Tagged Questions

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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85 views

Bounds on correlation to ensure covariance matrix is positive definite

UPDATED: I am constructing a correlation matrix for an MA(1) process, which would look something like... $$ C = \left( \begin{array}{cccccccccccccccccc} 1 & \rho & 0 & 0 & 0 & 0 ...
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0answers
20 views

Can the law of total covariance apply to variables from different sample spaces?

Wikipedia says this about the law of total covariance (http://en.wikipedia.org/wiki/Law_of_total_covariance): In probability theory, the law of total covariance,[1] covariance decomposition ...
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1answer
58 views

Robust estimates of the covariance matrix in the big data space

I am trying to compute the robust estimates of the covariance matrix (and also the mean) in the big data space. I am aware of FastMVE and FastMCD (Minimum Covariance Determinant and Minimum Volume ...
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0answers
23 views

Efficient calculation of selected diagonals (bands) of a covariance matrix?

I'm looking for an algorithm that can calculate a select few diagonals of a covariance matrix. Here's the problem: I have an $m\times n$ data matrix $X$ where $m$ is the number of features, which is ...
2
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0answers
23 views

Regression variable conversion

There is a question that I cannot solve. They may be solved by variance and covariance but I couldn't. So I thought there should be another way to solve. Question: A researcher has a sample of 43 ...
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1answer
40 views

Conceptual questions: Variance of a process

Wikepedia, at Variance of Autoregressive model, mentions an expression of variance for an AR(1) process. I am learning signal processing (beginner level) and facing difficulty in understanding some ...
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0answers
23 views

How to find the covariance matrix? [duplicate]

I have some trouble understanding the concept of a covariance matrix. For instance, I'm going over this question that says: assume that we have U1, U2 and U3 as independent zero-mean, unit-variance ...
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0answers
11 views

How to decorrelate residuals in r from the covariance matrix

I fit a geeglm model with clustered data and now I would like to decorrelate the residuals of the model in order to run model diagnostics. I read that if I can obtain the covariance matrix of the ...
2
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1answer
27 views

Covariance for Two Variables when Divided by a Third Variable

In a Bioinformatics article appendix, I found the equation : $$ Cov(X/Z, Y/Z) = Cov(X, Y)E(1/Z^2) + E(X)E(Y)Var(1/Z) $$ when supposing that the random vector $(X, Y)$ with covariance $Cov(X, Y)$ is ...
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0answers
22 views

What is variance and co variance related to time series?

I'm trying to understand the Mahalanobis distance method which makes use of a covariance matrix. However i am not clear about the idea of variance and covariance with respect to time series. And also ...
0
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1answer
15 views

Pretest-Posttest comparison

I am having a hard time doing this on Stata. I have a group of 32 students. All perform a pretest and are scored. Next, half of them are randomized to receiving an intervention and the other half ...
2
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0answers
20 views

asymptotic covariance between mean and standard deviation

I am trying to estimate the asymptotic covariance between mean and standard deviation. I know the following $$\sqrt n \hat \mu \xrightarrow{d}N\left( {\mu ,{\sigma ^2}} \right),\sqrt n \hat \sigma ...
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1answer
58 views

What are the implications of estimating a covariance matrix from a correlated sample?

Given a sample of $n$ independent observations $x_1,...,x_n$ (where $x_i$ are $p$-dimensional column vectors), the $p \times p$ sample covariance matrix is defined as ...
1
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1answer
30 views

Sample covariance mean-corrected vector proof

Prove that $$(n-1)S = X^TX -{1\over{n}}(X^T\vec1)(\vec1^TX) = X^TX-n\vec{\bar x}\vec{\bar x}^T$$ My attempt so far goes like this $$S = {1\over{n-1}}X_m^TX_m$$ Edit: Where $X_m$ is the ...
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0answers
18 views

Is it possible to calculate the covariance btwn data and a subset of the data?

