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.

learn more… | top users | synonyms

0
votes
0answers
7 views

Covariance function and autocovariance function of a fractional Brownian motion?

What is the difference between covariance function and autocovariance function of a fractional Brownian motion? Page 6: Eq 1.6 /1.7 http://www.columbia.edu/~ad3217/fbm/thesisold.pdf Second: How can ...
0
votes
0answers
10 views

Adaptive Kalman Filter for INS/GPS

I am trying to implement the paper Adaptive Kalman Filter for INS/GPS and there are a couple of expressions where the paper says that it comes from the Standard Kalman Filter theory, but I can't see ...
0
votes
0answers
8 views

Whitening transform for the case of two stochastic processes

The classic whitening transform allows us to find a linear transformation for a given random process X, yielding a new process Xw with a unity Covariance matrix. Given an extended problem with two ...
2
votes
0answers
23 views

Can the determinant of the covariance of the union of sets be described as a function of the sum of the determinants of the covariance of subsets

Assume my data can be partitioned into subsets such that all my data $ w= \bigcup_{i \in N} w_i $, where $w_i$ are disjoint and so $ \bigcap_{i \in N} w_i = {\emptyset}$. How can I write the $\det\...
1
vote
0answers
35 views

How does R function summary.glm calculate the covariance matrix for glm model?

I would like to know how the covariance matrix of estimated coefficients is actually calculated. The code uses QR-decomposition and inversion of some sort. I have an idea that it would go something ...
0
votes
0answers
8 views

Modeling correlations between multiple Gaussian processes

I have several gaussian processes and I am looking for a way to model correlations between these GPs. Essentially, the predictions of each GP are dependent on the other GPs. Is there a way to impose ...
0
votes
0answers
6 views

Outputting quartile pairs from covariance/correlation matrix

Assume I have a mid-sized covariance/correlation matrix of, say 100x100. Obviously one way to deal with this amount of data is to, for example, output a heatmap. But a 100x100 heatmap is still a bit ...
0
votes
0answers
20 views

Using log returns with pearson's correlation, sensible?

I recently stumbled across the very sensible sounding concept of applying log-returns to financial data (https://mathbabe.org/2011/08/30/why-log-returns/). Being a bit of a stats newbie, I'm not ...
0
votes
0answers
12 views

Can you sum the Standard Errors in mm across related studies?

So, I have read three papers: One is describing a scanning procedure of a liver. Current scanning techniques vary by a Standard Error of 0,7mm. A voxel might vary by this magnitude, indicating that a ...
3
votes
0answers
10 views

Distance for Clusterization: Similarity Measure via Covariance Structure

I work with time series - vectors of length of $10^6$ real numbers. I have a lot of these vectors and use some algorithms that have $O(n^2)$ time complexity (n - number of samples), so if I will try ...
0
votes
1answer
27 views

Covariance between two random variables with different number of values?

Let's say I have a RV $X$ with values $100$, $200$ and their associated probabilities, and some RV $Y$ with values $35$, $47$ and $862$ with associated probabilities. What does it even mean to find ...
5
votes
1answer
110 views

Understanding that $\operatorname{COV}(X,X) = \operatorname{VAR}(X)$ intuitively

I just saw this question and the wonderful accepted answer in this forum. I was then triggered to try understanding intuitively why division of $S_xS_y$ is normalizing the covariance: $$\frac{\...
2
votes
0answers
18 views

Intuition for why $\left|\sigma_{XY}\right|\leq \sigma_X \sigma_Y$ (|covariance| $\leq$ product of standard deviations)?

The population correlation coefficient is defined as $$\rho=\frac{\sigma_{XY}}{\sigma_X \sigma_Y}.$$ I was looking for an intuitive explanation for why $\rho\in[-1,1]$. Or equivalently, why $$\left|...
9
votes
2answers
332 views

Are a sum and a product of two covariance matrices also a covariance matrix?

Suppose I have covariance matrices $X$ and $Y$. Which of these options are then also covariance matrices? $X+Y$ $X^2$ $XY$ I have a bit of trouble understanding what exactly is needed for ...
0
votes
0answers
15 views

What is the conceptual difference between the marginal variance and the range in the Matérn covariance?

The Matérn covariance kernel is given by: $$ C_\nu(d) = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{p}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{p}\right) $$ My question is, what ...
1
vote
0answers
15 views

SEM: Modification indices suggest covariance between an exogenous variable and an endogenous variable. Is there a way to make this work?

I am fairly new to Structural Equations Modeling and I’m trying to fit a simple path model with observed variables only. My original model is as follows (using code from the R package lavaan): ...
2
votes
1answer
55 views

Intuition (geometric or other) for $Var(X+Y) = Var(X)+Var(Y) + 2 \ Cov(X, Y)$

Is there any way to make sense out of this formula intuitively? I rederived it algebraically (took me a while...), which made me happy because I used to be incapable of doing that kind of stuff, but ...
4
votes
0answers
21 views

Get covariance from conditional covariance for lognormal (and other) observations?

