Covariance is a quantity used to measure the strength and direction of the linear relationship between two variables. The covariance is unscaled, & thus often difficult to interpret; when scaled by the variables' SDs, it becomes Pearson's correlation coefficient.

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Intuitive explanation for when Pearson correlation coefficient equals 1

Whenever I teach a complicated formula, I always demonstrate to the class that it works intuitively in extreme cases. For example, when all the data values are equal, the standard deviation is 0. Not ...
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7 views

On the use of the autocovariance generating function

The autocovariance generating function is defined as: $$g_X(z) = \sum_{h = -\infty}^{\infty} \gamma(h)z^h.$$ Where $\gamma(h)$ is the autocovariance function of the considered process $X$. I can ...
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40 views

Is the covariance of standardized variables the correlation?

I have a basic question. Say I have two random variables, $X$ and $Y$. I can standardize them by subtracting the mean and dividing by the standard deviation, i.e., $X_{standardized} = \frac{(X - ...
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15 views

Convert principal components based on covariance into principal components based on correlation

I am wondering if there is any way to mathematically express the change in direction of the principal components from the $2\times2$ covariance matrix to the correlation matrix. In other words, if ...
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31 views

Puzzling average of correlated measurements [on hold]

I want to average correlated measurements. Given a set of measurements: x = ( x1, x2, x3 ) whose internal correlations are given by the covariance matrix C. A ...
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13 views

Calculation of coefficients in multiple regression [duplicate]

I am trying to calculate the coefficients of a linear regression using 2 or more independent variables. Case for 1 independent variable: Using one independent variable X and one dependent variable ...
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1answer
13 views

Marginal Likelihood of a Gaussian Process Model, Duplicate entries in kernel matrix

I am trying to fit a Gaussian process model using the toolbox and I got stuck in the following problem. Assuming that I have some duplicated data points in my training data, then those will map to ...
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3answers
70 views

Covariance between variables

When looking at correlation, some variables may have a higher correlation but others may have a stronger relationship between them and a lower correlation. What do you do to figure out which variables ...
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1answer
32 views

Is a covariance matrix composed of matrixes derived from separate samples guaranteed to be positive definitive?

I have two samples that partially overlap on the variables they describe. The samples are taken from more or less the same population, and show similar values on the overlapping variables. Based on ...
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8 views

Covariance structure of a 3-level hierarchical model with random slope and intercept

I would like someone to confirm if I am getting correctly the covariance matrix in a 3 level mixed model with random slope and intercept. I have random intercepts at level 2 and 3 and a random slope ...
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1answer
33 views

What can be inferred from “covariance matrix of residuals” and “correlation matrix of residuals” after VAR?

I have this VAR: summary(VAR(V6CADModelSt45obs1D.df[,c(5,3,2,6,1,4)], p=5, type="none", ic="SC")) The following is the result of this VAR: ...
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23 views

Correlation between two quadratic forms of Gaussian random vectors

I want to approximately calculate the correlation between two quadratic forms of two Gaussian random vecotrs (of course these are in fact non-Gaussian densities). Does anyone know the derivation of ...
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21 views

Matlab: Help in mitigating the problem: Non-positive definiteness of a matrix

EKUKF toolbox contains a file ut_sigmas.m I am facing a frustrating problem with the covariance matrix $P$ of the measurement when using Cholesky decomposition. ...
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6 views

In proc glimmix covariance parameter estimates, what is “scale”? Is it equivalent to residual error?

I conducted data analysis using proc glimmix for my proportional data. Below is the sas code I used and covariance parameter estimates from the output. The experiment was conducted by split plot in ...
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8 views

drawbacks of Soft and Hard thresholding operators

I am reading about Soft and Hard thresholding operators for the estimation of very large covariance matrices. I noticed that another operator such the "Smooth Clipped Absolute Deviation (SCAD)" is ...
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25 views

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 indepenent given all the other variables. For real-world data samples, when is an ...
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7 views

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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14 views

Matlab FACTORAN error on line 162: a covariance matrix is not positive definite [closed]

I have a data set called Z2 that consists of 717 observations (rows) which are described by 33 variables (columns). The data is standardized by using ZSCORES. Additionally, there is no case for which ...
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1answer
49 views

Decomposing a random variable with random mean into a sum

I have two random variables: $X\sim \mathcal{N}(0,\sigma^2+1)$. $Z$, a gaussian with mean $X$, distributed so that $E_{X,Z}[(X-Z)^2]=s^2.$ We know that: $$s^2\geq\sigma^2+1 \Leftrightarrow Z ...
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1answer
29 views

assuming independency between independent variables in multiple regression?

