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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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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352 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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37 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
68 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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35 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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1answer
41 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
89 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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53 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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34 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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33 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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81 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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25 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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11 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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138 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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1answer
45 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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30 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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1answer
24 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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45 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
45 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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33 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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104 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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2answers
47 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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74 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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38 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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23 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
84 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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26 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 ...
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60 views

Problems estimating covariance matrices with small $n$, smaller $p$?

It is well known that estimating large covariance matrices from small samples is problematic. For instance, the $p \times p$ sample covariance matrix $\Sigma_n$, estimated from $n$ samples, is not ...
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1answer
40 views

Covariance of conditionally independent random variables

I want to find the covariance between two random variables $X$ and $Y$, which are independent given another random variable $M$. I thought the calculation would be $$cov[X,Y]=E_M[cov[X,Y|M]] ...
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135 views

Intuition on the definition of the covariance

I was trying to understand the Covariance of two random variables better and understand how the first person that thought of it, arrived at the definition that is routinely used in statistics. I went ...
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28 views

How to create a new covariance function using GPML toolbox in Matlab?

I want to create my own covariance function based on squared exponential or Matern that treats each dimension differently i.e. having a hyperparameter for each dimension, not just ell. How do I need ...
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44 views

Covariance between factors in CFA (Amos)

I ran a Confirmatory Factor Analysis (CFA) in Amos and obtained a model with high covariance between the factors. James described this covariance in his video as alarming. What can I do to tackle ...
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22 views

Covariance of ARMA(2,1) series

Consider the ARMA(2,1) time series $$ x_t−0.1x_{t−1}−0.06x_{t−2}=w_t−0.5w_{t−1} , $$ where $w_t$ is white noise with mean zero and variance $\sigma^2_w$. Find the expectation of $$ ...
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26 views

Is eigen-gene the principal components of gene dataset just like eigen-faces in face recognition?

I have read about eigen faces and eigen voices etc in signal processing which involves PCA. They used to refer to the principal components selected to process for image recognition etc. But now I ...
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2answers
108 views

How to determine the confidence interval or significance of a covariance estimate

I was wondering if there is a way to determine the significance of a covariance? So, if have two vectors of returns, and then calculate the covariance, how do I determine if the sample covariance is ...
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1answer
42 views

factor analysis: can irrelevant factors be identified by tweaking FA options?

So I ran a factor analysis (principal components method) on a dataset, first using the correlation matrix and then using the covariance matrix, both with varimax rotations. The results of both factor ...
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28 views

AR(1) correlation matrix

Suppose we have the following AR(1) correlation matrix: $$ \begin{bmatrix} 1 & \alpha & \alpha^{2} \\ \alpha & 1 & \alpha \\ \alpha^{2} & \alpha & 1 \end{bmatrix}$$ How many ...
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1answer
29 views

Question on Portfolio Risk & Covariance

Good evening, I was reading a stats textbook where I came across the following statement: In order to reduce the portfolio risk of a portfolio involving two risky investments, we should choose ...
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1answer
48 views

Covariance help

Consider an experiment in which a fair coin is tossed 10 times in a row (the tosses are independent of each other). Let X denote the number of heads observed and let Y=X^2. Find the covariance between ...
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1answer
120 views

Removing structure with a known functional form from the covariance matrix

I have a set of timeseries data $X^B = \begin{bmatrix} {X^B_1},{X^B_2},\dots, X^B_n \end{bmatrix}$ consisting of observations recorded at different spatial locations. There is crosstalk between the ...
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1answer
89 views

Covariance between OLS estimates in a non-standard linear model

Consider a variant of the classical linear model: $y_i=a + b\left(x_i-\bar{x} \right)+e_i $ Because $e_i \sim N \left (0, \sigma^2 \right)$, $y_i \sim N \left(a+ b \left(x_i -\bar{x} \right), ...
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1answer
63 views

AR(1) Covariance [duplicate]

So I'm trying to derive the covariance between $z_t$ and $z_{t-1}$ in the $AR(1)$ model: $z_t =\phi z_{t-1} + a_t$. Can anyone give me some advice on where to start?
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72 views

What is $\text{cov}$$(\hat{y_i}, \hat{y_i}^*)$?

Using the simple linear regression model: $ {y_i}= {\beta_0} + {\beta_1}x_i + \epsilon_i$, where E[$\epsilon_i$]=0 and var[$\epsilon_i] = \sigma^2$... If $ \hat{y_i}= \hat{\beta_0} + ...
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40 views

What is ${\rm cov}(e_i, \hat y_i)$ in simple linear regression?

The model is $y_i = \beta_0 + \beta_1x_i + \epsilon_i$ What is ${\rm cov}(e_i, \hat y_i)$? What is ${\rm cov}(\epsilon_i, \hat \beta_1)$? What is ${\rm cov}(e_i, \epsilon_i)$? For 1, I am writing ...
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224 views

In simple linear regression, what is the covariance between the error term and the residual?

In simple linear regression, what is the covariance between the error term and the residual? Model: $y_i = \beta_0 +\beta_1 x_i + \varepsilon_i$ What will be the $\rm {cov}(\varepsilon_i,\ e_i)$, ...