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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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Is the covariance matrix a diagonal matrix with variances on the diagonals?

I am a geophysicist learning about geophysical inverse problems. In many papers, the authors discuss the "covariance matrix" as it applies to the inverse problem. In most geophysical applications, ...
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A non-ANCOVA situation for correcting slope heterogeneity with a covariate

I have a statistics problem than I have spent hours trying to tackle with no success. I know what I want to test, I just can't figure out what the method is. I am looking at the relationship of ...
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If one dimension of the data is scaled by a factor, how would it affect the probability of the Gaussian distribution?

I have fitted a maximum likelihood Gaussian distribution $N(\mu, \Sigma)$ on a multidimensional data set $X$. I wonder how would $p(X)$ change if one dimension of $X$ is scaled by a factor? It's ...
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Why is distance covariance defined squared, while covariance is not?

I am dealing in a data science project with correlation analyses using pearson and distance correlation. While trying to understand the differences between them, I learned about the differences by ...
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21 views

Covariance between a variable and a non-linear transformation of it

Suppose $\epsilon \overset{\text{iid}}{\sim} N(0, \sigma^2)$ Can we make any assumptions about Cov$(\epsilon, \frac{\epsilon^2}{1 + \epsilon^2})$?
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Should I remove co-varying factors before clustering?

I have a data set of around 850 factors representing 150 geographical areas. I am looking to cluster these geographical areas, and I am intending to use a K-means clustering algorithm to do this. My ...
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16 views

Covariance of a matrix concatenated from two matrices

Let there be two matrices $X$ and $Y$ with dimensions $n$ x $2$ and $p$ x $2$ respectively. Also, let their $2$ x $2$ covariances be known such that $cov(X) = K_X$ and $cov(Y) = K_Y$. If another ...
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can I calculate a new related annual statistic from annual and lifetime data?

If I know that 45% of the population will suffer from a particular strain of influenza over their lifetime (and 55% never will) and I also know that in any one year 20% of the population have this ...
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21 views

Why is correlation proportionate to the area of the intersection of the areas under the curves of the series?

I've been staring at this diagram from Wikipedia, trying to understand (particularly autocorrelation): I see that the autocorrelation seems to be proportionate to the overlap between the graphs of ...
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1answer
18 views

How to compute the proportion of variance explained by a variable while accounting for covariance?

I am trying to do a manual calculation of the proportion of variance explained by one variable, relative to the total explained variance. However, this variable is correlated with another variable ...
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1answer
52 views

Multivariate statistics sampling and standard error problem

I have a group of workers listening to audio from many various audio files and typing it out. I want to compare the average transcription accuracy of these old workers with 2 new workers. I have a ...
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1answer
20 views

Calculating a mixed model using pdIdent var-cov matrix

I have a question in mixed models. I'm quite new at this field and I'm trying to calculate a model in which "y" is predicted according to multiple covariates (Age,gender,BMI) and the random variable "...
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30 views

Is a kernel a correlation or a covariance function?

I am reading this paper on multi-fidelity optimization, where I came across an introductory section on kriging a.k.a. Gaussian Process regression (see Figure below). It confused me about the notion of ...
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How to measure the relation between two random variables that are not linearly correlated? [duplicate]

Sometimes it appears that the covariance is not sufficient to express how much two random variables are related. Bellow, I have draw random samples from a multivariate normal distribution with two ...
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20 views

Simulate an image giving its covariance

I want to simulate atmospheric effect to an image with, e.g., m rows n columns. The input data I have are covariances between pixels, i.e., Cij between pixel i and pixel j (for all i, j = 1 to mxn). ...
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23 views

How to calculate standard deviation of 2D points along an arbitrary axis?

I have a data set $\vec P$ which consists of two arrays of coordinates $X$ and $Y$. I can calculate $\sigma X$ and $\sigma Y$, and derive the standard deviation of the vector lengths by $\sqrt{\sigma ...
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1answer
42 views

Proper way of estimating the covariance error ellipse in 2D

I am aware of this question but my issue is about two competing ways of obtaining the 2D covariance error ellipse in two competing answers over at StackOverflow. The first answer obtains the width ...
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1answer
16 views

Equal Covariance in Linear Discriminant Analysis?

