Questions tagged [correlation]

For questions about correlation of two random variables. Correlation is a statistical technique that can show whether and how strongly pairs of variables are related. Use it with [tag: random-variables] and [tag: probability].

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7 views

Completing a positive semidefinite matrix; filling in the blanks

I have a (very) partially known correlation matrix. Is there an easy way to fill in the blanks and get a) the/a positive semidefinite matrix with the maximal sum of the covariances, and b) the/a ...
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How does r and $r^{2}$ both describe the strength of correlations of variables? [closed]

I am currently going through Witte Statistics 11th edition and have come to an error in either the book or my understanding. Earlier in the book it says r gives the strength of correlations and $r^{2}$...
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Is there a general method for generating correlated random variables from independent random variables?

For generating correlated normal random variables from independent normals, I know that you can use Cholesky/SVD. Is there a general method that applies for other random variables, e.g., uniformly ...
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Range of correlation between two random variables

Suppose $X, Y, Z$ have mean zero and variance $1$. The correlation between each pair of random variables is $\rho$. What is the range of $\rho$. I have solve this problem in several ways: (1) $$ E[(X^...
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Prove the equations for variance over the conditional-expectation

If $a = \frac{cov(X,Y)}{var(Y)}$ and b = $E(Y) - \frac{cov(X,Y)}{var(X)}\cdot E(X)$ prove the equation: $\frac{var(E(Y|X))}{var(Y)} = (corr(X,Y))^2$ For the general case of E(Y|X) prove: $\frac{var(E(...
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Relationship between conditional probability and co-occurrence and cross correlation

What is the relationship between conditional probability and co-occurrence and cross correlation? I have this example dataset of daily number of occurrences of events A and B: A alone = 100 B alone = ...
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1answer
43 views

Given the following joint density function; find the expectation of $h(x,y)=2x+5y$

Let $f(x,y) = c(2x+y)$ ; $0<x<2$; and $0<y<3$ and $0$ otherwise Calculate: $(i)$ Value of $c$ $(ii)$ Obtain Marginal PDF's of both $X$ and $Y$ $(iii)$ find the expectation of $h(x,y)=2x+5y$...
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Is there a range or minimum value of correlation dimension to determine chaotic or not?

I am trying to understand how to test if an observed time series is chaotic or not. Matlab's correlation dimension is one such test. However, reading the papers and the documentation it is not clear ...
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Finding the correlation function of the output process at zero and checking if it is WSS

Assume two systems for which the following differential equations hold between their input and output signals. $$a \dfrac{dv(t)}{dt}+b v(t)=x(t)$$ $$\dfrac{dy(t)}{dt}=v(t)u(t)$$ Also, assume that the ...
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1answer
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Two correlated AR(1) series [closed]

I have two AR(1) series that are correlated. $$X_{t,1} = \rho_1X_{t-1,1} + e_{t,1}$$ $$X_{t,2} = \rho_2X_{t-1,2} + e_{t,2}$$ and $corr(X_{t,1}, X_{t,2}) = \rho.$ I want to generate at each time $t$ a ...
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50 views

Correlation between union of correlated Bernoulli processes

Let $X_{1,1},X_{1,2},X_{2,1},X_{2,2}$ be identically distributed r.v.s with distribution $\sim Be(p)$ and equally par-wise correlated, with pair-wise Pearson correlation coefficient $\rho$, e.g. $Corr[...
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1answer
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Correlation between 2 independent Brownian motions

I have been playing with some Brownian motion simulations (Euler scheme), when I discovered some strange behavior. When simulation 2 classical Brownian motions, the (Pearson) correlation between them ...
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58 views

Dudley's integral convergence for stationary gaussian process

I am now studying "Gaussian processes" course at my university and i've faced a difficult problem for me: Imagine we have a Gaussian stationary process with expectation equal to zero, ...
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15 views

Is a pearson correlation matrix Symmetric Positive Definite?

Let's suppose I have a table of size m x n. Is a correlation matrix SPD? Why / Why not? (By correlation matrix I mean a matrix M of size mxm where M(i,j) = Pearson correlation between i-th column and ...
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Probability that random set is in an up-set?

Let $X$ be a finite ground set and $\mathcal{A}\subset \mathcal{P}(X)$ be an up-set. That is, a family of sets such that if $A \in \mathcal{A}$ and $A \subseteq B$, then $B\in \mathcal{A}$. Now let $...
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Why is the scatter plot of two uncorrelated random variables in circle shape?

I am trying to understand why the scatter plot of two random variables $X_1$ and $X_2$ is in a circle shape when $corr(X,Y)=0$. Is there any intuitive explanation for this?
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Consider $n$ random variables with the same pairwise correlation $\rho$. What is the range of $\rho$?

