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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Implied Correlation from Gaussian Copula
I am building a spreadsheet model that allows marginal distributions to be correlated together using a Gaussian copula (with prescribed correlation matrix).
The inputs into the model are the ...
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Universality of correlators in $\beta$ Hermite ensembles
Recently I got interested in the world of Random Matrix Models and I bumped into some generalizations of the usual random matrix theories classified by Dyson whose probability density functions are:
$...
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Solving questions about correlation with symmetry
I try to determine if $X$ and $Y$ are (un)correlated and (in)dependent, when $X = \sin(\theta)$, $Y = \cos(\theta)$ and $\theta \sim N(0,1)$.
I know about the following formula:
$corr(X,Y) = \frac{...
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Confusion with Complex Gaussian process with Auto-covariance
I have a complex sequence $z(t)$ in time which I know to be a Gaussian process. I read that the complex Gaussian process is not only characterized by the covariance, but also the pseudo-covariance ...
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Compute channel frequency resopnse of a channel with fixed bandwidth using Python [closed]
I want to make a visualization of how a channel with a fixed bandwidth (40MHz, 160MHz) changes a channel impulse response. My very simple channel impulse response consists of 3 dirac impulses and is ...
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Maximize $ \frac{\sum x_i r_i}{|x_i|}$
Given a vector $r$, Maximize $ \frac{\sum x_i r_i}{|x_i|}$
My intuition is that the solution is when $x_i = k r_i$ for a constant $k$. But I cannot prove it.
I have been trying to rearrange it to form ...
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Express $\rho_{X+U,Y}$ , in terms of, $\rho_{X,Y}$ ,$\rho_{U,Y}$ ,$\sigma{X}$ and $\sigma{U}$ where $X$ and $U$ are independent [closed]
I need to express the correlation of $(X+U)$ and $Y$, in terms of the correlation between $X$ and $Y$ , the correlation between $U$ and $Y$ and the standard deviations of $X$ and $U$.
I know that $\...
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Three independent variables and correlations as cosines
Suppose we have three independent random variables $X$, $Y$ and $Z$ so the correlation among them is zero. However, I'm struggling to express them using a chart and the fact that correlations are ...
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Range of correlation between *Stacked* Variables
Similar questions have been asked before where we have Cor(X,Y) and Cor(X,Z) and we want to find out the range of correlations of sum linear combination X & Z with Y. Or a third correlation: (X,Z)....
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Estimate null hypothesis for correlation of linear combinations of variables? [migrated]
Suppose I have a variable $x$ of length $n$ and I have another $p$ variables $y_1, y_2, \dots, y_p$, where $y_i$ is also of length $n$.
Based on the y's, I can make a linear combination to estimate $x$...
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Uncorrelated but not independent random variable
I have a uniform discrete RV defined as $$X \sim Uni \{-1, 1\}$$. Now, I want to create another uniform random variable from $X$ defined as $Y =X^2$.
Am I right to say the following?
$$E[XY] -E[X]E[Y]...
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How does autocorrelated noise models converge to white noise models when considering kinematic tracking models?
In the article: A jerk model for tracking highly maneuvering targets
see
http://eprints.iisc.ac.in/2710/
The jerk model is explained.
What happens with the process noise matrix Q if the limit is taken ...
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Cross - correlation between two complex signals
I want to calculate the degree of similarity of two complex signals with different lengths in the time domain using python, so I’ll get a scalar value representing the degree of similarity.
I thought ...
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Contrapositive of independence --> uncorrelation statement
So I know that the statement "If two random variables are independent, then the two random variables are uncorrelated" is true.
Does that mean its contraposition "If two random ...
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Correlation of X and Y, when E[X] = 0
X and Y are random variables and it is known that E(X) is zero. Then how can I prove that corr(X,Y)is zero?
Here is what I tried:
The covariance between two random variables X and Y is defined as: Cov(...
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Pearson Correlation of the Principal Curvatures and their Derivatives
In continuation to a previous question, consider that I want to extend the calculation, and also calculate and numerically evaluate the correlation between the principal curvatures and their ...
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Determine the error on f(x) knowing R² and best fit for a set of points
As a concrete example, I have four points:
x | y
---+------
1 | 1854
2 | 2174
3 | 2258
4 | 2953
The best linear fit for these four points is ...
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Methodology to determine similarity between two signals
I'm working on the following problem:
I have an object detection model that detects bounding boxes of objects in a point cloud. I convert the point cloud to a bird-eye-view perspective, i.e. I create ...
