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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tell whether they are covariance matrix [on hold]

If A and B are co-variance matrix, is A+B a co-variance matrix? Is A^2 a co-variance matrix? Is AB a co-variance matrix? I know the necessary properties of covariance matrix, but I did not know the ...
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2answers
112 views

Covariance of two time series driven by a restricted VAR(1) model

Suppose that I have two time series $X_n$ and $Y_n$ where: $$ X_n = \rho_x X_{n-1} + \epsilon_n $$ and $$ Y_n = \rho_y Y_{n-1} + \rho_{xy}X_n +z_n $$ Here, $z_n,\epsilon_n$ are independent random ...
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13 views

Decorrelation of variables without Cholesky

To correlate variable together, for example to achieve a desired covariance, you can use either Cholesky decomposition or eigenvectors and eigenvalues. To decorrelate correlated variables, (in my ...
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17 views

How to compute the variance-covariance of log binomial distributions

I have some problems computing the variance and covariance of log binomial distributions. If $A \thicksim binomial(\theta,n)+1 $ and $B \thicksim binomial(p,A)+1$ (where $+1$ is added to avoid ...
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1answer
26 views

Multivariate posterior probability

This is a 2-dimensional pattern recognition system that I am working on. It is known that the distribution between the two classes are $1/2$ and $1/2$ respectively for class $\omega_1$ and class ...
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21 views

Online weighted covariance

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: ...
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3 views

How to use weight and correlation options of the nlme–package to define complex covariance structures?

At the moment I try to explore the various options of the gls-function which is implemented in the nlme-package. The data that will be analysed with it are repeated measures with random intercepts, ...
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25 views

Covariance in binomial distribution

Is the covariance between number of success and failure in a binomial distribution with parameters n and p, the same as the covariance between two binomial variables, which is -np(1-p)?
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23 views

Do I distort information stored in covariance matrix if I normalize the matrix to range of $[-1 , 1]$

I am using covariance matrix to maximize mutual information between samples that I selected and sample that I did not selected. The optimization algorithm that I use is sfo_greedy_lazy. It computes ...
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1answer
15 views

“weigh” different variance covarance matrixes

so my question if I have a set of weights which sums to 1 (say: [0.2,0.2,0.6]) which would represent my states of the world and I have forecasted 3 different variance covariance matrixes (all of which ...
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2answers
93 views

PCA on prices or returns

This question has been addressed here: Can Principal Component Analysis be used on stock prices / non-stationary data? In his answer Jon Egil wrote please make sure you analyse returns not ...
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0answers
29 views

Intuitive explanation for covariance and inverse covariance

Say we are given n observations from the same multidimensional distribution $D$. I am trying to understand what is the intuition behind the following norms: $\sqrt{x^TC_nx}$ and $\sqrt{x^TC_n^{-1}x}$ ...
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52 views

Variance of a portfolio

In this blog post the author mentions the following: [... Given several arrays holding returns for a portfolio], one should calculate the standard deviation via the following function call: (1) ...
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17 views

Correlated Normal Random Variables with Circulant Matrix

Let $\Sigma \in \mathbb{R}^{n \times n}$ be a circulant, symmetric, positiv definite matrix. To generate correlated random variables $Y$ with the covariance matrix $\Sigma$, one has: $$ Y = C X $$ ...
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15 views

Reducing the co-variance of data for denoising, with minimum change in the mean

Assume $X$ and $Y$ are two sets of $n$ dimensional feature vectors from two different multivariate Gaussian distributions with the covariance of $\Sigma_X$ , $\Sigma_Y$ and the mean of $\mu_X$, ...
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1answer
173 views

Are Cov(A-B) and Mean(A-B) equal to Cov(A)-Cov(B) and Mean(A)-Mean(B)?

If $\mathbf A$ and $\mathbf B$ are real value matrices (sets of vectors) and $\textrm{cov}(\mathbf A)$, $\textrm{cov}(\mathbf B)$ and $\textrm{cov}(\mathbf A - \mathbf B)$ exists, are these equations ...
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3answers
79 views

Why does a regression tree not split based on variance?