Correlation coefficient can be interpreted as the cosine of the angle, $\theta$ between the centered random variables in a vector space. When $\cos \theta = \rho = 1$, we have $\theta = 0$. When $\cos ...
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How good do two datasets agree?

Short story: I have two vectors (random variables) $A$ and $B$ and want to compute how much they agree. Each entry is either 1 or 0. I would like to compare how often they agree to how often we would ...
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Why square of pearson correlation does not match with r2 score?

As per link of correlation and R2 , it is mathematically shown that square of pearson correlation is equal to r2. However, I am trying to replicate these results with my data in python and I do not ...
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Correlation Coefficient of two Order Statistics [duplicate]

My problem is exactly the same as asked in here with a change in the notation of the two order statistics. Reframing the question: If $\left(X_1,X_2,…,X_n\right)$ are a random sample from Uniform(0,1)...
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If $\mathbb{E}(X_n)=\mu$ and $\mbox{var}(X_n)=\sigma^2$, show that the correlation between $X_n$ and $X_m$ converges to 1 as $m,n\rightarrow\infty$.

a) Let $(X_n : n \ge 1)$ be a sequence of random variables which converges in mean square. Show that $\mathbb{E}([X_n-X_m]^2)\rightarrow 0$ as $m,n\rightarrow \infty$. b) If $\mathbb{E}(X_n)=\mu$ and $...
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Why is there no change to the correlation coefficient and gradient?

For the question below, I know there is no change to the Pearson product-moment correlation coefficient after the $x$ values increase by 5 and the $y$ values decrease by 4, and I know that the ...
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1answer
27 views

Correlation of IID normal RV's

If we know that $corr(X,Y)=\rho$ and $Y,Z$ are independent and also $X,Y,Z\sim N(0,1)$ then can we deduce anything about $corr(X,Z)$?
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How to calculate a sports team cohesion by how often a team plays together

I’m not sure where to start with this so here goes: I’m researching cohesion within a team environment. Using soccer as an example. There are 20 teams in a league. Each team has a squad of 30 players ...
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Correlations estimator is invariant by space and location

I need to check that Is invariant by space and location. Do not even know where or how to start exactly.
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Can correlation of two merged dataset be bigger than correlation of each?

if I have data set (x1, y1) with 500 data points and correlation 0.05 and (x2,y2) with the same number of data points and correlation, can cor(x,y) = 0.1 where x and y are result of union of two ...
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Is it correct to compute the coefficient of correlation from fiducials/mean into Fisher's formalism

A simple question : In a Fisher's formalism context, I have to know the correlation coefficient $\rho_{xy}$ of 2 random variables $X$ and $Y$. My code generates 5 values for $\sigma_x$ and $\sigma_y$ :...
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Sum of squares of errors identity through correlation coefficient

Let $X $ and $ Y $ be random variables with standard deviations $\sigma_X $ and $\sigma_Y$. Consider the linear model $Y_i=\beta_0+\beta_1 x_i +\epsilon_i$ where $\epsilon$~$N(0,\sigma$). If the ...
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1answer
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Finding correlation between U and W

Two investments $X$ and $Y$ give returns as follows (expectation and variance): $E(X)=0.5$,$E(Y)=0.4$ and $V(X)=2$, $V(Y)=1$. The correlation between X and Y is ρ (X,Y) = 0.6. (a) Find the ...
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51 views

Expression of multiple covariance terms in the Delta method for a 2D function

I am looking for the expression of variance of a 2D function which is is actually $f(x,y) =\dfrac{x}{y}$. Let's do a Taylor expansion for the 2 function of 2 variables $X$ and $Y$ : $$ \begin{array}{c}...
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State a relationship about this correlation

I have a set of test scores with X being test #1 and Y being test #2. After doing all the calculations to find the correlation I got .35 (which I know is correct). The question is what is the ...
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Independence between $2$ Pearson Correlation Coefficients

Suppose that $X_1, X_2, Y$ are three scalar Gaussian distributed variables. Suppose we know that $X_1$ independent from $X_2$. However, $X_1, Y$ could be correlated. Similarly, $X_2, Y$ could be ...
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Order statistics: Distribution of order statistics for dependent variates

Suppose that we have a sequence of $n$ i.i.d. random variates $X_1, X_2, ..., X_n$ with cdf $F_X(x)$ and pdf $f_X(x)$. Now define $Y$ as the sum of $k$ consecutive realizations of $X$ $Y_i = \sum_{j=i}...
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Best way to determine cross-correlation between two time series of a source in two different spectral regions

I am working with x-ray and ultraviolent time series of astronomical sources, and want to study the correlation between the two for any given source. Namely, I want to examine whether there is any ...
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How is the correlation coefficient computed for time-series cross-correlation?

