Questions tagged [covariance-matrix]

A $k\times k$ matrix of covariances between all pairs of $k$ random variables. It is also called variance-covariance matrix or simply the covariance matrix.

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17 views

How to find covariance matrix [duplicate]

I have some trouble understanding the concept of a covariance matrix. I want to find the covariance of a and b Cov(a,b). I have a random vector y=(y1, y2, y3)' with mean vector and covariance matrix ...
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How to obtain the covariance matrix from two regression model (for estimating 95% CI using delta method)?

I am trying to estimate the 95% CI for a function using delta method. Let say this function is t1/t2 In the case that t1 and t2 come from the same regression model, I can estimate the 95% CI using the ...
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If we have a covariance matrix $X$, what does $(X'X)^{-1}$ result in? [closed]

As the title states. Is there any significance behind this matrix? Is there a name? Does it give any information that is not present in the first matrix?
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how to understand the implications of the SVD matrix of the covariance $C_{XY} = X Y^T$

Given an $m \times n$ data matrix $X$, the SVD of its covariance matrix $$C = XX^T = ULU^T$$ provides the orthogonal unit vectors that maximize the variance in these directions. In the case of an $m \...
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If normally distributed random vector $X$ has a PD covariance matrix, then any conditional distribution induced by $X$ has a PD covariance matrix?

Suppose I have a random vector $X$ whose distribution is joint normal. I know that the covariance matrix of $X$ is positive definite. I wonder if I partition $X$ in any way: e.g., $X=(X_1, X_2)$, then ...
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How to test the significance of plm estimation used for diff-in-diff application

Good morning, I used panel linear model estimation taking into account a generalized diff-in-diff approach. Using R and the function plm I set a two way fixed effects regression with time and unit ...
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Covariance of the means of $x_t$ and $y_t=(x_t-\bar{x})^2$

Given a sample of a real-valued time series, $x=\{x_t\}_{t=1,...,T}$, let $\bar{x}$ be the sample mean of $x$ and set $y_t=(x_t-\bar{x})^2$. Then, $\bar{y}$ estimates the variance of $x$. Question: ...
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How does eigendecomposition form principal components? [duplicate]

I have a few questions regarding how specifically principal components are formed: What is the relevance of the magnitudes of the covariance when it comes to eigendecomposition? How does ...
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Is it ever acceptable to ignore the "leading minor of order 1 is not positive definite" error when plotting ellipses onto a NMDS?

I have a dataset that is evaluating the impacts of various factors on the composition of metabolites within trees. As part of this analysis I am running NMDS with metaMDS in vegan and plotting with ...
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How to find matrix of residual variances and covariances in a reticular action model of a SEM matrix specification?

I am currently trying to create a matrix specification of a structural equation model (SEM) using the reticular action model (RAM) from an example in Gerrard & Johnson's (2015) Mastering ...
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Covariance matrix with diagonal elements only

Let's imagine an arbitrary data set in $ D=3 $, is it possible that the covariance matrix consists of diagonal elements only? I'd say such a data set can't exist. Or would all data points be incident ...
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Alternatives to point biserial correlation discrete variable with more than 2 levels

I am trying to estimate the covariance between two variables, one is continuous and the other is discrete. The discrete variable has 3 levels (Red, Blue, Green). I know point biserial correlation will ...
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covariance between continuous and discrete variable

I am interested in estimating the variance-covariance between few variables in my dataset. The variables are a combination of continuous and discrete. I am curious how covariance can be estimated ...
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Covariance Matrix Estimation for the Generalized Method of Moments

I am solving and empirical exercise on the Generalized Method of Moments. It's a classical application/test of a famous model in Economics. There are 2 parameters $(\beta, \gamma)$ to be estimated ...
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Asymptotic covariance matrix of $\bar{\pmb x}$

In a text I'm reading it says that we define $$ \begin{align} \bar{\pmb x}= \begin{bmatrix}\bar x_1 \\ \bar x_2 \end{bmatrix} \end{align} $$ And then immediately says the asymptotic covariance matrix ...
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How to show this equality holds?

Sorry for the vague title, I couldn't find anything better suited. Let $X_{n\times 2}$ be a data matrix (with $i$th row vector $x_i$ and mean vector $\overline x$), with corresponding sample ...
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Gaussian Process for noise free data

In a Gaussian Process model, the covariance function is given as follows: $$ C\left(y_{*} \mid y_{d}\right)=K\left(x_{*}, x_{*}\right)-K\left(x_{*}, x_{d}\right) K\left(x_{d}, x_{d}\right)^{-1} K\left(...
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How can nuisance parameters in Fisher matrix can deteriorate the useful constraints?

