivariate normality test for large data sets in R

I have a data set of about 260,000 observations of 50 variables. Although I highly suspect it's multivariate distribution to be non-normal, I still need a proof of it. I tried ...
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Dependent variables meet different test assumptions - Should I treat them all the same, or individually?

I'm conducted pair-wise analyses. I have a binary independent variable and 17 dependent variables on a continuous scale. To explain the basic premise: An assessor watched an event in person and scored ...
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MLE and MOM estimates coincide in Normal distribution

Let $(X_1,Y_1),...,(X_n,Y_n)$ be a sample form $N(\mu_x,\mu_y,\sigma^2_x,\sigma^2_y,\rho)$ population. If $n\geq 5$ and $\mu_x$ and $\mu_y$ are unknown, I want to conclude that the estimates of all ...
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How to fit a mixture of 2D Gaussian in BUGS/JAGS?

I am trying to estimate the parameters of a mixture of 2D Gaussian distribution using JAGS. I first created two components from a multivariate normal distribution and then combined them to get a ...
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Confidence interval for $\mu_i$ (multivariate normal): some $\mu_j$ are known, unknown $\sigma^2$ and an estimation of the correlation structure?

Say we have a multivariate normal with m dimensions (let's say that m=1,000), the mean vector $\mu$ is known only for the first 100 elements, but unknown for the remaining 900. We have a single ...
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Is there an intuition about the matrix operations in the exponent of the multivariate normal distribution?

In the exponent of the multivariate distribution, there are 2 vectors and a square matrix multiplied together to get a scalar result: $$(\mathbf{x} - \mu)^{\text{T}}\Gamma^{-1}(\mathbf{x} - \mu)$$ ...
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How to solve the summation part of the log likelihood of multivariate normal distribution?

The probability density function of $X$ is \begin{equation} f(x)= (2\pi)^{-\frac{m}{2}} |\Sigma|^{-\frac{1}{2}} \exp (-\frac{1}{2}(x-\mu)^T \Sigma^{-1} (x-\mu)) \label{pdf} \end{equation} where $X$ is ...
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MANOVA MLE proof

I checked several textbooks and papers, they both give the result directly without any proof, so I'm trying to prove the MLE for MAOVA myself. For the following model, where $\epsilon \sim N(0, \Sigma,...
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How to find the variance(s) of a bivariate normal density such that 95% of the mass is within a certain radius from the mean defined by a point A?

I would like to find the variance of a bivariate normal density (BND), centered at the mean M, such that 95% of its mass is within a certain radius, which depends on the position of a point, A. (Note: ...
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Definition of Gaussian process

Definition of a GP: a stochastic process $\{X_t, t\in T\}$ such that for every finite set of realization times $t_1, \dots, t_k \in T$, the joint $(X_{t_1}, X_{t_2}, \dots, X_{t_k})$ is a multivariate ...
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Multivariate normal distribution -testing a quadratic form of mean vector

Suppose $X_{1},X_2,...,X_n$ are i.i.d.observations from a multivariate normal distribution $N(\mu,\Sigma)$ where $\Sigma$ is known. Assume that $a$ and $b$ are given vectors. Use the likelihood ratio ...
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Prove the joint distribution of an AR(1) process is Multivariate Gaussian

I'm struggling a bit with the proof of this if anyone can help! I keep ending up going in circles with the conditional probabilities and I don't know what are the right steps to take. For an $AR(1)$ ...
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Estimating Means of a Bivariate Normal Distribution where some parameters are known

I am trying to figure out how to estimate means of a bivariate normal distribution from a sample when some of the parameters are already known. let $$ \boldsymbol{x} = \begin{bmatrix} x\\ y\\ \end{...
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Multivariate canonical exponential family

Consider the canonical d-dimensional exponential family with densities $$p(x)=exp\left(\langle\theta,T(x)\rangle-A(\theta)\right)h(x),\theta\in\Omega$$ with $\Omega\subset\Omega_0=\{\theta:A(\theta)&...
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Partial derivative of multivariate cdf with respect to coefficients

I want to take the partial derivative of this multivariate gaussian cumulative distribution function with respect to $\beta_1$ (which is a single element of the $\beta$ vector). $X_1$ is a n $\times$ ...
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Covariance matrix in multivariate is Wishart random variable

I'm working on sample variance distribution, on page 111 of Methods of Multivariate Analysis, it says "The joint distribution of these $p(p + 1)/2$ distinct variables in $W =(n−1)S = \sum_i(y_i − ...
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Incremental assumption to go from uncorrelated and (marginally) normal to independent errors

Take the following simple linear model in which for simplicity, we'll assume $x_i$ to be non-random: $$y_i=\beta_0+\beta_1x_i+\epsilon_i,$$ $$E[\epsilon_i]=0$$ Suppose we additionally assume that the ...
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Jointly complete and sufficient statistics for multivariate normal distribution

Consider the random sample X from the multivariate normal distribution where xi are i.i.d as N(µ,Σ). *Show that the sample mean x̄ and Sample covariance matrix S are jointly complete and sufficient ...
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Weighted Inverse Wishart Distribution

Assume $X \in \mathbb{R}^{n \times p}$ ($p<n$). If the rows of $X$ are i.i.d. $N(0,I_p)$, we know that $$ (X^{\rm T} X)^{-1} \sim \text{inverse-Wishart}(n, I_p). $$ Let $W$ be a diagonal matrix ...
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Kullback-Leibler divergence between multivariate t and the multivariate normal?

