**1.030 covariance questions.**

In my field, some scientists look for relationships between a dependent variable, y and a covariate, x1, while controlling for a ...

I am using the statsmodel.ols module to compute an omnibus (ANOVA) F-test for three within-subjects factors; 2*3*2 levels design. The Cond. No. of the omnibus test (...

When computing the covariance matrix of a sample, is one then guaranteed to get a symmetric and positive-definite matrix?
Currently my problem has a sample of 4600 observation vectors and 24 ...

I am reading Simple Regression Model from this book, Section 6.5 (page 267 in downloaded pdf, 276 if viewed online).
The author starts with below equation for a simple linear regression model,
$$...

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:
$$
\Sigma_{ij} = Cov( X_i,X_j).
$$
...

Let's start with a simple linear model
$$Y_t = β_0 + β_1*X_t + ϵ_t$$
If $ϵ_t$ is serially correlated and heteroskedastic, I can calculate N-W standard error of $β_1$.
I have also read in the ...

I have three independent random variables X, Y and Z, uncorrelated between each other. Y and Z have zero mean and unit variance, X has zero mean and given variance. Do you know how to compute the ...

For instance, in linear regression, we have that
$$
Cov(e,\hat Y) = 0
$$
That is, the residuals and fitted values are uncorrelated.
Is this always true in any sample realization of residuals and ...

In time series context, let $\gamma_j=E[(y_t-\mu)(y_{t-j}-\mu)]$ denote population autocovariance, where $\mu$ is population mean of $y_t$, assuming covariance-stationary. Then, $\gamma_j$ goes to $0$ ...

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 been working on the following problem:
Given you have $\text{Var}(X) = 1$, $\text{Var}(Y) = 4$, and $\text{Var}(Z) = 25$, what is the minimum possible variance for the random variable $W = X + ...

I am finding the value of a,b and c for minimising the variance of the following equation (four variables are correlated):
$$
Var(\Delta V)=V ar[\Delta S-a\Delta F_1 - b\Delta F_2 -c\Delta F_3]
$$
...

I'm a bit confused on how to start calculating by hand the covariance and intraclass correlations for mixed effects models. For example, in the particular example below:
$$
y_{ijk} = \beta'\...

Currently I am at the stage where:
$Cov(Z^2,Z_1) = E(Z^2*Z_1) - E(Z^2)E(Z_1)$
$=E(Z^2*Z_1)$, because expectations of normal, or combination of normal variables is zero.
After this I have no idea ...

The Yule-Walker estimator of an AR(1) process has some well-known asymptotic properties ... except there are TWO results governing these properties. I am not sure which one to use?
The first result ...

Hi, I have a question.
The scatter plot doesn't show any type of correlation and there is an outlier.
If the outlier was to be removed, would the correlation:
Increase dramatically
Increase ...

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 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 ...

Given 2 processes
$$
Z_t = \epsilon_t + \theta\epsilon_{t-1}
$$
$$
Z_t' = \epsilon_t' + \theta^{-1}\epsilon_{t-1}'
$$
where
$$
\epsilon_t \overset{iid}{\sim}\mathcal{N}(0, \sigma^2)
$$
$$
\epsilon_t' ...

I've got a few measurements $\vec{x}$ for some real-world value $\hat{x}$. These measurements have some uncertainty, and are correlated. Given these estimates, and their covariances, I want to take ...

How to calculate the Variance and Covariance of $exp(X_{t})+2X_{t-1}$? where X is an i.i.d. normal random variables with mean zero and variance one?

I'm given
$$
Y_t = p(t) + \epsilon_t
$$
where $\epsilon_t$ is a stationary series with covariance $\gamma_t$. Also given
$$
p(t) = \sum_{r=0}^kK_rt^r
$$
where the $K$s are constants, for the ...

I need to show that the covariance and autocorrelation functions of a stationary time series are symmetric around zero. From my understanding, this entails
$$
\gamma(h) = \gamma(-h)
$$
$$
\rho(h) = \...

Suppose that you take $M$ samples with replacement from numbers $\{1,2,..., N\}$. Denote the number of time each number is sampled by $\{K_1, ..., K_N\}$. Is it possible to say something about the ...

For a "simple" chi-squared test statistic $\chi^2 = \sum_i (x_i - \mu_i)^2 / \sigma^2$, it's clear that the domain is positive since both the numerator and denominator of every term in the sum over ...

