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)

Filter by
Sorted by
Tagged with
0 votes
0 answers
19 views

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) ...
user avatar
1 vote
0 answers
14 views

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 ...
user avatar
  • 531
1 vote
1 answer
40 views

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 $\beta_x$ ...
user avatar
0 votes
0 answers
9 views

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 ...
user avatar
  • 531
0 votes
0 answers
26 views

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 ...
user avatar
  • 423
3 votes
1 answer
39 views

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 ...
user avatar
0 votes
1 answer
47 views

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, ...
user avatar
1 vote
0 answers
9 views

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. ...
user avatar
0 votes
0 answers
17 views

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' ...
user avatar
  • 1,063
0 votes
1 answer
34 views

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 (...
user avatar
  • 556
1 vote
1 answer
38 views

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 ...
user avatar
1 vote
1 answer
47 views

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 ...
user avatar
0 votes
0 answers
42 views

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 ...
user avatar
1 vote
1 answer
70 views

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 ...
user avatar
  • 11
0 votes
0 answers
29 views

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 ...
user avatar
  • 1
0 votes
0 answers
29 views

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 (...
user avatar
  • 184
2 votes
1 answer
114 views

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 ...
user avatar
0 votes
1 answer
51 views

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 ...
user avatar
  • 1,303
0 votes
0 answers
29 views

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 ...
user avatar
  • 1
0 votes
0 answers
26 views

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 ...
user avatar
0 votes
0 answers
58 views

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 ...
user avatar
  • 1
0 votes
0 answers
18 views

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 $k-$...
user avatar
  • 249
1 vote
0 answers
17 views

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 ...
user avatar
  • 249
2 votes
0 answers
68 views

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 ...
user avatar
  • 3,546
2 votes
1 answer
85 views

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 ...
user avatar
  • 3,546
0 votes
0 answers
18 views

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?
user avatar
1 vote
1 answer
60 views

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, $\Sigma$).
user avatar
0 votes
0 answers
29 views

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 ...
user avatar
  • 3,546
1 vote
1 answer
90 views

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}...
user avatar
  • 25
0 votes
1 answer
57 views

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 ...
user avatar
  • 598
0 votes
1 answer
38 views

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 ...
user avatar
  • 421
1 vote
1 answer
29 views

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
user avatar
0 votes
0 answers
25 views

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,\...
user avatar
3 votes
1 answer
99 views

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 ...
user avatar
0 votes
0 answers
35 views

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,...
user avatar
  • 1
0 votes
0 answers
23 views

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 ...
user avatar
0 votes
0 answers
21 views

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 ...
user avatar
0 votes
0 answers
15 views

Probability bound for normal distribution based on independence

Let's say we have a multivariate normal distribution $x \sim N(m, S)$ with $x \in R^{n_x}$ and some thresholds $x_{min}, x_{max}$. Is the following inequality, which is based on statistical ...
user avatar
  • 117
0 votes
0 answers
70 views

Generate simulated dataset for multi class classification

I want to generate multi class classification data in R. Suppose, I want to create 4 classes and $X|Y=y \sim \mathcal{N}(\mu_y,\Sigma_y)$. I know that we will use ...
user avatar
  • 1
0 votes
0 answers
8 views

Generate multi class data from Simulations

I want to generate multi class classification data in R. Suppose, I want to create 4 classes and $X|Y=y \sim \mathcal{N}(\mu_y,\Sigma_y)$. I know that we will use mvrnorm function in R to create ...
user avatar
0 votes
0 answers
21 views

whether $T_3$ has a chisquare distribution subject to a multiplicative constant

Suppose $X_1,...,X_n$ are random samples from $N(0, \sigma^2)$, and $\bar X_n = n^{-1}\sum_{i=1}^{n}X_i$. Let $Y =$ $Y_1 \choose {Y_2}$, where $Y_1 = X_1 - \bar X_n$, $Y_2 = X_2 - \bar X_n$, be a ...
user avatar
  • 21
1 vote
1 answer
50 views

Rank-deficient transformation of gaussian vector

Suppose we have a gaussian $n$-dimensional vector $X \sim \mathcal{N}\left(\mu, \Sigma\right)$ I know that for a non-singular linear map $A$, its image $Y = AX$ will also be a gaussian vector with ...
user avatar
2 votes
1 answer
112 views

PCA as Pre-Processing before Clustering through GMM

Suppose I start with a data matrix $X \in \mathbb{R}^{N \times D}$, where each row $x_i$ is $D$-dimensional sample. I would like to cluster this data through a Gaussian Mixture Model (GMM). If I pre-...
user avatar
1 vote
1 answer
106 views

What priors are typically used for the correlation parameter in a bivariate normal?

The Bivariate normal distribution contains a correlation parameter. I want to implement an MCMC sampler to sample from the posterior distribution of the parameters of the bivariate normal distribution....
user avatar
0 votes
0 answers
9 views

Single draw from multivariate normal with constant mean and variance looks like univariate normal

The random vector Z of length p follows a multivariate normal distribution: $$Z \sim MVN(\boldsymbol{\mu}, \boldsymbol{\Sigma})$$ Now when all means and variances are equal (e.g. $\boldsymbol{\mu} = \...
user avatar
  • 1,534
2 votes
1 answer
52 views

Correlation matrix for 2d normal variables with components constrained by $y_1 + y_2 = x_1 + x_2$

I'm following this tutorial on Canonical Correlation Analysis, and had a question about an example from that tutorial. Question: On Page 4 of that document, the following example is given: "...
user avatar
1 vote
1 answer
67 views

In SEM, use WLS or transform data?

In SEM, we many times face non-normal data. Some software like Lisrel can normalize the data with just a click (and the formula of normalizing could be available in some articles of Joreskog). You ...
user avatar
  • 79
1 vote
1 answer
64 views

Distribution of Variance of draws from Multivariate Normal Distribution

Let the vector $\boldsymbol x$ be a draw of $n$ values from a multivariate normal distribution with zero mean. $$ \boldsymbol x \sim \mathcal{N}(\boldsymbol 0, \Sigma) $$ It may be assumed that $\...
user avatar
  • 504
0 votes
0 answers
13 views

Size of a sample needed for good estimation of multivariate normal distribution [duplicate]

I'm currently dealing with a problem where an underlying assumption is that, in general, subsets of my data follow somehow distinguishable multivariate normal distributions. I'm estimating via MLE ...
user avatar
  • 249
0 votes
0 answers
29 views

Understanding jointly Gaussian distribution [duplicate]

I am currently trying to understand to get an expression for $p(X,Y) = p(X) p(Y|X)$ where $X \sim \mathcal{N}(x;\mu_1,\sigma^2_1)$ $Y|X \sim \mathcal{N}(y;X,\sigma^2_{2})$. I assume the joint $X,Y$ is ...
user avatar

1
2 3 4 5
13