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
10 views

Covariance of ARMA (2,1)

I am preparing for an exam and need help. Consider the following estimated ARMA(2,1) model, y(t) = 0,05 + 0,83y(t-1) + 0,13y(t-2)- 0,15e(t-1) + e(t): Given the unconditional variance and 1st order ...
0
votes
1answer
16 views

How do you define the structure of a Covariance Matrix

I am reading a paper by Ledoit and Wolf,2001 on Improved Estimation of the Covariance Matrix of Stock Returns. I am a little confused by some points, and will appreciate a broader explanation. The ...
1
vote
0answers
8 views

Fit multiple regression model with pairwise deletion (or on a correlation/covariance matrix) in R

I'm trying to fit a multiple regression model with pairwise deletion in the context of missing data. lm() uses listwise deletion, which I'd prefer not to use in my ...
1
vote
0answers
9 views

slope comparisons given by independant measurements

With the data used in a previous question matplot/ggplot2 ...
3
votes
1answer
60 views

Is $\text{Cov}(|a|,|b|)\geq \text{Cov}(a,b)$?

The above seems intuitively true, (where $|a|$ refers to the absolute value of $a$), but I'm struggling to prove it - would be very grateful for either a proof or a reference.
1
vote
0answers
14 views

Computing covariance between normal and uniform distributions

I think that this question is related: Covariance of a compound distribution, but it's too abstract for me (I believe that what I have to do must be simpler). Here's the given: machines $A$ and $B$ ...
1
vote
0answers
20 views

Testing for structural break in the covariance

I estimated a bivariate VAR(p) model and assume that there exist two covariance regime $\Sigma_1$ for the period 1 to $T_B$ and $\Sigma_2$ for the period $T_B+1$ to $T$. I am now interest in testing ...
3
votes
0answers
60 views

Does a “pruned” i.i.d. multivariate sample behave as the i.i.d. sample?

Let $z_1,\cdots,z_n$ be $n$ points drawn i.i.d. from $\mathbb{C}N_p(0,\Sigma_n)$. The distribution of the covariance $S_n=\frac1n\sum_{i=1}^n z_i z_i^*$ is well known in the limit as $n,p(n)\to\infty$ ...
1
vote
1answer
30 views

How do i show multiplication of covariances?

i am trying to show the following $$Cov(a,b) = \frac {Cov(a,x) * Cov(b,x)} {Var(x)}$$ I am a bit lost on how to expand the numerator term, so i wrote the following $$Cov(a,b) = \frac ...
0
votes
0answers
21 views

Multifactor Covariance Matrix

hanks for taking a look. I am struggling to understand a rather simple concept. I ran a simple linear regression of the form $$A= \alpha+ \beta X + E$$ $$C = \alpha +\beta X + E$$ Then i ...
0
votes
0answers
10 views

On the correlation function of a stationary time series (spectral analysis)

I am following a proof of the following fact of which I do not understand only the last step. I will post it entirely for the sake of completeness but do not hesitate to just look at my question at ...
1
vote
0answers
6 views

Distribution of the maximum vector (over i.i.d. set) and its dot product with the eigenvectors spanning the rest.

Let $z_1,\dots,z_n$ be i.i.d. draws from $N(0,\Sigma)$, where $\Sigma$ is a $p\times p$ matrix. Assume that $p>n$. Suppose (up to re-labeling) that $z_n=\max_i \|z_i\|_2$. Consider the eigenvectors ...
1
vote
0answers
10 views

Distribution of correlation coefficient in compound symmetric covariance model

Suppose $X_i \overset{iid}{\sim} N_m(0, \Sigma)$ for $i=1, \dots, n$ for the case that $$\Sigma = \sigma^2((1-\rho)I_m + \rho \mathcal{1}\mathcal{1}^\prime),$$ where $\mathcal{1} \in \mathbb{R}^m$ is ...
2
votes
2answers
42 views

Covariance of Categorical variables

I know that the following is true for one categorical variable $X \in \{1,...,k\}$ $cov(X=i, X=j) = -p_i p_j$ However, I don't have intuition behind this and cannot find a proof for this. Is there a ...
0
votes
1answer
29 views

estimators with singular covariance matrix

Suppose I have 2 vectors of random variables $\boldsymbol\theta_1 \in \mathbb{R^n}$ and $\boldsymbol\theta_2 \in \mathbb{R^m}$ with asymptotic covariance $\Sigma_1$ and $\Sigma_2$ respectively. I want ...
0
votes
2answers
44 views

Sample covariance matrix and its inverse

Assume we have the sample covariance matrix $S_1 = XX'/k$ which is not positive definite (in fact it is positive semi-definite) and not well conditioned in very large dimension (large $p$, small $k$). ...
0
votes
0answers
11 views

How to find covariance matrix from correlation if mean is not given?