I want to calculate the Kullback Leibler divergence between a multivariate t distribution and a multivariate normal distribution, for different values of the degrees of freedom $\nu$. However, this ...
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Covariance matrix in Multivariate Gaussian Process

I am working on Gaussian Processes Regression and having some trouble understanding how to properly state the covariance matrix of 3 random vectors. Say I have an input space of 3 dimensions, $X$, $Y$ ...
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Likelihood of position random variable belong to a certain region

I have a motion model which operates with state estimates and covariances, e.g. outputs from a Kalman filter, and I need to be able to describe the likelihood of the tracked object existing in a ...
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"Invalid Parent Values' in Multivariate Normal approximation of Multinomial (JAGS)

I have a transition matrix, describing the probability of an entity moving from one state to another in a time-period. I use this transition matrix to generate a series of "flow matrices", $...
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What calculation is implied by this Gaussian term?

I'd appreciate this community's help in understanding the calculation implied by the Gaussian term in this equation: $$w_{ik} = \alpha_{k} \mathcal{N}( \mathbf{y}_i | \mathbf{X}_i \mathbf{\beta}_k , \...
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Bias of mean in toy Monte Carlo sampling of $A\times B$ for $A=a\pm\sigma_A$ and $B=b\pm\sigma_B$

I am trying to do some toy Monte Carlo sampling, to calculate the uncertainty of the product $A\times B$ of two random variables $A=a\pm\sigma_A$ and $B=b\pm\sigma_B$. I also assume that these ...
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Loss Calculations: Multivariate Normal Distribution

The company could be vulnerable to serious problems defined as $Event_1$, $Event_2$ or $Event_3$ (equally likely events, independent). It can be assumes that they follow log-normal distribution: $ln(...
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multivariate gaussian integral

I am looking for the solution of multivariate gaussian integral over a vector x with an arbitrary vector a as upper limit and minus infinity as lower limit. The dimension of the vectors x and a are p $...
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Aligning multivariate Gaussian distributions

I have two multivariate Gaussian variables $\mathbf x_0\sim\mathcal N(0,\Sigma_0)$ and $\mathbf x_1\sim\mathcal N(0,\Sigma_1)$, which are generated by almost the same process. Specifically $\mathbf ...
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Gaussian Process conditioning $X|Y \sim N(\mu_{X}+\sum_{XY}\sum_{YY}^{-1}(Y-\mu_Y), \sum_{XX}-\sum_{YY}^{-1}\sum_{YX})$

I found this formula in a post about Gaussian Proccess: https://distill.pub/2019/visual-exploration-gaussian-processes/ $X|Y \sim N(\mu_{X}+\sum_{XY}\sum_{YY}^{-1}(Y-\mu_Y), \sum_{XX}-\sum_{YY}^{-1}\...
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How is covariance matrix affected if each data points is multipled by some constant?

I have a 2D multivariate Normal distribution with some mean and a covariance matrix. While fitting the function I had normalized the data.so the mean and covariance I have are for the normalized data. ...
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Hotelling T squared seemingly useless at detecting a mean shift

I am trying to develop a single-sample Hotelling $T^2$ test in order to implement a multivariate control chart, as described in Montgomery, D. C. (2009) Introduction To Statistical Quality Control, ...
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Multi-variate normal distribution covariance matrix

Let x ~ CN (0,a) and y ~ CN(0,b), with a, b positive constants. Let z = x + y. I need to find the distribution of the vector t=[x; z]. My basic understanding is that vector t is also complex Gaussian ...
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Assumptions of Path Analysis - Multivariate Normality inspite Univariate Non Normality

I am currently checking if my data meets the assumptions for path analysis: mainly multivariate normality of the three endogenous variables $m_1,m_2,y$ (As recommended by e.g. Streiner, 2005). I ...
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What is the joint distribution $(\hat{\mu_1},\hat{\mu_2},\hat{\Sigma})$?

Let $X_1,...,X_{n_1}$ be an i.i.d. sample from $N_p(\mu_1,\Sigma)$ and let $Y_1,...,Y_{n_2}$ be an independent sample from $N_p(\mu_2,\Sigma)$, for some $\mu_1,\mu_2 \in \mathbb{R}^p$ and some ...
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Multivariate Normal Distribution: Divide each random variable by its standard deviation

If $X$~$Normal(\mu,\Sigma)$, and I divide each random variable in $X$ (the marginals) by its standard deviation, what will happen to the covariance matrix $\Sigma$?
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Derivation of Pearson's Product Moment Correlation Coefficient's (PPMCC) distribution from bivariate normal variables?

I'm interested in reading through the derivation of the probability density function that PPMCC follows when it's input is a bivariate normal variables. Mathworld gives the following equalities: $$P(r)...
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EM algorithm for MLE from a bivariate normal sample with missing data: Stuck on M-step

I'm trying to understand applying the EM algorithm to compute the MLE in a missing data problem. Specifically, suppose $(x_1,y_1),\ldots,(x_n,y_n)$ is a random sample from the bivariate normal ...
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How to test normality of errors in multiple linear regression? [duplicate]

In the linear model we make the assumption that the vector of errors follows a multivariate normal distribution: I want to test this assumption for a given data set. How can I do this? I was ...

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