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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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Compare how close two covariance matrices resemble each other

Say that my model assumes vector $v_t$ has covariance matrix $\Sigma$. i.e $E[v_t v_t^T] = \Sigma$. But the data I actually feed into the model has covariance matrix $\Omega$. i.e $E[ w_t w_t^T] = \...
Taylor Fang's user avatar
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Is covariance estimation via the inverse hessian method generalizable (or possible) for loss functions other than least squares?

I know from other resources such as here that the scaled inverse hessian of your least squares loss can be used to estimate your model's parameter uncertainty (specifcally, covariance), but I can't ...
Will's user avatar
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Expectation of reciprocal residual sum of squares

Consider an IID sample $X_1 , \cdots, X_n \in \mathbb{R}^d$, then what can we say about the expectation of the reciprocal residuals when projecting onto every other point? That is can we compute $$ E \...
mather's user avatar
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Covariance of multivariate negative binomial with random effects

I am fitting a negative binomial-2 regression model where there is a multivariate normal random effects term. I would like to find an equation for the covariance of two outcomes. In "the ...
Nick Link's user avatar
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What are the implications of setting off-diagonal elements of estimated covariance to 0?

I have sometimes seen in published work that when estimating covariance matrices, off-diagonal elements are set to 0. For example, in this paper, $N$ neurons are recorded and authors wish to use the $...
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Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances? [closed]

Is the variance of the mean of a set of possibly dependent random variables less than or equal to the average of their respective variances? Mathematically, given random variables $X_1, X_2, ..., X_n$ ...
HappyFace's user avatar
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How do I calculate the error on the extrapolation of a double natural log fit? [duplicate]

I am writing software in Python that tries to fit a data set $t, y$ to the function $y = a \ln(pt) - b \ln(qt)$ and solve for the value of $y$ at $t=30$, denoted $y_0$, and its error $\sigma_{y_0}$. ...
ohshitgorillas's user avatar
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Correlation between two Gaussian Processes

I have a space-time series, so it is in 2D. So, the signal model $\mathbf{S}$ is a matrix. If I fix the space, the time series at that point in space is a complex GP: $$ \mathbf{S}[x, :] \sim \...
CfourPiO's user avatar
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Equivalence of $\sum_{i=1}^3 \sum_{j=1}^3 k_j Cov[y_i, x_j]$ and $\sum_{i=1}^3 k_j\sum_{j=1}^3 Cov[y_i, x_j]$ [duplicate]

We define two correlated random variables $Y_i$ and $X_j$ and say we have this sample, $Y_i:\{y_1,y_2,y_3\}$ and $X_j:\{x_1,x_2,x_3\}$ for the convenience of illustration. I want to calculate $Cov[\...
Roger Jia's user avatar
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Relation between first three moments of distribution

Consider a random variable $X$ with distribution described by the first three moments: $$\mathbb{E}(X) = \mu$$ $$\mathbb{E}(X-\mu)^2 = \sigma^2$$ $$\mathbb{E}(X-\mu)^3 = \gamma$$ Is there a nontrivial ...
Luca Gi's user avatar
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Decomposing Returns into Systematic and Idiosyncratic Components Using Eigenvectors of a Known Covariance Matrix

I am working with a factor model where stock returns are given by the equation: $r=\beta^Tf+\epsilon$ where $r$ is an n-dimensional vector of returns $f$ is a k-dimensional vector of factor returns $\...
Chechy Levas's user avatar
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Variable weighted PCA

I have seen a lot of "weighted PCA" but they are really all on "observations". For example Weighted principal components analysis if you have K variables, N observations, the ...
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Ancova or other test?

I am struggling to select the correct type of test for my data and would like to eventually run it in R. The design is as follows: A treatment vs control group of plants From each plant the following ...
Allyssa Hinkle's user avatar
4 votes
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What is the difference between a matrix normal distribution and the multivariate gaussian distribution?

$\newcommand{\vec}{\operatorname{vec}}$Consider a set of $N$ matrices $X_1, X_2, \ldots, X_N$. I want to estimate the distribution of these matrices represented by the mean and covariance. I address ...
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Proportion of variance in a linear regression model with a covaring predictor

Given a model: \begin{align}Y_{i}=Z_{i}*\beta * X_{i} + Z_{i} + \epsilon_{i}\tag{Eq. 1}&\end{align} I am interested in a closed formula for the proportion of variance explained by the predictor ...
CafféSospeso's user avatar
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Why does R robustbase and rrcov covMcd compute reweighted step trimming adjustment with actual fraction of outliers?

covMcd and CovMcd in R robustbase and rrcov compute by default a reweighting step. Reweighting in MCD and similar computes the Mahalanobis distances and the uses a cutoff using the chisquare ...
Josef's user avatar
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Different measures of covariances

Suppose I have two random time series variables, $X$ and~$Y,$ with a particular realization $x_{t}$ and $y_{t}.$ Suppose I find that $cov\left(X,Y\right)>0.$ What does this imply for i) $cov(\...
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Path analysis with perfect fit

