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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49 views

General formula for finding covariance of monomials of multivariate random variables

Suppose that we have independent random variables $X_1,X_2,X_3$ which are gaussian multivariate distributed with a mean of zero vector and a diagonal covariance matrix. $X=[X_1,X_2,X_3] \tilde{} N(0, ...
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2answers
34 views

Notation: covariance on scalars?

I've recently seen the degrees-of-freedom for K-nearest neighbors regression specified like so: $\frac{1}{\sigma^2}\sum_{i=1}^NCov(y_i,\hat{y}_i)$ But what does $Cov(y_i,\hat{y}_i)$ mean? ...
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8 views

Co-variance matrix and Factor graph representation

I have problem in fundamental understanding of factor graph representation: I am trying understand the relation between the Co-variance matrix and associated factor graph, of a Normally distributed ...
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0answers
9 views

binomial error calculated with formula

i have uploaded a 2D histogram ,, actually it is filled by dividing two 2Dhistograms with same bins.i want to calculate manually each error which is shown with each value i.e binContent.Please ...
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0answers
19 views

Expected Value and Variance of a summation of correlated sample means

Let $\gamma_i$ be constants and $f(\gamma_i)$ be normally distributed random variables. More specifically, $f(\gamma_i)$ is the sample mean of the population $\gamma_i$. Now I have a sum of the ...
3
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1answer
71 views

What is the covariance called when it is not divided by N?

I noticed that in signal processing they have this term called cross-covariance. The cross covariance function produces covariances of two functions with different lags. At the center of the vector ...
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13 views

Suppose X_1 X_2, …, X_n are n independant variables, is their Covariance matrix sigma diagonal?

Suppose I have n variables X: X_1, X_2, ..., X_n that are independent from each other. Which means that: if i≠j, then Cov(X_i, X_j) = 0 As a consequence, I'm wondering if their Covariance Matrix ...
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1answer
30 views

How does the covariance matrix of a fit get computed?

I often have to fit data of physical experiments as a student. I always use (python's) functions like numpy.polyfit or scipy.optimize.curve_fit for that purpose. They also allow you to retrieve the ...
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0answers
18 views

Looking for a principled/systematic procedure for discarding features

I have a collection of $M_i \times N$ matrices $X_i$ whose rows are (raw) feature vectors (from a common $N$-dimensional feature space). MATLAB reports that most of the covariance matrices $C_i := ...
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1answer
34 views

When is it necessary to estimate the covariance matrix instead of calculating it directly?

The covariance of two random variables $(X_i,X_j)$ is given by $$Cov(X_i,X_j) = E[X_iX_j] - E[X_i]E[X_j],$$ where $E[X_i]$ is the expected value of $X_i$, or its mean. The covariance matrix $\Sigma$, ...
1
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1answer
23 views

Auto-covariance in a stationary time series

I am trying to calculate the auto-covariance of time series $Z_t$. Given a weakly stationary process $Y_t$; $t \in \mathbb{Z}$: $$Z_t=a+Y_t$$ Now, my goal is to show that $Z_t$ is also a stationary ...
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0answers
5 views

Covariance structures and two-way ANOVAs

I have been working with mixed-models and ANOVAs in R for the past year, and am just now getting into the extensive information on covariance structures. Using a mixed-model with repeated measures on ...
0
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1answer
35 views

Finding covariance matrix for weight priors for bayesian regression with feature space mapping of inputs

I want to implement Bayesian regression which returns the MAP estimate for given aggregation of columns of design matrix mapped into the feature space $\Phi(X)$, responses as a column matrix $y$ ...
2
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1answer
79 views

Intuitive explanation for when Pearson correlation coefficient equals 1

Whenever I teach a complicated formula, I always demonstrate to the class that it works intuitively in extreme cases. For example, when all the data values are equal, the standard deviation is 0. Not ...
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0answers
20 views

On the use of the autocovariance generating function

The autocovariance generating function is defined as: $$g_X(z) = \sum_{h = -\infty}^{\infty} \gamma(h)z^h.$$ Where $\gamma(h)$ is the autocovariance function of the considered process $X$. I can ...
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1answer
46 views

Is the covariance of standardized variables the correlation?

