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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$3\sigma$ region of locally non-parabolic cost function
Consider you have an optimization problem where your are given:
Dataset $(\mathbf{x}, \mathbf{y})$
$\mathbf{x}$ is perfectly known
$\mathbf{y} = \mathbf{y}_{\text{true}} + \mathbf{n}$ with $n_i \...
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Expectation of constant matrix sandwiched between two random matrices
I have a $N\times P$ random matrix $X$ with i.i.d. coefficients from a standard normalized Gaussian $\mathcal{N}(0, 1)$. The corresponding Wishart matrix is
$$W = \frac{1}{P}X X^{\top}$$
Calculating ...
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Weighted average of covariance matrixes
My issue is as follows:
In my model there are 4 different states, which each have a calculated probability of happening. I also have calculated covariance matrixes for my variables in each of these ...
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Variances explained by each feature on PC in PCA
I came across this article with an associated Python codebase.
In brief, there is a section "Understanding How Features Contribute to PCs
", where...
One method for understanding which ...
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Calculating Mean Vector and Covariance Matrix of Mixture of Multivariate Normal Distributions [duplicate]
In an effort to better understand multivariate normal distributions, I am attempting to derive the mean vector and covariance matrix of multivariate random vector defined by a mixture distribution. ...
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Reference about the comparison between covariance matrices
Suppose we have 2 symmetric matrices $A$ and $B$.
Then, we say that $A \succeq B$ if $A - B$ is a positive semi-definite matrix.
I was wondering about the intuition and interpretation of $A \succeq B$,...
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Measuring level of uncorrelation from correlation matrix?
This question might not be completely well defined mathematically, but if so I am hoping that people can help point me in the right direction, since this is not a part of mathematics I have ever had ...
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How to compute the covariance matrix for a mixture model estimated by the EM algorithm
I am trying to compute the observed Fisher information matrix for a mixture model estimated by the EM algorithm. My original thought is to simply compute the second derivative of a mixture density. ...
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Sample covariance of t distribution and degree of freedom
If $X$ is a P by N size matrix, $X_{ij} \sim N(0,\sigma_i^2)$ if I standardize this X matrix with sample mean and sample variance (assuming I don't have access to the population mean and variance) I ...
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PCA with gram matrix produces different results from PCA done using covariance matrix?
I was trying PCA on a dataset (#samples=24, #dims=42) via eigendecomposition using numpy. I read that for matrices where the number of features exceeds the number of samples, we should use the gram ...
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comparison of covariance matrices with unit that allows interpretation
I have fitted Gaussian distributions (3 dimensions) and need to compare their "variance".
I found that there are multiple approaches, such as
The $\text{trace}(\Sigma)$ (which does not take ...
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How to calculate covariance matrix in nonlinear least squares
I am fitting a nonlinear model to observations by using least squares to estimated the model parameters.
Theoretically, the covariance matrix of the parameters can be estimated by inverting the ...
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Question on the normality assumption of the predictor variable and the covariance matrix of the likelihood
Suppose you are trying to fit a continuous variable y depending on some input x ( es y= ax+b) and you are using a multivariate distribution as likelihood for the fitting.
Does using a multinormal ...
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Need help defining the theoretical covariance matrix between a parameter and two distinct estimators
I have a problem that resembles SEM or factor analysis, but the indicators are estimators of the parameter, not empirical observations of random variables. The model is a one-factor model with two ...
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How to calculate covariance matrix properly using Bartlett's formula
Batlett's formula for asympthotic covariance matrix $C$ is given as:
$$c_{ij}= \sum_{k=1}^{\infty} (\rho_{k-i}+\rho_{k+i}-2\rho_{k}\rho_i)(\rho_{k-j}+\rho_{k+j}-2\rho_{k}\rho_j)$$
where $\rho$ is the ...
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Confidence interval for covariance matrix
I am trying to get a understanding of how good the estimation is of my covariance matrix. Suppose I have random variables X1,..., X10 and they are all iid $N(\mu, \Sigma)$ where $\Sigma$ is a NxN ...
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How to Implement Newey-West covariance matrix properly for MDE estimation
I am trying to implement the MDE method for GARCH given by Baillie and Chung '01 (Estimation of GARCH Models from the Autocorrelations of the Squares of a Process, Jrl. of Time Series Analysis) but I ...