I'm interested in writing an algorithm to compute the cross-correlation of two time-series. So, something like this. The series in question are either Gaussian noise, or Gaussian noise with a "...
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Confidence interval for $\mu_i$ (multivariate normal): some $\mu_j$ are known, unknown $\sigma^2$ and an estimation of the correlation structure?

Say we have a multivariate normal with m dimensions (let's say that m=1,000), the mean vector $\mu$ is known only for the first 100 elements, but unknown for the remaining 900. We have a single ...
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1answer
23 views

Show that if $x_{t+l}=Ax_t$, then $ \rho(l)=1$ if $A>0$, and $\rho(l)=−1$ if $A<0$.

Given a single stationary series $x_t$ with zero mean and autocovariance function $\gamma(h)$ and autocorrelation function $\rho(h)$ Show that if $x_{t+l}=Ax_t$, then $ \rho(l)=1$ if $A>0$, and $\...
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Covariance/Correlations between circular and linear random variables?

Assume that I have $N$ samples $(x_1,\theta_1),...,(x_N,\theta_N)$ from two scalar random variables $X$ and $\Theta$, where $X$ is defined on a linear space, and $\Theta$ is defined on a circular ...
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Squared of series containing delta function

In spectral density theorem we often found spectral density in a form of series with delta function $$J(\omega)=\sum_{k=0}^\infty \frac{c_k^2}{m_k \omega_k}\delta(\omega-\omega_k)$$ If I want to ...
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Real-world example where two positively correlated pairs (X, Y ) and (Y, Z) of random variables give a negative correlation for (X,Z).

Let X, Y and Z be random variables with finite means and finite variances. Suppose that the correlations ρ(X, Y ) and ρ(Y, Z) have unspecified positive values. Then without additional information on ...
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Calculate $\text{Cov}(\, \, \mid X\mid \, \, , \, \, \mid Y\mid \, \, )$ where $X$ and $Y$ are jointly normal

Let two random variables $X$ and $Y$ are jointly normal as follows: \begin{eqnarray} \left( \begin{array}{c} X \\ Y \end{array} \right) \sim \text{Normal} \left( \left( \begin{array}{c} \mu_x \\ \...
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1answer
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Normally distributed and uncorrelated does not imply independent wiki page

In https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent, it states To say that the pair $(X,Y)$ of random variables has a bivariate normal distribution means ...
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11 views

Generation of spatially correlated Poisson random fields

I know how to generate spatially correlated Gaussian fields using e.g. spectral methods. What I need is to generate Poisson random fields with prescribed spatial correlations $$\operatorname{cor}(N(\...
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1answer
48 views

Is there a good reason to use covariance and not correlation?

Correlation is a normalization of covariance by the standard deviation of each variable. So is there a good reason (and example) when we should (and have) to use <...
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1answer
11 views

Why is the correlation between two data series negative when the top half and bottom half are both strongly correlated?

I have two time series of data variables with each having 111 data points. The data series should be strongly correlated given the nature of what is being explored. However, the correlation between ...
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1answer
31 views

Show the eigenvalues of this submatrix are all less than $1$

Let $C$ be a positive definite correlation matrix partitioned as $$C=\begin{bmatrix} I_{k_1} & A \\ A' & I_{k_2} \end{bmatrix}$$ How can I show that the eigenvalues of $AA'$ are all less than $...
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1answer
19 views

Integration of autocorrelation function when $\int_0^1 f(t) dt = 0$

When $\int_0^1 f(t) dt = 0$ and $f(t)=0$ for $t\in \mathbb{R} \setminus [0,1]$, my conjecture is that $$ \int_{\mathbb{R}} R_{ff}(\tau)\,d\tau = 0, $$ where $$ R_{ff}(\tau) = \int_{\mathbb{R}} f(t+\...
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1answer
45 views

Wiener-Khinchin theorem and fluctuations in flicker noise

Suppose that we have a flicker noise process, that takes values $d(t)\in\{-1,+1\}$. We discretize time into $t\in[0,dt,2dt,\ldots, (N-1)dt]$, and $dt = T/N$ with $T$ being the total length of the '...
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20 views

Time-lagged correlation between two factors

I have the following problem: I have a large dataset of a health facility where every (emergency) call was recorded. Besides this calls, I have other data about the weather, air quality and other ...
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1answer
14 views

Pearson coefficient may not be bounded by 1

I am reading this paper. On the paragraph below Theorem 2.2, it says However, without the normality assumption, Pearson coefficient, ρ, may be problematic. Indeed, as shown in Frechet (1957), ρ may ...

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