Questions tagged [correlation-matrix]

A $k\times k$ matrix of correlations between all pairs of $k$ random variables. All its diagonal elements are equal to one.

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What is the null and alternative hypothesis of a chi square test whether a single matrix is an identity matrix?

Say I had a correlation matrix: $$ M = \begin{bmatrix} 0.8 & 0.1 & 0.1\\ 0.3 & 0.7 & 0.1 \\ -0.1 & -0.2 & 0.9 \end{bmatrix} $$ I want to show that it is approximately an ...
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Independent copula vs Student-$t$ copula with zero correlation matrix?

Suppose I have the random variables $X_1, \dots, X_n$ with the marginal distributions are not normal (in fact, unknown marginal distribution). Will there be any difference between the assumption $X_1, ...
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When to use Simple Linear Regression over Multiple Linear Regression

I am fairly new to the world of statistics and approaching it as I learn more about machine learning. I have a fairly firm grasp on regression analysis so far but not necessarily on nuances and best ...
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Difference between a multivariate normal regression and multiple regressions with shared random effect

Let's $Y_1$, $Y_2$ be two random variables representing two outcomes and $X$ a covariate. I want two regress $Y=(Y_1,Y_2)$ on X, but by taking into account the potential correlation between $Y_1$ and $...
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What came first, the covariance or the correlation matrix?

Covariance can be calculated from correlation and correlation can be calculated from covariance. Is it sufficient to refer to them as only transformations of one another? Is one considered more of a ...
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How can one derive the original data from the correlation- or covariance matrix of that data?

How can one derive the original data from the correlation- or covariance matrix of that data? I know the way a new, reduced dataset can be calculated from the correlationmatrix and it’s eigenvectors ...
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Clustering data based on correlation

I have a dataset where each row represents a sample and each sample is described by its chemical composition. You can see the 10 first rows of the dataset in figure 1. Figure 1 - Each row represent a ...
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Bounds for common correlation of uncorrelated random variables with another random variable

Suppose we have a random variable $y$ and a collection $(x_1,\dots,x_n)$ of $n$ random variables that are all uncorrelated: $\operatorname{corr}(x_i,x_j) = 0$ $\forall i \ne j$ and that all have the ...
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Correlation matrix: decline in correlation

I performed a correlation matrix with Python. The following variables were included total_equity,net_income,...
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Calculate unbiased standard errors for a regression from latent factor correlations

I have a SEM where various latent variables are correlated with each other. I want to use a subset of these correlations to run a regression. This can be done easily enough to find point estimates, ...
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How to determine mse of estimate from correlation matrix of estimate error?

I have a model of an information transmission system Y = XH + N, where X is a diagonal matrix with the transmitted "symbols" (known), H is a column vector which distorts the transmitted symbols and N ...
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Eigenvectors for correlation and covariance matrix PCA

I know the generally reasons of using correlation matrix vs a covariance matrix when doing PCA (and visa versa) however when thinking about the eigenvectors (principal components of the data) of each ...
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Change in eigenvalues due to perturbation to a correlation matrix

Let $A$ be a $m \times n$ matrix defined as $ A = \Big[\frac{a_1}{\|a_1\|} \cdots \frac{a_n}{\|a_n\|}\Big]$ and $a_k \in \mathbb{R}^{m\times 1}$ where $k \in [1,\dots,n]$. Now, we define a ...
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How to simulate random correlation matrix containing off diagonal structures

I want to simulate a correlation matrix which has some off-diagonal structures and also should have some hierarchical structures. For simulating correlation matrices which contain hierarchical ...
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Simulate correlation matrix using a given structure [duplicate]

I want to generate correlation matrix such that it follows the below structure $$\Sigma = B \Lambda B^T $$ where $\Lambda$ is a diagonal matrix containing positive elements, $\Sigma \in R^{n \times n}...
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How do I simulate a random valid correlation matrix of ordinal variables given a list of marginal probabilities?

I am trying to use R to simulate random variations in a real dataset with a known number of categorical and continuous predictor variables, as well as known marginal probabilities for each ordinal/...
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Is it possible that 3 vectors have all negative pairwise correlations?