I have a regular and complete time-series vector and have created a subset of this vector based on a particular sampling algorithm (say every 10th value). To calculate the error variance of this ...
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0answers
38 views

Minimum variance for sum of three random variables

I have been working on the following problem: Given you have VarX = 1, VarY = 4, and VarZ = 25, what is the minimum possible variance for the random variable W = X + Y + Z, or min Var(X+Y+Z)? My ...
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1answer
25 views

Multivariate Regression on Multicollinear Categorical Variables

I am working on a dataset with a continuous response (which could be dichotomized), one continuous covariate, and multiple categorical variables. The continuous covariate (weight) is directly ...
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0answers
13 views

Finding ACVF and two random variables

let $X_t = 0.5X_{t-1} + Z_t$ where $Z_t$ ~ $ WN(0,\sigma^2)$ I want to find the ACVF of both $X_t$ and $Z_t$, but I am a little bit confused. Say for $X_t$ $$\gamma(h) = Cov(0.5X_{t-1} + Z_t, ...
0
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1answer
45 views

Covariance of a compound distribution

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 ...
0
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0answers
29 views

Covariance between normalised correlation functions

If I have a set of correlated random variables $\mathbf{X}=\{X_1,\dots,X_n\}$ that have been sampled $N$ times, I can calculate the correlation function for pairs of variables as $$ ...
2
votes
1answer
27 views

Stationary function

I am reading Karl Rasmussen's book on Gaussian processes and in the introductory chapter he has the following statement: ...
2
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0answers
60 views

Variance along the regression line

I have two random variables $X$ and $Y$ of unknown distribution which I sample $n$ times to get a set of $n$ random points. I do simple linear regression on this set and arrive at an equation for a ...
2
votes
1answer
59 views

What does the covariance of a quaternion *mean*?

If I have a set of Euler angles (representing the orientation of an object) and I find the covariance of those angles then I have some intuition that $\sigma^2$ is in units of $\text{rad}^2$ and I can ...
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0answers
6 views

Convergence of sample concentration matrix

I'm interested in the Frobenius or $\infty$ norm convergence rate bound for the sample inverse convariance (concentration) matrix. That is, suppose: $$ Y \sim \mathcal{N}_p\left(0, \Sigma\right) $$ ...
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0answers
14 views

Weighted mean and covariance

I have a variable $O$, depending on two other variables $x$, $y$ as $O=\frac{x}{y}$. I have several measurement (each one with its uncertainty) of $x$ and $y$ and for both of them I have computed a ...
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0answers
13 views

Derive covariance of coefficients in simple linear regression [duplicate]

i just need help showing my work to show that Cov(b0,b1)=-xσ^2(sxx) i know that Cov(b0,b1)=E(b0b1)−E(b0)E(b1)=E(b0b1)−β0β1 and i know that var(b0,b1)=σ^2*(X'X)^-1
2
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1answer
71 views

Identifying the coefficients of a principal component

Suppose that a two-dimensional random variable $X$ has a covariance matrix given by $$ \Sigma = \pmatrix {1 & -2\\ -2 & 4}$$ One of the three linear combinations below corresponds ...
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0answers
21 views

Interpreting the inverse covariance matrix: $S^{-1}x$ and $x^T S^{-1}x$

Let $S$ be the covariance matrix of some data set. $S^{-1}$ is the inverse covariance matrix, also called the precision matrix. Question: In practice, then, what does $S^{-1}x$ mean for a data point ...
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0answers
10 views

What if the variance-covariance matrix of a sum of two random vectors? [duplicate]

If X is a px1 random vector with mean Mu(x) and variance-covariance matrix sigma(x) and if Y is a qx1 random vector with mean Mu(y) and variance-covariance matrix sigma(y). If p=q, what would be the ...
0
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2answers
26 views

Understanding COVARIANCE when using same scale

Assuming that an experiment has 3 variables. Time, Temperature inside, Temperature outside. Also, considering that both the temperatures are measured using the same scale say Degree Celsius. If the ...
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0answers
23 views

How to calculate the lag 1 autocovariance for the difference of two variables from the individual autocovariances of the two variables

Is it possible to calculate the auto-covariance of the difference of two variables, from the auto-covariances of two variables being differenced? I have a situation where: Y=βx x is 3*3 matrix of ...
1
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1answer
53 views

Is it possible to get a covariance matrix of fitted values for a GLM model in R?

I would like to get a covariance matrix of fitted probabilities for a logistic regression model in R. I would like to do this because I want to find the variance of the difference between the two ...
1
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0answers
29 views

Weighted variance of the weighted variance in frequency transition matrix?

