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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Expected Covariance from Graph Traversal

I have a problem in which I am interested in taking a matrix of positive integer weights, including zero, where the matrix has dimensions nrow x ncol and the columns always sum to the same arbitrary ...
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How to generate a random positive definite matrix in R of a given dimension [closed]

I am writing a function in which the input is a correlation matrix and I need to generate a random positive definite matrix of the same dimensions. I am unsure what the code is for this?
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Can $Cov(X,Y^2)=0$ if $Corr(X,Y)$ is negative?

Basically the title. I have fiddled around with the various forms of $Cov(X,Y^{2})$ but to no avail. At this point if I had to guess, I would say it's possible [?} Thanks. EDIT: the original question ...
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Intuitive explanation of Minimum Covariance Determinant (MCD)

I am an undergrad working on Anomaly Detection on an 8 dimensional dataset, with PYOD, which relies on the MCD in the sklearn's MinCovDet. I tried reading Minimum Covariance Determinant and Extensions,...
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Should allowing multiple DVs to covary in SEM influence beta coefficients?

I am running a replication study to test a model with 8 IVs and 3 DVs (all variables are continuous). In the initial study, I had a moderate sample (≈ 200), and thus relied upon multivariate multiple ...
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Covariance of derivative of Gaussian Process Regression

There are a quite a few questions and answers which discuss how to calculate the gradients/derivatives of the posterior of Gaussian Process Regression (see here, here). These include the equations for ...
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Plotting histogram for all features from dataframe [closed]

I wrote the following code to plot histograms of all the features in my dataframe dff. My code snippet: ...
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Covariance of a non-stationary AR(2) process

I have the following AR(2) process: $(1+B^{2})X_t=Z_t$ where $X_0=X_1=0$ and $t=1,2,3...$ This is clearly not stationary since the roots are i,-i and therefore have modulus of 1 (i.e on the boundary ...
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Comparison of values by group with covariance

I have 3 groups of different sample sizes with covariance I need to include. I wish to evaluate the value dependence of the groups compared to group 1 (group 2, group 3 values compared to group 1). I ...
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giving the sample covariance formula I want to show that they are equal [closed]

I want to understand how to move from the first formula of sample covariance in the image attached to the one below as indicated by the arrow
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How can you find the correlation coefficient without joint distribution?

$Let X ∼ N(0, 1), Y := I_{X>1} − I_{X<−1}$ How can you determine the correlation coefficient of X and Y? I've tried to first calculate $COV[X, Y] = E[XY] - E[X]E[Y]$, but since $E[X] = 0$, $COV[...
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Time-series Auto-Covariance vs. Stochastic Process Auto-Covariance

My background is more on the Stochastic processes side, and I am new to Time series analysis. I would like to ask about estimating a time-series auto-covariance: $$ \lambda(u):=\frac{1}{T-u}\sum_{t=1}^...
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Proof problem in Correlation and Covariance: given, $Z=aX+bY$ and $W=cX-dY$ | to prove: $σ_z σ_w=(a^2+b^2)σ_x σ_y (1-r^2)^{1/2}$ [closed]

If $Z=aX+bY$ and $W=cX-dY$ and if correlation coefficient between $X$ and $Y$ is $r$ but $Z$ and $W$ are uncorrelated, show that $σ_z σ_w=(a^2+b^2)σ_x σ_y (1-r^2)^{1/2}$ where $σ_z$, $σ_w$, $σ_x$ and $...
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Linear Regression, $\mathrm{Cov}(\hat{y},e)=0$, correct Argument?

I am trying to reproduce the equality of $R^2 = r_{y, \hat{y}}^2$ from this site. The author uses the equation $cov(\hat{y}, e) = 0$, which is what I am trying to explain. Notation: $X$ invertible ...
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Understanding Covariance Matrix- Formula

I watched Andrew NG's lecture on PCA and happened to come across this formula for computing Covariance Matrix which I don't comprehend . I feel there's no need for a summation(sigma) over ranging ...
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distribution marginal associations in linear model with random design

We have a vector $x \in \mathbb{R}^d$, and it's a random vector which has some underlying distribution $p(x)$. We are also given a fixed vector $\beta \in\mathbb{R}^d$. Suppose we have $n$ vectors i.i....
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Examples of positive definite periodic covariance matrices

My aim is to find a few examples of positive definite covariance matrices $\pmb{R} = \{R(s,t)\}_{s,t=1}^n$ that satisfy $$R(s,t) = R(s+T, t+T),~~~1\leq s,t \leq n-T,$$ where $T$, $1\leq T\leq n-1$, is ...
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Why is the correlation between independent variables/regressor and residuals zero for OLS?

