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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Co-Variance of Zero and Non-Zero mean random variables

Is the value of co-variance function for non-zero mean random variables different from the value of co-variance function when random variables have a zero-mean? I think yes based on this: For a ...
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73 views

Covariance of order statistics

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Note that I'm relatively new to stack exchange and I have already posted this on ...
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23 views

What is the variance of the sum of Yi's

Seems a simple enough question, and I presume that, if Yi are normally distributed, Var(Sum(Yi)) = Sum(Var(Yi)) This feels like I'm jumping to the wrong conclusion though. Any help would be ...
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11 views

Correlations between non-independent variables and x-y [closed]

In a biological system in one group X and Y are not significantly correlated (assumed relatively independent) and both X and Y correlate significantly with X ± Y (r > 0.6 in both cases). This makes ...
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1answer
40 views

What is the difference between the anti-image covariance and the anti-image correlation?

What is the difference between the anti-image covariance and the anti-image correlation? How are the matrices of these coefficients computed, and what is the meaning of their elements?
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1answer
24 views

Prove that $Var(\hat {Y_i})=\sigma^2h_{ii}$

I just got a simple question. In general linear model, we have $$\hat Y=HY$$ where $H=X(X^TX)^{-1}X^T$ and the residual $$E=Y-\hat Y.$$ Now I want to prove that $$Var(\hat {Y_i})=\sigma^2h_{ii}$$ ...
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35 views

$c(n)$ is trend, $r(n)$ is fluctuation. Should $\text{cov}[c(n),r(n)]/\text{var}[r(n)]$ be close to zero?

Suppose $y(n)$ is a random time series given as function of the discrete-time variable $n$. Suppose we can decompose it into $y(n) = c(n) + r(n)$, where $r(n)$ is a strict stationary residual ...
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12 views

Using Covariance and Correlation for Similarity matching

I am trying to find if a particular pattern exists in a time series. I have found that I could try using Covariance or correlation for the task. I have used a sliding window technique for doing this. ...
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32 views

Is there any point to Reverse Engineering the Fisher Information Matrix from an Inverse Covariance Matrix?

Would there be any advantage in deriving a Fisher Information Matrix backwards from an inverse covariance matrix? I've discovered that this is much easier to do on the SQL Server platform I use than ...
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7 views

longitudinal correlations matrices (structural covariance)

I am interested in doing structural covariance analyses in a longitudinal manner. In structural covariance analyses in our field, we correlate grey matter volumes of various regions of the brain with ...
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3 views

Calculate covariance of slow and fast variables

Say you have two time series $X_t$ and $Y_t$ where $X_t$ is given by an $AR(1)$-process and $Y_t$ is a deterministic function of $X_t$: $$Y_t = f(X_t).$$ Also assume that the fluctuations of (the ...
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23 views

What's the relationship between covariance, shared variance, and common variance?

I've generally assumed that shared variance and common variance were the same thing. However, here it is written that "Common variance is the realm of total collinearity. On the other hand, the term ...
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7 views

Transformations and covariances

I am looking for some potential results regarding the relationship between the covariance of $X,Y$ versus the covariance of $T_1(X),T_2(Y)$, where $T_i$ is some map. I know the relation when $T_i$ is ...
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46 views

understanding Batch Normalization

I'm trying to make sense of the Batch Normalization paper, in Section 3.2, it says We could have also normalized the layer inputs u, but since u is likely the output of another nonlinearity, the ...
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0answers
33 views

Splitting up the variance of Z for Z = X*Y

$Z$ is a function of two dependent random variables, e.g. $X \cdot Y$. Here it is shown that $$var(Z) = var(XY)=(cov(X^2,Y^2)+E[X^2]E[Y^2])-(cov(X,Y)+E[X]E[Y])^2$$ I am interested in a metric that ...
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20 views

Covariances of a random variable and its subset

Suppose event B is a subset of event A. $P(A) = p$ and $P(B) = q$. What is the Covariance of indicator functions from A and B. $$ \mathbb{1}_{A}(\omega) = \begin{cases} 1 & \omega \in A\\ 0 ...
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1answer
21 views

Covariance Decomposition

I have the returns of three stocks, $R_{1t}$, $R_{2t}$, $R_{3t}$, with 100 monthly observations for each return series. Lets suppose that I create a portfolio consisting of stocks 1 and 2, ...
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1answer
72 views

Proving that $y_t = \beta_1 x_t + \beta_2 y_{t-1} + u_t$ parameters are biased when $u_t$ is autocorrelated

How do you prove the result that for equation: $$y_t = \beta_1 x_t + \beta_2 y_{t-1} + u_t$$ the beta parameters are biased when $u_t$ is autocorrelated? In other words, that$$ \text{Cov}(u_t, ...
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17 views

What is a good estimator for the reciprocal of covariance?

