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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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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Compound Distribution Question

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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24 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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24 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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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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Is covariance matrix always eigendecomposition? [closed]

I heard that for eigendecomposition, N by N Matrix should have n linearly independent vectors. Then, N by N Covariance matrix always have N linear independent vectors?
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1answer
55 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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3 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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14 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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11 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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63 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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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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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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26 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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16 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 ...
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1answer
43 views

Is it possible to get a covariance matrix of fitted values for a GLM model in R?

I would like to get a covariance matrix of fitted probabilities for a logistic regression model in R. I would like to do this because I want to find the variance of the difference between the two ...
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27 views

Weighted variance of the weighted variance in frequency transition matrix?

Assume I have a $m\times n$ Transition Matrix $A$ with $m$ different observations and $n$ represents a discrete state space. Each column counts the frequency that state $j\in [1...n]$ was visited from ...
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periodic covariance function

"Gaussian processes can be completely defined by their second-order statistics.Thus, if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the ...
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34 views

Estimate the covariance matrix of a normal distribution if the mean vectors is given by a linear rule

Let $X=(x_1,\ldots,x_n)^\top\in\Bbb{R}^n$ be a random vector that follows a multivariate Gaussian distribution with known mean vector $\mu=(\mu_1,\ldots,\mu_n)^\top\in\Bbb{R}^n$. The covariance matrix ...
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38 views

Difference between identity and diagonal covariance matrices

thanks in advance for the help. Suppose I am training a linear model. What are the conceptual differences between using a diagonal covariance matrix and the identity? It is clear to me that the ...
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1answer
18 views

Comparing and Interpreting covariances

I had a discussion about covariance recently and it would be nice to hear your feedback about this. Let's say we have a dataset of $n$ samples with $d$ attributes. For simplicity, let's say 3 of ...
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1answer
39 views

SLR: Variance of a residual

I am having problems calculating the variance of a residual in an SLR setting, ie $\text{var}$$(y_i- \hat{y_i})$. Here is what I have thus far. If $ \hat{y_i}= \hat{\beta_0} + \hat{\beta_1}x_i$ ...
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15 views

How to compare two matrices (A given matrix and a scaled up one)?

I have a matrix with 0.25 million rows and 50 columns. I have scaled up this matrix to 1.5 million rows and 50 columns using a Method A. I would like to measure the quality of the method I have ...
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11 views

Fisher information and asymptotic covariance matrix [duplicate]

I am reading the Categorical Data analysis by Dr. AGRESTI. Here, it explains "The liklihood function of for the GLM also detemines the asymptotic covariance matrix of the ML estimator Beta_hat. This ...
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1answer
54 views

How to proof relationship between inverse covariance matrix and linear regression coefficients?

Edited: I would like to work out the above relationship, more precisely: Let $(Y_{1}, ..., Y_{m})$ be a zero-mean vector with covariance matrix $\Sigma$, and let $S \subset \{1, ..., m\}.$ The ...
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24 views

Covariance Matrix using Delta Method

I am trying to get the variance-covariance matrix of a transformation using the Delta Method. Originally, I have coefficients (let's call them X1, X2, etc...) and their variance-covariance matrix. ...
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37 views

Covariance of $cov(5W_7+6W_9,W_7)$ where $W_t$ is a standard Brownian motion

I'm having trouble deducing the value for the problem in the title. Here is what I have done so far. (Given a standard Brownian motion (BM) $W_t, t\geq0 $ with $W_0 = 0$ and $\sigma^2=1$) The ...
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13 views

Univariate fixed effect Vs Multivariate model -Negative Covariance, positive parameter estimate, but why?

I am trying to compare the results of two models. The first model looks at y with x as a fixed effect. The second looks at the covariance between x and y. Both models have repeated measures for x ...
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78 views

Variance-covariance matrix

$\DeclareMathOperator{\var}{Var}$ How to compute prediction bands for non-linear regression? In the above link, you have mentioned about the variance-covariance matrix of the estimates. What is the ...
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63 views

Inference of Pearson's rho from distribution perturbation

I would like to infer the correlation between random variables $Q$ and $R$, however, I have access only to the distribution of $Q$ and the distribution of $P=Q+R$. We can see how Pearson's $\rho$ ...
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16 views

Estimating a vector from a rank-one symmetric matrix plus scaled identity

I have a problem regarding estimating a $M\times 1$ vector from a given $M\times M$ symmetric matrix. The known matrix is a scaled identity matrix with a rank-one update. I have some idea how to ...
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21 views

expected value of least squares parameters

I'm having some trouble with this equation from the least squares model. $$E[\parallel \mathcal {\hat {B}} \parallel ^2] = [E\parallel \mathcal {\hat {B}} \parallel] ^2 + trace ( \text {cov} ...
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46 views

why is the denominator of the correlation coefficient the SD of X multiplied by SD of Y?