Given three vectors $a$, $b$, and $c$, is it possible that correlations between $a$ and $b$, $a$ and $c$, and $b$ and $c$ are all negative? I.e. is this possible? \begin{align} \text{corr}(a,b) < ...
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What is the best way to Show a Correlation Matrix as a Cluster/Network Graph in Python? [closed]

I'm struggling because while I want to show the interrelationship of correlation between my fields, I realize that trying to plot nodes in terms of distance away from each other based on correlation ...
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Method to generate random correlation matrices with specified structure.

Within the social sciences there is a popular technique called Factor Analysis and I am interested in generating random correlation matrices that uniformly sample all the space parameterized by one ...
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Help needed with intuition of eigenvalue spectrum of correlation matrix

I wish to get a better understanding of the meaning of the eigenvalues of a correlation matrix I am studying. I have a correlation matrix of noise levels for 10 cells in a wireless network over time....
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How to generate a set of random variables with specific correlations and correlational distributions

I apologize if this question is a bit confusing. Suppose we get to generate an $n \times n$ matrix, $M$, whose values are all between 0 and 1. How can we generate a set of $n$ random variables, $\{...
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Do the Determinants of Covariance and Correlation Matrices and/or Their Inverses Have Useful Interpretations?

While learning to calculate covariance and correlation matrices and their inverses in VB and T-SQL a few years ago, I learned that the various entries have interesting properties that can make them ...
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Cluster analysis using the posterior distribution of a Bayesian correlation matrix

Background and Problem I recently ran a Bayesian multivariate epidemiological meta-analysis on prevalence estimates for several disorders. This analysis included a probit-based model to deal with the ...
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Does using a covariance matrix of scaled and centered variables compare with using a correlation matrix?

I have some data with features which have different units of measurements. Here, by data, I'm trying to say that the row represents the observations and column the features. There are correlations ...
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Correlation matrix for biostatistics [closed]

I want to convert the expression of the genes from an array to a gene correlation matrix, to know the correlation of each gene with the other genes. I have 6 samples, 3 controls and 3 test, is it ...
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correlation matrix test: is this code correct or is it missing a multiple comparisons correction?

I have $m$ variables $x_1,\dots,x_m$, measured in $N$ independent tests $\{x_{i1},\dots,x_{im}\}_{i=1}^N$, leading to the design matrix $X$. I noted that the demo function ...
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Check whether a sample correlation matrix is valid [duplicate]

Is it possible to have the following sample correlation matrix for $x$, $y$, $z$? $\begin{pmatrix} 1 & 0.8 & 0.2 \\ 0.8 & 1 & 0.7\\ 0.2 & 0.7 & 1\end{pmatrix}$ Where a 3 by 3 ...
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Reporting Fisher-transformed Pearson's correlations when the magnitude of difference is relevant

My field regularly demonstrates a certain type of result with pairwise Pearson's correlation matrices between predicted and measured data. As soon as such correlations become high, Fisher-transforming ...
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13 votes
3 answers
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Is there a serious problem with dropping observations with missing values when computing correlation matrix?

I have this huge data set with like 2500 variables and like 142 observations. I want to run a correlation between Variable X and the rest of the variables. But for many columns, there are entries ...
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How to compare diagonal elements of precision matrix (the inverted correlation matrix)?

Let $$C=\begin{pmatrix}C_{11} & C_{12}\\ C_{21} & C_{22}\end{pmatrix}$$ be a $p\times p$ correlation matrix with positive entries, where $C_{11}$ is a $q\times q$ matrix. Define $D=C^{-1}=(d_{...
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How to randomly generate a positive semidefinite matrix subject to Loewner constraint?

For real, symmetric, positive semidefinite matrices $A$ and $B$, let $\leq_L$ denote the Loewner partial order: $A \leq_L B$ iff $B-A$ is positive semidefinite. Suppose $A$ and $B$ are fixed PSD ...
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21 votes
6 answers
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Completing a 3x3 correlation matrix: two coefficients of the three given

I was asked this question in an interview. Lets say we have a correlation matrix of the form \begin{bmatrix}1&0.6&0.8\\0.6&1&\gamma\\0.8&\gamma&1\end{bmatrix} I was asked to ...
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2 answers
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With an R function that expects a covariance matrix, can I give it a correlation matrix?