Consider lognormal random variables $X_1$ and $X_2$ with correlation coefficient $ρ$ and a partial observation sample of them of length N, the sample being partial because it only contains occurrences ...
1
vote
0answers
12 views

Expected value of a semi-partial correlation

Say I have 4 random variables. $X^{(1)}$ and $X^{(2)}$ are jointly multivariate normal with mean 0 and covariance $\Sigma_X$, and $Y^{(1)}$ and $Y^{(2)}$ are jointly multivariate normal with mean 0 ...
0
votes
0answers
72 views

factor model and large covariance matrices

Imagine the number of variables $\textbf{p}$ is 5 times larger than the number of observations $\textbf{n}$ and the sample covariance matrix is almost block diagonal (with low off block diagonal ...
0
votes
0answers
9 views

Which average to use when calculateing covariation in one pass?

I want to calculate covariance in one pass through my data set. John Coook has a recipe for one pass calculation of variance. It goes like this: Initialize $M_1 = x_1$ and $S_1 = 0$. For ...
-6
votes
1answer
210 views

How to define the covariance for a finite set of vectors in an inner product space space V? What object is it? [closed]

Motivation: This question is motivated by a type of problems in medical imaging and computer vision as follows: suppose we've a set $A$ of points ("shapes") $\{x_1, ...x_d\} $in a Riemannian ...
0
votes
0answers
8 views

Covariance of GMM-distributed vector

Let $X\sim~GMM(\pi,\theta)$, i.e. $$p(X) = \sum_i \pi_i g(X;\theta_i )$$ where $g(X;\theta_i)$ is a Gaussian density with parameters $\mu_i,\Lambda_i$ and $\sum \pi_i=1$. Is there a formula for the ...
1
vote
1answer
11 views

R- is it safe to use the package sandwich for instrumental variable or GMM models?

Both of the following approaches should lead to the same results in my opinion. This is a modified example from ?ivreg where I wanted to use ...
2
votes
0answers
37 views

Generating samples from high-dimensional multivariate Gaussian with few training samples

Say I have a $n\times d$ dataset $D$ where $n\ll d$ ($n$ number of observations, $d$ number of dimensions). Currently, if I want $m$ samples from $D$ assuming it is multivariate Gaussian, I can do ...
3
votes
1answer
41 views

Covariance matrix through bootstrapping - close to zero determinant

I have a set of 260 sets of measurements (for each set of measurements there is an amplitude measured as a function of 8 radii). Since I do not get measurement errors and I am interested in the ...
3
votes
2answers
102 views

Are there ways in which perfect correlation (i.e., $\text{Cov}(X,Y) = 1$) can still be a deceiving statistic?

Are there ways in which $\text{Cov}(X,Y) = 1$ can hold and yet "something surprising" can still also be true? For example: it is possible for the mean depth of a pool to be only $10 \text{ ft}$, yet ...
1
vote
0answers
58 views

Euclidean distance between the unbiased covariance estimator and the true covariance

Let X be a n*p matrix whose p columns are sampled from a centred normal multivariate distribution with true covariance matrix $\Sigma$ and let $\Sigma_{n}$ be its unbiased sample covariance estimator ...
4
votes
1answer
74 views

Covariance of two variables that are products of shared random variables

How to analytically express cov(X,Y), when: X=C*A/(A+B) and Y=C*B/(A+B) Here C, A and B are independent variables with ...
0
votes
0answers
13 views

How to make a covariance matrix from multiple observations of different objects?

I have $N$ objects. From each object, I sample $M$ values $(x,y)$ like so: ...
0
votes
0answers
12 views

plots of covariance with jackknifed confidence intervals in OPLS

The SIMCA-P software is able to produce plots of covariances with jackknifed confidence intervals for OPLSDA models. Is there any R package implementing the same feature? Or is there other means of ...
0
votes
1answer
44 views

Covariance and correlation in multivariate random variables

I have this experiment where there are two random vectors $P_1 = (x_1,y_1)$ and $P_2 = (x_2,y_2)$. These two vectors represents two measurements for the location of two nearby points ($10$ meters ...
0
votes
0answers
23 views

Covariance matrix of complex random variables

Let's say we have datapoints $x_i \in \mathbb{C}^N$, like this: $x_1 = \begin{pmatrix}z_1 & z_2 & ... & z_N \end{pmatrix}^T$ and $x_2=\begin{pmatrix}z_1 & z_2 & ... & z_N \end{...
3
votes
1answer
41 views

How to prove that $Cov(\hat{\beta},\bar{Y}) = 0 $ using given covarience properties