I heard that multiple regression assumes that the independent variables are correlated somehow. So when we convert the multiple regression into SEM diagram, we see covariance arrows are drawn ...
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23 views

Instances of sparse covariance matrices

I am trying to find large datasets with inherently sparse covariance matrices, to be tested with our algorithm. Basically, we will take the sample covariance matrix and enforce some structured ...
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1answer
40 views

Finding $Cov(2X+7, X^2 +3X - 12)$

So I have this pdf, $f(x)=3x^2$ for $x\in (0,1)$ and I need to find $Cov(2X+7, X^2+3X-12)$. My main concern about how I answer this is, what is the joint pdf for these two distributions? I guess ...
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1answer
55 views

What is the demonstration of the variance of the difference of two dependent variables?

I know that the variance of the difference of two independent variables is the sum of variances, and I can prove it. I want to know where the covariance goes in the other case.
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sparse covariance/correlation thresholding

In our project, we would like to do some optimization on sparse matrices. The idea is to scrape massive amounts of data, form a covariance/correlation matrix, and form a sparsity pattern basically by ...
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3answers
111 views

Random variables have non-zero covariance but expected sample covariance is zero? (intuition)

This post asks "why a familiar and widely used estimator of sample covariance has expected value zero, in a situation where the variables involved are characterized by non-zero and equal pair-wise ...
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1answer
30 views

Covariance definition

Why is covariance defined the way it is? $$\sigma(x,y)=\mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])]$$ How do we know that this definition behaves in the following way? Covariance is a measure ...
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Why shouldn't the denominator of the covariance estimator be n-2 rather than n-1?

The denominator of the (unbiased) variance estimator is $n-1$ as there are $n$ observations and only one parameter is being estimated. $$ ...
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27 views

Relation between raw and central moments

This question arose when reading Johansen's likelihood-based inference in cointegrated VAR models, the 2009 reprint, page 146. I will do my best to make my post self-contained. Let $Z_{0t}=\Delta ...
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1answer
15 views

Calculating covariance matrix with different time periods

Instead of using a covariance matrix that was calculated by using the entire sample to simulate results, I am interested in possibly using a covariance matrix that was calculated by using a selected ...
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8 views

Testing for covariance on a fixed scale?

I'm struggling on finding a way to analyse my data, that may be fixed just with a basic statistics insight. I have a dataset that has a large number of variables (100+) some of which tend to covary ...
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26 views

Necessary conditions for causality

E. Tufte writes: 'Probably the shortest true statement that can be made about causality and correlation is "Empirically observed covariation is a necessary but not sufficient condition for ...
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1answer
33 views

how to plot a 2D covariance error ellipse?

I have a location of landmark in 2D. According to Extended Kalman Filter EKF- SLAM, if the robot re-observes the same landmark, the covariance ellipse will shrink. I collected the necessary ...
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1answer
52 views

Generating independent random variables from correlated random variables

I have 2 standard normal, bivariate correlated random variables, $corr \ (X_1, X_2)=\rho$. I want to generate two independent standard normal random variables from these 2. I tried to use what I ...
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474 views

How does the formula for generating correlated random variables work?

