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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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Correlation between $y_t$ and $y_{t - T}$

Let $y_t = 0.8y_{t-1} + \epsilon_t$ where $\mathbb{E}[y_t]$ and $\mathrm{Var}[y_t]$ are constants (or $\mathbb{E}[y_t] = \mathbb{E}[y_{t-1}]$ and $\mathrm{Var}[y_t] = \mathrm{Var}[y_{t-1}]$) and $\...
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Finding correlation between CDF of two normal distributions

Suppose that $X\sim N(0,1)$, $Y\sim N(0,1)$ with correlation $(X, Y) =ρ$ where $ρ ∈ (−1, 1)$. Show the following, Correlation $(Φ(X),Φ(Y))=\dfrac6π \arcsin\dfrac ρ2 $. Here $Φ(X)$, and $Φ(Y)$ ...
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Zero Mean Normalized Cross Correlation in Einstein Notation

I try to formulate the Zero-Mean Normalized Cross-Correlation in Einstein Notation. Thus without the Sigma's and with indices. I came up with the following but im not sure if the is correct and ...
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Inter-rater reliability tests

I’m writing a research project to form part of my grade for my final year at university and as part of it me and some of my classmates have categorised some of our previous exam questions using the ...
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Quadratic equations with multiple variables

I cannot think out a accurate expression to describe this question, but I think this is a quadratic problem with multiple variables or kind of an auto-correlation problem. The question is as follows. ...
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Finding any correlation between curves

Say i have 2 curves A/B. I would like to find out the points where at least twice (a pattern) when something happen in A, it affect B. I dont care what it is, i just look for points of correlation ...
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The correlation between two random variables

Let $X_{1}$, $X_{2}$, and $B$ be independent random variables and $P(B=-1)=P(B=1)=\frac{1}{2}$. Let $Y_{1}=B\left\vert X_{1}\right\vert$ and $Y_{2}=B\left\vert X_{2} \right\vert$. Let $Y_{1} \sim N(0,...
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Regression - slope doubt

The gradient of the regression line $x$ on $y$ is $-0.2$ and the line passes through $(0,3)$. If the equation of the line is $x = c + dy$, find the value of $c$ and $d$. What I did: As per my ...
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Canonical correlation and within variable covariances

I am applying canonical correlation analysis (CCA) to a problem. In particular, I have random vectors $X \in \mathbb{R}^{10}, Y \in \mathbb{R}^{8}$ $cov(X) = E(X - E(X))(X - E(X))^{T} = \Sigma_{1}$ ...
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Funky change in correlation

Context: The figure below plots the record-high yearly precipitation in each state against the state's record-high 24-hour precipitation. Hawaii is a high outlier, with a record-high yearly record of ...
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Is $\mathbb{E}[f(\vec{X})g(\vec{X})] \ge \mathbb{E}[f(\vec{X})]\mathbb{E}[g(\vec{X})]$ for $f,g$ bounded, non-decreasing?

For a random variable $X$ and bounded and non-decreasing functions $f,g : \mathbb{R} \to \mathbb{R}$, we have: $$ \mathbb{E} [f(X) g(X)] \ge \mathbb{E} [f(X)] \mathbb{E} [g(X)]$$ In case I am given ...
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Autocorrelation and cross-correlation of sampled sinosoids

Can someone help to find the autocorrelation and cross-correltion for a sampled data. I have: $$u(n)= \sin \left( \frac{2 \pi n}{10}+\phi \right)$$ Where $\phi$ is a uniform random variable from $0 \ ...
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Correlation analysis of proteins to clinical variables of individuals

I want to do a correlation analysis of proteins to a clinical variable. e.g. correlation of a protein to weight For many individuals I have: 4 time points with different weights 4 protein ...
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How do they simplify the autocorrelation function?

