For questions about correlation of two random variables. Use it with [tag: random-variables] and [tag: probability].

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13
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2answers
13k views

Generating correlated random numbers: Why does Cholesky decomposition work?

Let's say I want to generate correlated random variables. I understand that I can use Cholesky decomposition of the correlation matrix to obtain the correlated values. If $C$ is the correlation ...
11
votes
2answers
34k views

Determining variance from sum of two random correlated variables

I understand that the variance of the sum of two independent normally distributed random variables is the sum of the variances, but how does this change when the two random variables are correlated? ...
6
votes
1answer
508 views

Covariance, covariance operator, and covariance function

I am trying to get my head wrapped around this article in Wikipedia. The first definition given there is the covariance of a probability measure $\mathbf{P}$: $$\mathrm{Cov}(x, y) = \int_{H} \langle ...
6
votes
1answer
211 views

Correlations between neighboring Voronoi cells

For a sequence $X_1,X_2,X_3,\ldots$ of random variables, what it means to say $X_1$ is correlated with $X_2$ is unambiguous. It may be that the bigger $X_1$ is, the bigger $X_2$ is likely to be. If, ...
6
votes
4answers
922 views

Going back from a correlation matrix to the original matrix

I have N sensors which are been sampled M times, so I have an N by M readout matrix. If I want to know the relation and dependencies of these sensors simplest thing is to do a Pearson's correlation ...
6
votes
1answer
166 views

Weighing correlation by sample size

I'm a scholar in the humanities trying to not be a complete idiot about statistics. I have a problem relevant to some philological articles I'm writing. To avoid introducing the obscure technicalities ...
6
votes
3answers
3k views

Correlation Coefficient and Determination Coefficient

I'm really new to linear regression and am trying to teach myself. In my textbook there's a problem that asks why $R^{2}$ in the regression of $Y$ on $X =$ the sample correlation between X and Y the ...
5
votes
3answers
1k views

Is correlation (in some sense) transitive?

If we know that A has some correlation with B ($\rho_{AB}$), and that B has some with C ($\rho_{BC}$), is there something we know to say about the correlation between A and C ($\rho_{AC}$)? Thanks.
5
votes
1answer
10k views

Generate Correlated Normal Random Variables

I know that for the $2$-dimensional case: given a correlation $\rho$ you can generate the first and second values, $ X_1 $ and $X_2$, from the standard normal distribution. Then from there make $X_3$ ...
5
votes
2answers
33 views

correlation between $\sum_{i=1}^{98}X_i$ and $\sum_{i=3}^{100}X_i$

Let $X_1,...,X_{100}$ be iid $N(0,1)$ random variables. The correlation between $\sum\limits_{i=1}^{98}X_i$ and $\sum\limits_{i=3}^{100}X_i$ is equal to (A) $0$ (B) $\dfrac{96}{98}$ (C) ...
5
votes
1answer
644 views

Asymptotic correlation between sample mean and sample median

Suppose $X_1,X_2,\cdots$ are i.i.d. $N(\mu,1)$. Show that the asymptotic correlation between sample mean and sample median (after suitably centering and renormalization) is $\sqrt{\frac{2}{\pi}}$.
5
votes
2answers
840 views

Eigenvalue decomposition of block covariance matrix for Canonical Correlation Analysis (CCA)

Edited: My question is related to a tutorial I was reading. The covariance matrix is a block matrix where $C_{xx}$ and $C_{yy}$ are within-set covariance matrices and $C_{xy} = C_{yx}^T$ are ...
5
votes
1answer
164 views

Find $\operatorname{argmax}_x \operatorname{corr}(Ax, Bx)$ for vector $x$, matrices $A$ and $B$

This is similar to, but not the same as, canonical correlation: For $(n \times m)$ matrices $A$ and $B$, and unit vector $(m \times 1)$ $x$, is there a closed-form solution to maximize the correlation ...
4
votes
1answer
284 views

Uncorrelated but not independent random variables

Is it possible to construct two random variables $X, Y$ both of them assuming exactly two non-zero values which are uncorrelated, i. e. $\mathbf{E}[X \, Y] = \mathbf{E}[X]\,\mathbf{E}[Y]$, but not ...
4
votes
3answers
1k views

Bounds on off-diagonal entries of a correlation matrix

Assume that all the entries of an $n \times n$ correlation matrix which are not on the main diagonal are equal to $q$. Find upper and lower bounds on the possible values of $q$. I know that the ...
4
votes
1answer
216 views