I'm given autocorrelation function of gaussian random process: $$ R_x(\tau) = 3e^{|-\tau/3|} $$ Now I should find covariance matrix. I know the formula and solutions, where $$ C_{xx} = R - E[X]^2 $$ ...
0
votes
0answers
16 views

Interpretation of (diagonalized) inverse covariance matrix

There are several threads here about covariance matrix and inverse covariance matrix interpretation (here, here or here). However, I was wondering how to interpret the inverse covariance matrix (or ...
1
vote
1answer
68 views

When the sample covariance matrix becomes singular

Assume a data set $X$ which contains $k$ iid random vectors of size $p$. Denote by $S$ the sample covariance matrix. Really I have some questions and I need your very appreciated opinions: 1) Ledoit ...
0
votes
3answers
43 views

How to generate a sparse inverse covariance matrix for sampling multivariate Gaussian vectors?

I need to generate a sparse 100x100 precision matrix to sample multivariate Gaussian random vectors using the inverse of it as the covariance matrix. To be a valid precision matrix, the matrix I ...
3
votes
2answers
30 views

Covariance estimation of overlapping time series

I have two series $x_t,z_t$, and compute the differences like $\Delta_h x_t=x_t-x_{t-h}$. What is a good estimator of the covariance of changes? $$Cov[\Delta_h x_t,\Delta_h z_t]$$ The intervals are ...
2
votes
0answers
36 views

Regularization parameter to generate inverse covariance matrix

My data consists of approx. 5 Million binary strings (n) and every string is 2788 characters long. My goal is to find out if position i is correlated with position j. I approximated a covariance ...
2
votes
1answer
38 views

What is the duality relationship between eigensystems of X X' vs X' X?

I want to know the relationship between the eigensystems of two non-negative-definite (covariance-like) matrices. Both are derived from X which is a T-by-K real matrix (wlog say K > T). I avoid ...
1
vote
0answers
33 views

Calculating the covariance of effect sizes for multiple-treatment multiple-enpoint studies

I'm interested in finding the covariance of effect sizes for multiple-treatment multiple-outcome design given the scenario below. Gleser and Olkin (2009) provided the formula for the estimating the ...
0
votes
0answers
5 views

In an analysis of covariance table why do the sums of squares for the treatment, covariate and residual not add up to the total sum of squares?

I am trying to understand why this is the case, since in an analysis of variance table the sums of squares in the rows prior to the total row add up to the total sum of squares?
2
votes
1answer
28 views

GARCH and (non) covariance stationarity

I am trying to model two series of financial returns using a bivariate BEKK GARCH estimation. From the output of the statistical software, I see that covariance stationarity is not ensured in my case ...
4
votes
1answer
62 views

Shrinkage estimation of Efron and Morris (1972)

I read this article: Article1 and which was refined by the second article Article2 that was considered as a generalization of the James-Stein estimator. In article 1 for example, they considered the ...
2
votes
0answers
33 views

Where is the dominated convergence theorem being used?

I am trying to fully understand the proof of a theorem, I only have a problem with the application of the dominated convergence theorem. For the sake of completeness I will upload the whole statement ...
3
votes
1answer
27 views

Covariance of 2 sequences of random variables

We have a random number generator that generates random integers independently uniformly distributed from 1 to 5, inclusive. This random number generator is used to generate a sequence on $n$ ...
2
votes
1answer
37 views

Incremental Gaussian Process Regression

I want to implement an incremental gaussian process regression using a sliding window over the data points which arrives one by one through a stream. Let $d$ denote the dimensionality of the input ...
2
votes
1answer
59 views

Property of the autocovariance function in time series

In the framework of time series analysis Why does $\lim_{n \rightarrow \infty} n^{-1} \sum_{|h| <n} |\gamma(h)| = \lim_{n \rightarrow \infty} 2|\gamma(n)| $? The LHS (left hand side) sequence of ...
5
votes
1answer
53 views
-1
votes
1answer
63 views

General formula for finding covariance of monomials of multivariate random variables

Suppose that we have independent random variables $X_1,X_2,X_3$ which are gaussian multivariate distributed with a mean of zero vector and a diagonal covariance matrix. $X=[X_1,X_2,X_3] \tilde{} N(0, ...
0
votes
2answers
34 views

Notation: covariance on scalars?

