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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Finding the expected value of two correlated RVs

$\newcommand{\Exp}[1]{\mathbb{E}\left[#1\right]}$ I am interested in understanding wether the following approach holds when calculating the expectation of two correlated random variables. Suppose $X\...
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Is there a definition of correlation for multiple random variables?

I know that there is a definition of independence of random variables for finitely many random variables, and this is NOT equivalent to pairwise independence (although it implies pairwise independence)...
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Bounding Correlation

I need to find min/max possible values of $q$ in the following situation. "Consider random variables $x, y$, and $z$. Suppose it is known that $\operatorname{Corr}[x, y]=\operatorname{Corr}[x, z]=...
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Sampling from a Poisson distribution

I am currently working on a thesis for my final year and am stuck with a problem involving the poisson process. I was wondering if someone could help me with it. I am trying to simulate a one ...
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If a correlation matrix is a band matrix with value 1 on the band, is this equivalent to correlation matrix with all 1?

Say I have a stationary sequence ${x_1,x_2,\cdots,x_n}$ and if two elements of the sequence are less than or $m$ apart from each other, we say they are correlated. This is called an m-dependent ...
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Time series analysis on ACF and PACF plots

So I have a non stationary time series that is hourly, daily and monthly recorded for a year and I have the ACF and PACF plots for the serie. ACF and PACF I applied the the Ljung-Box test and for ...
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If there is a high correlation between two variables, should we expect the high linear regression coefficient? [closed]

I have a data set with multiple features, let suppose $x_1,x_2,x_3,x_4$ and my dependent variable is $y$, when I compute the correlation matrix for $y,x_1,x_2,x_3,x_4$ then imagine the correlation ...
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"Leave-on-out Correlation" between Matrices

I'd like to enforce a special constraint in my optimization problem. The solution to my problem is a set of matrices $Q_1, ..., Q_N \in \mathbb{R}^{G \times K}$ and I'd like to make sure that: For ...
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Practical correlation metric for a large number of vectors

I am dealing with a timeseries consisting of input flow sampled every 5 minutes over 441 days. My aim is to find any possible correlation from data coming from: The same day of the week The same ...
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Autocorrelation of a random process

Let X be a random process. X(t) = A*Cos(wt+θ) ; where A and w are constants. The only random thing is θ. Lets say θ has a probability density function, f(θ)= 1/2pi for 0<θ<2pi and zero elsewhere....
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Can the chi-square statistic in Kruskal-Wallis test be compared to determine the most appropriate to distinguish the groups?

I have a dataframe in R which format is similar as follows: v1 v2 v3 group 1 3.5 100 a 3 5 200 a 10 5.5 150 b 8 7.5 210 b 4 4.5 300 c 9 2.5 200 c ... My ...
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Maximum number of random variables to make pairwise correlation coefficient less than C

Assuming N random variables $X_1, X_2, ..., X_N$, each has $m$ samples, (e.g. $m=3$). what is the maximum N such that all pairwise sample correlation coefficients $\rho_{X_i, X_j}$ are less than some ...
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Range of correlation

$A, B, X$ are random variables. If we know $corr(A, X)=0.03, corr(B, X)=0.04, corr(A, B)=0$ What's the range of the correlation between an arbitrary linear combination of $A, B$ and $Y$ i.e. $corr(...
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Joint distribution of correlated gamma random variable

I'm dealing with two correlated random variables $d_1^2 = || r_1 - r_0 ||^2$ and $d_2^2 = ||r_2 - r_0||^2$, where $|| \cdot ||$ is the standard Euclidean distance ($L^2$ norm) and $r_\cdot \sim \...
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Hypothesis testing on sample mean when observation are correlated

How can I test whether the mean of a sample is significantly larger than the population mean, when observations are not independent, but the exact correlation matrix is known? I have a variable $X$ ...
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Let $X,Y,Z$ be three random variables such that the correlation coefficients $\rho_{XY}=0.2, \rho_{YZ}=0.2$, what values can $\rho_{XZ}$ take?

Let $(X,Y,Z)^T$ be jointly normal variable with zero mean such that the correlation coefficients $\rho_{XY}=0.2, \rho_{YZ}=0.2$, what values can $\rho_{XZ}$ take? Prove that there exists a ...
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Can I subtract a correlation factor from another correlation factor to remove noise?

I’m trying to understand how spending on digital advertising affects things like first installs of a product. I have many channels where spending has occurred, but all with wildly different amounts. ...
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Intuition of R-squared in a quantile

I assume, R-squared (correlation of determination) can be used as a measure of the goodness-of-fit in a quantile plot (QQ-plot). In a QQ-plot we measure between empirical quantiles (the ranked sample) ...
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time-shifted signals correlation

imagine you have the 2 following signals: s=[0,0,0,1,1,1,0,0,0,3,3,3,0,0,0] p=[1,1,1] The correlations for each shift (considering only lags for which the shifted ...
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How general is this property about correlation and the sum of two normal RVs?