Mean density of the nontrivial zeros of the Riemann zeta function

As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: ...
4
votes
1answer
72 views

PCA vs Correlation

What is the relationship between (first) principal component(s) and the correlation matrix or the average correlation of the data. For example, in an empirical application I observe that the average ...
4
votes
2answers
148 views

Correlation in errors

I'm not good in statistics, so please excuse my noob question. We want to ask a question from people (say what is $2+2$). They might make mistake. We assume that they give the correct answer with the ...
3
votes
4answers
8k views

Correlation between three variables question

I was asked this question regarding correlation recently, and although it seems intuitive, I still haven't worked out the answer satisfactorily. I hope you can help me out with this seemingly simple ...
3
votes
2answers
107 views

Independence of Random Variables (kernel ICA)

In the paper Bach, F. R., & Jordan, M. I. (2002). Kernel Independent Component Analysis. Journal of Machine Learning Research, 3(1), 1-48. doi:10.1162/153244303768966085 I stumpled upon ...
3
votes
3answers
640 views

Correlation between variables

I asked this question on stats SE but did not find a suitable answer so far. Maybe someone can help. Given n random variables x1,...,xn (one-dimensional). The following is known (corr() = Pearson ...
3
votes
2answers
132 views

Correlation of Proportions

To introduce my question, here is a small simplification for consideration: Let $X,Y$ be independent random variates, each with finite mean and variance. Interestingly, ...
3
votes
1answer
640 views

Necessary and sufficient conditions for a matrix to be a valid correlation matrix.

It's not too hard to see that any correlation matrix must have certain properties, such as all entries in the range -1 to 1, symmetric, positive semi-definite (excluding pathological cases like ...
3
votes
3answers
4k views

Correlation between two linear sums of random variables

I understand how to create random variables with a prespecified correlational structure using a Cholsesky decomposition. But I would like to be able to solve the inverse problem: Given random ...
3
votes
2answers
108 views

Covariance$(X,Y) \geq 0$ if $X,Y \geq 0$?

I was wondering if you can say something about the covariance of two positive variables $X$ and $Y$?
3
votes
2answers
161 views

Proving that the magnitude of the sample correlation coefficient is at most $1$

How can you show that the magnitude of the sample correlation coefficient is at most $1$? The formula is huge, I'm not even sure how to approach this. Can anyone point me in the right direction? ...
3
votes
3answers
80 views

Correlation of uniform variables

Let $X$ and $Y$ be independent random variables, $X,Y \sim unif(0,1)$. Let $U = \min \{X,Y\}$ and $V = \max\{X,Y\}$. Find the correlation coefficient of $U$ and $V$. I think we can assume that $U = ...
3
votes
2answers
302 views

Correlation and squared variables

According to my textbooks if two variables are uncorrelated, they are not necessarily independent (unless they are normally distributed). My question is, are 2 variables still not independent if they ...
3
votes
1answer
1k views

quadratic relationship

Detection of linear relationship is possible with correlation coefficient. If absolute value of correlation coefficient is 1, then the relationship is linear. Is there any way for detecting quadratic ...
3
votes
1answer
164 views

Autocorrelation of wrapped Wiener process

Let $\phi(t)$ be a Brownian Walk (Wiener Process), where $\phi\in[0,2\pi)$. As such we work with the variable $z(t)=e^{i\phi(t)}$. I would like to calculate $$E(z(t)z(t+\tau)).$$ This is equal to ...
3
votes
1answer
4k views

What is the standard deviation of multiple correlated random variables subtracted from another random variable?

Wiki states that standard deviation of $X-Y$ is: $$\sigma_{x-y} = \sqrt { \sigma_x^2 + \sigma_y^2 - 2\rho\sigma_x\sigma_y }$$ I have a number (say 3) correlated random variables to be subtracted ...
3
votes
1answer
67 views

Covariance of 1-D random process is $n\times n$!!!!

I'm reading a tutorial on stochastic processes. There is an example in the tutorial as follows: General Moving Average random process given as $X[n]=\frac{(U[n]+U[n-1])}{2}$ where $E[U[n]]=\mu$ ...
3
votes
1answer
112 views

How to curve fit an unknown function?

