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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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 ...
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31 views

Hayashi yoshida estimator for correlation not coming between -1 to 1

I took two time series data with 141 data points in total with time stamps. i found out actual correlation between them which is about 0.97. Now i find the Hayashi Yoshida estimator for correlation. ...
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Covariance of the empirical distribution function

Let $X_1, \dots, X_n$ be iid with cdf $F$. Let $\hat{F}(x) = \frac{\sum I(X_i \leq x)}{n}$ be the empirical distribution function. Suppose $x < y$ and compute $Cov(\hat{F}(x),\hat{F}(y))$. This is ...
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33 views

Calculating expectation function and covariance function

Let $E_n(t)$ denote the empirical cdf based on iid uniform $u[0,1]$ random variables $U_1,...,U_n.$ The corresponding uniform empirical process $(e_n(t),0\leq t\leq 1)$ is given by ...
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18 views

How to normalize by the covariance matrix? [duplicate]

I am trying to understand an image processing research paper [1] that calls for normalizing a distance between an object's center point and the center of a cluster of points by the covariance matrix ...
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1answer
114 views

What is the correlation between X and X+Y?

If $X$ and $Y$ are two random variables, how do I calculate the correlation of $X$ and $X+Y$ in terms of $\rho$, $σ_x^2$ and $σ_y^2$ given that the $\text{Variance}(X)= σ_x^2$ and ...
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1answer
87 views

Clarification: The covariance of intercept and slope in simple linear regression?

Help me understand this relatively simple (I think) concept: The covariance of the intercept ($\beta_0$) and the slope ($\beta_1$) in simple linear regression. Furthermore, what range of values ...
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1answer
22 views

Regression: Covariance table in spss vs. mplus

I want to use regression output (b, se B and cov of several predictors) as input for a new analysis. One example: I want to compute: SE^2(b1) + SE^2(b2) + 2 COV (b1,b2). If I do a basic regression ...
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2answers
26 views

Direct parametrization of Cholesky decomposition of spatial covariance matrix

In spatial data analysis, a simple way to model the covariance stucture between spatial observations is via a covariance function like $cov(y_i,y_j) = C e^{-rD_{ij}}$, based on some (euclidean) ...
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165 views

Expected value of a product of two compound Poisson processes

I'm working on my master thesis now and I've been struggling with a problem for some while now and no one seems to be able to help me or point me in any direction. So now I reach out to see if someone ...
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43 views

How do I solve this stochastic differential equation?

So I have a second order stationary process $Y(t), \infty < t < \infty$ which has a continuous sample function, mean $\mu_Y = 1$ and covariance function $r_Y(t) = e^{-|t|}, -\infty < t < ...
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1answer
38 views

In a two equation system, what is the meaning of the assumption “exogenous X is uncorrelated with ε1ε2”

Assume a triangular system such as \begin{eqnarray*} Y = X'\beta_1 + D\gamma_1 + \varepsilon_1 \\ D = X'\beta_2 + \varepsilon_2 \end{eqnarray*} with $Y$ and $D$ as observed endogenous variables, $X$ ...
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9 views

Fisher information matrix with negative eigenvalues, fix through singular value decomposition?

I have the following Fisher information matrix (just an example): ...
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1answer
18 views

minimizer weighted linear regression

In a regression problem, with $y=X\theta+\epsilon$ and $X$ is an $n$ by $p$ matrix the ‘weighted least squares estimate is the minimizer $\theta^{*}$ of ...
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1answer
91 views

(Co)variance of product of a random scalar and a random vector

Given a random scalar $ x \in \mathbb{R} $ and a random vector $ Y \in \mathbb{R}^n $ that are independent, can it be said that: $$ {\rm cov}(xY) = {\rm var}(x){\rm cov}(Y) + {\rm var}(x)E[Y]E[Y]^T + ...
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80 views

Why does the density function has product of variance and covariance for higher model order time series

In my previous question Density function for AR model, the density function of AR model has the covariance-variance matrix given as $\sigma^2 *V_p$. In multivariate Gaussian distribution, the pdf ...
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1answer
41 views

Finding regression coefficient only with matrix correlation

How do I find Regression coefficient if data provided is only matrix correlation table? Here is example for x1 matrix correlation. ( I also have x2,x3,x4, but only provided x1 in here) for the sake ...
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12 views

