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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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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31 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
16 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
20 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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12 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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10 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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24 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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17 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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36 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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6 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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73 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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60 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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10 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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1answer
20 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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1answer
38 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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13 views

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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1answer
144 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
64 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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64 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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1answer
368 views

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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42 views

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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5 views

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
73 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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45 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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43 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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3answers
92 views

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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21 views

Standard Error of the Correlation Coefficient

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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73 views

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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37 views

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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37 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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89 views

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 ...
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27 views

Stock Returns Covariance Prediction - Number of Principal Components

I am working on the following problem. Given N days of stock returns, I compute the covariance matrix for stocks. I then use Probabilistic PCA to "shrink" the covariance matrix. I am trying different ...
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13 views

Covariance matrix specification in multivariate probit

Im having trouble with a multivariate probit model with partial observability/sample selection (written in GAUSS). In this model there is a probit at each of multiple stages, and only one of the two ...
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152 views

How to calculate the covariance between two observations of the same variable?

I'm getting confused about what I read about covariance. I know how to calculate covariance between two different variables, but not between two observations of the same variable. Imagine you have ...
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33 views

Regression problem

Regression was estimated using OLS. We get y=a0 + a1x1 + a2x2 + error. We know covariance matrix ∑ of our estimator. 1. How to get confidence interval for a1/a2 ratio? 2. In what case would the ...
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48 views

Variance of sample mean for dependent samples

Suppose I have two discrete independant random variables $X$ and $Y$, and that I'm interested in the expected value of the random variable $W$, where: $$ W= \text{sign}(X-Y). $$ So, W is 1 if ...
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31 views

Random vector times random matrix

$$ Y_i = \sum_j B_{ij} X_j$$ $$ covar(X_i, X_j) = V_X = \delta_{ij} var(X_i)^2 $$ $$ covar(B_{ij},B_{kl}) \neq 0 $$ $$ V_Y = ? $$ I know, that if $B$ was fixed, it is straight forward, but I would ...
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30 views

Univariate Normal Converted to Multivariate Normal: Covariance Derivation

I am reading the paper available at this link: https://drive.google.com/file/d/0B2_rKFnvrjMARnU1QjB4anR3RDA/edit?usp=sharing I am having trouble understanding section 5.1 (page 2741). Essentially ...
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50 views

Weighted sample covariance

I have read the Wikipedia article, and know that the unbiased weighted sample covariance matrix for the row vector $\mathbf{x}_i$ is $$\Sigma=\frac{1}{\sum_{i=1}^{N}w_i - 1}\sum_{i=1}^N w_i ...
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1answer
59 views

Why is error variance important in CFA?

I am reading the book related to SEM (Byrne, 1998) and it is stated that regression of the observed variables on the factor, and the variances of both the errors of measurement and the factor, as well ...
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34 views

Sample autocovariance of Durbin–Watson test

I understand Durbin–Watson test, but I can't understand this sentence. I cannot prove it. The Durbin-Watson test statistics is asymptotically equivalent to (rootT*C), where C is the sample ...
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30 views

When is the determinant of a covariance matrix is 0? [duplicate]

Any covariance matrix $A$ must be non-negative definite or semi-positive definite. This means that its deteraminant should always $|A|\ge0$. In case $|A|=0$, what would happen? or what does this mean ...
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125 views

Background subtraction for signal and error analysis

I use a CCD to see the split of a energy level due to Zeeman effect. I have a 1 dimensional CCD of 7926 pixel of 7μm each one. My CCD analyze a region 2 dimensional, and then it steps forward 200 ...
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50 views

Gaussian sampling in high dimension

I have a covariance function $f(x)$, where $x = (x_1, x_2, x_3)$ is a point in three-dimensional space. I need to generate a Gaussian field with given covariance function on a 3D grid of points, that ...
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90 views

LDA - Why differents formulas to calculate covariance and pooled covariance matrix

Reading materials from differents sites some questions have risen about covariance and the pooled covariance matrix calculation to implement LDA: Definitions Ci - covariance matrix of group i (C1 and ...
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40 views

Correlation between two variables measured in separate experiments in R

I'm trying to show a correlation between growth in a petri dish of some fungi and its effect on a plant. I have ten strains of fungi which I tested in the plant and in petri dishes. I can put data ...
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26 views

confidence interval for co-variance

is there a way to compute the confidence interval of a co-variance matrix in Matlab.. Suppose I have 2 vectors of stock returns. If I compute a co-variance matrix (2x2) the sample co-variance will not ...
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1answer
85 views

Covariance between f(x+y) and f(x)

Is there some theorem that allows us to work out: $Cov\big( f(x+y), f(x)\big)$ We now the $Var\big(f(x)\big)$ and $Var\big(f(x+y)\big)$ and also the $Var(x)$, $Var(y)$ and the $Covar(x,y)$ Our ...
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1answer
31 views

Mean of covariance matrices

I'm trying to generalise a formula that takes the mean of some variances to it work with vectors. I'm not sure it makes sense to take the variance between a bunch of vectors, rather it is more suited ...