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Covariance as a function of explanatory variable
Suppose we have a collection of bivariate random variables $X_{1i}$ and $X_{2i}$ indexed by a continuous variable $t$ such that, for the vector ${\bf{X}} = (X_{1i} \ X_{2i})^T$ we can assume
\begin{...
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How to write the likelihood for a multivariate gaussian linear model
I have a lasso-like bayesian graphical model where we try to estimate precision matrices between two conditions (0 and 1), $\Sigma_0^{-1}$ and $\Sigma_1^{-1}$, respectively. The model can be ...
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The covariance of a data matrix
Please let me know if the below statement is valid or not ;
Suppose that $X$ is an $n\times p$ data matrix with $p$ features and $n$ data samples.
Suppose further that each feature(column) is zero ...
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Is this algorithm for robust estimation of the covariance matrix sensible?
I have a high dimensional dataset $\bf{X} \subset \mathbb{R}^d$, which is multimodal and has outliers. I want to estimate a robust measure of association, something like the correlation between two ...
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Bounding the error of a quadratic form based on distance between two quadratic form matrices
In survey statistics, it is common to estimate sampling variance for a sum using a quadratic form. For example, I want to estimate the number of people in the U.S. who have been diagnosed with a ...
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Subtilities of MCMC method and more generally about covariance matrix and Samplers [duplicate]
i have difficulties to better understand about what we commonly called a sampler, especially how to produce a covariance matrix between parameters during a MCMC code run.
In MCM, I know that we start ...
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Precision matrix and conditional uncorrelatedness
Let $\mathbf{u}_t = (u_{1t},u_{2t},\ldots,u_{pt})^{\prime}$ be a $p \times 1$ (stationary) random vector, and let $\mathbf{\Sigma}_{u}$ be the covariance matrix of $\mathbf{u}_t$. Further, denote the ...
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Collapse of sampled Mahalanobis distance
Let $\{ x^{(1)}, \ldots, x^{(M)}\}$ be $M$ samples from a $n$-dimensional multivariate Gaussian distribution $\pi_{X} = \mathcal{N}(\mu, \Sigma)$.
We recall the definition of the squared Mahalanobis ...
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Is the sum of the diagonal elements of a covariance matrix always equal or larger than the sum of its off-diagonal elements?
For any given covariance matrix, will the sum of the diagonal elements always be bigger than the sum of the off-diagonal elements?
Let $\sigma_i$ be the standard deviation of the $i^\text{th}$ term of ...
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Options for transforming the variance-covariance matrix generated by the survreg() function to the original scale of the Weibull distribution?
I'm working with the survreg() function of the R survival package, and I understand that the default scale parameter for the ...
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Sensitivity analysis of a fuzzy cognitive model
I have a fuzzy cognitive model of inter-organizational collaboration that is represented by a 27x27 matrix. I want to analyze the effects of individual variables. I've read I can do this best with a ...
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Cross-correlation for univariate GARCH models
I have a strange and maybe stupid question about cross-correlation.
Let's imagine having 2 times series, for example, asset A and ...
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Having difficulty finding regression function and conditional variance [duplicate]
The random vector distribution X = (Y, X, Z) is Gaussian with mean µ = (1, 2, 4)T and a covariance matrix Σ is equal to:
\begin{pmatrix}
2 & 3 & 1\\
3 & 5 & 2\\
1 & 2 & 6
\end{...
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Properties of the generalized centering matrix [closed]
Let $L_{\boldsymbol{\Delta}}\in \mathbb{R}^{M \times M}$ be the generalized centering matrix given by:
$L_{\boldsymbol{\Delta}} = \boldsymbol{\Delta} - \frac{1}{\text{tr}(\boldsymbol{\Delta})} \...
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Eigenvalues/Eigenvectors of Correlation and Covariance matrices
Suppose $\Sigma$ is a covariance matrix $P$ is its corresponding correlation matrix. Let $\lambda_1, \dots, \lambda_p$ and $\tau_1, \dots, \tau_p$ denote the ordered eigenvalues of $\Sigma$ and $P$, ...
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Bessel correction for the variance of dependent sample
Assuming a sample $X_1, X_2, ..., X_n$, the sample variance is calculated as
$s^2 = \frac{1}{n-1} \sum (X_i-\bar{X})^2$
The fact that there is $n-1$ in the denominator instead of $n$ is called the ...
