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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How do I calculate the expected values if the correlation coefficients are -1/0/+1? [closed]

How do I calculate the expected values if the correlation coefficients are -1/0/+1? I know that -1 means backwards correlation and so on but I just need an example.
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Determining Values of Parameters so that Observation Equation is Stationary

Consider a system process given by $x_t=-0.9x_{t-2}+z_t$,$t=1,2,…,n$ with observation $y_t=x_t+v_t$ where ${z_t}$ and ${v_t}$ are independent white noise with variances $σ^2$ and $σ_v^2$. Assume that ...
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Non-parametric measures of covariance?

We have non-parametric measures like Spearman's correlation and Kendall's correlation in addition to Pearson's correlation. What about covariance? I guess one can normalize the variables to have zero ...
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Adjust SEM model for age, BMI, and year of education in Lavaan [closed]

I am doing SEM in Lavaan. I have the following latent factors (all continuous) IL6 Brain IQ My outcome is centered_POMS_total (continuous) This is my unadjusted model #Structural model ...
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How do I calculate the weighted variance, $\sigma^2$, of a set of $N$ random variables considering their correlation $\rho$? [duplicate]

In a finance textbook of mine, there is an equation for calculating the variance $\sigma^2$ of a portfolio of two risky assets (i.e. random variables) $X$ and $Y$ by considering the correlation $\rho$ ...
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Covariance matrix of multivariate normal when negative values are made zero

Let $x$ be $n$ dimensionally multivariate normally distributed with mean $\mu$ and covariance matrix $\Sigma$. Now let $y$ be random variables defined by \begin{equation} y_i= \begin{cases} 0, ...
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Detecting multivariate outliers with Minimum covariance discriminant and mahalanobis distance

I've read in some papers (such as this) and CrossValidated questions (such as this, that people are using mahalanobis distance based on robust estimations of location and scatter using minimum ...
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Computing the Autocorrelation Function of a Particular Moving Average Series

For a moving average process of the form $$ x_t = w_{t-1} + 2w_t + w_{t+1}$$ where $w_t$ are independent with zero means and variance $\sigma^2_w$, determine the autocovariance and autocorrelation ...
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Infer relationships between coefficients given covariance information

Quant Interview Question: Given $y$ = $a_1$$x_1$, $y$ = $b_1$$x_1$+$b_2$$x_2$, $cov(x_1,y)$ != 0, $cov(x_1,x_2)$ > 0, $cov(x_2,y)$ = 0. Compare $abs(a_1)$ and $abs(b_1)$. Which one is larger and ...
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"Re-standardization" of variables in factor analyses?

I'm currently reading the textbook, "Latent Variable Models" by JC Loehlin and AA Beaujean. Today I've encountered the following regarding whether to analyze covariances or correlations in ...
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Significant interaction in repeated measures ANOVA becomes insignificant with a covariate (SPSS)

I'm finishing my masters' project and have some results that are confusing me! I ran a repeated measures anova as 2 (valence - neutral or negative) x 4 (Task 1-4) This showed a positive main effect of ...
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$\mathrm{E}\left[u u^{\mathrm{T}}\right]=\sigma^{2} I_{n} \text { is untrue } \iff \text{heteroskedasticity?}$

$\mathrm{E}\left[u u^{\mathrm{T}}\right]=\sigma^{2} I_{n} \text { is untrue } \iff \text{heteroskedasticity?}$ I know heteroskedasticity $\implies \mathrm{E}\left[u u^{\mathrm{T}}\right]=\sigma^{2} I_{...
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Expectation given pairwise covariances

I have 4 variables A,B,C,D over {-1,1} (Rademacher variables) and know that ...
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Sufficient conditions for positive, symetric, bivariate function to be a covariance

Let $f:\mathbb{R}_+^2\rightarrow \mathbb{R}$ be a symmetric function (i.e., $f(t_1, t_2) = f(t_2, t_1)$ for all $t_1, t_2 \in\mathbb{R}_+^2$). Are you aware of any extra condition that $f$ should ...
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$cov(X,f(X))\neq 0$ and $E(X f(X))\neq 0$

Take a random variable $X$. Is it true that (1) $cov(X,f(X))\neq 0$ for any function $f$? (2) $E(X f(X))\neq 0$ for any function $f$? I believe the answer to both questions is no. However, can you ...
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Ratio of first two moments of sample covariance eigenvalues

Suppose we form $X$ by stacking $b$ examples drawn from some distribution in $n$ dimensions as rows and look at $\{\lambda_i\}$, the eigenvalues of $\Sigma=X^TX$. Consider the following random ...
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How to derive covariance between Y and random effect in linear mixed model

In Searle et al.'s book Variance Components, the covariance between the response variable Y and the random effect u is described as follows (highlighted with red rectangle): I am not sure why cov(y, ...
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Request a geometric proof that difference of sample-means has additive variance larger than the components?

