**1.016 covariance questions.**

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 ...

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 ...

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 ...

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).
...

Suppose you have vectors X and Y with covariance matrix
$V = \left( \begin{array}{cc}
A & B \\
B^T & C \end{array} \right)$. This Wikipedia article says that $Var(X | Y) = A - BC^{-1}B^T$, ...

This is a very simple question: how does one get the standard error for the covariance estimate in R? I estimate the covariance using the cov function but there ...

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 ...

Say $X \in \mathbb{R}^n$ is a random variable with covariance $\Sigma \in \mathbb{R}^{n\times n}$. By definition, entries of the covariance matrix are covariances:
$$
\Sigma_{ij} = Cov( X_i,X_j).
$$
...

I’ve independent measurements of 4 quantities, A,B,C,D for a list of objects. Some mutual correlations exist between the quantities, but I'm interested in exploring the correlation between A and B (...

I have a question about how to get the standard errors of the coefficients in my GLM model. I have the fisher information matrix which I calculated by hand, but it is unscaled.
How can I scale the ...

I'm trying to calculate a covariance matrix using weighted data in a single pass, and I'm not sure that I'm doing it correctly. I found a wikipedia article which gave the following python* code:<...

I have a data set that consists of 717 observations (rows) which are described by 33 variables (columns). The data are standardized by z-scoring all the variables. No two variables are linearly ...

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 \...

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 ...

It is well known that correlation is the normalized covariance, i.e. $\ Cor(X, Y) = Cov(X, Y)/\sqrt{Var(X)Var(Y)}$.
These two related measures describe the linear relationships in the data.
Is ...

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 $...

I am having trouble generating a set of stationary colored time-series, given the covariance matrix (their PSDs and CSDs).
I know that, given two time-series $y_{I}(t)$ and $y_{J}(t)$, I can ...

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 ...

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:
...

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 ...

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 ...

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 ...

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})...

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 ...

I have been trying to find an expression for the covariance of two normally distributed variables X and Y if cov(x,y)=c then cov(x,xy)=?
I would greatly appreciate any help. Probably it must be ...

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 ...

What are the main differences between performing principal component analysis (PCA) on the correlation matrix and on the covariance matrix? Do they give the same results?

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 ...

I'm trying to fit a multilevel model for a repeated measures design with three levels:
Subjects - conditions - trials. Each subject passes the test in two conditions, and there are 21 trials in each ...

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_{...

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.
...

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 ...

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 ...

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} = \...

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 ...

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 $...

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 ...

It might seem obvious that the covariance is a "deeper" property of the data generation process (DGP), since normally the specification of a joint distribution is done in terms of its mean vector and ...

In order to improve the convergence of a MLP, one should do "mean removal", "decorrelation" and "covariance equalization".
I know how to do mean removal, but how do you do decorrelation and ...

I am not sure what the formula is for the covariance of an AR(2) process
$$
X_t = \phi_1 X_{t−1} + \phi_2 X_{t−2} + \epsilon_t
$$
where $\{\epsilon_t\}$ is white noise process (Gaussian) $N(0, \sigma^...

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)=...

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 ...

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?

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 ...

If $\mathbf {Z}$ is random vector and $A$ is a fixed matrix, could someone explain why $$\mathrm{cov}[A \mathbf {Z}]= A \mathrm{cov}[\mathbf {Z}]A^\top.$$

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 ...

When $x\sim N_k(\mu,\Sigma)$ is a multivariate normal distribution, $A$ is a symmetric matrix, how can I show that
$$\text{cov}(x, x^TAx) = 2\Sigma A\mu$$

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 ...

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 ...

I am practicing deriving proofs and I cant seem to yield the correct answer for the covariance of an AR(1) model:
$$X_t=pX_{t−1}+e_t.$$
Would greatly appreciate if someone could tell me where I am ...

- correlation
- variance
- covariance-matrix
- self-study
- regression
- mathematical-statistics
- time-series
- normal-distribution
- r
- random-variable
- matrix
- estimation
- pca
- multivariate-analysis
- expected-value
- independence
- probability
- gaussian-process
- distributions
- autocorrelation
- mixed-model
- machine-learning
- linear-algebra
- bayesian
- stationarity

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