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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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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3answers
72 views

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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49 views

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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1answer
25 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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10 views

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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1answer
33 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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0answers
17 views

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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39 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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1answer
31 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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1answer
57 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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1answer
26 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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1answer
35 views

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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1answer
34 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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1answer
36 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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1answer
35 views

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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2answers
54 views

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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21 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 ...
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55 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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31 views

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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37 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
115 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
100 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
31 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
29 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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195 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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44 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
42 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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15 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
20 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
126 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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84 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
44 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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13 views

Kernel matrix is a covariance function

How to prove that the kernel matrix is actually a covariance matrix?
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41 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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104 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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36 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
94 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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30 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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25 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
71 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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16 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
32 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
22 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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64 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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1answer
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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20 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 ...