Assume I have a $m\times n$ Transition Matrix $A$ with $m$ different observations and $n$ represents a discrete state space. Each column counts the frequency that state $j\in [1...n]$ was visited from ...
0
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0answers
14 views

periodic covariance function

"Gaussian processes can be completely defined by their second-order statistics.Thus, if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the ...
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37 views

Estimate the covariance matrix of a normal distribution if the mean vectors is given by a linear rule

Let $X=(x_1,\ldots,x_n)^\top\in\Bbb{R}^n$ be a random vector that follows a multivariate Gaussian distribution with known mean vector $\mu=(\mu_1,\ldots,\mu_n)^\top\in\Bbb{R}^n$. The covariance matrix ...
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2answers
43 views

Difference between identity and diagonal covariance matrices

thanks in advance for the help. Suppose I am training a linear model. What are the conceptual differences between using a diagonal covariance matrix and the identity? It is clear to me that the ...
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vote
1answer
20 views

Comparing and Interpreting covariances

I had a discussion about covariance recently and it would be nice to hear your feedback about this. Let's say we have a dataset of $n$ samples with $d$ attributes. For simplicity, let's say 3 of ...
2
votes
1answer
43 views

SLR: Variance of a residual

I am having problems calculating the variance of a residual in an SLR setting, ie $\text{var}$$(y_i- \hat{y_i})$. Here is what I have thus far. If $ \hat{y_i}= \hat{\beta_0} + \hat{\beta_1}x_i$ ...
0
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0answers
16 views

How to compare two matrices (A given matrix and a scaled up one)?

I have a matrix with 0.25 million rows and 50 columns. I have scaled up this matrix to 1.5 million rows and 50 columns using a Method A. I would like to measure the quality of the method I have ...
2
votes
1answer
75 views

How to proof relationship between inverse covariance matrix and linear regression coefficients?

Edited: I would like to work out the above relationship, more precisely: Let $(Y_{1}, ..., Y_{m})$ be a zero-mean vector with covariance matrix $\Sigma$, and let $S \subset \{1, ..., m\}.$ The ...
0
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0answers
48 views

Covariance Matrix using Delta Method

I am trying to get the variance-covariance matrix of a transformation using the Delta Method. Originally, I have coefficients (let's call them X1, X2, etc...) and their variance-covariance matrix. ...
3
votes
1answer
39 views

Covariance of $cov(5W_7+6W_9,W_7)$ where $W_t$ is a standard Brownian motion

I'm having trouble deducing the value for the problem in the title. Here is what I have done so far. (Given a standard Brownian motion (BM) $W_t, t\geq0 $ with $W_0 = 0$ and $\sigma^2=1$) The ...
0
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0answers
27 views

Univariate fixed effect Vs Multivariate model -Negative Covariance, positive parameter estimate, but why?

I am trying to compare the results of two models. The first model looks at y with x as a fixed effect. The second looks at the covariance between x and y. Both models have repeated measures for x ...
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0answers
90 views

Variance-covariance matrix

$\DeclareMathOperator{\var}{Var}$ How to compute prediction bands for non-linear regression? In the above link, you have mentioned about the variance-covariance matrix of the estimates. What is the ...
3
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0answers
68 views

Inference of Pearson's rho from distribution perturbation

I would like to infer the correlation between random variables $Q$ and $R$, however, I have access only to the distribution of $Q$ and the distribution of $P=Q+R$. We can see how Pearson's $\rho$ ...
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0answers
19 views

Estimating a vector from a rank-one symmetric matrix plus scaled identity

I have a problem regarding estimating a $M\times 1$ vector from a given $M\times M$ symmetric matrix. The known matrix is a scaled identity matrix with a rank-one update. I have some idea how to ...
1
vote
1answer
21 views

expected value of least squares parameters

I'm having some trouble with this equation from the least squares model. $$E[\parallel \mathcal {\hat {B}} \parallel ^2] = [E\parallel \mathcal {\hat {B}} \parallel] ^2 + trace ( \text {cov} ...
1
vote
1answer
55 views

why is the denominator of the correlation coefficient the SD of X multiplied by SD of Y?

I don't quite understand what is going on in the correlation coefficient formula. In the numerator we have the covariance, and in the denominator we have the standard deviation of variable x ...
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0answers
15 views

Standardizing variables at item or index level: Does it make a difference?

I'm running some multiple group CFA models comparing covariance structure by race/ethnicity and have survey data from 6th, 8th, 10th and 12th graders. My supervisor has told me to combine 6th and 8th ...
2
votes
1answer
155 views

Covariance of a random vector after a linear transformation

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