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Questions tagged [correlation]

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

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Covariance of assets with different probabilities for each scenario.

There are three assets given and for each asset there are three scenarios with their respective probabilities. Asset 1: $$\begin{array}{c|c|c|} & \text{Return} & \text{Probability} \\ \...
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Ordinal scrambling quantification upon placing and retrieving labeled spheres to and from a cylindrical container.

Trying to derive a formula to quantify the degree to which objects in an original order are scrambled upon some amount of repeated random handling. Say there are $n$ spheres labeled with labels $1$ ...
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What is the name of this Correlation matrix?

$ \left[ {\begin{array}{cc} 1 & p & p^{2} & ... & p^{n}\\ p & 1 & p & ... & p^{n-1}\\ p^{2}&p&1&...&p^{n-2}\\ ...\\ p^n&p^{n-1}&...&...
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Does a square root matrix of a circulant correlation matrix with positive entries also have all positive entries?

I have a circulant correlation matrix that has only positive entries. (Because it is a correlation matrix, it is symmetric with diagonal entries of 1.) I am wondering about the entries of the square ...
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Bound for type of correlation measure

Assume you have a finite, discrete probability distribution for a joint random variable and such that $P(X=i,Y=j) = p_{i,j}$ for $i \in \{1, \dots, |X|\},j \in \{1, \dots, |Y|\}$. The marginal ...
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Relationship Between Independence, Covariance, and Correlation

Can someone verify if the conditional/bi-conditional relationships below are correct? $x \perp\!\!\!\perp y \Rightarrow E[XY]=E[X]E[Y] \iff COV(X,Y)=0\iff X,Y uncorrelated$
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Cross correlation coefficient for 2 matrices (two 2d arrays)

I'm trying to manually calculate cross correlation coefficient for 2 matrices. But I'm not sure how to apply the formula and how it works. Here are my example matrices: Matrix 1 Matrix 2 Any help ...
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20 views

Correlation between 2 signals of uneven dimensions

As a part of my work, I am trying to correlate the audio signal in a video with the pixels of each frame. The steps I follow are: 1] Audio sampling rate and frame rate of the video are known. So, ...
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29 views

Autocorrelation in a Brownian model

I have the following brownian model: $$ \dot{x}=v_0cos(\theta(t))+\sqrt{2D_t}\xi_x(t) \\ \dot{y}=v_0sin(\theta(t))+\sqrt{2D_t}\xi_y(t) \\ \dot{\theta}=\sqrt{2D_r}\xi_\theta(t)\\ $$ with $v_0$ ...
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25 views

Correlated Chance Constraint

If we want to satisfy the following constraint with the probability of 1-$\epsilon$: $$ \mathcal{P}(a \leq x) \ge 1 - \epsilon $$ where $x$ is the decision variable, $a$ is a random variable with a ...
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Linear fit for correlated data [migrated]

Assume I have a set of data points $(x_i,y_i)$ that I want to fit to some model function $f$ which depends on a couple of fit parameters, i.e. $y=f(x;a_1,a_2, ...)$. Usually I'd define a "chi-square" ...
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Using Blomqvist beta as a correlation test

I am currently writing my medical thesis (MD) about the topic "A pilot study:training and recovery monitoring in swimming using artificial neural network geometric optimisation". I am doing it ...
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33 views

Relationship between Pearson's Correlation Coefficient and distance of data points from line of best fit

I am trying to clarify the concept of Pearson's correlation coefficient, r. I am aware that correlation coefficient is a measure of the strength of a linear association. But, is it correct to say the ...
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41 views

Joint Distribution of two dependent Bernoulli Random Variables for $\rho=1$

Say we have two random variables $X\sim B(p_1),\ Y\sim B(p_2)$ where $B(p)=$ Bernoulli with probability $0\le p\le 1$. I am interested in the case when the correlation $\rho$ of $X,Y$ tends to $1$. ...
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Definition of correlation functions

The correlation of two quantities $A(t)$ and $B(t)$ is usually given as $\left< A(t)B(t')\right>$, not specifying what one is supposed to integrate over. My first guess would be to integrate ...
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Equation of an exponential correlation from coords on a scatter plot

How can I find the equation of an exponential correlation from coordinates on a scatter plot? I can't find this anywhere but I'm probably just not using the appropriate terms. For example, how to ...
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Simplifying an equation with covariances of random vectors

Let $I=\begin{pmatrix}I_1\\\vdots\\ I_n \end{pmatrix}$ and $X=\begin{pmatrix}X_1\\\vdots\\ X_p \end{pmatrix}$ be two random vectors and $\Omega_I$ a random variable. I am looking for A such that: $$\...
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how to use correlation function main definition?