Suppose that we have integer random variables $X>0$ and $Y>0$ and constant number $a$. We have: $X+Y < a$. Can we say that the covariance of these random variables is less than or equal to ...

let's say I have n correlated variables, from which I would like to sample. I know there are several packages, like mvrnorm, ...

In the paper, "Data analysis recipes: Fitting a model to data" (Hogg, Bovy, Lang), individual data point variances are found and used for subsequent statistical analysis. The data and corresponding ...

I have found an equation for the entropy of a $p$-variate Cauchy distribution here [page 70]:
$H(X,R) = \frac{1}{2}\log(\det(R))+f(p)\,,$
where $X=(X_1,X_2,\dots,X_p)$ is vector of random variables ...

Question: Can the correlation coefficient (r or r²) or the covariance (Cov(X,Y)) be somehow translated into a probability distribution for the dependent variable?
Example: If r²=0.8 for two random ...

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 ...

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, ...

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 ...

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 ...

In one post it was written that:
You tend to use the covariance matrix when the variable scales are
similar and the correlation matrix when variables are on different
scales.
What does scale ...

In this blog figure 4 shows that the principal components of a random walk are sinusoidal with increasing frequency for decreasing eigenvalue. Is there an intuitive way of understanding this?
If I ...

I am a little stuck with my project.
In the calculations of my project, I need to calculate the spread of some random variables. Up to now, there was no special difficulty to analytically calculate ...

I am trying to find the covariance of a compound distribution.
Given $X=x$, where $X \sim \mathrm{Uniform}(0,1)$, $Y$ is (conditionally) normally distributed with mean $x$ and variance $x^2$.
I ...

Suppose $\epsilon \overset{\text{iid}}{\sim} N(0, \sigma^2)$
Can we make any assumptions about Cov$(\epsilon, \frac{\epsilon^2}{1 + \epsilon^2})$?

In general, how many points are needed to estimate a p-dimensional covariance matrix? Does it depend on how the data are spread out across the different dimensions? Does it depend on the true ...

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 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 ...

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 ...

In unconstrained optimization, covariance matrix $C$ is parameterized in terms of its Cholesky ($C=LL^\prime)$. In other words, the parameter vector $\theta$ involves elements of the lower triangular ...

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 ...

The denominator of the (unbiased) variance estimator is $n-1$ as there are $n$ observations and only one parameter is being estimated.
$$
\mathbb{V}\left(X\right)=\frac{\sum_{i=1}^{n}\left(X_{i}-\...

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 see that lots of machine learning algorithms work better with mean cancellation and covariance equalization. For example, Neural Networks tend to converge faster, and K-Means generally gives better ...

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 ...

Imagine we have one sample (n=40) from a population on 2 variables, for which we estimate the covariance (say, 0.1). If we would have another sample (n=10) from the same population, and we estimate ...

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

- Toll The Dead never does damage, how to call your DM out?
- Short story about a space salvage tug called the Dido, which salvages a passenger ship
- How recoverable is Magical Ammunition?
- How can a person be kept alive while being periodically drained of blood?
- "Slow" teleportation
- Lightning: <table> with dynamic column count, want fixed width columns and scrolling instead of automatic resizing
- Is there any combination of two airports that are connected via taxiways?
- Have figures and tables under sections in table of contents
- Do the seat rows across the aisle from the Cathay Pacific 777 300 ER 77H aircraft line up?
- Why do we need a Builder class when implementing a Builder pattern?
- A man on a work project in West Africa asks me for 2000
- Are there cases of Democrats engaging in voter suppression?
- Why is the Space Stone wrapped in a cube and called Tesseract?
- Reuse the `apt up` part of the `apt update` and `apt upgrade` commands to execute both in sequence in just one line
- What would be a seemingly fair, but still unfair way of dividing council seats?
- Work done by a gas
- Either background is overexposed or the foreground is underexposed. Are there possible solutions/approaches?
- Could dark matter consist of the supermassive black holes at the centers of galaxies?
- Psychology of Long Life
- Is this animation showing the moon's librations a real capture or generated?
- Are there any truly 'unrecoverable' situations?
- Regeneration slaves
- How to communicate with a device that has an assigned “0.0.0.0” IP address?
- Under what circumstances would 5V@2.1A be lethal?