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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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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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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46 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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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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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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16 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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10 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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7 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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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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21 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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48 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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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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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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28 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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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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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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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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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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35 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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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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55 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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48 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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24 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. ...
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How to get covariance matrix of the coefficient estimates of compound regression model in SPSS

AFter running regression model in SPSS, I get the compound model: ln(diamter)=1.126*height +3.102. I would like to have the covariance matrix of the coefficient estimates of this model and get the ...
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94 views

Caclulating standard errors for generalized least squares without raw data (just sample means and covariance)

I have a question very similar to the question asked here: is it possible to calculate standard errors (specifically, the standard error of the intercept) for generalized least squares regression ...
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79 views

Show that $\widehat{Cov(\hat{\mu},Z_i)}$ is always zero even $Cov(\mu,Z_i)$ is not always $0$

I will state the question first then my work. Q: We have a regression model, $Y_i=\beta_0+\beta_1X_i+\mu_i$ where $Cov(\mu_i,X_i)=0$ is not guaranteed. Suppose that $Z_i$ is an instrumental ...
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centering two variables X and Z makes cov (X,XZ) = 0

I've read that centering two normal (or symmetrical) variables $X$ and $Z$ affects correlation of centered $X$ with interaction term $X\cdot Z$ in such way, that this correlation $cor(X-EX, X\cdot Z)$ ...
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76 views

Show that the sample covariance converges in probability to the $Cov(X,Y)$

Suppose we are given $[(Y_i,X_i)]^{n}_{i=1}$ which is a random sample from the joint distribution of $(Y,X). Show that the sample covariance converges in probability to the $Cov(X,Y)$ My thought ...
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Covariance greater than Variance?

It is straightforward to verify that for two random variables $X$ and $Y$ with variances $\sigma^2_X \neq \sigma^2_Y$, we have that $$\Big|{\rm Cov}(X, Y)\Big| \leq \max\{\sigma^2_X,\, \sigma^2_Y\}$$ ...
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If $E[Y|X]=a$ for some constant $a\neq 0$, then does $cov(X,Y)=0$?

I'm currently working on the following problem: Q: If $E[Y|X]=a$ for some constant $a\neq 0$, then does $cov(X,Y)=0$? Now I am quite lost as to how to do this problem as the question does not ...
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49 views

Gaussian Process: Using partitions of a Cholesky decomposition solution for conditional deduction

If I define a GP over observed values, $y$, of a sensor reading over time, $t$, as (for simplicity assuming discrete time series e.g lets say readings after every 5 mins) : $y=f(t)+\epsilon$ where ...
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Find the bounding rectangle of a covariance matrix based on mahalanobis distance

I'm trying to develop an algorithm that makes use of the Mahalanobis distance from an arbitrary test point to assign a score to each observation in a dataset. I want to only consider observations ...
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46 views

Posterior covariance from GPML toolbox

I am currently using the GPML toolbox to perform regression. Generally, after learning the hyperparameters we can extract the posterior mean and variance by using the function in the toolbox as ...
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Analyze estimated noise covariance matrix

In image processing, an algorithm was created for the simultaneous Bayesian estimation of the unknown correlated noise covariance matrix. How can it be proven that the mean of the computed estimates ...
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50 views

effects of Box-Cox transformation on covariance

I'm trying to synthesize data for a Monte Carlo simulation. I have a stationary random process $x$ and can readily estimate its covariance matrix $S$. I know that if the increments of the process are ...
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81 views

Examples of marginal independence, conditional dependence

I am interested in finding "real-world" examples of when variables might exhibit marginal independence but are conditionally dependent given some other variable. It seems to me that the converse ...
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83 views

Find the expectation and covariance of a stochastic process

The problem is: Let $W(t)$, $t ≥ 0$, be a standard Wiener process. Define a new stochastic process $Z(t)$ as $Z(t)=e^{W(t)-(1/2)\cdot t}$, $t≥ 0$. Show that $\mathbb{E}[Z(t)] = 1$ and use this result ...
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30 views

Question regarding covariance

I'm trying to prove a theorem, where it is given that each $X_i$ is independent and identically distributed with mean $\mu$ and variance $\sigma^2$. Within this theorem, I have multiple sub-results to ...
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What is the asymptotic covariance matrix?

Is it true that the asymptotic covariance matrix is equal to the covariance matrix of parameter estimates? If not, what is it? And what is the difference between the covariance matrix and the ...
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50 views

Expectation of covariance in derivation of Kalman filter

I'm working through the derivation of the Kalman filter equations from this paper (or alternative source here) and I'm unsure of the derivation of the state prediction covariance (equation 2 in the ...
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51 views

Example of dependence with zero covariance

This is a constructivist question. Please provide a bi-variate distribution or density/mass function of two absolutely continuous/discrete (but not mixed-type) random variables, which (may) have ...
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partitioning variances, higher order products and multivariate skewness

If A,B and C all contribute to Z as: $$ Z = A*B*C $$ we can get the contributions of ABC to the variance in Z as: $$ Var(Z) = Var(ABC) $$ getting independent contributions of A, B and C and ...
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Is covariance between two dummy variables zero?

Here is a problem I am facing: I need to test a hypothesis (t test), the formula for which is $t = \frac{\hat{B_1} - \hat{B_2}}{se(\hat{B_1} -\hat{B_2})}$ Now, we know that the bottom isnt actually ...
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27 views

Meta analysis on multiple endpoints and unknown covariance

I am doing meta-analysis on intervention studies on human subjects where a number of measures were obtained before and after the intervention in a treatment and a control group. We categorize the ...