In my book they state that ...the autocorrelation function can be defined as $$r_X(s,t)=\frac{\text{Cov}[X(s),X(t)]}{\sqrt{\text{var}[X(s)]\text{var}[X(t)]}},\tag 1$$ where $X(t)$ and $...
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Correlation for boolean variable

I am new to statistics and had this doubt. I am trying to understand the correlation between two variables. One of the variables is a boolean i.e 1 or 0. Can the same correlation algorithms like '...
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Clustering using Hidden Markov Model

We are using Hidden Markov Model to cluster gene expression data where every observation has univariate gaussian emission based on latent state. We have chosen to have three latent states and mean and ...
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symmetry of Regression line when SDx equals SDy and non intutive results

This question is taken from Freedman In a certain class, midterm scores average out to $60$ with an SD of $15,$ as do scores on the final. The correlation between midterm scores and final ...
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Correlation coefficient in terms of standard units: intuition

Correlation coefficient $$r = \frac{1}{n}\sum_{i=1}^n\frac{(x_i-\bar x)(y_i-\bar y)}{\sigma_x\cdot\sigma_y}$$ But for a given data point $x_i$ and predicted value $y_p$, $$\frac{(y_p-\bar y)}{\...
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1answer
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Correlation of a 2D array and its transpose

I was working on a tabular data and was thinking about the correlation meaning in here: We are taking a sample of several Whisky drinks and their corresponding rating on various taste criteria. ...
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Correlation coefficient and regression line : Geometric intuition

correlation coefficient $$r = \frac{1}{n}\sum_{i=1}^n\frac{(x_i-\bar x)(y_i-\bar y)}{\sigma_x\cdot\sigma_y}$$ may be thought of as cosine of angle between two $n$-dimensional vectors $$ (x_1- \bar ...
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Time-dependent correlation of random walk and integrated random walk

consider an ideal white-noise process $X0 = \epsilon(t)$ with a constant double-sided PSD, which is approximated by $\sigma^2 * dt$ when observing the process in intervals of $dt$. Next, consider ...
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Hybrid Multivariate Normal Distribution (Folded axis, Warped axis and Truncated axis) Random vector generation

I'm looking to describe a multivariate normal distribution in cylindrical coordinates with truncated/bound height, i.e. it has 3 axes $r$, $\theta$ and $h$. Obviously angle $\theta$ is a warped ...
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Sampling error of correlation coefficient (Phi coefficient) for binary variables

Suppose I have two correlated binary variables (A and B) with known probabilities ($p_a$ and $p_b$) and correlation (Phi) coefficient in population - $\rho$ . Is there any analytic function for ...
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Coefficient of Determination and Correlation between observed and fitted value in Multiple Linear Regression.

Consider Multiple Linear Model $$y= X\beta + \epsilon$$ Then using Ordinary Least Square, we get estimate of $\beta$ as $$\hat{\beta} = (X^{'}X)^{-1}X^{'}y$$ And $$\hat{y} = X\hat{\beta}$$ $$SS_{Res}= ...
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how can i compute the correlation between two matrices?

In Wireless communication, we define a channel matrix H to be a matrix of size N by M, where ...
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94 views

$d_{TV}$ VS correlation coefficient.

Consider two RV $X,Y$. If $d_{TV}(X,Y)=0$ you may couple them in such a way that $$\rho_{XY}=\frac {\operatorname{COV}(X,Y) }{\sigma_X\sigma_Y}=1.$$ So, is there any formula to bound $d_{TV}$ in ...
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Formula for copulas of bivariate mixed (discrete and continuous) data

I have a question as I am new to copulas. I have bivariate data (X,Y), one is discrete and one is continuous distributed. Can I use the usual formula for the common density function given by $$f_X(x) \...
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Is the autocorrelation value affected by the order of the time series values?

Let's say we have a timeseries value such as 5 2 3 6 8. Will the value of autocorrealtion be changed when we change the order the time series like ...
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Given the correlation between shares of all $\frac{n(n-1)}{2}$ pairs among $n$ different companies. How to acheive maximum diversification?

I have the historical data about the stock prices of the companies and can easily calculate the historical correlation between any two of them. Given this information, how should I go about finding ...
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Calculate correlation coefficient for discrete random variable

From a population consisting of the numbers: $\lbrace 1,2 \ldots 10 \rbrace$, two samples are chosen from it without replacement. If the random variable denoting the first choice is X and the second ...
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Can a stochastic process has time dependent mean and time independent autocorrelation?what is the autocorrelation of the process through a LTI system?

I have this process: $$X(t)=A\cos(2\pi f_0t+\Phi)\ast \operatorname{rect}\left[\frac{t-T/4}{T/2}\right]$$ where: $A$ and $\Phi$ are uniform random variable in $[0,\pi]$ statistically independent $\...
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Mean and correlation of product of two random processes

I have two random process: $$A(at)$$ $$cos(2\pi f_0t+\Phi)$$ with these hypothesis: $a$ and $f_0$ are constant $\Phi$ is uniformly distributed in $[0,\pi)$ $A(at)$ is WSS I must calculate the ...
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Standard measures derived from autocorrelation functions

Setting: Let $(x_i)_{i=1}^N$ be a discrete signal of length $N$. This question is concerned with discrete finite data sets for simplicity, but they can also be raised in more general contexts. The ...
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could I only use the correlation value of two variables to say that one variable has negative or positive effect on another variable?