When choosing each split, recursively, in a regression tree, I understand that you want to measure the spread, in each side of the split, essentially. So, in some sources, including this one at 6 ...
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2answers
48 views

Covariance of linear combinations of correlated random variables

I am trying to predict the covariance of two linear combinations of normal random variables: $\newcommand{\N}{\mathcal N}$ \begin{align} X &= w\N(u_1,\sigma^2_1)+(1-w)\N(u_2,\sigma^2_2) \\ Y ...
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101 views

How to test if a cross-covariance matrix is non-zero

The background of my study: In a Gibbs sampling where we sample $X$ (the variable of interests) and $Y$ from $P(X|Y)$ and $P(Y|X)$ respectively, where $X$ and $Y$ are $k$-dimensional random vectors. ...
3
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1answer
25 views

Standard error for the sum of regression coefficients when the covariance is negative

I have a question about appropriately calculation the standard error for the sum of two coefficients in a linear regression model. My question is similar to this and this, but I can't seem to solve ...
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29 views

Uncertainty in Peak Value of Spectrum (Standard Error or Parameter Error)

I want to extract the position of a peak from a spectrum (energy spectrum of scattered photons). To do so, I am using scipy.optimize.curve_fit to fit a Gaussian to the region of the spectrum that ...
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32 views

Simple variance / covariance example [duplicate]

I'm trying to refresh myself on some basic stats concepts and came across the following problem, which I can't wrap my head around. Any suggestions or tips would be appreciated! The problem is: ...
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20 views

Find the equilibrium point and its variance in stochastic data?

I have some data of the form I need to find the point where it starts reaching equilibrium (as I can see from the graph around 1000 iteration) and how much fluctuating the data points are. Is there ...
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1answer
32 views

Can ANCOVA be used with a dichotomous dependent variable? If not, what have these authors done?

As stated above, if and how can ANCOVA be used with a dichotomous variable as the dependent/outcome variable? I found an article of which I wish to replicate some analyses ...
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14 views

Computing the covariance matrix in PCA [duplicate]

PCA method finds the covariance between data vectors, where each data vector includes observations of different variables (dimensions). So if the data matrix has variables in columns and observations ...
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6 views

Test that constant input corresponds to constant output

This may be a silly question... I have 2 times series: S1 and S2 The assumption is that the values of S2 are dependent on the values of S1. When I graph S1 and S2 for a period of time when S1 is ...
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24 views

How do the elements of a covariance matrix differ from individual covariances?

In particular, I get this when I use numpy: ...
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10 views

combining two linear fits

I have two linear fits, in this case taken from two calibrations of an instrument, one before and one after a field project. For the two linear fits I have the covariance matrix, but not the original ...
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34 views

Ill-conditioned covariance matrices in EM

I am currently working with the Expectation-Maximization algorithm. I have some pre-clustered sets of 3D points and am trying to run the algorithm. However I've seen that most of my covariance ...
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8 views

Multiple Spatial Dependence Weighting Matrices

Is it possible to use multiple weighting matrices in spatial error models? A possible use case might be when there are more than one way that areas could be related. For instance, states in the U.S. ...
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38 views

Expressing conditional covariance matrix in terms of covariance matrix

Suppose we have two multivariate random variables $\mathbf{X}$ (of dimension $n_x$) and $\mathbf{Y}$ (of dimension $n_y$). The covariance matrix $C_{X,Y}$ can be written as the following block-matrix ...
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44 views

Efficient, stable inverse of a patterned covariance matrix for gridded data

I have computed a stationary covariance matrix defined for data on a grid. The data y are regularly spaced in 3D, lexicographically ordered in the covariance matrix, and I'm using a using a square ...
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112 views

Why is Covariance Useful?

