Questions tagged [precision-matrix]
Use the precision-matrix tag when the analysis or discussion relates to the inverse of a covariance matrix.
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Random correlation matrices
Suppose that we simulate random $n\times n$ correlation matrices by assigning iid $U(-1,1)$ random variables to all off-diagonal entries and accept matrices $\boldsymbol\Sigma$ that are positive ...
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Graphical Lasso for estimating words network
I have a matrix whose columns are words and rows are different speeches by a person. Therefore, the i,j element of the matrix is the count of occurence of a word in a speech. I would like to estimate ...
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Gaussian Markov Random Fields - Conditional distribution from jointly gaussian with given precision matrix
Suppose I have jointly normal random vectors $[\bf{v_1}, \ldots, \bf{v_K}]$' with mean $ \bf{M}$ and joint block tridiagonal precision matrix $ \bf{P}$:
$$ \bf{M}= \begin{bmatrix}\bf{\mu_1} \\\ldots \\...
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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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Asymptotics of the Sample Inverse Covariance Matrix
This paper provides the asymptotic normal distribution of the sample covariance matrix $\hat\Sigma$ under some fairly general conditions. Are there any resources which discuss similar results for the ...
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Creating a random sparse precision matrix?
In my current project, I want to create a random sparse precision matrix $\boldsymbol{P}=\boldsymbol{\Sigma}^{-1}$ (the inverse of a covariance matrix $\boldsymbol{\Sigma}$). My current procedure ...
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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. ...
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What is the analog of precision matrix for cross-covariance matrices?
For a covariance matrix, I am aware of the precision matrix, the covariance matrix inverse. What's the analog for that for a cross covariance matrix, i.e. $E[XY^{\top}]-E[X]E[Y^{\top}]$ for two random ...
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When should we decompose the precision matrix as opposed to the covariance matrix to generate correlated variables?
We can take a covariance matrix $\Sigma$ and decompose this into a lower and upper triangular matrix $\Sigma = U^T U$ where $U$ is the Cholesky matrix. This matrix can be used to transform ...
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R package to solve Gaussian MLE under conditional independence constraints
Is there any R package or function to solve Gaussian MLE under conditional independence constraints?
Suppose we have $y_i\overset{i.i.d}{\sim}\mathcal{N}(0,\Sigma_{p\times p})$, $i = 1,2,\ldots,n$. We ...
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graphical lasso with known precision matrix structure
I wonder if there is a way to estimate the precision matrix when certain elements are restricted to be zero?
Suppose data are from $N(\mu,\Omega)$, where $\Omega=V^{-1}$, i.e. the precision matrix. ...
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This representation of the precision matrix (Inverse of the covariance matrix) confuses me
I am currently reading the book titled "Generalized Least Squares" by Takaeki Kariya and Hiroshi Kurata. In one section, a General linear regression model of the form
\begin{equation}
y=X\...
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What if zero mean assumption is relased in graphical LASSO?
I am working on a graphical LASSO (GLASSO) shrinkage of the variance-covariance matrix of financial log-returns data for 10 years. The objective of the graphical LASSO is:
$$\ell(0,\Sigma) = {-\text{...
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Decomposition of a Gaussian Markov random field in independent subfields
A zero-mean GMRF (i.e., a multivariate normal distribution whose precision matrix is sparse) with precision $Q \in \mathbb{R}^{n \times n}$ and covariance $\Sigma = Q^{-1}$ is eigendecomposed as $Q = ...
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Recover full covariance matrix from covariance diagonal and precision off-diagonals
Consider an $N$-by-$N$ covariance matrix:
\begin{equation}
Σ =
\begin{bmatrix}
Σ_{11} & Σ_{12} & \dots & Σ_{1N}\\
Σ_{12} & Σ_{22} & \dots & Σ_{2N}\\
\vdots & \vdots & ...
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Asymptotics of 2 x 2 precision matrix
Edited to give the answer... but I still don't understand where it came from!
Suppose we have
$$X_1, X_2,..., X_n \overset{i.i.d.}{\sim} N(0, \Omega^{-1})$$
where $\Omega \in \mathbb{R}^{2 \times 2}$ ...
user272429
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How to calculate ± deviation for precision and recall?
I found a precision and recall report table like as below
Precision Recall
.470±.009 .934±.013
.239±.010 .610±.013
I need the guidelines for ±.009 and ±.013 ...
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estimate precision matrix with given spatial sparsity pattern
I have a set of $n$ measurements of $p$ variables $\xi_i$. I am interested in the inverse covariance or precision matrix $P$ of the variables, but because $p \gg n$ and because of limited storage ($p$ ...
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Problems with Graphical Lasso
I'm trying to use the Graphical Lasso algorithm (more specifically the R package glasso) to find an estimated graph representing the connections between a set of nodes by estimating a precision matrix....
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Scaling precision matrix
I have little knowledge about algebra, and need to rescale a precision(not variance-covariance) matrix. Suppose I have two variables, X1, X2, and their precision matrix is [5, .7, .7, .2]. Because the ...
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What are good metrics for evaluating inverse covariance matrix?
For real datasets, where it's impossible to know the true inverse covariance, what are the methods of evaluating your inverse covariance estimator?
Possible answers:
If the number of features is ...
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Can near zeros in precision matrix be treated as zeros?
A zero entry in the precision matrix (the inverse of the covariance matrix) means the corresponding variables are independent given all the other variables. For real-world data samples, when is an ...
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Stability precision matrix under small changes in covariance
I am trying to understand how the precision matrix changes under the influence of small changes in the covariance matrix. I have several similar datasets: the differences in standard deviation for the ...
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Find an unbiased estimator of $\Sigma^{-1}$
Suppose $ X_1,\dots, X_n$ be a random sample from $N_p(\mu, \Sigma), \Sigma > 0$. Find an unbiased estimator of $\Sigma^{-1}$.
I know the unbiased estimator of $\Sigma$ is $\dfrac{1}{n-1} \sum_{j=...
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How to interpret an inverse covariance or precision matrix?
I was wondering whether anyone could point me to some references that discuss the interpretation of the elements of the inverse covariance matrix, also known as the concentration matrix or the ...
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