Kernel matrix is a covariance function

How to prove that the kernel matrix is actually a covariance matrix?
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36 views

different definition of compound symmetry in SAS

I have two questions about the covariance structure in SAS (proc mixed). I realize the compound symmetry structure in SAS allows the covariance term to be negative. This is different from the ...
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94 views

Bounds on correlation to ensure covariance matrix is positive definite

UPDATED: I am constructing a correlation matrix for an MA(1) process, which would look something like... $$ C = \left( \begin{array}{cccccccccccccccccc} 1 & \rho & 0 & 0 & 0 & 0 ...
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25 views

Can the law of total covariance apply to variables from different sample spaces?

Wikipedia says this about the law of total covariance (http://en.wikipedia.org/wiki/Law_of_total_covariance): In probability theory, the law of total covariance,[1] covariance decomposition ...
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1answer
79 views

Robust estimates of the covariance matrix in the big data space

I am trying to compute the robust estimates of the covariance matrix (and also the mean) in the big data space. I am aware of FastMVE and FastMCD (Minimum Covariance Determinant and Minimum Volume ...
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25 views

Efficient calculation of selected diagonals (bands) of a covariance matrix?

I'm looking for an algorithm that can calculate a select few diagonals of a covariance matrix. Here's the problem: I have an $m\times n$ data matrix $X$ where $m$ is the number of features, which is ...
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23 views

Regression variable conversion

There is a question that I cannot solve. They may be solved by variance and covariance but I couldn't. So I thought there should be another way to solve. Question: A researcher has a sample of 43 ...
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1answer
62 views

Conceptual questions: Variance of a process

Wikepedia, at Variance of Autoregressive model, mentions an expression of variance for an AR(1) process. I am learning signal processing (beginner level) and facing difficulty in understanding some ...
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24 views

How to find the covariance matrix? [duplicate]

I have some trouble understanding the concept of a covariance matrix. For instance, I'm going over this question that says: assume that we have U1, U2 and U3 as independent zero-mean, unit-variance ...
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14 views

How to decorrelate residuals in r from the covariance matrix

I fit a geeglm model with clustered data and now I would like to decorrelate the residuals of the model in order to run model diagnostics. I read that if I can obtain the covariance matrix of the ...
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1answer
29 views

Covariance for Two Variables when Divided by a Third Variable

In a Bioinformatics article appendix, I found the equation : $$ Cov(X/Z, Y/Z) = Cov(X, Y)E(1/Z^2) + E(X)E(Y)Var(1/Z) $$ when supposing that the random vector $(X, Y)$ with covariance $Cov(X, Y)$ is ...
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23 views

What is variance and co variance related to time series?

I'm trying to understand the Mahalanobis distance method which makes use of a covariance matrix. However i am not clear about the idea of variance and covariance with respect to time series. And also ...
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1answer
19 views

Pretest-Posttest comparison

I am having a hard time doing this on Stata. I have a group of 32 students. All perform a pretest and are scored. Next, half of them are randomized to receiving an intervention and the other half ...
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24 views

asymptotic covariance between mean and standard deviation

I am trying to estimate the asymptotic covariance between mean and standard deviation. I know the following $$\sqrt n \hat \mu \xrightarrow{d}N\left( {\mu ,{\sigma ^2}} \right),\sqrt n \hat \sigma ...
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1answer
61 views

What are the implications of estimating a covariance matrix from a correlated sample?

Given a sample of $n$ independent observations $x_1,...,x_n$ (where $x_i$ are $p$-dimensional column vectors), the $p \times p$ sample covariance matrix is defined as ...
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32 views

Sample covariance mean-corrected vector proof

Prove that $$(n-1)S = X^TX -{1\over{n}}(X^T\vec1)(\vec1^TX) = X^TX-n\vec{\bar x}\vec{\bar x}^T$$ My attempt so far goes like this $$S = {1\over{n-1}}X_m^TX_m$$ Edit: Where $X_m$ is the ...
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19 views

Is it possible to calculate the covariance btwn data and a subset of the data?