I have a Fisher matrix $F$ which has the matrix blocks form like this : $$ F=\begin{bmatrix} A & B\\ C & D \end{bmatrix} $$ The block $A$ is the most important block, in the sense the ...
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SVD on demeaned matrix

I'm trying to understand the effect of de-meaning with SVD. Suppose I have matrix $WW^T = \sum_{i=1}^n w_iw_i^T$ where $W$ is $n \times m$ and $w_i$ are its columns. Running SVD on this yields ...
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How to get variance-covariance matrix for dispersion part of hglm model

I'm wanting to fit a 2-phase model to assess whether there is a change in within subject variance due to experimental group in this pre-post study design. Therefore, I am interested in the interaction ...
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Does the correlation matrix always have a smaller condition number than the covariance matrix?

In my experience and the experience of others (for example: https://stats.stackexchange.com/a/287737/193216), a covariance matrix $\textrm{COV}=\langle {\bf x}_t{\bf x}_t'\rangle$ has a larger ...
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Less data for one feature than other features in a regression?

Say you have a 10 timeseries features, where 9 of them have 10 years of data and 1 of the features has 5 years of data. In this scenario, more data would lead to better estimates of the true ...
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Choice of covariance matrices in Kalman filtering?

I am attempting to learn about Kalman filtering. I understand the state vector, call it $\mathbb{x}$, comes with a covariance matrix call it $P$. I can initially choose my $\mathbb{x}$ by my pre-...
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How to calculate univariate conditional distribution of a trivariate gaussian [duplicate]

I am trying to find the conditional distribution of a trivariate gaussian. So here is a hypothetical trivariate gaussian: $$\mathcal{N}(\mu_{ABC},\Sigma_{ABC}),\;\mu_{ABC}=\begin{bmatrix}\mu_A \\ \...
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Covariance Matrix with Block Structure is still PSD after setting Offdiagonals to 0?

Consider a matrix $A\in R^{MF\times NF}$ which is known to be PSD. The matrix $A$ consists of submatrices $A_{m,n} \in R^{F\times F}$. As $A$ is PSD it is also symmetric, i.e. $A_{m,n}=A_{n,m}$. $A=\...
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convergence of sample covariance matrix in case sample size depends on dimesion

Let $X_1,X_2,\dots,X_n$ be random sample from $\mathcal{N}_p(\mathbf{0},\mathbf{\Sigma})$ and put $\mathbf{S}=\frac{1}{n}\sum_{i=1}^nX_iX_i^t$, which is sample covariance matrix. If $p<n$, it is ...
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Variance of regression coefficients difference - approximating covariance

I have a following question. Let’s assume that we have a following linear model: $y = b_1x_1 + b_2x_2 + … + b_nx_n + b_m$ I would like to find a difference between coefficients with its accompanying ...
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VAR model and robust estimators of covariance matrix

I have a VAR(2) model which has autocorrelations (since lag = 8 mostly), even when number of lags for this model are bigger. I got and advice that robust estimators of covariance matrix will help with ...
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Inner Product for Geometric Interpretation of Multivariate Random Vectors

I was looking into the geometric interpretation of random variables as random vectors in a vector space. The textbook I'm referring to defined $\operatorname {Cov}(X,Y)$ as the inner product for any ...
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Concentration of top eigenvectors

Let $M$ be a $d \times d$ symmetric matrix with rank $k < d$, write $M = U \Lambda U^T$. Define $\hat{M} = M + Z$, where $z_{ij} \sim N(0, 1 /n )$. Suppose we try to estimate $U$ by taking the top $...
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Integral of the error function

Assume a pair of normal variables $a,b\sim N(\mu, \Sigma)$, with $\rho_{ab}\neq0$. We know their joint distribution in the (shorthand) form: $$f_X(a,b)=\frac{1}{2\pi\sqrt{|\Sigma|}}exp(-\frac{1}{2}(x-\...
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How to simplify fallowing martrices' Expected value elements in the equation

I have a matrix equation with four variables inside, $x^1_{00}$, $x^1_{tt}$ and $x^2_{tt}$, $x^2_{tt}$. $x^1_{00}$, $x^1_{tt}$ come from the same distribution , they are only shifted by timelag $t$. ...
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Why would SAS proc mixed produce residual variance estimates under AR1 covariance structure, but none for unstructured covariance?