I've recently seen the degrees-of-freedom for K-nearest neighbors regression specified like so: $\frac{1}{\sigma^2}\sum_{i=1}^NCov(y_i,\hat{y}_i)$ But what does $Cov(y_i,\hat{y}_i)$ mean? ...
0
votes
0answers
10 views

Co-variance matrix and Factor graph representation

I have problem in fundamental understanding of factor graph representation: I am trying understand the relation between the Co-variance matrix and associated factor graph, of a Normally distributed ...
0
votes
0answers
11 views

binomial error calculated with formula

i have uploaded a 2D histogram ,, actually it is filled by dividing two 2Dhistograms with same bins.i want to calculate manually each error which is shown with each value i.e binContent.Please ...
0
votes
0answers
20 views

Expected Value and Variance of a summation of correlated sample means

Let $\gamma_i$ be constants and $f(\gamma_i)$ be normally distributed random variables. More specifically, $f(\gamma_i)$ is the sample mean of the population $\gamma_i$. Now I have a sum of the ...
3
votes
1answer
77 views

What is the covariance called when it is not divided by N?

I noticed that in signal processing they have this term called cross-covariance. The cross covariance function produces covariances of two functions with different lags. At the center of the vector ...
4
votes
2answers
62 views

Suppose $X_1 X_2, …, X_n$ are $n$ independent variables, is their Covariance matrix, $\Sigma$, diagonal?

Suppose I have $n$ variables $X: X_1, X_2, ..., X_n$ that are independent from each other. Which means that: if $i≠j$, then $\text{Cov}(X_i, X_j) = 0$ As a consequence, I'm wondering if their ...
0
votes
1answer
39 views

How does the covariance matrix of a fit get computed?

I often have to fit data of physical experiments as a student. I always use (python's) functions like numpy.polyfit or scipy.optimize.curve_fit for that purpose. They also allow you to retrieve the ...
1
vote
0answers
19 views

Looking for a principled/systematic procedure for discarding features

I have a collection of $M_i \times N$ matrices $X_i$ whose rows are (raw) feature vectors (from a common $N$-dimensional feature space). MATLAB reports that most of the covariance matrices $C_i := ...
0
votes
1answer
37 views

When is it necessary to estimate the covariance matrix instead of calculating it directly?

The covariance of two random variables $(X_i,X_j)$ is given by $$Cov(X_i,X_j) = E[X_iX_j] - E[X_i]E[X_j],$$ where $E[X_i]$ is the expected value of $X_i$, or its mean. The covariance matrix $\Sigma$, ...
1
vote
1answer
26 views

Auto-covariance in a stationary time series

I am trying to calculate the auto-covariance of time series $Z_t$. Given a weakly stationary process $Y_t$; $t \in \mathbb{Z}$: $$Z_t=a+Y_t$$ Now, my goal is to show that $Z_t$ is also a stationary ...
0
votes
0answers
10 views

Covariance structures and two-way ANOVAs

I have been working with mixed-models and ANOVAs in R for the past year, and am just now getting into the extensive information on covariance structures. Using a mixed-model with repeated measures on ...
2
votes
1answer
90 views

Intuitive explanation for when Pearson correlation coefficient equals 1

Whenever I teach a complicated formula, I always demonstrate to the class that it works intuitively in extreme cases. For example, when all the data values are equal, the standard deviation is 0. Not ...
0
votes
0answers
24 views

On the use of the autocovariance generating function

The autocovariance generating function is defined as: $$g_X(z) = \sum_{h = -\infty}^{\infty} \gamma(h)z^h.$$ Where $\gamma(h)$ is the autocovariance function of the considered process $X$. I can ...
1
vote
1answer
57 views

Is the covariance of standardized variables the correlation?

I have a basic question. Say I have two random variables, $X$ and $Y$. I can standardize them by subtracting the mean and dividing by the standard deviation, i.e., $X_{standardized} = \frac{(X - ...
0
votes
0answers
16 views

Convert principal components based on covariance into principal components based on correlation

I am wondering if there is any way to mathematically express the change in direction of the principal components from the $2\times2$ covariance matrix to the correlation matrix. In other words, if ...
0
votes
1answer
21 views

Marginal Likelihood of a Gaussian Process Model, Duplicate entries in kernel matrix

I am trying to fit a Gaussian process model using the toolbox and I got stuck in the following problem. Assuming that I have some duplicated data points in my training data, then those will map to ...
0
votes
3answers
76 views

Covariance between variables

When looking at correlation, some variables may have a higher correlation but others may have a stronger relationship between them and a lower correlation. What do you do to figure out which variables ...