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Questions tagged [covariance]

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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36 views

When covariance matrix will be identical to the correlation matrix [on hold]

Under what conditions will the covariance matrix ∑ be identical to the correlation matrix, whose (i, j ) entry gives the correlation between attributes Xi and Xj ?
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Prove stationarity

It can be easy question, but it is a part of bigger exercise. I have problem with lack of statement "uncorrelated" Prove that if ${X_t,t \in T}$ is stationary, then $Y_t = X_t - X_{t-1}$ is also ...
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Proof that if covariance is zero then there is no linear relationship

I get that a zero covariance doesn´t imply independence, but everybody says that if there is dependence and the covariance is zero then it is a non linear dependence. People base their interpretation ...
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Expected Value of Random Vector [on hold]

I've got this exercise but cannot really understand the "logic" behind it. I have a random vector $t$ where $t \thicksim N(0, \sigma^2 I_n)$ and I've been asked to give: $E[t_i]$, $E[t_i^2]$ and $E[...
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40 views

Example of random process with negative variation

I study about random processes. Let us have $\{X_1, X_2, \dots X_n\}$ observations. I learned, that in stationary time series the sample autocovariance function is defined as $$ \widehat{γ}(h)= \...
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Covariance matrix as a sum of two covariance matrices

Suppose that a random vector $\mathbf{n}$, $$\mathbf{n} = \mathbf{n}_A + \mathbf{n}_B \ , \tag{1}$$ can be written as a sum of two random vectors $\mathbf{n}_A$ and $\mathbf{n}_B$, that are ...
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Linear regression $y_i=\beta_0 + \beta_1x_i + \epsilon_i$ covariance between $\bar{y}$ and $\hat{\beta}_1$

I am currently reading through slides from Georgia Tech on linear regression and came across a section that has confused me. It states for $$ y_i=\beta_0+\beta_1x_i+\epsilon_i $$ where $\epsilon_i \...
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Covariance equation [closed]

we have the function for marginal rate of substitution as M_t+1. log of that is m_t+1 = ln B -γln((c_t+1)/(c_t)) We also have another equation. cov(ln((c_t+1)/(c_t)),lnRe_t+1) = 0.03 B = e^-0.03 In ...
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40 views

find the autocovariance function of the process $Y_t$

Consider the processes $X_t = \phi X_{t-1} + v_t$ and $Y_t = \phi Y_{t-1} + X_t + e_t$, in which $|\phi| < 1$ and $v_1$ and $e_t$ are non-correlated random errors with zero mean and variances equal ...
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Covariance Matrix for LDA

I have two classes of data: $X = (3,2),(2,3),(1,2),(2,1),(-2,-1),(-1,-2),(-3,-2),(-2,-3)$ $Y = 1,1,1,1,0,0,0,0$ Here, $Y$ is a binary label for each corresponding value of $X$. For example, the ...
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Expected Optimism 0-1 Loss with 0-1 Response

Want to show that $$ E_X op = \frac{2}{n} \sum_{i=1}^n Cov_X(g(x_i), Y_i)$$ For 0-1 loss function with 0-1 response. Want I've done $$op = l_{in}(g) - l(g)=\frac{1}{n}\sum_{i=1} ^n Loss(Y_i', g(...
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Sample covariance vs population variance of means [duplicate]

Was looking at this link and wondering why for population covariance, the denominator is n, while for sample covariance, denominator is n-1. How does this 1/n(n-1) replace the 1/n^2 in the proof in ...
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Degrees of freedom in covariance calculation [duplicate]

When calculating the sample covariance, why do we divide by $n-1$ instead of $n-2$? Don't we lose two degrees of freedom since we need to calculate two sample means? For example, when estimating the ...
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Law of total covariance with multiple conditionals

The law of total covariance states: $Cov(X,Y) = E(Cov(X,Y|Z)) + Cov(E(X|Z),E(Y|Z))$ If I condition on another variable $T$, does this still hold? E.g. $Cov(X,Y) = E(Cov(X,Y|Z,T)) + Cov(E(X|Z,T),E(Y|...
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Covariance Matrix (Long/Short Portfolio; Different Weightings)

I am attempting to calculate the expected one-day standard deviation of a portfolio in dollars. In other words, I am looking for the following: "I expect my portfolio to move _______ dollars on ...
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Computing ACF of Time Series Process

I need to compute the ACF of the following process: $X_t = \frac{1}{4}X_{t-1} + Z_t$, where $Z_t$ is a white noise process with variance 1. My work so far: $$Cov(X_t, X_{t+h}) = Cov(\frac{1}{4}X_{t-1} ...
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What is the quantile covariance?