I'm enrolled in a course off control where the formula for correlation was presented to us without any context . $RX(t1,t2) = \int\int x1x2 fX(t1)X(t2)(x1,x2)dx1dx2$ the problema than I'm facing is ...
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Is there any relationship between efficiency and correlation coefficient?

Let $t_1$ be the most efficient estimator and $t_2$ be the less efficient estimator with efficiency $e$ and let $r$ be correlation coefficient between the two estimator $t_1$ and $t_2$.Define ...
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For a set of functions, can we create a function with a maximal average correlation?

Let $F = \{F_1, F_2, \dots, F_n\}$ be a set of functions of the form $F_i: S \to \mathbb R$. Additionally, let $X$ be a random variable with state space $S$. The average correlation of $f: S \to \...
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Correlation distance

I'm computing the correlation distance with Matlab (https://de.mathworks.com/help/stats/pdist2.html#d120e570586). The purpose is to calculate the distance between two histograms. My source says, that ...
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Why is cross-correlation not defined in a normalized sense?

When correlation is defined in systems and signals, as well as in the statistical sense, it is defined as a normalized measure with respect to the Cauchy-Schwarz inequality. $\space$ In systems and ...
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63 views

Kendall's Tau of bivariate normal

Prove Kendall's tau of a bivariate normal is given by $$\rho_\tau (X_1,X_2)=\frac{2}{\pi}\arcsin\rho$$ I can derive the bivariate normal as $$F(x_1,x_2)=\frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{1}{2}...
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How can we study the correlation of two variables showing spatial and/or temporal autocorrelation?

Option 1: Can we eliminate the autocorrealtion of each variable, and then study the correlation? If so, how? Option 2: Whether are there methods that can directly model the correlation of the two ...
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Linear correlation between Distance and an Acceleration statistic

A soccer team has equipped themselves with gps & accelerometer devices. All the games they have played were used in this analysis. When plotting total distance vs the statistic called PlayerLoad, ...
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28 views

Correlation Coefficient r, Formula Explained Intuitively

I've seen several videos, Khan Academy included, explaining the correlation coefficient formula but none explain the "logic" behind the formula, not to my satisfaction anyways. The Formula: ...
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Expectation computation of correlated normal random variables

I'm struggling to prove the following maybe some of you can help me. Here is my problem Let $\mathbf{X}\sim\mathcal{N}(\mu,\Sigma)$ where $\mathbf{X}=\pmatrix{X_1\\X_2}, \mu=\pmatrix{\mu_1\\\mu_2}$ ...
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Prove $P(\{X\ge t\}\cup\{Y\ge t\})\le t^{-2}(1+\sqrt{1-r^2})$ where $r$ is the correlation of $X$ and $Y$ centered with unit variance

Let $\xi$ and $\eta$ be random variables with variances $\mathbb{D}\xi$ and $\mathbb{D}\eta$, and correlation coefficient $\rho$. Show that $$\mathbb{P}(\{\xi - \mathbb{E}\xi \geq \varepsilon \sqrt{\...
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Proving $\mathbb{E}\max\{\xi^2,\eta^2\}\leq 1 + \sqrt{1-\rho^2}$ [duplicate]

Let $\xi$ and $\eta$ be random variables and $\mathbb{E}\xi=\mathbb{E}\eta=0$ also $\mathbb{D}\xi=\mathbb{D}\eta=1$. Here $\rho=\rho(\xi,\eta)$ is a correlation coefficient. Need to show $$\mathbb{E}\...
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Proving the correlation $\rho$ is always $|\rho|\leq 1$ and finding when $\rho=-1$

Let $\xi$ and $\eta$ be random variables. $\mathbb{D}\xi >0$ and $\mathbb{D}\eta>0$ Let $\rho=\rho(\xi,\eta)=\frac{\mathbb{E}(\xi-\mathbb{E}\xi)(\eta-\mathbb{E}\eta)} {\sqrt{\mathbb{D}\xi \...
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Correlation of function of two random variables after resampling one

Consider a measurable function $g:\mathbb{R}^2\rightarrow [0,+\infty)^2$ that satisfies that for all $(x,y)\in\mathbb{R}^2$ at least one coordinate of $g(x,y)$ is 0. Call its coordinates $g_1$ and $...
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Autocorrelation of Geometric Process

I have this scenario where $S[n] = 0$ if an email does not arrive and $S[n] = 1$ if an email arrives in the $n^{th}$ minute. Therefore, $S[n]$ follows a Bernoulli distribution. Now $U[m]$ is the ...
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1answer
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Autocorrelation when $m = 0$

I am trying to understand something with respect to autocorrelation. If the process is iid, I can say: $R_{XX}[n, n + m] = E[x[n]] \times E[x[n + m]] \quad \textrm{if} \quad m \ne 0$ But if $m= 0$ ...
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Why is a pole-polar relation preserved under mappings?