Let $X,Y$ be random variables with unknown but nonnegative covariance. What is a good estimator for $1/\operatorname{Cov}[X,Y]$? Specifically, how does one deal with negative sample covariance when it ...
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0answers
15 views

Factor analysis Geometric interpretation of common variance [duplicate]

I am trying to understand the fundamental differences between PCA and Factor Analysis. PCA is straight forward in that you take the eigenvectors of the data's variance (and hence considering all the ...
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17 views

ANCOVA vs two-way ANOVA

I'm a little new to R and I haven't done stats in a while. I know a one way ANOVA is the same as a linear regression, but is there a difference between a two way ANOVA and a linear regression with two ...
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17 views

Aggregating covariances

Suppose I have 2 data sets too large to combine. Each data set has one row per user, $i$, and two columns $(x, y)$. Users can be in both data sets but have different column values, $(x_{i1}, ...
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1answer
47 views

Numerically stable correlation coefficient calculation

I have been trying to calculate the correlation coefficient $(\rho)$ of two variables, and noticed that in cases where either $var(X)$ or $var(Y)$ are very small, the correlation coefficient ...
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15 views

What test is appropiate to compare two variables having the same pattern?

we have an experiment with two variables that behave very similar (see attached picture). We think a correlation test would be suitable, but we are not sure which one. What would you recomend? ...
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26 views

Cov(y,z)? when z=min(Y)

For independently distributed normal random variables $y_i$ ~ $N(\mu_i, \sigma_i ^2 )$ Let $z =$ min$(Y) $, where $Y=${$y_1,y_2,y_3...y_n$}. How to calculate cov$(y_j,z)$ ? I tried to calculate it ...
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17 views

A function that measures covariance and size of random variables [closed]

I am trying to implement a function of two random variables that assigns higher values if the values of the variables are higher AND their (cross-)covariance or correlation is too. Does such a ...
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33 views

Why are principal components of the residuals from a multivariate regression correlated with the estimated coefficients?

Say I have some data that follows a general linear model: $$ Y = XB + E $$ for which: $Y \in \Re^{n \times m}$, $X \in \Re^{n \times p}$ and $B \in \Re^{p \times m}$ Further, let's assume (1) that ...
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25 views

Trasformation to Positive or Negative Quadrant Dependent Random Variables

Can we transform two statistically dependent random variables $X_1,X_2$ such that the covariance $Cov(X_1,X_2) \geq 0$? Can we transform $X_1,X_2$ such that $Cov(f(X_1), g(X_2))\ge0$ for all real ...
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22 views

Independence of Gaussian surrogate model

For a Gaussian process model, $$ f (x) ∼ GP(m(x), k(x, x' )) $$ $m$ is mean and $k$ is covariance function. The predictive distribution of unknown $x$ is, $$ \hat{y}(x) \sim N(\mu(x), \sigma(x)^2) ...
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1answer
31 views

Law of total covariance for products of random variables

I have two sets of random variables. $X_i \sim N( \theta, \sigma^2 )$, where $X_i$ are i.i.d. $Z_j$ which are simply iid binary random variables with success probability $p$. I want to find ...
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1answer
109 views

How to Normalize data?

In a study, the baseline data are recorded (blood pressure, BP1) prior to the experiment (watching a horror film). After the experiment, the data (BP2) are collected again. The problem is that not all ...
3
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1answer
30 views

Covariance between two random matrices

I have two random matrices (matrix-valued random variables) $X$ and $Y$, both of dimension $n \times n$. Is there a notion of covariance between the two random matrices, i.e., $\text{Cov}(X,Y)$? If ...
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14 views

Covariance between added harmonic timeseries

I am trying to construct a covariance matrix for an optimization between two added harmonic time series. In general, the basic equation looks like this: $Z(t)=Y(t)-X(t)$ When I break this down into ...
2
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1answer
22 views

Random effects model: Observations from the same level have covariance $\sigma^2$?