In page 4 of https://web.stanford.edu/~mrosenfe/soc_meth_proj3/matrix_OLS_NYU_notes.pdf, it states that the regressors have zero correlation with the residuals for OLS, but I don't think this is true. ...
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Mean and variance of $Y'\Sigma^{-1}Y-Y_1^2/\sigma_1^2$ when $Y\sim N_2(0,\Sigma)$

Let $\underline Y=(Y_1,Y_2)'$ have the bivariate normal distribution $N_2(\underline0,\Sigma)$, where $$\Sigma=\begin{pmatrix}\sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \...
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Eigenvalue bias in covariance estimation with limited number of samples

In the paper regularized discriminant analysis by Friedman, after introducing the sample covariance estimation as where the coefficient $W_k$ is related to the class priors in multi-class ...
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Can we simplify $Cov(XY,X)$?

Can we express $Cov(XY,X)$ in terms of moments of $X$ and $Y$ (instead of joint moments)? Is there an alternative simplification than $$Cov(XY,X)=E[X^2Y]-E[XY]E[X]$$ If not, is there perhaps an ...
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Autocovariance of a non-stationary process

I'm just going to apologize first thing, because I know my understanding of these topics is very lacking. I'm reading some lecture notes from what appears to be an econometrics course, and they are ...
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A linear process $x_{t}$ satisfies $\sum\limits_{j \in \mathbb Z}\lvert \gamma(j) \rvert < \infty$

A linear process $x_{t}$ is the weighted sum of white noise variates $(w_{t})_{t}$, i.e. $$x_{t}=\mu+\sum\limits_{k \in \mathbb Z}\psi_{k}w_{t-k}$$ such that $$ \sum\limits_{j \in \mathbb Z}\lvert \...
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Show that $x_{t},y_{t}$ are jointly stationary, and interpretation of CAcovF, $\gamma_{XY}(h)$ not being symmetric for lags $h$

Consider two white noise processes $(w_{t})_{t}$~$WN(0,\sigma_{w}^{2})$ and $(u_{t})_{t}$~$WN(0,\sigma_{u}^{2})$ that are also independent of each other such that $y_{t}=w_{t}-\theta w_{t-1}+u_{t}$ ...
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Multivariate Box-Cox Transformed Normal Generation?

My data sets are decidedly non-normal (mine have fat tails and sometimes have convex tails and other times concave tails) but correlated. For that reason, I think a Box-Cox transformation on each data ...
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Decomposition of Contribution to Total Variance

$C$ is a $N\times N$ covariance matrix of stock returns. Assuming $w$ is a vector of positions in each asset, the total variance of the portfolio is $$w^TCw$$ The contribution to total variance of the ...
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Using Joint covariances in Batch Normalization

I was going through the paper on Batch Normalisation, and wasn't able to understand what the author means when he says this: In the batch setting where each training step is based on the entire ...
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Is it enough for the ACF to be defined at $1$ for it to be defined at every other natural

I've read that a time series should be a weakly stationary for the Autocorrelation function (ACF) to make sense. The definition for weakly stationary series that I have is that all the observations ...
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Intuitive explanation of “invariance”

...assuming that I'm able to augment their knowledge about variance in an intuitive fashion Understanding "variance" intuitively and about covariance How would you explain covariance to ...
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Interpretation of covariance term in loss function

We have an $N\times 1$ vector containing some experimental values $y$, an $N\times 1$ vector $\hat{y}$ containing some predicted values, and an $N\times N$ covariance matrix $V_y$ for the experimental ...
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Variance/covariance computations in a simple probability weighted estimator

My recent homework included a problem where I had to to show that: $\frac{\delta^2}{p^2} Var(\hat{p}) - \frac{2 \delta}{p} Cov(\hat{\delta} + \frac{\delta}{p} (\hat{p} - p), \hat{p})$ < 0, where $\...
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Covariance matrices with exponential time decay

I am applying exponential time decay to financial time series to estimate their covariance matrices. The decay factor corresponds to a half-life equal to half of the estimation period. What I get is ...
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Algebraic simplification of a formula for the covariance between observed and true scores

Furr and Bacharach (2014) present on p. 115 the following equation for the covariance between observed and true scores: $$cov_{ot} = \frac{\sum(X_t+X_e+\bar{X}_t)(X_t+\bar{X}_t)}{N} $$ I understand ...
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Limiting variance of an AR(1) process