I have a regular and complete time-series vector and have created a subset of this vector based on a particular sampling algorithm (say every 10th value). To calculate the error variance of this ...
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50 views

Minimum variance for sum of three random variables

I have been working on the following problem: Given you have VarX = 1, VarY = 4, and VarZ = 25, what is the minimum possible variance for the random variable W = X + Y + Z, or min Var(X+Y+Z)? My ...
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1answer
38 views

Multivariate Regression on Multicollinear Categorical Variables

I am working on a dataset with a continuous response (which could be dichotomized), one continuous covariate, and multiple categorical variables. The continuous covariate (weight) is directly ...
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13 views

Finding ACVF and two random variables

let $X_t = 0.5X_{t-1} + Z_t$ where $Z_t$ ~ $ WN(0,\sigma^2)$ I want to find the ACVF of both $X_t$ and $Z_t$, but I am a little bit confused. Say for $X_t$ $$\gamma(h) = Cov(0.5X_{t-1} + Z_t, ...
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1answer
47 views

Covariance of a compound distribution

I am trying to find the covariance of a compound distribution. Given $X=x$, where $X \sim \mathrm{Uniform}(0,1)$, $Y$ is (conditionally) normally distributed with mean $x$ and variance $x^2$. I ...
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33 views

Covariance between normalised correlation functions

If I have a set of correlated random variables $\mathbf{X}=\{X_1,\dots,X_n\}$ that have been sampled $N$ times, I can calculate the correlation function for pairs of variables as $$ ...
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1answer
27 views

Stationary function

I am reading Karl Rasmussen's book on Gaussian processes and in the introductory chapter he has the following statement: ...
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62 views

Variance along the regression line

I have two random variables $X$ and $Y$ of unknown distribution which I sample $n$ times to get a set of $n$ random points. I do simple linear regression on this set and arrive at an equation for a ...
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1answer
60 views

What does the covariance of a quaternion *mean*?

If I have a set of Euler angles (representing the orientation of an object) and I find the covariance of those angles then I have some intuition that $\sigma^2$ is in units of $\text{rad}^2$ and I can ...
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8 views

Convergence of sample concentration matrix

I'm interested in the Frobenius or $\infty$ norm convergence rate bound for the sample inverse convariance (concentration) matrix. That is, suppose: $$ Y \sim \mathcal{N}_p\left(0, \Sigma\right) $$ ...
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16 views

Weighted mean and covariance

I have a variable $O$, depending on two other variables $x$, $y$ as $O=\frac{x}{y}$. I have several measurement (each one with its uncertainty) of $x$ and $y$ and for both of them I have computed a ...
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14 views

Derive covariance of coefficients in simple linear regression [duplicate]

i just need help showing my work to show that Cov(b0,b1)=-xσ^2(sxx) i know that Cov(b0,b1)=E(b0b1)−E(b0)E(b1)=E(b0b1)−β0β1 and i know that var(b0,b1)=σ^2*(X'X)^-1
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1answer
74 views

Identifying the coefficients of a principal component

Suppose that a two-dimensional random variable $X$ has a covariance matrix given by $$ \Sigma = \pmatrix {1 & -2\\ -2 & 4}$$ One of the three linear combinations below corresponds ...
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22 views

Interpreting the inverse covariance matrix: $S^{-1}x$ and $x^T S^{-1}x$

Let $S$ be the covariance matrix of some data set. $S^{-1}$ is the inverse covariance matrix, also called the precision matrix. Question: In practice, then, what does $S^{-1}x$ mean for a data point ...
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11 views

What if the variance-covariance matrix of a sum of two random vectors? [duplicate]

If X is a px1 random vector with mean Mu(x) and variance-covariance matrix sigma(x) and if Y is a qx1 random vector with mean Mu(y) and variance-covariance matrix sigma(y). If p=q, what would be the ...
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27 views

Understanding COVARIANCE when using same scale

Assuming that an experiment has 3 variables. Time, Temperature inside, Temperature outside. Also, considering that both the temperatures are measured using the same scale say Degree Celsius. If the ...
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34 views

How to calculate the lag 1 autocovariance for the difference of two variables from the individual autocovariances of the two variables

Is it possible to calculate the auto-covariance of the difference of two variables, from the auto-covariances of two variables being differenced? I have a situation where: Y=βx x is 3*3 matrix of ...