I'm trying to determine if I can display two regression models and the covariance between the dependent variables in one unified model using path analysis with lavaan in R. In the following (scaled) ...
BlueMarlin's user avatar
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Using bootstrap to estimate covariance of mean estimators from two distinct (dependant) populations

Given two samples : an i.i.d. $n_u$-sample $(u_{j})_{1 \leq j \leq n_u}$ and an i.i.d. $n_v$-sample $(v_{i})_{1 \leq i \leq n_v}$. Note : The populations of the two samples are disjoint (let's say we ...
Steve R. NUNES's user avatar
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Alternative method to deriving autocorrelation function of stationary AR(2) process [duplicate]

I have read this question/answer: Autocorrelation of a stationary AR(2) process How can we derive this using Expectation. Let $Y_t = \phi_0 +\phi_1 Y_{t-1} + \phi_2 Y_{t-2}+\epsilon_t$ I found the ...
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Explicit sampling with a random conditional covariance matrix

Let $\mathbf X = (x_1,\ldots,x_N)$ be a zero-mean random vector which has conditional covariance $$ \Sigma_\xi \equiv \operatorname{cov}[x_i,x_j| \xi]=\frac{1}{N}\left( \sigma^2 \delta_{ij} + \sum_{\...
Emmy B's user avatar
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In blmer, is sigma variable random-effect covariance, and why would cov(r,c)var(r) = var(c)?

As the title says, I have two questions regarding the blmer() models in R. I have tried to get firm understanding about the ...
Imsa's user avatar
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Does multiplying an exogenous variable reduce endogeneity?

Let $X$ be an endogenous variable, $Y$ be a non-negative exogenous variable, and $e$ the error term. Define $\operatorname{cov}(.,.)$ as the covariance. Then, $\operatorname{cov}(X,e) \neq 0$ and $\...
msc's user avatar
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Covariance matrix of sliding windows of a time series

I have a non-stationary time series $\{X_t\}$ (it is a stock price) and from this set I collect sliding windows of length 400 timesteps $\{Y^i_\tau\}$ where $i$ labels the window and $\tau \in [1,400]$...
apt45's user avatar
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1 answer
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Does Partial Correlation Affect Likelihood of Multivariate Normal? [duplicate]

Suppose I have a 3-dimensional multivariate normal distribution characterized by the following variance-covariance matrix $$ \begin{bmatrix} V_{X} & C_{XY} & C_{XZ} \\ C_{XY} & V_{Y} & ...
A Friendly Fish's user avatar
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Change in classwise distribution of hidden layer outputs given categorical crossentropy loss in a single layer linear neural network

Given a linear neural net with a single hidden layer, and a set of input samples $\mathbf X\in\mathbb R^{m\times n}$. Consider an input $\mathbf x\in \mathbf X$, such that output $$\mathbf z=\mathbf{...
Phoenix's user avatar
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Weighted Covariance [closed]

So X and Y are my random variables and they are vectors representing observations of actual production over predicted production ratios of two different factories. I want to weigh every observation by ...
rocky's user avatar
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Why or When is between-group variance equal to the covariance among members of a group?

In Introduction to Quantitative Genetics (Falconer) , one of the ways of estimating heritability of a trait is by using a full-sib/half-sib model. In a full-sib/half-sib model, a number of males are ...
S. Sasindran's user avatar
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Joint Distribution Formulation of a Spatial X, Spatial Y, and Spatial Error Model

Introductory Problem: I have $n$ points in 3-D space, where I know their X and Y coordinates (not Z), and therefore the distances between points in those 2 dimensions. Each of the three dimensions has ...
A Friendly Fish's user avatar
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Variance-Covariance Decomposition

I am trying to perform a variance-covariance decomposition of a variable X. X is the sum of three variables, A, B and C. My goal is to understand what percentage of the variance in X is arising from ...
Saunok Chakrabarty's user avatar
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9 views

How to use ICC with given data

I have data with columns like this: ENTITY Avg_Score_1 N_obs_1 Avg_Score_2 N_obs_2 With sample values: 001 | 0.997 | 900 | 1.13 | 905 002 | 0.890 | 250 | 0.96 | 251 For about 1000 unique ENTITY values,...
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1 vote
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Finding covariance structure for Bernoulli GLMM (Random Intercept)

How does one find the covariance structure theoretically for a Bernoulli GLM? For Normal LMMs ($y_{ik} = x_i^T\beta + \epsilon_i + u_k$) it's quite straight forward. For observations within the same ...
Maverick Meerkat's user avatar
1 vote
0 answers
45 views

Covariance matrix of autoregressive process

I am learning about autoregressive processes and there is something that I find unclear about the structure of their covariance matrix. Some sources (e.g. Box and Jenkins, 2016) describe the ...
Lester B. Barnett's user avatar
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1 answer
28 views

Estimating covariance matrix of irregularly updated time series

I would like to estimate the covariance matrix of returns from a set of time series that don't get updated regularly. To be precise, in my case all of the series fall into 2 classes. Class A gets ...
user408937's user avatar
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Dealing with covariance

Within my dataset, I employ a variable termed "Instagram reach," quantifying the audience size exposed to a particular post. Simultaneously, "engagement" denotes the count of ...
Thiago Cunha's user avatar
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Endogeneity Analysis without the access of raw data?