Let $\{x_t\}_{t\in\mathbb{N}}$ be a zero mean strictly stationary sequence of random variables and $c:\mathbb{N}\to\mathbb{R}$ the (auto)covariance function. If the process follows the AR(1) model $$...
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Covariance of Mahalanobis distances

Assume that $X_1,\ldots,X_n$ iid $N_2(0,\Sigma_x)$ and $Y_1,\ldots,Y_m$ iid $N_2(0,\Sigma_y)$ are independent random vectors and $\Sigma_x \stackrel{rot}{=} \Sigma_y$ (are the same covariance matrices ...
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covariance of mean estimate via Monte Carlo approximation

Please read the following explanation: Let's consider an example where we want to estimate the mean of a random variable $x$. Let's call this a Monte Carlo approximation $\hat{\mu}$. If we ...
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Shouldn't all values of the covariance matrix under homoskedacity be zero?

The following is an excerpt from Greene's Econometric Analysis, 8th Edition. In homoskedacity, the covariance matrix has zero values for the expected errors of all pairs of observations $(i,e)$ ...
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Throw out data from covariance/scatter matrix calculation when there is only one observation of a class?

If you only have one example of a class would it be better to throw out the data from the beginning (prior to covariance matrix calculation and feature reduction) and not consider it at all? I am ...
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two-way MANCOVA with two covariates in SPSS

I am running a two-way MANCOVA which needs to be adjusted by two covariates. Problem is, I am not entirely sure whether I clarified all assumptions correctly and how to finally deal with two ...
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Correlation between normal and log-normal variables

(This is not a homework question.) Let $(X_1 \sim N(\mu_1,\sigma_1), X_2 \sim N(\mu_2, \sigma_2))$ be a bivariate normal random variable with the correlation between $X_1$ and $X_2$ given by $\rho$. ...
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Is the covariance matrix almost always postive definite?

I understand a covariance matrix is always positive semi-definite, but it seems that the covariance matrix would almost always be positive definite (although theoretically is only guaranteed to be ...
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Compare two Image Datasets Distributions for Domain Adaptation - Say MNIST with USPS datasets - Dataset Shift/ Covariance Shift

I am new to statistics, Could you guys help me in solving the below questions: I need to prove that my Image Datasets have Domain Shift/ Covariance Shift/ Dataset Shift. Q-1: What are the ways to ...
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Finding a matrix $\mathbf{A}$ that projects a point to an eigenvector of $\mathbf{A}\mathbf{C}\mathbf{A}^T$

Suppose $\mathbf{b}=[b_1,b_2]'$ is $2\times 1$ and $\mathbf{C}$ is a full-rank symmetric $2\times 2$ matrix which both are real and given. Now, consider the problem of finding a $2\times 2$ matrix $\...
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Does the covariance of i.i.d. random vectors/multivariate random variables have any zero terms?

If we have i.i.d. random variables, $X$ and $Y$, then $\text{Cov}(X,Y)=0$. But let's say we have i.i.d. random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ ...
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Covariance matrix of the residuals in the linear regression model

I estimate the linear regression model: $Y = X\beta + \varepsilon$ where $y$ is an ($n \times 1$) dependent variable vector, $X$ is an ($n \times p$) matrix of independent variables, $\beta$ is a ($...
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sign symbol above or below the sigma operator

the below two expressions are the same according some references, so, my question is, why does it change the sign in $Y_{i+k}$ or $Y_{i-k},$ when there is a plus or a minus sign above or below the $\...
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Multiple imputation and covariance

I am calculating the ratio of predictive margins from ten imputed data set. What I am doing is: 1)Calculating a model for each 10 datasets 2)Calculating the predictive margins for a variable (ex ...
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Covariance of $X$ and $M$ [closed]

I was wondering how we compute the covariance of $r$$x$ and $r$$m$. Here is the problem: We know that: For stock $X: E(r_x) = 0.21$, and $Stdev(r_x) = 0.15$. For stock $Y: E(r_y) = 0.15$, and $...
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The covariance of two related variables each multiplied by a third independent variable

If given a known covariance, \begin{equation} cov(X,Y), \end{equation} what would the covariance, \begin{equation} cov(RX,RY) \end{equation} be, if R is an independant random variable with a variance $...
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What does covariance matrix in probability density function signify

I referred the literature and understood that in the image shown the sigma square multiplied by Identity matrix represents covariance matrix. But in many cases the distribution is given without ...

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