Edited to make this more concrete: Given a random vector $(X_1,X_2)$ that is jointly normal with means / sd's $\mu_1,\mu_2, \sigma_1,\sigma_2$ and correlation $\rho$, the sum of $S=X_1+X_2$ is ...
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Auto correlation

Consider the signal x(t)=u(t) where u(t)=1(t≥0), i.e. the Heaviside function. Find the signal y(t)=x(t)∗x(−t) My attempt: y(t)=x(t)∗x(−t) =u(t)∗u(−t) =∫∞−∞[u(τ)][u(−t+τ)] dτ since u(τ) exists from 0 ...
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Maximal Correlation of two random variables

I am trying to wrap my head around one problem that involves two identically distributed random variables $X$ and $Y$ with distribution $Bernoulli(p)$ assuming that $P(X=Y=1)=\theta$. It then asks a ...
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Intraclass correlation: Separating intra-rater variability from inter-examination variability.

I have a question regarding Intra-class correlation. Suppose we have 2 MRI-scans (2 trials) made on each of $n$ subjects (on the same MRI-scanner). All MRI-scans are analyzed by the same rater. The ...
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Is there a matrix algebra equivalent of finding the cross correlation between two vectors?

To calculate the cross correlation of two vectors we slide the vectors over each other multiply the corresponding elements and sum them. This is an example for two length 8 vectors giving a length 15 ...
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Joint Density Question

So the problem is $p(x,y) = 120xy(1-x-y)I \{x \geq0,y \geq0,x+y \leq 1 \}$ Now that $Z = Y - E(Y|X)$ What is the correlation coefficient of $Z$ and $X$ So here I First tried to get $E(Z)$, which $E(Z)...
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3 votes
1 answer
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What is the best way to measure similarity between two histograms

What is the best way to measure the similarity between two histograms? For example, in the following pictures, how can I tell if the distributions are similar enough? I now have 2 lists of values, ...
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Correlation on principal component analysis[PCA]

I am analysing a dataset on NBA players, using R. I am having some doubts on correlation in pca. I know the eigenvalues represent the variability in each PCA. Each eigenvalue has a eigenvector. I ...
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Is it possible that the correlation between $\hat{b}$ and $\hat{c}$ can be negative multiple linear regression?

Given the following linear regression model as following, with two explanatory variables $x_1$ and $x_2$ and response $y$ $$y_i=a+bx_{i1}+cx_{i2}+\epsilon_{i}$$ We say that $\hat{a}, \hat{b}, \hat{c}$ ...
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Kendall 's tau in high dimensional setting

Often I see that the Kendall's tau is defined for 2 random variables by using their bivariate copula $C(u_1, u_2)$ I wonder in the case of multidimensional setting (i.e. from 3 dimensions), how the ...
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2 answers
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Why does $H(X|Y)$ equal the "missing information" of $Y$ about $X$?

I've seen mentioned in (Horodecki, Oppenheim, Winter 2005) the fact that the conditional information equals the amount of information that Alice needs to send Bob in order for him to fully reconstruct ...
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Correlation of random variables after multiplication by a constant

Why is the following true for the covariance $cov(X, Y)$ and variances $Var(X)$, $Var(Y)$ of two rvs $X$ and $Y$: $$ \frac{Cov(c X, Y)}{\sqrt{Var(c X) Var(Y)}} = \frac{c Cov(X, Y)}{\sqrt{c^2 Var(X) ...
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Component-wise correlation on hypersphere

Let ($x_1,\dotsc, x_n$) be sampled uniformly from the $(n-1)$-sphere $\{\mathbf{x}\in \mathbb{R}^n\mid \|\mathbf{x}\|=\sqrt{n}\}$ of radius $\sqrt{n}$. I want to show that $\mathbb{E}[x_ix_j]=\delta_{...
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1 vote
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Negative correlation between $\hat{\beta}_1$ and $\hat{\beta}_2$

So today I've seen a lecture regarding multiple linear regression with response $y$, an intercept and two explanatory variables $x_1$ and $x_2$. That is, $$y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\...
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2 votes
1 answer
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Why Null Hypothesis for contingency tables is 'independent'?