There are a number of topics related to covariance on this site. What I am having trouble grasping: why is covariance a useful thing to calculate? As far as I see it, covariance is not a helpful ...
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Covariance of parameter estimates in Method of Moments

I have a $3\times1$ vector function $f(x_i;\theta)$ where $X$ is a rv and $\theta$ is $3 \times 1$ parameter vector, such that \begin{equation} E \, f(X;\theta) = {\bf 0}.\end{equation} If I have a ...
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10 views

Intuitive explanation for variance of vector

I am reading about variance and covariance of vectors. variance of vector X is defined as: (X_transpose * X)/(n-1) co-variance of vector X and Y is defined as: (X_transpose * Y)/(n-1) I could ...
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161 views

Can Principal Component Analysis be used on stock prices / non-stationary data?

I am reading an example given in the book, Machine Learning for Hackers. I will first elaborate on the example and then talk about my question. Example: Takes a dataset for 10 years of 25 share ...
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3answers
94 views

Correlation between four (more) variables

If A, B, C and D are four variables, the correlation matrix will be 4x4 matrix with elements, $$ \left[ \begin{matrix} 1 & Corr(A,B) & Corr(A,C) & Corr(A,D) \\ Corr(B,A) & 1 & ...
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2answers
69 views

In PCA, what is the connection between explained variance and squared error?

For an observations matrix $X$, I use PCA to reduce the dimension of the data to $L$. I know that in this case, PCA is guaranteed to minimise the mean reconstruction error. In Wikipedia notation, ...
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41 views

Understanding Autocovariance under Gaussian Random Process

I'm recently been trying to understand time series better,and would really appreciate if someone can show me this: I found this online under a lecture slide by J. McJames of Portland Univ., and I ...
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28 views

How to prove the unbiased and biased estimator of autocovariance function?

These two estimators are commonly referenced as sample autocovariance functions. I'm curious how you're to show the first is an unbiased estimator, while the second is a biased one. And how would ...
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28 views

How to bin correlated data?

Let's say I have 100 data points $\bf{x}$ with covariance matrix $\mathbf{C}$ where non-diagonal elements are non-zero. Now I have a binning operator $\mathbf{P}$ of dimension 10$\times$100, that ...
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1answer
16 views

Approximate estimation of a covariance involving ratios

I have three random variables (T, O, A) that are each approximately normal. These three random variables are combined to form two ratios: X=T/A and Y=O/A. I wish to get an estimate of the covariance ...
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1answer
36 views

Apparent Contradiction in Inverting Linear Regression

I believe this is a basic question, but I still can't seem to figure it out. In linear regression, if the standard deviation of the x-data and the y-data is 1, the slope of the best fit line is the ...
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1answer
20 views

adjust a non normally distributed numerical variable for a categorical covariate

Length of hospital stay was measured in 3 groups of patients (group 1,2,3) the length of stay is numerical continuous e.g. 4 days of 6 days one of the other variables is a confounder measured as a ...
3
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1answer
158 views

ML covariance estimation from Expectation-Maximization with missing data

Assuming a multivariate normal distribution with missing data, is there a straightforward way to find the maximum likelihood estimate for covariance using an Expectation-Maximization algorithm? NOTE: ...
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1answer
34 views

Compare covariance ratio

I am using a nonparametric item response theory method called Mokken scale analysis. One of the coefficients that is calculated is called the item scalability coefficient, and it is calculated as the ...
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1answer
25 views

Reducing variance of Monte Carlo estimates via a known relationship

Here's a simplification of the actual problem, but I believe it captures the essence of it. Say we are interested in estimating the expectation of a stochastic time series. We can simulate the time ...
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22 views

covariance of a normal distribution and a quadratic form [duplicate]

I'm working on this problem and am a little lost, can anyone lend their assistance? $$X\sim N_k(\mu, \Sigma)$$ show $$cov(x, x'Ax)=2\Sigma A\mu$$ I tried ...
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69 views

How can I calculate the root mean square error (RMSE) for two covariance matrices?

I want to compare different methods of estimating the covariance matrices on the basis of RMSE and will recommend having the minimum RMSE. I have a sample of, say, 356 weekly observations of 10 ...
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32 views

error covariance matrices of (vector) MMSE & LMMSE

Consider estimating $X$ based on observation of $Y$, with an estimator $\hat{X}$. The estimation error is $Err\triangleq\hat{X}-X$. If $X$ is a scalar R.V., the expectation of the squared estimation ...