Suppose that $X$ is a p-dimensional random vector and $Y$ is a random scalar. Then, Dodge and Whittaker (2009) indicate that the covariance of these two variables can be formulated as a minimization ...
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How to calculate covariance between stocks with different observation periods?

Let's say I have three stocks. Stock A and B I have daily prices from 1 Jan 2000 to 31 Dec 2018. Stock C I only have daily prices from 1 Jan 2005 to 31 Dec 2018. Is it possible to calculate the ...
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30 views

Probability, covariance, joint density

Let $f(x, y) =3\min(x,y)$ if x and y are values between $0$ and $1$. Find $\operatorname{cov}(x, y)$. Is the right way to solve it, finding the marginal density of $x$ and $y$? And then the expected ...
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Asymptotic Distribution of Covariance

I've seen a lot of questions revolving around the asymptotic distribution regarding the sample variance, such as $\sqrt{n} (s^2 - \sigma^2) \xrightarrow{d} N(0, \mu_4 - \sigma^4)$, however, what would ...
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Recovering dimensionality of shared subspace?

Suppose I have X random variable have form $\langle x1,0,x2\rangle$ and Y random variable have form $\langle y1,y2,0\rangle$. These variables have 1 dimension in common. Is it possible to determine ...
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Multivariate Gaussian distribution explanation needed [duplicate]

I am pretty new in statistics. I Googled the multivariate Gaussian distribution, but still have no idea how to solve this. I tried to make $\mu_{x} \rightarrow a\mu_{x} \ and\, \mu_{y} \rightarrow ...
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How do I get the covariance of a time series

I have the following time series: $$y_t = \mu_t + \sigma_tx_t$$ I want to get the covariance from this time series. How would one proceed with this? I have found the following formula: $$cov(x, y) = ...
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logic of autocovariance in time series

I am confused with the “autocovariance” in time series. In textbooks it says that it: “measures the linear dependence between two points on the same series observed at different times”. Why “two ...
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20 views

Linear model fitting with covariance, by group

I need to fit a linear model to a percent change value over time (categorical), for grouped data and need to include age covariance. I've tried to work with these models: ...
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Should you correct for group characteristics that are correlated to group membership

A little over a year ago, we have collected data by administering neurpsychological tests in several prisons (n=280). While this data is interesting by itself, we would like to compare these scores ...
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1answer
36 views

What is the variance of the estimator in ordinary least squares with correlated residuals

If we assumed that $y \sim N(X\beta,S)$ where S= $\sigma^2\begin{bmatrix} 1 & \rho & \rho &...\\ \rho & 1 & \rho &...\\ \rho & \rho & 1 &...\\ \rho & \rho &...
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How should I test for relationship between two variables if there exists a lag?

I have two time-series: Number of emails opened daily between January-March. Number of applications created daily between January-March. There is an expected time lag between someone opening an ...
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Fast inverse of positive definite matrix subtracted by its Nystrom approximation

Assume a positive definite symmetric covariance matrix $$C_{n,n}$$ and let its Nystrom approximation be $$\hat{C}_{n,n}=C_{n,q} C^{-1}_{q,q} C_{q,n}$$ for some $q<n$ Inverting $C_{n,n}$ is of $\...
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Interpreting predictive models in the presence of omitted variables

Suppose the best predictive model from a set of possible models is a univariable model, due to lots of moderate correlations with other variables for example. However, if I use this model for ...
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De-standardizing covariance matrices after applying PCA?

If I decide to use PCA to estimate a high-dimensional asset portfolio covariance matrix using reduced dimensions, I can use the following procedure to transform the low-dimensional matrix back to the ...
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1answer
40 views

covariance between continuous and discrete random variables

Let $Y \sim Ber(p)$ and $X \sim N(\mu,\sigma^2)$. I want to compute $Cov(Y,X)$ and joint distribution of $(Y,X)$. Are there any useful reference? Thanks.
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If two lognormal distributions have correlation $\rho$, what is the correlation between the log of those distributions?