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Quadratics and Correlation coefficients
I have a question about qudratic expressions and correlation coefficients. I know how to prove that if $y=mx+b$ then the correlation coefficent between $x$ and $y$ is 1. If $y$ is a quadratic function ...
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How to prove $\mathbf{1}^\top\mathbf{Q}^+\mathbf{Q}=\mathbf{1}^\top$, where $\mathbf{Q}$ is any element-wise squared correlation matrix?
Let $(X_1,…,X_n)$ be a random vector with $0<\prod_{j=1}^n\text{Var}(X_j)<∞$.
Let $\mathbf{Q}=(\mathbf{q}_{1},…,\mathbf{q}_{n})=(ρ_{jk}^2)_{n×n}$, where $ρ_{jk}$ is the Pearson correlation ...
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Proof of triangle inequality (where d is a metric described in the question)
Let $x, y, z$ be some three (real-valued) time series.
Prove that $$\sqrt{1/2-\rho(x,z)/2} \leq \sqrt{1/2-\rho(x,y)/2} + \sqrt{1/2-\rho(y,z)/2}$$
After this assuming that two time series cannot be ...
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Compare two groups of correlation coefficients (statistical test)
I am considering the following problem. I have 2 sets of correlation coefficients (measuring the correlations between behaviours and neural activities in 2 different regions of the brain). I would ...
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Prove that condition number of a matrix increases in its dimension
This is my first time asking a question here. Thus, I apologise in advance if it is not articulated correctly or something else turns out to be wrong with it. Before asking the question itself, ...
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Does uniform distribution on every square $[0,a]^2$ along diagonal imply uniform CDF on the entire $ [0,1]^2$
Let $F: [0,1]^2\to R $ be a continuous cdf with uniform marginals, i.e., $F(x,1)=x$ and $F(1,y)=y$. Suppose $F$ is symmetric, i.e., $F(x,y)=F(y,x)$.
Suppose we also know that $F(a,a)=a^2$ for all $a\...
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Covid Correlation Coefficients
New to statistics. Reading a intro book. I just finished a chapter on Correlation Coefficients, and I'm going over covid data. I think I grasp the relevant formula, but I think I'm making some ...
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Pearson Correlation of the Principal Curvatures
Consider the set of all possible smooth 3D surface patches. Now, let's say that I draw $K$ random points from that set, where each point is parametrized by its principal curvatures $\kappa_1$ and $\...
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Given descriptive stats for two sets $(x_1,y_1)$ and $(x_2, y_2)$, can $\rho^2$ be found for their union $(\{x_1 \cup x_2\}, \{y_1 \cup y_2\})$?
I have two sets of x-y coordinates: $(x_1,y_1)$ and $(x_2, y_2)$. I also have the $R^2$ value for each set of coordinates. (e.g. $(x_1,y_1)$ has the correlation coefficient of $R^2$)
Given the stats ...
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Finding a concise relation between $\operatorname{ns}\left(\frac{K(k)}{3},k\right)$ and $\operatorname{ns}\left(\frac{2K(k)}{3},k\right)$
Let $\operatorname{ns}\left(z,k\right)$ be one of the Jacobi's elliptic functions, and $K(k)$ the complete elliptic integral of the first kind. It's well-known that $\operatorname{ns}(mK(k),k)$ where $...
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Using correlation as a distance metric
I was trying to come up with a distance measure $d(X;Y)$ based on correlation $\rho_{XY}\in[-1;1]$ satisfying the following conditions:
$d(X;Y)\ge0$ $\forall X,Y$
$d(X;X)=0$
$d(X;Y)=d(Y;X)$
$d(X;Z)\...
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Auto correlational of a unit pulse function.
$$
p(t)=\begin{cases}
0& t<0\\\\
1& 0<t<1\\\\
0& t>1
\end{cases}$$
I need to compute $$ \int_{-\infty}^{\infty} p(\tau) p(t+\tau) d \tau$$
My thinking is :
First convert the ...
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Derive $E[\alpha_1\alpha_2]$ when they have correlation coefficient.
How can I analytically derive $E[\alpha_1\alpha_2]$, where $X_1=X_{1r}+j\,X_{1i}$ and $X_2=X_{2r}+j\,X_{2i}$. Further, $\alpha_1=|X_1|$ and $\alpha_2=|X_2|$?
Here, $X_{1r,},X_{1i},X_{2r,},X_{2i}\sim ...
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Can two random variables have linear relationship but still be uncorrelated?