The R function mvrnorm from the MASS package generates random numbers from a multivariate normal distribution. It expects a ...
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Is there a way to reconstitute the unit of measurement in PCA?

I have a data set composed of changes in financial asset prices. Because they are on different levels, e.g., one asset is trading in the 100s, the other in the 5s, the change their prices have vastly ...
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Exact meaning of correlation coefficient

What is the exact meaning of the entries of a correlation coefficients matrix? I have spent time researching this, but could find only approximate interpretations which give me no good understanding ...
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How the Correlation Matrix is built for PCA in Weka?

Just to give a context, I want to use PCA (Principal Component Analysis) to identify which attributes are similar to others, so I can use just one (or a subset) of them. The correlation matrix of n ...
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Covariance Matrix and Correlation Matrix - Singularity

If a covariance matrix is non-singular, does this implies that correlation matrix is also non-singular. My guess is it depends on mean vector in $K_{X} = R_{X} - m_X.{m_X}^H$ Not sure though.
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Correlation Matrix from given R output of Factor Analysis

I carried out a factor analysis of 5 variables using a single factor. How do I estimate the correlation matrix assuming the one factor model holds? The R output is:
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Does it make sense to use PCA when the determinant of the correlation matrix is (almost) zero?

I'm running a PCA over a data set of $N \times p$ size ($N\approx 1000$ being the number of measurements and $p\approx 200$ being the number of dimensions/predictors). I expect many of the predictors ...
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1 answer
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Off-diagonal elements of a correlation matrix after removing the first principal component

I have some data with more variables than observations, that I'd like to subject to a principal components analysis. For didactic reasons (to give an intuition for factor retention criteria under ...
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Is every correlation matrix positive definite?

I'm talking here about matrices of Pearson correlations. I've often heard it said that all correlation matrices must be positive semidefinite. My understanding is that positive definite matrices must ...
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2 votes
1 answer
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Fisher z-transformation and normality

I have 100 50x50 correlation matrices, which I have all Fisher z-transformed. I understood that this results in the all the entries of one matrix being approximately normally distributed. Questions ...
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4 votes
1 answer
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Matrix multiplication to find correlation matrix

In this book on matrix factorizations, the author states the following, which I don't find to be true empirically. Is it true and under what conditions? ADD: Trying to recreate the answer in R, what ...
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Does a correlation matrix of two variables always have the same eigenvectors?

I perform Principal Component Analysis using two variables that are standardized. This is done by applying a SVD on the correlation matrix of the concerned variates. However, the SVD gives me the same ...
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9 votes
2 answers
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Generate symmetric positive definite matrix with a pre-specified sparsity pattern

I am trying to generate a correlation matrix $p\times p$ (symmetric p.s.d) with a pre-specified sparsity structure (specified by a graph on $p$ nodes). The nodes that are connected in the graph have ...
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6 votes
2 answers
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Is a weighted average of two correlation matrices again a correlation matrix?

Let $R$ and $Q$ be two correlation matrices of the same size and let $p\in[0,1]$. I'm trying to show that $pR+(1-p)Q$ is still a correlation matrix. I claim that $\sqrt pX+\sqrt{1-p}Y$ is a random ...
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8 votes
3 answers
4k views

Is every correlation matrix positive semi-definite?

I am generating correlation matrix by some new algorithm. Generated matrix is non positive semi-definite matrix. I'm getting a few negative eigenvalues. The rest of eigenvalues are quite equal to the ...
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29 votes
5 answers
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How to generate a large full-rank random correlation matrix with some strong correlations present?

I would like to generate a random correlation matrix $\mathbf C$ of $n \times n$ size such that there are some moderately strong correlations present: square real symmetric matrix of $n \times n$ ...
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6 votes
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Diagonal elements of the inverted correlation matrix

Is it true that the diagonal elements of the inverted correlation matrix will always be larger than 1? Why?
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Quantifying how much "more correlation" a correlation matrix A contains compared to a correlation matrix B

I have 2 correlation matrices $A$ and $B$ (using the Pearson's linear correlation coefficient through Matlab's corrcoef()). I would like to quantify how much "more correlation" $A$ contains compared ...
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