Request a geometric proof that difference of sample-means has additive variance larger than the components? You are not allowed to use the formula quoted. In general, the variance of the sum of $\...
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Datasets with long tail of eigenvalues of the covariance?

In most datasets I use, the spectrum of the covariance matrix decays to 0 quite fast, meaning that they are more or less low-rank. My question is, whether there are setting or disciplines that are ...
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Prove OLS consistency

Consider the linear model $$ Y={\underbrace{X_i}_{K\times 1 }}^\top\beta+U_i $$ and assume (0) There is no intercept in the model (1) $E(X_i U_i)=0_K$ [orthogonality] (2) $E(X_i X_i^\top)$ has rank $K$...
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How to compute this moment of a bivariate normal distribution?

Consider that $X \sim N(\mu_X, \sigma_X^2)$ and $Y \sim N(\mu_Y, \sigma^2_Y)$ and $\text{Cov}[X, Y] = \sigma_{XY}$ where $\sigma_{XY} \ne 0$. How can an expression for $\text{E}[Y(X-\mu_X)^p]$ in ...
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What is the distribution of the Frobenius distance between two covariance matrices?

I am computing the Frobenius norm of the difference between two covariance matrices, \begin{align} |\mathbf{C}-\mathbf{C}'|_F=\sqrt{\sum_{i,j}\left(c_{ij}-c'_{ij}\right)^2}. \end{align} Each of these ...
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Covariance formula

I'm struggling to understand this presentation of covariance. It says: The variance-covariance matrix (or simply the covariance matrix ) of a random vector $\overline{X}$ is given by: $$Cov(\overline{...
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Covariance in Karhunen-Loeve transformation code

Which covariance formula does the following code use (code adapted from here) Why they call it covariance although it is not equal with the results of cov(A)? ...
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Covariance divided by variance interpretation

I came across a problem in which we are interested in the extent to which a trait is inherited from the parents by their offsprings. To do so, one can simply calculate the Pearson correlation of the ...
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Non-independence of observations

I have a latent random variable $X_i$ and observations $y_i$, for $1<i<N$. I want to write a model for $X_i$ that provides the density $f(X_i|y_{1:N})$, where $y_{a:b}$ means all observations ...
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Why calculate the residual covariances between variables with the same residuals in structural equation models?

I'm currently reading the book, "Latent Variable Models". Today, I've encountered the following. The pictures are the structural equation model and its solution. $A_1$ represents the ...
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Solving for auto-regression coefficients when covariance is singular?

I need coefficients of autoregressive model of $x$: matrix $B$ with zero diagonal which minimizes squared error: $$\text{argmin}_B E[||x - Bx||^2]$$ For non-singular covariance, I can get solution ...
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Statistical interpretation of diagonal of Cholesky decomposition?

Given a set of $m$ examples $x$ arranged as rows in $m\times n$ data matrix X, consider Cholesky decomposition of covariance matrix $X'X$. Is there a statistical interpretation of diagonal entries of ...
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Should this be an indicator or covariate in Latent Class Analysis?

I conducted a latent class analysis, using several indicators for vulnerability. One of those indicators was sex, which I then realised should be a covariate rather than an indicator, because many of ...
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What transformation(s) $g$ satisfy $\text{Cov}(g(X), g(Y)) \le c(g) \text{Cov}(X,Y)$?

Given two random variables (say standard Normals) that are not necessarily independent, are there non-linear functions for which $$ \text{Cov}(g(X), g(Y)) \le c(g) \text{Cov}(X,Y), $$ where $c(g)$ is ...
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How to set the the variance of a stationary Gaussian AR(p) process given the variance of the innovation?

In Cario and Nelson (1996), we find the following. Let {$Z_t$, t = 1, 2, ...} be a stationary Gaussian $AR(p)$ process. $$Z_t = \alpha_1 Z_{t-1} + \alpha_2 Z_{t-2} + ... + \alpha_p Z_{t-p} + \...
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How does adding a predictor lead to a higher covariance between prediction and actual values

I am trying to figure out what the best predictors are for a certain variable. I used gglasso to get the best predictors, resulting in two groups (two variables ...
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Options for investigating hierarchical covariance structures

In many distinct contexts in biology, one covariance structure will emerge in a dataset of some kind if another covariance structure is present. As an example, we may consider proteins produced by ...
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Why does a zero entry in the inverse covariance matrix of a joint Gaussian distribution imply conditional independence?