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Existence of stationary process with a given ACF
Consider the sequence
$$\gamma(\tau) = \begin{cases}
1 & \text{if } |\tau| ≤ K \\
0 & \text{if }|\tau| > K
\end{cases}$$
where $K$ is a positive integer. Is $\gamma(\tau)$ an auto ...
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Is it correct to estimate covariance matrix of null distribution of related random variables using independent groups as data points?
I will use an example of an antidepressant drug trial as an example to illustrate my question - hope this helps as I, too, am confused.
Suppose a study is testing the effects of some drug on people ...
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What's exactly inside the attribute 'Pars' in apVar of the lme object in R
In this example:
what is the third term in attr(,"Pars") within the apVar of the ...
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Covariance of stochastic process
I have the following stochastic differential equation
$dX_t=\kappa\left [ \theta-X_t\right ]dt + \Sigma d W_{t}$
I derived formula for $X_t$ which is in the following form
$X_{t}=\theta+e^{-\kappa t}\...
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In what way, the covariance of two variable shows their correlation or dependency? [duplicate]
I know that the covariance of two random variables, such as X and Y, is calculated as follows:
$$ Cov(X, Y) = \frac{\sum{(X - \bar{X})(Y-\bar{Y})}}{n} $$
where $n$ is the size of the sample and $\bar{...
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Using covariance matrix as data for linear discriminant analysis?
In PCA analysis, covariance matrix as input is often used. But for LDA, it seems that it's not so usual. Is there any reason why covariance as data is not used well as a standard for LDA?
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Covariance matrix of rotation and translation applied to a point
I'm trying to calculate a 2x2 covariance matrix in Cartesian coordinates that represents the amount of uncertainty when rotating and translating a point in 2D space,
$\Sigma = \begin{pmatrix} \sigma_{...
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Square-root matrix of coefficient var-cov matrix $\mathbf{V}(\boldsymbol{\beta})$ during simulation
I am performing simulations to obtain predicted yhat values based on fixed x-values and random coefficient values about their uncertainty. The matrix with random values of coefficients $(m \times p)$ ...
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How to build the covariance matrix with different weighted moments via GMM [closed]
I have two sets of moment conditions, one is IV moment with N observations but the second moment only has N_1 observations, N_1<N.
How to build the covariance matrix? Appreciate for any replies!
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Difference of the conditional variance-covariance matrices between lme4 and nlme
In ?lme4::ranef, it is stated:
condVar: a logical argument indicating if the conditional variance-covariance matrices of the random effects should be added as an ...
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Looking at how covariance/correlation between variables differs in two groups?
I have a few hundred variables representing different biomarkers. These variables have been measured in both cases and controls. The underlying units of measurement are not important, so I have ...
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Analyse clustered items from different scales?
I am a non stat person so pls be kind in your reply! What sort of statistic can I use to compare how the items of 3 different Likert scales covariate? My respondents sample is 150 ppl. Each respondent ...
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When to use 'unconditional = FALSE' in plot.gam()
I'm trying to figure out under what conditions one would make 'unconditional = FALSE' (in plot.gam and gratia::draw), because in my case 'unconditional = TRUE' shrunk the uncertainty bands around my ...
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Evaluate relative quality of covariance matrix relative to a set
My ultimate goal is a way to evaluate a group of "m" covariance matrices (all size n*n) so I can pick an arbitrary one and calculate "this one is tighter than the average covariance ...
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Gaussian copula: how to scale data back to get target covariance matrix (not correlation)
I would like to use a Gaussian Copula to simulate data with a given covariance matrix and given marginal distributions. I understand that the input to the copula cannot be the covariance matrix $\...
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Multivariate Log-Normal variables with given covariance
Given a symmetric positive definite matrix $\bf \Sigma \in \mathbb{R}^{n \times n}$, I want to find a matrix ${\bf \Gamma} \in \mathbb{R}^{n \times n}$ and a vector ${\bf m} \in \mathbb{R}^n$ such ...
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Generate nonnegative variates with mean 1 and specified variance-covariance
Problem
In several applications in surveys, it would be helpful to be able to generate a set of $R$ $n$-dimensional variates with the following properties:
Has mean vector $1$
Has a specified ...
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