To quote: It is well known that, if $W_1, ..., W_n, Z_1, ..., Z_m$ are random variables and $a_1, ..., a_n, b_1, ..., b_m$ are constants, then $Cov ( \sum_{i=1}^n a_iW_i, \sum_{j=1}^m b_jZ_j) = \...
0
votes
0answers
19 views

Error propagation with correlation

I have two different multidimensional objects in two different conditions, therefore four different vectors of observations: $$V1_{cond1} = v1_{1,1},v1_{1,2},\dots , v1_{1,n}$$ $$V1_{cond2} = v1_{2,1},...
0
votes
2answers
30 views

Co-Variance of Zero and Non-Zero mean random variables

Is the value of co-variance function for non-zero mean random variables different from the value of co-variance function when random variables have a zero-mean? I think yes based on this: For a ...
6
votes
2answers
117 views

Covariance of order statistics

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Note that I'm relatively new to stack exchange and I have already posted this on math....
0
votes
2answers
27 views

What is the variance of the sum of Yi's

Seems a simple enough question, and I presume that, if Yi are normally distributed, Var(Sum(Yi)) = Sum(Var(Yi)) This feels like I'm jumping to the wrong conclusion though. Any help would be ...
1
vote
1answer
117 views

What is the difference between the anti-image covariance and the anti-image correlation?

What is the difference between the anti-image covariance and the anti-image correlation? How are the matrices of these coefficients computed, and what is the meaning of their elements?
2
votes
1answer
27 views

Prove that $Var(\hat {Y_i})=\sigma^2h_{ii}$

I just got a simple question. In general linear model, we have $$\hat Y=HY$$ where $H=X(X^TX)^{-1}X^T$ and the residual $$E=Y-\hat Y.$$ Now I want to prove that $$Var(\hat {Y_i})=\sigma^2h_{ii}$$ ...
3
votes
2answers
37 views

$c(n)$ is trend, $r(n)$ is fluctuation. Should $\text{cov}[c(n),r(n)]/\text{var}[r(n)]$ be close to zero?

Suppose $y(n)$ is a random time series given as function of the discrete-time variable $n$. Suppose we can decompose it into $y(n) = c(n) + r(n)$, where $r(n)$ is a strict stationary residual ...
0
votes
0answers
13 views

Using Covariance and Correlation for Similarity matching

I am trying to find if a particular pattern exists in a time series. I have found that I could try using Covariance or correlation for the task. I have used a sliding window technique for doing this. ...
1
vote
0answers
38 views

Is there any point to Reverse Engineering the Fisher Information Matrix from an Inverse Covariance Matrix?

Would there be any advantage in deriving a Fisher Information Matrix backwards from an inverse covariance matrix? I've discovered that this is much easier to do on the SQL Server platform I use than ...
0
votes
0answers
8 views

longitudinal correlations matrices (structural covariance)

I am interested in doing structural covariance analyses in a longitudinal manner. In structural covariance analyses in our field, we correlate grey matter volumes of various regions of the brain with ...
0
votes
0answers
4 views

Calculate covariance of slow and fast variables

Say you have two time series $X_t$ and $Y_t$ where $X_t$ is given by an $AR(1)$-process and $Y_t$ is a deterministic function of $X_t$: $$Y_t = f(X_t).$$ Also assume that the fluctuations of (the ...
1
vote
0answers
42 views

What's the relationship between covariance, shared variance, and common variance?

I've generally assumed that shared variance and common variance were the same thing. However, here it is written that "Common variance is the realm of total collinearity. On the other hand, the term "...
0
votes
0answers
7 views

Transformations and covariances

I am looking for some potential results regarding the relationship between the covariance of $X,Y$ versus the covariance of $T_1(X),T_2(Y)$, where $T_i$ is some map. I know the relation when $T_i$ is ...
1
vote
0answers
88 views

Why do we batch normalize the pre-synaptic values rather than the activations when using batch normalization?

I'm trying to make sense of this batch normalization (1) paper, in Section 3.2, it says We could have also normalized the layer inputs u, but since u is likely the output of another nonlinearity,...
1
vote
0answers
42 views

Splitting up the variance of Z for Z = X*Y

$Z$ is a function of two dependent random variables, e.g. $X \cdot Y$. Here it is shown that $$var(Z) = var(XY)=(cov(X^2,Y^2)+E[X^2]E[Y^2])-(cov(X,Y)+E[X]E[Y])^2$$ I am interested in a metric that ...
1
vote
0answers
23 views

Covariances of a random variable and its subset

Suppose event B is a subset of event A. $P(A) = p$ and $P(B) = q$. What is the Covariance of indicator functions from A and B. $$ \mathbb{1}_{A}(\omega) = \begin{cases} 1 & \omega \in A\\ 0 &...