Suppose I have $X,Y$ which are lognormal with correlation, $\rho$, what is the correlation between $log(X)$ and $log(Y)$? I tried working it out analytically and I'm getting that you have to ...
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59 views

Can you use Kendall's Tau to compute covariance matrix?

I am working with multivariate archimedean copulas, and I am wondering how I can extract a covariance matrix out of them? I can get Kendall's Tau matrix of correlation so I was thinking that maybe I ...
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Multiplying by vectors to assess covariance is zero

I want to prove the following but am unsure how. Show that if: For all fixed vectors $c$, $Cov(X,c'Yc) = 0$ Where $X$ and $Y$ are matrices of random variables, then it must be true that: $Cov(X,Y)=...
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Inverse Wishart Prior for linear model

I know some bayesian methods employ an inverse wishart distribution for the prior distribution of the covariance matrix in a linear regression. I.e. for the model: $$Y=X\beta+\epsilon$$ Where $\...
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Are the conditional expectation values of y and f necessarily equivalent in Gaussian processes?

Suppose $y$ is a Gaussian process given by $y \sim f + \epsilon$, where $\epsilon$ is a Gaussian noise model with zero mean, and $f$ is a deterministic yet unknown mean function (or a Gaussian process ...
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26 views

Proof that covariance of RV and group average is less than variance of RV

I have a random variable $X_i$ and a group of $N_j$ other random variables that includes $X_i$. Let's just call this group $J$. There are no distributional assumptions made on these RVs (other than ...
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How to make a scree plot out of SVD data to validate PCA

After doing a singular value decomposition (SVD) of a data set, I'm left with three matrices: 1. An orthogonal Left Singular Vector (U) 2. diagonal matrix with elements in descending order (S) 3. ...
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Question on a proof in Bickel's covariance estimation paper

Recently, I am reading Peter J. Bickel's paper Regularized Estimation of Large Covariance Matrices. In that paper, the author tries to prove lemma A.3, which goes as follows: Let $Z_{i}$ be i.i.d. $\...
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In a repeated measures model where assessments are performed over time, should baseline data be excluded if baseline covariate is used?

In a model where subjects are evaluated over time and a baseline (time=0) covariate is used (eg, ...
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How does a covariance intensity function measure clustering?

I was taught in a class on spatial statistics that the covariance intensity function (defined below) measured clustering and inhibition in a point process, but isn't used because good test statistics ...
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Moderation and Ancova

I conducted a repeated measures anova with accent (British versus American) as 1 level and dress style (business, super casual, and business casual). My dv was how professional each was rated (1 to 7)....
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Covariance between two binomial random variables

Consider a binomial random variable $X$ with parameter $p$ and another binomial random variable $Y$ with parameter $q$. What is the covariance of $X$ and $Y$? How well does the proof generalize to $n$...
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28 views

What statistic to use in testing the variance of maximum likelihood estimators

(A physicist self-studying statistics here) I was previously confused about the meaning of the standard error in a maximum likelihood estimate. Certain stack exchange posts (linked below) have gone ...
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the accuracy of covariance between two high-dimensional vectors

Question Is the covariance between high-dimensional vectors less accruate than covariance between two vectors in low-dimensional vecotrs? I am asking this questio to check if there is a need for '...
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13 views

Covariance correction for linearly mixed signals

Consider the simple linear mixing model: $$ X = AS + v $$ where: $X$ is N-by-T, $A$ is N-by-M, $S$ is M-by-T, and noise $v\sim\mathcal{N}(0,\Sigma_v)$. Assume that we know the matrices $X, A,...
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43 views

What is the formula for the conditional variance when taking the derivative of a Gaussian process?

The formulae for the conditional mean and variance of a Gaussian process is given by equations (2.23) and (2.24): Also, the formula for the covariance of the derivative of a Gaussian process is given ...
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100 views

Is the spherical covariance function not positive definite for d > 3?

I read in a textbook (Japanese one) that the spherical covariance function is only valid for dimensions $d = 1,$ $2,$ and $3.$ I have the following questions: Does that mean the spherical covariance ...