I have a basic question. Say I have two random variables, $X$ and $Y$. I can standardize them by subtracting the mean and dividing by the standard deviation, i.e., $X_{standardized} = \frac{(X - ...
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0answers
16 views

Convert principal components based on covariance into principal components based on correlation

I am wondering if there is any way to mathematically express the change in direction of the principal components from the $2\times2$ covariance matrix to the correlation matrix. In other words, if ...
0
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1answer
18 views

Marginal Likelihood of a Gaussian Process Model, Duplicate entries in kernel matrix

I am trying to fit a Gaussian process model using the toolbox and I got stuck in the following problem. Assuming that I have some duplicated data points in my training data, then those will map to ...
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3answers
71 views

Covariance between variables

When looking at correlation, some variables may have a higher correlation but others may have a stronger relationship between them and a lower correlation. What do you do to figure out which variables ...
3
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1answer
37 views

Is a covariance matrix composed of matrixes derived from separate samples guaranteed to be positive definitive?

I have two samples that partially overlap on the variables they describe. The samples are taken from more or less the same population, and show similar values on the overlapping variables. Based on ...
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0answers
11 views

Covariance structure of a 3-level hierarchical model with random slope and intercept

I would like someone to confirm if I am getting correctly the covariance matrix in a 3 level mixed model with random slope and intercept. I have random intercepts at level 2 and 3 and a random slope ...
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1answer
47 views

What can be inferred from “covariance matrix of residuals” and “correlation matrix of residuals” after VAR?

I have this VAR: summary(VAR(V6CADModelSt45obs1D.df[,c(5,3,2,6,1,4)], p=5, type="none", ic="SC")) The following is the result of this VAR: ...
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34 views

Correlation between two quadratic forms of Gaussian random vectors

I want to approximately calculate the correlation between two quadratic forms of two Gaussian random vecotrs (of course these are in fact non-Gaussian densities). Does anyone know the derivation of ...
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0answers
27 views

Matlab: Help in mitigating the problem: Non-positive definiteness of a matrix

EKUKF toolbox contains a file ut_sigmas.m I am facing a frustrating problem with the covariance matrix $P$ of the measurement when using Cholesky decomposition. ...
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12 views

In proc glimmix covariance parameter estimates, what is “scale”? Is it equivalent to residual error?

I conducted data analysis using proc glimmix for my proportional data. Below is the sas code I used and covariance parameter estimates from the output. The experiment was conducted by split plot in ...
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27 views

Can near zeros in precision matrix be treated as zeros?

A zero entry in the precision matrix (the inverse of the covariance matrix) means the corresponding variables are indepenent given all the other variables. For real-world data samples, when is an ...
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0answers
7 views

Stability precision matrix under small changes in covariance

I am trying to understand how the precision matrix changes under the influence of small changes in the covariance matrix. I have several similar datasets: the differences in standard deviation for the ...
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0answers
23 views

Matlab FACTORAN error on line 162: a covariance matrix is not positive definite [closed]

I have a data set called Z2 that consists of 717 observations (rows) which are described by 33 variables (columns). The data is standardized by using ZSCORES. Additionally, there is no case for which ...
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1answer
55 views

Decomposing a random variable with random mean into a sum

I have two random variables: $X\sim \mathcal{N}(0,\sigma^2+1)$. $Z$, a gaussian with mean $X$, distributed so that $E_{X,Z}[(X-Z)^2]=s^2.$ We know that: $$s^2\geq\sigma^2+1 \Leftrightarrow Z ...
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1answer
31 views

assuming independency between independent variables in multiple regression?

I heard that multiple regression assumes that the independent variables are correlated somehow. So when we convert the multiple regression into SEM diagram, we see covariance arrows are drawn ...
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0answers
25 views

Instances of sparse covariance matrices

I am trying to find large datasets with inherently sparse covariance matrices, to be tested with our algorithm. Basically, we will take the sample covariance matrix and enforce some structured ...
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1answer
41 views

Finding $Cov(2X+7, X^2 +3X - 12)$

So I have this pdf, $f(x)=3x^2$ for $x\in (0,1)$ and I need to find $Cov(2X+7, X^2+3X-12)$. My main concern about how I answer this is, what is the joint pdf for these two distributions? I guess ...
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1answer
63 views

What is the demonstration of the variance of the difference of two dependent variables?