I have data which can be described by $y=f(x,z)$ where $z$ varies from 170 ~ 154. Now values given by $ks$ are known sample values that equals value given in the table header, $uks$ are unknown ...
3
votes
1answer
57 views

finding the unspecified ${\bf E}[X]$ and $\rm var(X)$ given the expectation of higher powers of $X$

Homework Problem: It is known that a for a standard normal random variable $X$, we have ${\bf E}[X^3]=0$, ${\bf E}[X^4]=3$, ${\bf E}[X^5]=0$, ${\bf E}[X^6]=15$. Find the correlation coefficient ...
3
votes
1answer
487 views

Use Pearson's correlation coefficient on a matrix

I have a problem to interpret the following formula which is said to be the Pearson's correlation coefficient: $$r = \frac{N \left(\sum XY\right) - \left(\sum X\right) \left(\sum ...
3
votes
3answers
219 views

Using Correlation for mouse gesture recognition

I am in need to build a mouse gesture recognition system which will compare given recognition to the the gestures in training data and will say where a given gesture best fits. I am planning to use ...
3
votes
0answers
24 views

Detecting camera shake

I have a bunch of data captured from a worm tracker that consists of a B&W camera that stares down at a few dozen worms for an hour at a time. The tracker captures the outline of each worm ...
3
votes
0answers
163 views

How to perform nonlinear regression with correlated errors?

I have a nonlinear least squares problem, but the errors are correlated. I could use R's nls function to do the regression if the errors were independent, but I don't know the right way to handle ...
3
votes
0answers
1k views

Correlation and Regression Question

Two separate tests are designed to measure a students ability to solve problems. Several students are randomly selected to take both tests and the results are: $$ \begin{matrix} \text{Test A}(x) ...
2
votes
3answers
144 views

Product of standard deviations

If $u=cx+dy$ and $v=cx-dy$ and $R$ is the co-efficient of correlation between variables $x$ and $y$ and variables $u$ and $v$ have 0 correlation, then how can I prove that ...
2
votes
1answer
71 views

Given series $A$ and a correlation, is it possible to randomly calculate a fitting series $B$?

With reference to the original thread on Stackexchange, my question is as follows. Usually, one would enter two value-series and a script or program calculates the correlation. For instance, with $x ...
2
votes
2answers
949 views

Intuitive meaning of Pearson Product-moment correlation coefficient Formula

I can't understand the intuition behind Pearson Product-moment correlation coefficient Formula for bivariate data. The formula is : $\rho$ = cov(X,Y)/($S_x$ * $S_y$) where cov is covariance. $S_x$ and ...
2
votes
1answer
34 views

Comparing ranking algorithms

If I have several different ranking algorithms and a 'correct' ranking, is there a good way of "scoring" the alternative rankings given by the algorithms against the reference one? For example: ...
2
votes
1answer
43 views

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$ is a convex risk measure, but it fails the subadditivity property in order to be called coherent. A mapping ...
2
votes
2answers
308 views

Relationship between variances in perfect correlation

I have two random variables $X$ and $Y$ with mean and standard deviation $(\mu_1,\sigma_1)$ and $(\mu_2,\sigma_2)$ respectively. I know that for perfect correlation the relationship is given by a ...
2
votes
2answers
191 views

Calculating the correlation between subsequent values in a stream of numbers

I have a stream of integer values being generated $V_1,\cdots, V_n$ and want to calculate how the value of $V_{n+1}$ is correlated to $V_n.$ I would also like to calculate this at run time as ...
2
votes
2answers
1k views

How to 'minimize' correlation between series

Hi fellow mathemagicians, let's say that I have 3 series of numerical results (they represent 'drawdowns') : ...
2
votes
2answers
61 views

Generating correlated random variables with discrete distribution

I would like to find a simple way to generate two correlated random variables under the condition that each r.v has a same discrete distribution (for example Bernoulli distribution) This link provides ...
2
votes
1answer
1k views

Maximum and minimum Correlation Coefficient

I have a question regarding the correlation coefficient. The inspiration is from a story where a student collected a set of $(X,Y)$ pairs, but lost the pairings. Hence, he is left with two sets of ...
2
votes
2answers
1k views

help understanding step in derivation of correlation coefficient

I'm looking to understand the starred step in the derivation below (also, if someone could help with the LaTex alignment, I'd appreciate it). The regression line is $y= b_0 + b_1 x$, where $b_0$ and ...