For contingency tables I don't understand why the Null Hypothesis is 'independent'. We calculate Sum(Observed^2 / Expected) - N and compare this with the chi-squared distribution table. Let's say: Sum(...
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1 vote
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How to find the autocorrelation of a stochastic process described by a 1st order ODE? [duplicate]

I have the stochastic process given by $$ \dot{x} = ax + v$$ where $v$ is a zero-mean, Gaussian white noise signal. What is the autocorrelation of $x$? I believe the process is to first find the ...
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What is meant by, "both variables follow a normal distribution" in Pearson's product moment correlation coefficient hypothesis tests?

I am teaching myself further maths A Level Year 1 statistics from the OCR book (A). Chapter $5$ is about Correlation and regression. I get that the ppmcc is given by $r,$ which is just a number with $\...
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Indepedent copy of Bivariate Normal

In this question, Correlated joint normal distribution: calculating a probability Most upvoted answer obtained independent copy using this equation, $\pmatrix{U\\V}=\Sigma^{-1/2} \pmatrix{X\\Y}$. I ...
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correlation and covariance of two random variables

I have two continuous random variable X and Y. In U- shape region we have $P_{XY}(X,Y)=\frac{1}{12}$ in other regions we have $P_{XY}(X,Y)=0$. How the correlation and covariance of these two variable ...
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1 vote
1 answer
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Based on multiple daily weather observations, how can I calculate the "most similar" day to any other given day?

This is definitely a case of "not even sure how to ask the question," but I am wondering if there is math available to solve a problem I have. I have several years of daily weather ...
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1 vote
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Inequality of fourier transform for $L^2$ functions

Suppose $f,g$ are $L^2$ functions over $\mathbb R$ such that $\|f\|^2\ge \|g\|^2$ and for any $y\in \mathbb R$, $$ \int_{\mathbb R} f(x) \overline{g(x-y)} dx = 0.$$ If $F,G$ are their respective ...
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1 vote
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majorization of polynomials over the unit circle

Suppose you have two sequences of complex numbers $a_i$ and $b_i$ indexed over the integer numbers such that they are convergent in $l^2$ norm and $a$ has norm greater than $b$ in the sense $$\infty&...
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2 votes
2 answers
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How to calculate median, variance and correlation on two-dimensional random variable. [closed]

Two random variables, X and Y, have the joint density function: $$f(x, y) = \begin{cases} 2 & 0 < x \le y < 1 \\ 0 & ioc\end{cases}$$ Calculate the correlation coefficient between X and ...
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1 answer
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Kendall's tau coefficent of Bivariate Normal [duplicate]

Let the joint distribution of (𝑿, 𝒀) be bivariate normal with mean vector $\begin{pmatrix} 0 \\ 0\end{pmatrix}$ and variance-covariance matrix $\begin{pmatrix} 1 & 𝝆 \\ 𝝆 & 1 \end{pmatrix}$...
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1 vote
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Correlation coefficient when variance of one variable is infinite?

Since the formula for calculating the correlation coefficient between two RV X and Y is $E[(X-u_X)(Y-u_Y)]/σ_Xσ_Y$. I wonder that if we have a finite value for $E[(X-u_X)(Y-u_Y)]$ but found that the ...
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Can I maximize the correlation between a linear combination of variables and some other variable?

For example, If I have 3 sets of data and I want to see what is the best combination between these 3 sets of data and a stocks price, is there a way to optimize a linear combination between the 3 ...
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Given that X and Y are RV supported on [2,3], If the correlation coefficient of X^t and Y^s is 0 for any s,t ∈ [2,3], then X and Y are independent.

I am doing a probability course HW and run into trouble with the following problem: Given that X and Y are RV supported on [2,3], If the correlation coefficient of X^t and Y^s is 0 for any s,t ∈ [2,3],...
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3 votes
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Pearson Correlation as a measure for non-linear dependence.

It is known that $\rho$, the pearson correlation, is a measure for the linear dependence of two random variables say $X$, $Y$. But can't you say just transform $X$ and $Y$ such that we have, $$ \rho_{...
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1 vote
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Correlations in implementations of random unitary channels

A random unitary channel (RUC) acting on a quantum system $S$ is any completely positive and trace preserving (CPTP) linear map $\mathcal{E}$ that can be expressed as \begin{align} \mathcal{E}(\...
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Uniform lower bound of the inner product of two positively correlated random variables

Suppose $W$ is a mean-zero random variable with unit variance, i.e., $E(W)=0,Var(W)=1$. Let $g(\cdot)$ be a non-constant increacing function such that $g(W)$ has zero mean and variance $\epsilon>0$,...
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Sample Mean of Correlated/Dependent Random Variables...

Suppose $\bar{U}$ and $\bar{V}$ are sample means from a highly correlated or highly dependent random process (e.g., waiting times from a queueing process) That is, let $$ \bar{U} = \frac{1}{n}\left(...
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