If we have 2 normal, uncorrelated random variables $X_1, X_2$ then we can create 2 correlated random variables with the formula $Y=\rho X_1+ \sqrt{1-\rho^2} X_2$ and then $Y$ will have a correlation ...
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1answer
25 views

covariance of squared terms

Assuming two variables $X1$ ~ $N(0,1)$, $X2$ ~ $N(0,1)$ with $Cov(X1,X2) = a$. Is it possible to derive analytically what the covariance between $X1^2$ and $X2^2$ would be? Empirically (I tried this ...
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1answer
33 views

Closed-form expression for autocovariance of random walk with drift

I am working through slides hosted at Basic Time Series Models, and am not sure how to mathematically derive a closed-form expression of the autocovariance of the "random walk with drift" model. The ...
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15 views

Expectation (covariance) between $X_j$ and $Y$ in logistic regression

Is there some way to explore the relationship of $E(X_jY)$ in a logistic regression? That is, $Y$ is a binary variable observed together with $p$-covariates $X$ (wlog centered). I am interested in ...
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Why does inversion of a covariance matrix yield partial correlations between random variables?

I heard that partial correlations between random variables can be found by inverting the covariance matrix and taking appropriate cells from such resulting precision matrix (this fact is mentioned in ...
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19 views

How to test the significance of covariance?

I'm using the Mutual Informacion covariance in RNA sequences and I want to know if there exits a way to test if some covariance is significant, let's say, an associated p-value. Thanks to all for ...
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9 views

Sub-space / latent-space covariance

I am not entirely sure what I should be googling for this in my present context. Basically I am working with a set of latent variable models such as the Gaussian Processes latent variable model ...
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1answer
21 views

Covariance function to draw an inverse function

In a Gaussian Process (GP), we know that choice of the covariance function determines the shape of function that can be drawn from the GP. eg. Constant : $\sigma _{o}^{2}$ Draws constant function ...
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1answer
36 views

Why is pure sample covariance a bad metric to understand the degree of correlation between two variables?

Covariance helps you understand how variables are linearly related. Would it be possible to have two pairs of variables in a deterministic relationship (i.e. linearly correlated variables) that have ...
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variance of data of correlated poisson means?

I want to find the mean and standard deviation for the data for the following question. We have 10 houses of known different sizes. The number of people living per square meter has a poisson ...
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Econometrics 2: Covariance (xt, xt+1) in Moving Average process of order 1,

Could you help me to show that if X(t+1)=e(t+1)+alpha(1)*e(t), Then Cov[x(t), x(t+1)]=alpha(1)*Var[e(t)] Thank you very much, your help is appreciated
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Asymptotic distribution of $\hat{B_1}$in simple linear regression

I am currently studying how to $\bf{identify}$ the parameter $B_1$ in a simple univariate regression model where we have $Y=B_0+B_1X+\epsilon$ with the usual assumption of $X$ being exogenous, ...
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1answer
34 views

Linear regression coefficients from covariance matrix [duplicate]

In case of two random variables $y,x$ we have that the best linear fit $y = \beta x + \epsilon$ satisfies $$ \beta = \frac{\mathrm{Cov}(x,y)}{\mathrm{Cov}(x,x)}. $$ That is, if I known covariance ...
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Loss function for rank deficient covariance matrices?

I'm trying to compare the efficiency of different estimators of the covariance matrix of a particular type of multivariate normally distributed data. This comparison, as well as the estimation process ...
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1answer
58 views

How do I generate two correlated Poisson random variables?

How would I simulate observations from a bivariate Poisson distribution such that they have a nonzero covariance? The hint I was given is that I need to use the fact that the sum of two Poisson random ...
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1answer
54 views

Covariance of natural logarithms — how to estimate $\sigma (\mathbf X, \mathbf Y)$ from $\sigma (\ln{\mathbf {(X)}},\ln{\mathbf {(Y)}})$?

I have $$\operatorname{Cov} (f{\mathbf {(X)}},f{\mathbf {(Y)}})$$ where $\operatorname{Cov}$ denotes the covariance and $f(\mathbf X)$ is a nonlinear function, i.e. $f(\mathbf X) = \ln(\mathbf X)$. ...
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1answer
33 views

Mixed model analysis: random vs repeated statement

I have data from a longitudinal parallel groups study where there are 46 subjects randomized to 1 of 5 treatment groups, each subject with roughly 13 observations over time on a given outcome measure. ...