When $X_1, X_2, \ldots, X_n$ are random variables jointly following a Gaussian distribution, let $A$ be the inverse of the covariance matrix ($A=\Sigma^{-1}$). I am wondering how to prove following ...
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If $X\in\{0,1\}$, then $\frac{cov(X,Y)}{Var(X)}=\mathbb{E}(Y|X=1)-\mathbb{E}(Y|X=0)$

If $X\in\{0,1\}$, then $\frac{cov(X,Y)}{Var(X)}=\mathbb{E}(Y|X=1)-\mathbb{E}(Y|X=0)$ I have no idea what to address with the conditional expectation part. Thank you for any comments, someone has ...
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Aggregating Multiperiod DCC-GARCH Forecast Covariance Matrices

Say I fit a $DCC$-$GARCH(1,1)$ model to a dataset of weekly returns for four assets. I forecast the covariance matrix for the next month (so four weekly steps ahead). This gives me four $4 \times 4$ ...
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Variance, Covariance, Autocovariance

could anyone explain the differences between the Variance, Covariance and Autocovariance, please? The general topic is time series analysis - ARIMA models + vector models (VAR)
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Covariance between the first elements of the samples

There are two non-independent samples $a_1, a_2, ... ,a_n$ and $b_1, b_2, ..., b_n$. In Python I can calculate with np.cov($\{a_1, ..., a_n\},\{b_1, ..., b_n\}$ ) сovariance between these samples. But ...
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t-day covariance matrix from daily return data

How do I get t-day covariance matrix from daily return data? I have an idea of how to calculate t-day variance from daily return data. From 14.6, 14.7 in Options, Futures and Other Derivatives by John ...
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Building model for SEM: the problematic of highly correlated variables (SEM + Covariance)

I have recently come across articles suggesting we should avoid conducting SEM when variables are highly correlated. I have a model in which X1 and X2 predicts Y through M1 and M2. Which gives me the ...
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How are bivariate lognormal parameters related to the underlying normal parameters?

If $Y∼N(μ,σ^2)$ is normally distributed, then $X=\exp(Y)$ is lognormally distributed. The parameters for a univariate distribution, $\mu$ and $\sigma$ of this lognormal distribution are given by $$\mu{...
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Correlation of multivariate distributions without "slope"

Wikipedia has this image showing different correlations: It says: The correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (...
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Demonstration and Interpretation between a Fisher matrix and its dual space which is covariance matrix

I have a simple (maybe not) issue about the interpretation of the link between Fisher information matrix and its inverse which is the covariance matrix. How to formulate that a line of Covariance ...
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The Variance Covariance Matrix of an Estimator Stacking Two OLS Estimators

I am looking for how to derive the variance covariance matrix (henceforth, VCOV) of an estimator stacking two OLS estimators. Suppose that we have two OLS estimators: $$\hat{\alpha}\sim N(\alpha,\;\...
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When does $cov(X,Y)\neq 0$ imply $cov(XZ,ZY)\neq 0$?

For random variables $X$,$Y$,$Z$, when does $cov(X,Y)\neq 0$ imply $cov(XZ,ZY)\neq 0$? It would be great if you could provide some sufficient conditions or give some special cases such that this hold.
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Estimate the mean vector and the covariance matrix using the simple returns

I would appreciate help with how to to estimate the mean vector and the covariance matrix using the simple returns in R. I have historical (weekly) values of five stocks from a capital market for a ...
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Relationship between $Var(X)$, $Var(Y)$ and $Cov(X,Y)$ for random variables with zero mean

I have two correlated random variables $X$ and $Y$, both with zero mean. Are there any relationship/constraints between $Var(X)$, $Var(Y)$ and $Cov(X,Y)$, apart from the obvious $Var(X) > 0$ and $...
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Use of PCA to model variance for dependent variables

I am working on a math problem with some friend and there is some disagreement on the meaning of what we are doing. We have 3 independent variables measured tens of thousands times and we have a model ...
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Find the principal component and the proportion of the total population variance explained by each when the variance covariance matrix is given

I can understand the part where we have to find the principal component from the variane covariance matrix- find eigen values, make eigen vector and normalise. The principal component would be ...

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