I'm trying to understand what the following means (or how it's displayed): For a one way random effects model: $$Y_{ij}=\mu+\alpha_i+\epsilon_{ij}$$ $\alpha_i \sim N(0,\sigma_A^2)$, $\epsilon_{ij} ...
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9 views

Checking mean ergodicity of random field given the covariance function

Given a homogeneous 2D random field with a known covariance function, what is the easiest way to check if it is mean-ergodic? In my case the covariance function is rather complex and given by: ...
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1answer
48 views

Misguiding data relationship between Y and X?

Background: I am currently surveying a number of articles related to subject of urban sprawl. One of the things that I have come across multiple times in the literature (a thing which seems to bother ...
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1answer
52 views

variance inflation factors: `vif.mer` verus group/factor level `vif()`

Imagine we have a data set with a y, x z and group1 and ...
3
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0answers
19 views

Covariance of two distinct quadratic forms

Let's say that I have two random vectors, that should belong to the same distribution family, but are distinct. I.e. $\mathbf{Y_{1}} = (Y_{11}, ..., Y_{n1})^{T} \sim D(\mu_1, \Sigma_1)$ and ...
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0answers
23 views

How to calculate errors/confidence intervals of parameters from least squares fitting which have high correlation

I'm no statistician but I want to be more honest about my error bars. Let's say I have a dataset which is described by some model $f(x,{params})$ where ${params}$ is a vector of many parameters for ...
3
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1answer
67 views

Capturing correlation failure between two time series

I have two time series as shown in the figure. The step curve in black is the trigger that causes the blue curve. As you can see first two peaks in both the series occur in the same time interval, ...
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2answers
84 views

Minimum / Maximum and other Advanced Properties of the Covariance of Two Random Variables

Are there any advanced results established regarding the behavior of the Covariance of two random variables other than the bounds on the correlation and independence when it is zero etc. which are ...
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0answers
18 views

Evaluating variance efficiently

Let $f(x,y)$ be a scalar function and let $X$ and $Y$ be independent random variables with known distribution functions. I would like to evaluate the variance $$ \sigma^2 ...
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37 views

Joint Density and Covariance between Two Random Variables with the same Mean and Variance

This seems like a deceptively simple question, (and it perhaps is and I am missing something) but I could not find anything on this. Q1) Are there any general results / relationships to get the ...
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1answer
68 views

Mixed model repeated measures in R - specific questions

I am using a mixed model repeated measures analysis and have a few questions in terms of how to structure the model. My setup is 24 individual plots, grouped into 6 blocks. There are 4 treatments in ...
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31 views

SAS HLM--why is my UN(2,2) zero?

I am estimating a random intercept and slope model using sas proc mixed. I've already plotted the data and looked at the ICC and decided that a random intercept and possibly a random slope make sense. ...
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13 views

Conditional independence and product of variables

I am reading a paper and the paper states the following: "Suppose that e1 and e2 have the standard model error properties of being mean zero and have conditionally independence of each other, so e1 ...
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0answers
12 views

Quantifying relationship between many variables

Let's assume we have 4 independent random vectors that have values $\in [0.0, 1.0]$ and a lot of elements. Label them: $\{r_1,r_2,r_3,r_4\}$. Now, let's consider a simple case of 3 variables and try ...
4
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62 views

Covariance for three variables

I am trying to understand how covariance matrix works. So let's suppose we have two variables: $X, Y$, where $\text{Cov}(X,Y) = \mathbb{E}[(x -\mathbb{E}[X])(y-\mathbb{E}[Y])]$ gives the relation ...
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21 views

How to analyze a Semantic Differential?

1) How do we statistically analyze data from a semantic differential? I have seen the results rendered as a graphic joining each point as such
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1answer
55 views

Looking for the distribution of the difference of two Gaussians in a weird relationship

I have a covariate $B$ (let's say age) and two different responses $T_1$ and $T_2$. The bivariate distributions of $B,T_1$ as well as $B,T_2$ are bivariate normal and known: $$ ...