I am working on a program in SAS that seeks to extract r2 based on the residual variance produced by covariance parameter estimates in PROC MIXED. The specification ...
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Distribution of solution to linear system

I have a random symmetric matrix $ A \in \mathbb{R}^{M \times M}$, and random vector $b \in \mathbb{R}^M$. I also have access to expressions for the mean and variance of each element of $A$ and $b$ (...
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Maximum Likelihood estimate of Covariance Matrix of the Multivariate Gaussian Linear Model

Consider the following linear model \begin{align} \mathbf{y}_l = \mathbf{H}\mathbf{x}_l + \mathbf{v}_l, \quad l = 1, 2, \cdots, L \end{align} where $\mathbf{y}_l$ is $N \times 1$ measurement ...
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Compute Conditional Equicorrelation Matrix (DECO-GARCH) from DCC-GARCH estimation in R or Python

I am working on the conditional diversification benefit (CDB - Christoffersen et al. 2014), to do so I need to develop the DECO GARCH Model. Unfortunately due to my lack of algebra I find it difficult ...
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Find the variance-covariance matrix for a linear combination of multiple bivariate normal distribution?

I have an arbitrary number of independnet bivariate normal distributions with $\mu_i = [x_i,z_i]$ & $ \Sigma_i= \left(\begin{array}{cc} \sigma^2_{x_i} & \sigma^2_{x_i,z_i}\\\ \sigma^2_{x_i, ...
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Covariance of log-transformed variable

Consider a model with parameters $a,b$. We know the priors for the parameters are $b\sim\mathcal{N}(\mu_b,\nu_b^2)$ and $g=log(a)\sim\mathcal{N}(\mu_g,\nu_g^2)$. Using observed data we produce maximum-...
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Fisher information as negative log likelihood

The Fisher Information is defined as the covariance matrix, or $E_{y \sim P(y;\theta)}[ \nabla_{\theta} ln(p(y;\theta)) \nabla_{\theta} ln(p(y;\theta))^T]$. It can also be defined as $E_{y \sim P(y;\...
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Variance covariance matrix of parameters in logistic regression?

The given below image is taken from book Introduction to Linear Regression Analysis (Douglas C Montgomery) My apologies in advance for not typing , I just want to understand the concept. (1) First ...
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limit behaviour of a quadratic form

I'm reading a book on portfolio optimization and risk management, and I wanna clarify what the author wants to say. Let $\mathbf{X}=[X_1,...,X_n]$ be a random vector with mean $\mathbf{\mu}=E\{\mathbf{...
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Variance introduced by dropout

I just read this paper Sampling-free Epistemic Uncertainty Estimation Using Approximated Variance Propagation and stumbled upon two points at the end of section 3.1 that I don't understand. Here, the ...
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Correlation of the sample as the estimation of parameter of Gaussian copula

Given 3 variables $X, Y, Z$ and I assume that the multivariate distribution of them is a Gaussian copula. Now I need to estimate the correlation matrix of the Gaussian copula. As in many textbooks, i ...
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What is the covariance matrix of a fixed matrix multiplied by a vector of random variables? [duplicate]

Suppose we have a random variable $\epsilon\sim N(0,\sigma^2)$, then we know that if a constant, say $a$, is multiplied by $\epsilon$, then \begin{equation} a\epsilon\sim N\left(0,a^2\sigma^2\right) \...
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How to obtain the variance-covariance matrix from a multinomial logit model

I want to find the variance-covariance matrix for a multinomial logit model from the prior coefficients (Beta) and the attributes (X) of each alternative.
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Trouble with obtaining constant matrix to find variance-covariance matrix of regression parameters

I have been working on an exercise from Applied Linear Statistical Models - 5th edition- by Kutner. The question is asking me to obtain the variance-covariance matrix for a polynomial regression of ...
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Does the Cholesky decomposition of a covariance matrix lead to a lower triangular matrix with positive diagonals?

We know that an $N\times N$ covariance matrix $\Sigma$ is symmetric positive definite, and can be factorized using Cholesky decomposition as follows \begin{equation} \Sigma=LL' \end{equation} where $L$...
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Covariance of the sum of two random vectors

This is the situation. I have an estimation of the position $(x_t,y_t)$ of an object with its covariance $\Sigma_p$ and an estimation of its speed $(v_x, v_y)$ with its covariance $\Sigma_v$. Actually,...
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42 views

Prior for covariance matrix?

Given a set of data $\{(x_i\pm e_{x,i},\,y_i\pm e_{y,i})\}_i$ (with uncorrelated uncertainties), I want to model it as a multivariate Gaussian function with an unknown mean $\boldsymbol{\mu} $ and a ...
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Is there a way to use the covariance matrix to find coefficients for multiple regression WITHOUT intercept?

Given: $$ y=\alpha + \beta x $$ The problem on how to get regression coefficients $\alpha, \beta_0, \beta_1,...,\beta_n$ from the covariance matrix is solved here: Is there a way to use the covariance ...

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