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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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Pooled Covariance Matrix with very different amount of samples per class

I have a dataset with 10 classes, and want to estimate the covariance. It turns out that due to numerical stabilitiy, it is much better to use a pooled covariance matrix. Suppose I have $N$ samples ...
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Using variance-covariance matrix of mixed-effects logistic regression to obtain p-values for custom contrasts

My question is a follow-up to this question, following through on @Isabelle Ghement's excellent series of responses. I just want to run this past some people in the know to see if what I am doing is ...
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Question about the regularization on Deep learning book (Dr. Goodfellow) [closed]

In Chapter 8 page 228, I don't understand why the component of $w^\star$ that is aligned with the $I$-th eigenvector of $H$ is rescaled by a factor of $\lambda_i/(\lambda_i + \alpha)$. And in the ...
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Adjusted Mean of Variable given Single Covariate with Weak/Moderate Relationship

Say I have two variables X and Y, each a data set with corresponding data points 1 through n. These two variables have some casual, small but significant relationship (low r value). Then I am unsure ...
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25 views

Covariance of two uncorrelated variables multiplied with the same random variable [duplicate]

I have a problem where I am faced with the term $\operatorname{Cov}[XY,XZ]$. However, I do not know what to do with this term. I may assume that $Y$ and $Z$ are independent and that $X$ is independent ...
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Inferences about leverage score and Mahalanobis distance

I have some inference given below: Given the design matrix $\textbf{X}$, the leverage score is defined as $\textbf{H}_{ii}$ where $\textbf{H} = \textbf{X} (\textbf{X}^T \textbf{X})^{-1} \textbf{X}^T$ ...
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23 views

The covariance of 2 independent and identically distributed

I am currently researching a paper and they have the following set-up: " $(\epsilon_{1}, \epsilon_{2})iid \sim N(\mu, \xi)$. captures the collective biases that in-vestors may have about d, is ...
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Moving average process - stationarity

If we consider a moving average process of order 1, is that stationary? Because, although, the mean will remain the same for Yt and Yt+k, the variance and co-variance will change if you calculate ...
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1answer
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How to show sample correlation is sample covariance for standardized values?

Given a matrix $X$ and the resulting sample correlation matrix $R$, consider the standardized observations: $$\frac{(x_{jk} - \bar x)} {\sqrt{S_{kk}}} \quad k=1,2,...,p \quad j=1,2,...,n$$ Show that ...
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29 views

How to calculate Mean adjusted by Covariate?

I need to calculate the mean of a variable, adjusted by another variable. Both variables are ratio scaled. I found this online: https://ideas.repec.org/c/boc/bocode/s344803.html which does what I want,...
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27 views

Why does my SEM ('lavaan') return covariance estimates for all pairwise variable combinations that were not specified in the model?

I specified the following model for SEM analysis using the 'lavaan' package in R. I want to specify a covariance between two observed variables (livestock and human occupancy). This is the only ...
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33 views

What is the difference between covariate and confounding variables?

What do covariate and confounding variables have in common and how do they differ? And what are their specific effects in causal inference? (in statistics and causal inference)
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Variance of a Sum of Random Variable [duplicate]

Unsure about my teachers solution as she was not very convincing when she presented the solution and was quite confused herself. $$Y = a+bX+cZ \\ X \sim \mathcal{N}(4,\,\sigma^{2}_x)\\ Z \sim \...
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1answer
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Simplifying covariance expression, don't understand steps in between

Can anyone explain the steps in between? In particular I don't understand: 1) How the $n$ became part of the final result. Does this $n$ just denote the total number of observations? Does it iterate ...
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What is the best way to combine “Price” and “Volume” in stock prediction?

I am trying to use LSTM network to predict stock prices. I know in real world there is a relation between the stock price and the trade volume. So I am looking a way to see if is it possible to ...
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23 views

Common covariance matrix explanation (LDA and QDA)

I'm looking for a layman's explanation of the "common covariance matrix" assumption in LDA because I don't think I understand it. I understand that a common covariance matrix (as assumed in LDA for ...
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Create covariance with blocks structure (artificially)

I would like to create some artificial variables that their covariance matrix will have blocks structure. Any idea how to do this? By saying "their covariance matrix will have blocks structure" I ...
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How to show that the error variance of the best linear predictor is inferior to the proportional predictor?