In an online course, we are working through some linear discriminant analysis and I've been given an example. I am having trouble with the language used by the professor as it seems I am ...
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1answer
44 views

Covariance in terms of independent random variables

If I understood everything in statistics correctly, the covariance of two random variables should be given by $$\mathbb{E}[\phi(X_1)\phi(X_2)] = \iint \phi(x_1)\phi(x_2) p(x_1, x_2) \mathrm{d}x_1 \...
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84 views

Is covariance of two random variables from the same distribution the same as the variance of this distribution?

Sorry that this might be a very simple question, but I got confused: say we have a Binomial distribution $Bin(n, p)$, and two random variables, $X$ and $Y$, drawn from it. Is the covariance between $...
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19 views

Covariance between nominal and continuous variables?

I have an nominal independent variable (two groups) and a continuous dependent variable. A two-tailed Mann-Whitney test yields a significant difference between the two groups (p < 0.03; my data is ...
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29 views

Covariance in an error propagation leading to negative variance

Does it ever make sense that, propagating the error from a set of data, the covariance term being very negative makes the variance go to a negative value (which by itself makes no sense)? Context: ...
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37 views

Positive definiteness of Grammian with respect to Gaussian process' covariance function

A Gaussian process indexed by $T \subseteq \mathbb{R}^d$ is a collection of random variables $\{ X_t : t \in T\}$, for which each finite subset is distributed as a multivariate Gaussian. Let $G$ be a ...
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36 views

Geometric interpretation of covariance [duplicate]

Sorry it's not typeset, but hopefully it's readable. I've recently been making a Geogebra file to show linear regression and correlation coefficient. https://www.geogebra.org/m/kuzw2hyk I included ...
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1answer
26 views

What is a good measure of correlation between stocks in an equity portfolio?

I am trying to analyse the degree to which the stocks in the MSCI AC World index are correlated with each other. As there are thousands of stocks in the index, I would like a single measure of ...
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exponential of Covariance of two variables in logs, cov(X,Y)=exp(X)*exp(Y)*cov[ln(X),ln(Y)] [duplicate]

I am stuck in calculating steady state in a model that has covariances in logs. I am wondering in general if the following accurate. cov(X,Y)=exp(X)*exp(Y)*cov[ln(X),ln(Y)] if that is accurate ...
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6 views

Balanced Mixing for Target Covariance

Let $\mathbf{X}$ be a vector of i.i.d. random variables. Let $\mathbf{C}$ be a desired covariance matrix for which I like to determine mixing matrix $\mathbf{A}$ such that $$ var(\mathbf{A}\mathbf{X})...
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23 views

Multivariate Transformations of Random Variables

If we can write a covariance matrix $ W $ as $$ W = D^{1/2} R D^{1/2} $$ and $ W $ is distributed Wishart$_q (m, \Sigma) $ then how is the Jacobian of thet transformation from $ W $ to $(R,D)$ equal ...
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65 views

Cleaning Up the Data with Mahalanobis Distance [duplicate]

I have a data set and I want to cleaning up my data set from the ouliers, so I decide to use the Mahalanobis distance to find the outliers. But I have a problem here since my covariance matrix isn't ...
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51 views

Calculating the covariance between 1-D arrays for incorporation into propagation of uncertainty

I have four 1-D arrays of dependent variables. They contain hundreds of data points but I have cropped them to 20 in this example. Each point represents a grid cell on a map. ...
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26 views

Sample covariance matrix notation

I do not understand this notation for the sample covariance matrix (from Artificial Intelligence: A Modern Approach, Peter Norvig and Stuart J. Russell, Section 20.3, EM algorithm): $\Sigma_{i} = \...
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What is the covariance between X and Y, when $Y>0$ if $X<0$ and $Y<0$ if $X>0$?

Let $X$ and $Y$ be two random variables, and $Y>0$ when $X<0$ and $Y<0$ when $X>0$, but can we conclude Cov$(X,Y)<0$? If the question only states $Y>0$ when $X<0$, then the ...
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63 views

Requesting Intuitive Explanation to Covariance, Correlation and Standard Deviation

First I would like to state that I am not from a mathematical background. I am studying about change in price of products. So I have to understand about Correlation , Covariance and Standard Deviation ...
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How do I find the “elliptical confidence region” from columns of a matrix that follows the Wishart distribution?