I don't quite understand what is going on in the correlation coefficient formula. In the numerator we have the covariance, and in the denominator we have the standard deviation of variable x ...
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Standardizing variables at item or index level: Does it make a difference?

I'm running some multiple group CFA models comparing covariance structure by race/ethnicity and have survey data from 6th, 8th, 10th and 12th graders. My supervisor has told me to combine 6th and 8th ...
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149 views

Covariance of a random vector after a linear transformation

If $\mathbf {Z}$ is random vector and $A$ is a fixed matrix, could someone explain why $$\mathrm{cov}[A \mathbf {Z}]= A \mathrm{cov}[\mathbf {Z}]A^\top.$$
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1answer
66 views

Correlation between two normally distributed variables

Let a~$\mathcal{N}(\mu_a,{\sigma_a}^2)$,b~$\mathcal{N}(\mu_b,{\sigma_b}^2)$ and c~$\mathcal{N}(\mu_c,{\sigma_c}^2)$. We construct two normal variables x~$a-b$ and y~$a-c$. Can we find the ...
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70 views

How to 'read' (understand ) an expected value equation (example inside)

I have just come across expected values and they are giving me a bit of grief trying to understand them. e.g. for covariance the equation is $\text{E}\left((x - \bar{x})(y - \bar{y})\right)$ ...
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Does every semi-positive definite matrix correspond to a covariance matrix?

It is well-known that a covariance matrix must be semi-positive definite, however, is the converse true? That is, does every semi-positive definite matrix correspond to a covariance matrix?
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What does Determinant of Covariance Matrix give?

I am representing my 3d data in covariance matrix. I just want to know what the determinant of a covariance matrix gives. If the determinant is positive, zero, negative, high positive, high negative, ...
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10 views

Question about long vectors and covariance

I have a dataset with triplets of vectors $\{v_1,v_2,v_3\}_{i=1\dots n}$. Each of them can be understood as a time series (or as a window of a time series). In which way can I give a number "similar" ...
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Covariance matrix of distance matrix from an estimated squared distance matrix

I would like to perform fuzzy clustering on a data set $\boldsymbol{X}$ that contains missing elements. The fuzzy clustering algorithm requires the computation of covariance matrix $\boldsymbol{C}$ of ...
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1answer
75 views

Why cov(AX)=A cov(X) A'

I cannot verify the following theorem. Maybe I am doing something wrong, but I don't know what?! Additionally, I'm not sure about the meaning of a constant matrix in the theorem. Theorem: ...
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56 views

Understanding repeated covariance types in SPSS?

I am working in SPSS on a repeated measures linear mixed model and I am having a really hard time wrapping my head around how to select a "repeated covariance type". The options are: ...
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44 views

correlation of two sums of random variables

Imagine two random variables $X$ and $Y$ which are correlated with $\rho = 1$. Both have a mean of $100$ and a standard deviation of $40$. Two other random variables $U$ and $V$ are correlated at ...
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Correlation coefficient: If $\rho = 0$, then $r$ is normally distributed with mean 0. Why?

From this source, the estimation of the coefficient of correlation is $$r = \frac{\Sigma (X_i-E[X])(Y_i-E[Y])}{\sqrt { \Sigma (X_i-E[X])^2 \Sigma (Y_i - E[Y])^2}}$$ If the coefficient correlation is ...
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Standard Error of a linear regression

As defined here, the estimation of the coefficient of correlation is $$r = \frac{\Sigma (X_i-E[X])(Y_i-E[Y])}{\sqrt { \Sigma (X_i-E[X])^2 \Sigma (Y_i - E[Y])^2}}$$ and the standard error of $r$ is ...
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Does a high value for covariance suggest a strong relationship?

I have a data set consisting of about 100,000 rows and two columns. The values in one column range from 0 to 10^9 the values in the other column range from 0 to 245. I've calculated the covariance ...
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Covariance update from Jacobian of transition function

In this paper on particle filtering with gradient descent, the authors sample Xk+1 through gradient descent, then update the covariance matrix P associated with Xk+1 as follows: Pi+1(k + 1|k + 1) = ...
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38 views

Covariance of random vector multiplied with a random matrix

For a random vector $x$ multiplied by a non-random matrix $A$, $y=Ax$ the covariance matrix of $y$ is given by $\Sigma_y = E[Ax (Ax)^T] = E[Ax x^T A^T] = A E[x x^T ]A^T = A \Sigma_x A^T$, where ...
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Why does $r^2$ between two variables represent proportion of shared variance?

Firstly, I appreciate that discussions about $r^2$ generally provoke explanations about $R^2$ (i.e., the coefficient of determination in regression). The problem I'm seeking to answer is generalizing ...