If $P$ and $p$ are pole and polar with respect to a polarity with matrix $C$; then $P’$ and $p’$, their images under a collineation, will be pole and polar with respect to the polarity with matrix $C’$...
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Auto-correlation function for a time signal

For a time signal $x(t)= \cos^2(2\pi t)$, solve for the Auto-correlation Function. Basically I have been following a guide on how to compute the auto-correlation function and am getting hung-up on ...
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cross correlation of monotonic signal

I have two monotonic signals and I want to calculate the time delay between them. Say for example my signals are: s1 = [1 2 3 4 5] s2 = [2 3 4 5 6] To me these appear to be monotonic signals, with ...
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Spearman rank correlation- level of ranks

I have a question concerning the calculation of the Spearman rank correlation. When calculating the formula you need to subtract the different ranks one from another. Why can we do that? What is ...
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1answer
27 views

How to show Sub-independent Random Variables are uncorrelated.

I want to prove the following: If two RVs $X, Y$ are sub-independent, i.e., $\phi_{X+Y}(t) = \phi_X(t)\phi_Y(t), t\in\mathbb{R}$ then $X, Y$are uncorrelated. Keep $Cov(X,Y) = E(XY)-E(X)E(Y) = 0$ in ...
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A research paper presents an equation for 'correlation distance between between two vectors'. But I cannot find information its derivation.

Hallo Mathematics StackExchange, I currently trying to pick apart a research paper titled 'Accounting for Label Unscertainty in Machine Learning for Detection of Acute Respiratory Distress Syndrome' (...
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1answer
25 views

Correlation and Covariance on Standardized X

I am stuck on the following problem: Let $Z_X$ be the standardized $X$, $Z_X=(X-\mu_X)/\sigma_X$, and let $Z_Y$ be the standardized $Y$, $Z_Y=(Y-\mu_Y)/\sigma_Y$. Show that $Corr(X,Y)=Cov(Z_X,Z_Y)=E(...
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Calculate covariance and correlation

$X$ is to $Ber(1/2)$ and $Y$ is to $N(0,1)$. Assume $X$ is indep to $Y$. We define $Z=X+Y, W=X-Y$. Find $Cov(Z,W)$ and $Corr(Z,W)$. Firstly, we know some information: $E(X)=1/2, E(Y)=0$ $Var(X)=1/...
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A negative correlation property in a random matrix

I am trying to prove the following negative correlation property. (where neither FKG or the BK inequality apply) Any input/idea is much appreciated: Suppose each row of an $n\times n$ matrix is ...
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Addition of correlated probabilities?

Consider a set of random variables $X,Y,Z$. The random variables represent different measurements of an observable $O$ which can e.g., take on three different values $O\in\{1,0,-1\}$. The measurements ...
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39 views

Does negative correlation survive monotone transformation?

Let $X$ and $Y$ be two non-negative random variables and be negative correlated, i.e., $$\mathbb{E}[XY] \leq \mathbb{E}[X]\mathbb{E}[Y].$$ Let $h(\cdot)$ and $g(\cdot)$ be two non-negative, monotone ...
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102 views

Calculate boundries within which $r$ remains above a critical value given correlation coefficient $(x,f(x)) = r$

I'm experimenting with a trading indicator based on the correlation of the closing price of a day and the simple moving average of P periods. When the correlation is above certain threshold, I need to ...
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How to find Cross Correlation of two series over time containing periodic trends?

Considering the data in the series is real time in nature and there are periodic trends within the series , how to do a Cross-Correlation in real time so that each periodic trend can be identified?
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How is the auto-correlation of vectors defined?

Suppose $v$ is an $n$-ary vector with entries from the set $\{0,1\}$ (i.e. a vector of ones and zeros). A paper I am reading defines the "auto-correlation sequences" $$v*v$$ where $*$ denotes the ...
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1answer
52 views

Find variable that is uncorrelated but not independent

I am given PMF of random variable X. P(X=0) = P(x=1) =0.5. Now there is another RV Y such that Y = XZ. I have to find Z independent of X such that X and Y are uncorrelated but not independent. My ...
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Relation between Vector Auto-regressive models and correlation matrix

I am generating a multivariate time series using Vector Autoregressive Models- $$X(t) = AX(t-1) + \epsilon$$ where $X \in R^{n \times 1}$, $A \in R^{n \times n}$ and $\epsilon \in R^{n \times 1}$ is a ...
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2answers
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Confusion in Relationship between regression line slope and covariance

In simple linear regression model between RVs $(X,Y)$, the slope $\hat\beta_1$ is given as $$ \hat\beta_1 = \dfrac{\sum_i^N(x-\overline{x})(y - \overline{y})}{\sum_i^N(x - \overline{x})^2} \tag{1} $$ ...