The subject is about the sample mean and the sample covariance estimators and their respective confidence regions for the estimated parameters. Suppose that $n$ samples are taken from a $p$-variate ...
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What is Cov(X,Y) when X,Y is a F-distribution (a,b) and (c,d) ,in case X,Y Not independent [closed]

I want to know Covariance of random variable between $X$ and $Y$ when $X$ is a F-distribution with degree of freedom $a$ and $b$ and $Y$ is a F-distribution with degree of freedom $c$ and $d$,in case $...
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36 views

Pooled Covariance Estimate if means are unequal

Assume that $X_{i,j} \sim [\mu_i, \Sigma], i=1,...n; j=1...m$ and we have realisations $x_{i,j}=X_{i,j}(\omega)$. Is the formula: $\frac{1}{(n-1)m}\sum_{i=1}^n\sum_{j=1}^m[x_{i,j}-\overline{x_{i}}][x_{...
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23 views

High covariance data for linear model, and PCA in R

I have 4 set of data Y X1 X2 X3 in same length where I need to perform a model selection of linear regression on Y~ X1+X2+X3-1. However, there are significant covariance between X2 and X3. $Cov(X2,X3)=...
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40 views

What happens with Mahalanobis-Distance, when the assumption of equal Covariance-Matrices breaks down

Assume that we want to compare the forecast quality of various forecasters $f$ on $n$ values such as stock-market prices or whatever. We could then define a "Mahalanobis-Distance" (MD) (or rather ...
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1answer
116 views

Covariance of random variable with product of random variables

I have the following covariance term: $Cov(x,yz)$ with $x$, $y$, and $z$ being random variables with a mean and variance. I found a paper by Bohrnstedt and Goldberger from 1969, On The Exact ...
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127 views

What's the value of $\text{cov}(x, x^TAx)$, when $x$ follows a normal distribution

When $x\sim N_k(\mu,\Sigma)$ is a multivariate normal distribution, $A$ is a symmetric matrix, how can I show that $$\text{cov}(x, x^TAx) = 2\Sigma A\mu$$
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Is there any correlation between forecasting errors from forecasting done at different origins?

Let $\ e_{T+l|T} = Z_{T+l} - \hat{Z}_{T}(l)$ be the forecasting error $\ l$-steps ahead when the forecasting origin is time $\ T$. Now, let $\ e_{T+l-j|T} = Z_{T+l-j} - \hat{Z}_{T-j}(l)$ be the ...
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covariance matrix vs correlation matrix for multiple signal analysis

I'm dealing with a set of +100 input signals, and one output. I want to explore how each of the signals affects the output. Should I focus on covariance matrix, or correlation matrix, and why? I ...
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40 views

Covariates and moderator variables

I have a 4 * 3 (both the IVs are categorical) factorial design with two covariates (both are continuous) to run ANCOVA. Can I do a moderation analysis?
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1answer
48 views

Why does Fisher use covariance when only variance is needed?

With reference to the following image from here: (can not inline it due to unsupported format) https://wikimedia.org/api/rest_v1/media/math/render/svg/9af8aa035642689bb2004047416b069a15406447 If we ...
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1answer
42 views

Correlation and Covariance

Quick question that would really help me when studying for probability exam. A fair die is rolled $9$ times. with $S(k)$ denoting the total number of appearances of labeled $k$, where $k = 1,...,6$. ...
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Prove that $\text{Corr}(X^2,Y^2)=\rho^2$ where $X,Y$ are jointly $N(0,1)$ variables with correlation $\rho$

Consider jointly distributed random variables $X,Y\sim N(0,1)$ that have $\text{Corr}(X,Y)=\rho$. Show that $\text{Corr}(X^2,Y^2)=\rho^2$. (Hint: Consider $X,U\sim N(0,1)$ where they are ...
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Concurrent Time Series

I have a 3D spatial region of size X, Y & Z where each pixel (or voxel) in location $x$,$y$,$z$ has a time series of size $T\times 1$. Time series are highly (cross-)correlated with one another ...
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51 views

Covariance of Gaussian process?

Problem: Consider the random process defined by the Ito integral $$ X_t = \int_0^t f(\tau)\, dB_\tau $$ where $f(\tau)$ is a deterministic real-valued function and $B_\tau$ denotes the canonical real-...
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28 views

The lack of correlation determines the second-degree cross-moments (covariances) of a multivariate distribution?

It is given in the following image that lack of correlation determines the second-degree cross-moments (covariances) of a multivariate distribution,while in general statistical independence ...
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1answer
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Need handy formula for $\text{Cov}[\max(V_1-K_1,0), \max(V_2-K_2, 0)]$

In a recent post, I asked for help deriving a computable formula for $\text{Var}[\max(V-K,0)]$ based on the approach on p. 262 of ths book. $V$ is a lognormally distributed random variable and $K$ is ...