I know that the variance of the difference of two independent variables is the sum of variances, and I can prove it. I want to know where the covariance goes in the other case.
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81 views

sparse covariance/correlation thresholding

In our project, we would like to do some optimization on sparse matrices. The idea is to scrape massive amounts of data, form a covariance/correlation matrix, and form a sparsity pattern basically by ...
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3answers
117 views

Random variables have non-zero covariance but expected sample covariance is zero? (intuition)

This post asks "why a familiar and widely used estimator of sample covariance has expected value zero, in a situation where the variables involved are characterized by non-zero and equal pair-wise ...
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1answer
32 views

Covariance definition

Why is covariance defined the way it is? $$\sigma(x,y)=\mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])]$$ How do we know that this definition behaves in the following way? Covariance is a measure ...
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4answers
998 views

Why shouldn't the denominator of the covariance estimator be n-2 rather than n-1?

The denominator of the (unbiased) variance estimator is $n-1$ as there are $n$ observations and only one parameter is being estimated. $$ ...
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0answers
47 views

Relation between raw and central moments

This question arose when reading Johansen's likelihood-based inference in cointegrated VAR models, the 2009 reprint, page 146. I will do my best to make my post self-contained. Let $Z_{0t}=\Delta ...
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1answer
18 views

Calculating covariance matrix with different time periods

Instead of using a covariance matrix that was calculated by using the entire sample to simulate results, I am interested in possibly using a covariance matrix that was calculated by using a selected ...
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0answers
8 views

Testing for covariance on a fixed scale?

I'm struggling on finding a way to analyse my data, that may be fixed just with a basic statistics insight. I have a dataset that has a large number of variables (100+) some of which tend to covary ...
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0answers
27 views

Necessary conditions for causality

E. Tufte writes: 'Probably the shortest true statement that can be made about causality and correlation is "Empirically observed covariation is a necessary but not sufficient condition for ...
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1answer
65 views

how to plot a 2D covariance error ellipse?

I have a location of landmark in 2D. According to Extended Kalman Filter EKF- SLAM, if the robot re-observes the same landmark, the covariance ellipse will shrink. I collected the necessary ...
4
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1answer
55 views

Generating independent random variables from correlated random variables

I have 2 standard normal, bivariate correlated random variables, $corr \ (X_1, X_2)=\rho$. I want to generate two independent standard normal random variables from these 2. I tried to use what I ...
13
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3answers
490 views

How does the formula for generating correlated random variables work?

If we have 2 normal, uncorrelated random variables $X_1, X_2$ then we can create 2 correlated random variables with the formula $Y=\rho X_1+ \sqrt{1-\rho^2} X_2$ and then $Y$ will have a correlation ...
4
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1answer
26 views

covariance of squared terms

Assuming two variables $X1$ ~ $N(0,1)$, $X2$ ~ $N(0,1)$ with $Cov(X1,X2) = a$. Is it possible to derive analytically what the covariance between $X1^2$ and $X2^2$ would be? Empirically (I tried this ...
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1answer
39 views

Closed-form expression for autocovariance of random walk with drift

I am working through slides hosted at Basic Time Series Models, and am not sure how to mathematically derive a closed-form expression of the autocovariance of the "random walk with drift" model. The ...
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15 views

Expectation (covariance) between $X_j$ and $Y$ in logistic regression

Is there some way to explore the relationship of $E(X_jY)$ in a logistic regression? That is, $Y$ is a binary variable observed together with $p$-covariates $X$ (wlog centered). I am interested in ...
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0answers
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Why does inversion of a covariance matrix yield partial correlations between random variables?

I heard that partial correlations between random variables can be found by inverting the covariance matrix and taking appropriate cells from such resulting precision matrix (this fact is mentioned in ...
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0answers
21 views

How to test the significance of covariance?

I'm using the Mutual Informacion covariance in RNA sequences and I want to know if there exits a way to test if some covariance is significant, let's say, an associated p-value. Thanks to all for ...
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10 views

Sub-space / latent-space covariance

I am not entirely sure what I should be googling for this in my present context. Basically I am working with a set of latent variable models such as the Gaussian Processes latent variable model ...