I currently have the correlation/covariance matrix for a set of variables, as well as the output from a regression analysis, but lack access to the underlying raw dataset. Given these constraints, ...
Harshavardhana Srinivasan's user avatar
1 vote
0 answers
14 views

Correlation between variable and variable conditioned on the sign agreement [closed]

Suppose I have two variables X1 and X2, following a bivariate standard normal distribution with a correlation coefficient of 0.2, if I create a new variable X3 that is equal to X2 if X1 and X2 have ...
user99's user avatar
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2 votes
2 answers
69 views

Is $E((\Sigma_i^n u_i)^2) = 0$ or $n\sigma^2$ (in OLS)?

Consider an OLS estimator, $$y= \beta_0 + \beta_1 x_i + u_i $$ I think $E((\Sigma_i^n \: u_i)^2) $ is equal to zero because simply $\Sigma_i^n \: u_i$ is always zero. But my professor in class showed ...
ragul-n's user avatar
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Including covariates in simultaneous vs. hierarchical regression

If you are computing a regression, want to know if specific IVs predict the DV when controlling for covariates (one included because age was different between IV groups, another because it is ...
Erika's user avatar
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0 answers
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Clarification on covariance matrix for multidimensional Gaussian distributions

It is a well known property of Gaussian distributions that if $Y = (Y_1, \ldots, Y_n)$, where each $Y_i$ is a real Gaussian random variable, then the components of $Y$ are independent if and only if ...
CBBAM's user avatar
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0 answers
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Parameter estimation of a WSS process

As my research revolves around parameter estimation from signals that evolve in time in a random fashion, I am curious to know what features/ retrievals people normally use to determine the parameters ...
CfourPiO's user avatar
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1 vote
2 answers
94 views

Covariance of Best Linear Unbiased Estimators and arbitrary LUE

I'm working on a problem involving two linear unbiased estimators $T$ and $T'$ of a parameter $\theta$, defined from a sample $\{X_1, \dots, X_n\}$ with mean $\theta$ and finite variance. I aim to ...
Taha Rhaouti's user avatar
2 votes
0 answers
30 views

Is the Between-Groups Variance a Covariance?

I am currently working through a book/class in quantitative genetics, and in Falconer and Mackay's Introduction to Quantitative Genetics, the following line stumped me: "The between-group ...
The_Scientist___'s user avatar
1 vote
0 answers
150 views

Can I covary exogenous and endogenous latent variables in SEM in lavaan

I am editing the question as it was too meandering. I am sorry about that. I am running an extended Theory of planned behavior (TPB) model of digital piracy and I have a question that boils down to - ...
Miljan's user avatar
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0 answers
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Decompose covariance matrix into uncorrelated and correlated part

I have an almost diagonal covariance matrix, I would like to decompose it in an uncorrelated and correlated part: $$ \Sigma = \Sigma_U + \Sigma_C $$ where all the matrices above are covariance ...
Ruggero Turra's user avatar
4 votes
1 answer
281 views

Adding and interpreting covariates in logistic regression

I have a dataset and I want to do a logistic regression between the continuous variable "A" and the categorical variable "B". However, I also wanted to include "age" and &...
Erfan Naghavi's user avatar
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0 answers
24 views

Matern covariance increasing with smoothness

Consider the Matern covariance function with the following parametrization: $$C_{\nu,\phi,\sigma}(h) = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{h}{\phi} \right)^\nu K_\nu (\sqrt{2\...
Tommy Tang's user avatar
0 votes
1 answer
33 views

Prove that the equality holds [closed]

How to prove that for any random variables $X$, $Y$ and $Z$ with finite variances, we have $Cov(X,Y)=E(Cov(X,Y|Z))+Cov(E(X|Z),E(Y|Z))$?
Amirhossein's user avatar
8 votes
1 answer
315 views

Imaginary numbers in PCA output

Using PCA manually on correlation matrix, I'm getting imaginary numbers in both eigenvalues and eigenvectors. Is this expected behavior? I understand that when interpreting a matrix as a linear ...
jbuddy_13's user avatar
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3 votes
1 answer
82 views

If $A^2$, and $B^2$ are DEPENDENT random variables, will $A$, and $B$ be necessarily DEPENDENT too?

I know that if $A$, and $B$ are independent, the independence is preserved for $A^c$, and $B^c$, where $c$ is a constant. I am wondering if the same applies to the case where the random variables are ...
Roberto's user avatar
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