If two random variables does not have linear relationship then they are uncorrelated even if they might be dependent. I am unable to come up with an example where two random variables have linear ...
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Correlated equilibrium proved by convexity (Games Theory)
I am finding very hard to understand how convexity works and why a correlated strategy if a convex linear combination of Nashes is a correlated equilibrium.
(Both concepts of Correlated strat. and ...
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What does is mean to center variables?
I was reading this answer that is talking about properties of $AA^T$.
If you center columns (variables) of $\bf A$, then $\bf A'A$ is the
scatter (or co-scatter, if to be rigorous) matrix and $\...
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Issue in finding auto correlation coefficients
I found it to be 0.74 for k=1 and 0 for the rest. I am unable to obtain the following solution to this problem
Question
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Amount of change in Pearson correlation by new cases
Given Pearson correlation coefficient:
$$r = \frac{\displaystyle {}\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y})}
{\displaystyle \sqrt{\sum_{i=1}^{n} (x_i - \overline{x})^2(y_i - \overline{y}...
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Calculate the correlation coefficient
Every day man puts 10 1 £ coins and 8 2 £ coins in his pocket. Each day, in the morning he losses 8 coins in the afternoon he losses 6 coins. Find its morning and afternoon loss correlation ...
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Constant conditional expectation without independence
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of $(-1,\infty)$-valued random variables. Is it possible to have
$$
\forall \, n \colon \ \mathbb{E}[X_n \, \vert \, X_1, \ldots, X_{n-1}] = 0
$$
without ...
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Why is the dot product of a sine wave with anything a sine of the phase shift?
I'm trying to wrap my head around the fact that multiplying anything on a sine wave then calculating the sum of products is a sine of the sine wave's phase.
This is literally the foundation of the ...
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E[XY] where X and Y are the **sign functions** of standard normal distributions
The following question is from the book: "150 Most Frequently Asked Questions on Quant Interviews" By Stefanica, Radoicic, and Wang.
Let $X$ and $Y$ be standard normal variables with joint ...
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Bounding the inner product of a vector of correlations
Suppose $X$ is a Gaussian random variable and $Y$ is a Gaussian random vector of length $n$ (they are also jointly Gaussian).
Let $z$ be a vector with entries $z_i = \frac{\mathrm{Cov}[X, Y_i]}{\sqrt{\...
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Can the Wiener-Khinchin theorem be correctly applied to a periodic sound signal (such as a sine wave)?
The theorem speaks about a wide-sense stationary random process. Is, for example, a sine wave with a period 1/400 s considered a WSS (or, in general, a periodic sound signal with multiple frequency ...
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Expected value, dispersion and correlation coefficient of losses during the day
Every day a kid puts $7$ £1 coins and $9$ £2 coins in his pocket. Each day, in the morning he loses $3$ coins and in the evening of the same day he loses $5$ coins. Find his morning and evening losses'...
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Fourier transform of a (correlation) function with absolute value in the exponent
I have the following correlation function in time domain defined as
$$
\left<\xi(t)\xi(s)\right> = \frac{D}{\tau}e^{-|t-s|/\tau}
$$
I wish to take the fourier transform of the correlation ...
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correlation of the sums of samples
Given variables $X$ and $Y$ are positively correlated, if $n$ samples are randomly selected from $X$ and $Y$, respectively, and summed to obtain new variables $X_1$ and $Y_1$, are $X_1$ and $Y_1$ ...
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Additivity of Correlation
Given that random variables $X$ and $Y$ are positively correlated, and random variables $Z$ and $W$ are also positively correlated, can it be concluded that $X+Z$ is positively correlated with $Y+W$? ...
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Omitted Variable Bias
I have a question regarding the omitted variable bias. Its properties of it:
the omitted variable is a determinant of the dependent variable,$y$
$X$ is correlated with the omitted variable.
I ...
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Is there true randomness?
In a paper titled "Quantum Randomness: From Practice
to Theory and Back", Cristian S. Calude concludes that there is no true randomness in numbers:
"The “magic” of the quantum ...
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When should you use Mutual Information over Pearson Correlation?
I have a dataset with ~1000 features. I know that some of those features are leaking information from my target variable.
I'd like to find those features by checking similarity between all features ...
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Spearman's rho for tied ranks
I know how to prove that Spearman's $ \rho / r_s$ can be written as $r_s = 1-\frac{6\sum\limits_{i=1}^nd_i^2}{n(n^2-1)}$ with $d_i=R(x_i)-R(y_i)$ when there are no ties.
Now I came across the formula ...
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