Let's consider the 1D case. How do we prove that the error variance of the Best Linear Predictor (BLP) is inferior than the Proportional Predictor (i.e. the Linear Predictor without the intercept)? ...
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How to test correlation of variances?

Assume I ask several people two walk 100 steps several times. After each of their walks, I keep track of how far apart their footsteps were and the total distance they traveled. Assuming these people ...
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26 views

Explanation of covariance matrix of polynomial parameters [duplicate]

I'm asked to find the covariance matrix of $\alpha$, $\beta$, and $\gamma$ for: $$y_i=\alpha+\beta(x_i-\bar{x})+\gamma[(x_i-\bar{x})^2-\zeta^2]+\epsilon_i$$ where all the errors have equal variance $...
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What does it mean that the variance is equal to covariance?

I am reading a paper that says that for forecast unbiasdness it is necessary to asusume that Cov(x,x̂)=Var(x̂), where x̂ is the estimate and x is the true value. How should I interpret this assumption,...
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How can i calculate Covariance between numeric attributes and Binary attributes?

I want to calculate covariance between binary attribute and numeric attribute For example: if x is numeric attribute, y is binary attribute. Then how can i calculate cov(x,y)?
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Confidence intervals for multiple output values of a Markov process using covariance?

I have a Markov process (specifically the Wang-Landau algorithm) that produces a set of "weights" $w_i$ for $i=1,2,...,N$. The weights by themselves are actually meaningless, and what is actually ...
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1answer
33 views

Impact of sample size on sample covariance

From the formula of sample covariance, we see $$\operatorname{Cov}(X,Y) = \frac{1}{m} \sum_{i=1}^m (x_i - \bar{x})(y_i - \bar{y})$$ where $\bar{x}, \bar{y}$ are the sample means. It seems like the ...
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1answer
36 views

Sum of product as product of sums

Assuming I have two random non-independent vectors $A,B$ which are within [-1,1]. I want to approximate their sum of product by product of sums (everything is a dot product), i.e. $\sum_{i=1}^NA_iB_i ...
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1answer
19 views

relationship between correlation, covariiance and conditional distribution

What are the relationships between correlation and conditional distribution. For instance, given three dependent variables, X1, X2 and ...
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Does mean centering reduce covariance?

Assuming I have two non-independent random variables and I want to reduce covariance between them as much as possible without loosing too much "signal", does mean centering help? I read somewhere that ...
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1answer
29 views

Doubt about Covariance

The covariance measures the tendency of 2 random variables (for example) to have the same sign. Suppose that we choose 2 random variables that tend to have discordant sign, but the few times they are ...
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Calculating value of a variable from correlated variables [closed]

I have 4 different variables (3 known and 1 unknown) correlated among each other. Y,Z,W are the known variables and X is the unknown variable. I have calculated the correlation coefficients of ...
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0answers
34 views

Joint Probability Distribution and covariance

If $$f(x,y)=1/4 $$ $x=-3,y=-5; x=-1,y=-1; x=1,y=1; x=3,y=5. $Find cov (x,y). I know the formula for cov (X,Y) but I'm stuck at finding E (x) and E (y).
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1answer
60 views

Covariance of products of dependent random variables

I have four random variables, A, B, C, D, with known mean and variance. As well: Cov(A,B) is known and non-zero Cov(C,D) is known and non-zero A and C are independent A and D are independent B ...
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Linear fit for correlated data

Assume I have a set of data points $(x_i,y_i)$ that I want to fit to some model function $f$ which depends on a couple of fit parameters, i.e. $y=f(x;a_1,a_2, ...)$. Usually I'd define a "chi-square" ...
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28 views

Does the variance of a sum equal a double sum of covariances? [duplicate]

Say I have a collection of $n$ random variables $X_i$ that don't necessarily have any special properties like independence or identical distribution. Is it true in general that $\text{Var}\left(\sum_{...
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31 views

Variance of bivariate normal distribution plus normal distribution

Problem: $W = -27 + 0.3X + 0.45Y + E$ The pair $\begin{bmatrix} X \\ Y \end{bmatrix}$ behaves like a bivariate normal with vector of averages $\begin{bmatrix} 156 \\ 86 \end{bmatrix}$ and ...
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29 views

Variational Autoencoder and Covariance Matrix

Why does the encoder from a variational autoencoder map to a vector of means and a vector of standard deviations? Why does it not instead map to a vector of means and a covariance matrix? Is it ...
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Why correlations are different when they are calculated from latent variables (CFA), than when they are calculated from composite scores?