Fro example, for variable A and B, I have their time series respectively for 1000 long.Then using those to calculate their correlation is 0.8. Could I say that increase of A would also increase B. A ...
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Find correlation/covariance of two distributions with shared marginal distributions

I have two distributions which represent the sum of smaller distributions. Say distribution X is made up of distributions a, <...
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Correlated probabilities

I'm interested in playing daily fantasy sports but want to calculate my risk beforehand. Let's say I have determined the three best lineups possible but only have enough capital to bet on two of the ...
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1answer
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Is there any logic to adding a threshold to see if two variables are related?

I have two variables $X$ and $Y$ given as tuples of $(x, y)$, and I want to see if there is a relationship between the two variables. I can do so by finding the correlation coefficient. However, I ...
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1answer
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How to find correlation between different tests in the same sample? [closed]

I did several tests in a sample of 90 individuals to check some behaviors. I got the following results: 10% of my sample has the behavior A 65% of my sample has the behavior B 80% of my sample has ...
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Expectation of max(X, Y) where X and Y follow a joint normal distribution with correlation of $\rho$

As stated in the title, I am looking for the solution to this expectation, possibly as a function of $\rho$. One relevant solution I find is Lord Shark the Unknown's answer, from which I can derive $E(...
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36 views

Which transform makes correllation a multiplication?

In Fourier analysis, a central theorem for the Fourier Transform states: $$\mathcal F\{(f*g)(t)\}(\omega)=\mathcal F \{f(t)\}(\omega)\cdot \mathcal F\{g(t)\}(\omega)$$ In other words, convolution ...
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3answers
54 views

Correlation between two linear combinations of random variables

Let $X_1, X_2, X_3, X_4$ be independent random variables with $\operatorname{var}(X_i)=1$, and $$U = 2X_1+X_2+X_3$$ $$ V = X_2+X_3 + 2X_4$$ Find $\operatorname{corr}(U, V)$ In general, how can I ...
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Contrasting effects on correlation between two variables with increase or change in dataset

I am trying to analyze two variables: Currency exchange rate and Stock price of a company. I am computing the correlation between them to determine whether the currency exchange rate has some effect ...
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Sums of realizations of dependent variables become independent

I have empirically noticed and interesting phenomenon. Suppose we have two continuous random variables $X$ and $Y$ which are dependent but not correlated. For instance: $X \sim \mathcal{N(0,1)}$ $Y =...
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Mathematical proof that white noise has zero autocorrelated

I would like to prove, using the definition of autocorrelation, that Gaussian and uniform white noise have zero autocorrelation. I am working on the continuous case (but I think discrete shouldn't be ...
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How can I calculate the correlation in these 9 androgen-dependent areas?

I know that the correlation bethween CAG Repeats and hair growth in 13 studied areas is r=-0.538. What I also know is that the correlation of CAG Repeats with hair score only holds for 9 of the 13 ...
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how can I normalize the correlation between two multi-dimension vector?

I have $X,Y \in R^d$. I would like to measure the correlation between the two vectors element-wise, but with respect to other dimensions information. Therefore the maximum correlation should be $d$ (1 ...
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Conditional probability with n correlated random variables

I encounter this problem in one of my research projects in economic/applied statistics decision making. Suppose there is a random variable $\omega \in \{1,0\}$ with $\Pr (\omega =1)=q\in (0,1).$ ...
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Why is the correlation of two random markov chains so large?

A very simple python code shows that the correlation coefficient and spearmann rank between two datasets of uniform random numbers drops proportionally to the square root of the number of points in ...
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Finding pearson correlation coefficient between two variables using the coefficient of one of them and a third veriable

I have been given three variables: x,y and z. y is a linear transformation of x: y=15x+20. The frequency distribution for x is also given. the pearson correlation coefficient between x and z is given ...
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Velocity Auto-correlation on Spinning Disk

For a 2D disk of diameter $l$ spinning with angular velocity $\omega$, what is the velocity auto-correlation in the y-direction averaged over the disk? So far, I have that the y-component of velocity ...