When I am doing a statistical analysis, first I take the measures of each of the constructs I am measuring and I create an average score. With these averages, I calculate the correlation between ...
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28 views

Simplifying a covariance expression

Let $I=\begin{pmatrix}I_1\\\vdots\\ I_n \end{pmatrix}$ be a random vector, and $\Omega$ and $\Omega_I$ two random variables. I am trying to simplify the following equation (which worth $\frac{\rho_{I\...
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14 views

Weighted covariance with different reliability weights?

Suppose observables $X$ and $Y$ possess different reliability weights $w_{x,i}$ and $w_{y,i}$ for the possible elements $x_i\in X$ and $y_i\in Y$ respectively. Considering a sequence of consecutive ...
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1answer
48 views

Expectation, variance and autocorrelation of a “complex” AR(1) function

I'm preparing the exam for "stochastic models" and I encountered this exercise which is giving me a lot of problems: Let $$X_t=\phi X_{t-1}+\epsilon_t, ~~~~~~~~~~\epsilon_t \sim WN(0, \sigma^2)$$ ...
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0answers
18 views

Covariance specification in hierarchical model

I am currently working on hierarchical models and try to get my head around the following question: What influence has the prior choice of the covariance matrix in the 2nd stage, especially when ...
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2answers
67 views

Proof expression for the autocovariance function of AR(1)

The representation for the model AR(1) is the following: $Y_t=c+ϕY_{t-1}+ε_t$ where $c=(1-ϕ)μ$ ($c$ is a constant). I want to understand the calculations that there are behind the general formula ...
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1answer
75 views

Why does $\operatorname E(\varepsilon\mid x) = 0 \implies \operatorname{cov}(\varepsilon,x) = 0$?

I understand the intuition behind the question but I'm trying to prove it to myself with math.
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1answer
126 views

Convergence of the Matérn covariance function to the squared exponential

The Matérn covariance function converges to the squared exponential covariance function. Many sources, amongst them the GPML book and Wikipedia, state this result. None of them provide details. I ...
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1answer
46 views

$V(X|Y)=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$

We know that the conditional variance of a multivariate normal vector $(X,Y)$ is equal to the Schur complement: $$V(X|Y)=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$$ However, $\Sigma_{XX}-\...
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1answer
33 views

Conditional covariance of a multivariate normal vector

We know that the conditional variance of a multivariate normal vector $(X,Y)$ is the Schur complement: $$V(X|Y=(y_1,...,y_n))=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$$ I have the intuition ...
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1answer
27 views

RCT analysis using ANCOVA for rates

I have a question based on the following approach for the analysis of RCT's. The following works well for the outcome (and baseline) being continuous with normal errors. Expanding upon this, I was ...
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14 views

What can be interpreted from (xi-xbar)(yi-ybar) [duplicate]

what is the interpretation of each term in the numerator? why do we multiply the two terms why not add them? what is the physical interpretetation of the these terms?
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1answer
86 views

Can one usefully specify a multilevel-model with a partially-nested, partially non-nested structure?

Background Gelman and Hill's Data Analysis Using Regression and Multilevel/Hierarchical Models includes an example in section 13.5 of how to model non-nested data. The second example in this section ...
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2answers
301 views

Quantifying dependence of Cauchy random variables

Given two Cauchy random variables $\theta_1 \sim \mathrm{Cauchy}(x_0^{(1)}, \gamma^{(1)})$ and $\theta_2 \sim \mathrm{Cauchy}(x_0^{(2)}, \gamma^{(2)})$. That are not independent. The dependence ...
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12 views

Identifying variables which differentiate 2 groups the most

I have 2 groups of patients (having and not having a disease A), respectively 41 and 19 patients. Set of about 25 different parameters was measured at those patients (some continuous variables and a ...