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

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85 views

Why is $R^2=\rho^2$

Considering $y_i=\beta_1+\beta_2x_i+\epsilon_i$ $\bar y_i=\hat\beta_1+\hat\beta_2\bar x_i+\bar\epsilon_i$ a linear equation of least square used when it seems that there is a like between two data, $...
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21 views

Formula for the n'th order correlator $\langle \Gamma(t_1)\Gamma(t_2)…\Gamma(t_n)\rangle$ of Gaussian white-noise?

Is there a closed form formula for the n'th order correlator $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\langle \Gamma(t_1)\Gamma(t_2)...\Gamma(t_n)\rangle$, of Gaussian white-noise $\Gamma(...
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2answers
51 views

Show $X_1$ and $X_2$ are negatively correlated

Consider $n$ independent tosses of a die. Each toss has probability $p_i$ of resulting in $i$. Let $X_i$ be the number of tosses that result in $i$. Show that $X_1$ and $X_2$ are negatively correlated....
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19 views

Correlation matrix

I have this example of a correlation matrix in my notes: $$R = E[XX^T]$$ $$n = 3, E[X_i X_{j}^*] = 2^{|i-j|}$$ $$R = \left[ \begin{matrix} 1 & 0.5 & 0.25 \\ 0.5 & 1 & ...
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21 views

Is the sum of all cross-correlation samples representative of target existence likelihood?

Answers to this question take the peak in the cross correlation as the measure to the likelihood of the trigger signal exist in the received signal - this is pretty much text book. My question is ...
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0answers
47 views

What is the correlation of two normal distributions with equal mean and known relation between variance

If I take the sample mean of a scaled $\chi$ distribution of $N$ samples, the distributions of these means should lead to a normal distribution $\mathcal{N}(\mu,\sigma)$. This procedure is repeated $M$...
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2answers
144 views

Conditions for the convergence of two sorted vectors of samples

Let $X$ be a random variable and $X_1, X_2, \ldots, X_n$ be a sample of size $n$ and $X_{(1)},X_{(2)},\ldots,X_{(n)}$ the corresponding order statistics, which are obtained by sorting the values $X_1, ...
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53 views

Conditional Expectation of random sum of independent random variables (when $N$ and $X_i$ are dependent)

In this question, $Y=X_1+X_2+\dots+X_N$ where $X_1,X_2,\dots,N$ are jointly independent random variables, $X_1,X_2 ...$ identically distributed continuous random variables with finite expectation, ...
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1answer
39 views

Given three sets of data A, B and C. Can A correlate with C, and C with B, but not A with B

Here are three sets of data: A B C 0 0 4.84 2 0 4.28 1 1.73 2.74 You can check that the (product moment) correlation coefficient between A ...
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0answers
41 views

What is the correlation between these two vectors?

Assume we have constant positive numbers $k$ and $N$ and a vector of the following form $$ \tag 1 a(\theta_1) = \frac{1}{\sqrt{N}}[ 1, e^{jk \sin(\theta_1)},e^{jk \sin(\theta_1)}, \cdots, e^{j(N-1)k\...
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1answer
24 views

Covariance and correlation, and how are they related?

I get that corellation is the covariance divided by the multiplie variance of the two, uh, things. What i don't get is why they are divided by the multiplied variance, and why that limits the value ...
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0answers
40 views

Is there any measure for detecting quadratic or cubic relationships?

Correlation coefficients is a useful measure for detecting linear relationships. Is there any measure for detecting linear relationship between one dependent variable and more than one independent ...
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0answers
63 views

Correlation function of an asymptotically stationary AR process

I have a great confusion with the autocorrelation function of an AR process. Its derivation usually follows in this way (Haykin, 2007): The difference equation for an AR(M) process, $u(n)$, is \...
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1answer
23 views

Correlation coefficient and orthogonality

In the book Matrix Analysis and Applied Linear Algebra, the author describes the coefficient of linear correlation as $$\frac{(x-\mu_xe)^T(y-\mu_ye)}{||x-\mu_xe||\cdot||y-\mu_ye||}$$ where $x,y\in \...
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1answer
115 views

Linear correlation in 3D?

What's the name of a statistical method used to determine the goodness of fit of a series of points in 3D space that are to be fitted on a regression line ? I can calculate a regression line and the ...
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0answers
23 views

Generating random variables with specific higher order correlations

I would like to generate a series of random variables {$X_1,X_2,...$} with the following properties: 1) $\langle X_i\rangle=1$ for all i. 2) $\langle X_i \cdot X_j \cdot X_k \cdot ...\rangle=0$ ...
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2answers
25 views

Identifying modules in a correlation matrix

I am a biologist with some maths background... but not enough knowledge to solve this problem, so I would be really grateful if someone could help (and explain it at a level that a biologist might ...
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0answers
17 views

Autocorrelation of a signal known only on an interval of finite size

Let's consider we have a continuous random signal ${ t \in ] - \infty \,;\, + \infty [ \mapsto b (t)}$. We assume this signal to be stationary, so that when ensemble-averaged, one may introduce the ...
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0answers
47 views

Limit of Covariance and Correlation of Random Walk

Let $S_n=X_1+...+X_n$, $n\geq1$ be a random walk, where $EX_k=\mu$ and $Var(X_k)=\sigma^2$, $0<\sigma^2<\infty$. a)Find the covariance $Cov(S_n,S_m)$ and the correlation coefficient $\rho(S_n,...
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1answer
51 views

Is there a simpler way to calculate correlation?

Let's consider that a variable y constructed from x $x_i ∈ \left\{1,3,5,7,8\right\}$ $f(x_i)=2x_i+1$ $y_i=f(x_i) + ε_i, ∀i∈ \left\{1;...;5\right\} $ where $ε_i$ is a identically ...
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0answers
21 views

Variance of estimating coefficients by correlating a sequence

I have a sequence $$ r[n] = a_1.t_1[n] + a_2.t_2[n] + a_3.t_3[n] + ... $$ where $t_1, t_2, t_3,...$ are uncorrelated, two-level (+A/-A), zero mean, pseudo-random sequences. To estimate $a_1$, $r[n]$...
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18 views

Why are the Real and Imaginary parts of a field in k-space uncorrelated?

I am in the process of generating a (real) Gaussian random field $\delta(\vec{x})$ from a given power spectrum $P(k)$. The way I define the power spectrum is, in Fourier space, $\left\langle \delta(\...
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22 views

Properties between Pearson correlation of two function and the correlations of their derivatives

For two functions, $f(t)$ and $g(t)$, I can calculate the Pearson correlation $Corr$. Does the correlation, $CorrD$, between the derivative functions, $f'(t)$ and $g'(t)$, has some properties with ...
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1answer
136 views

The relationship between diagonal entries and eigenvalues of a diagonalizable matrix

Let $\mathbf{C}$ be an $n\times n$ Hermitian matrix. Let $\dagger$ indicate a matrix conjugate-transpose. Let $\mathbf{V}\mathbf{D}\mathbf{V}^\dagger$ be the eigendecomposition of $\mathbf{C}$, where ...
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32 views

similarity between two ranked sequence

How can I measure similarity/distance between two sequences of ranked numbers/letters. The two sequences are of different length, and only have some elements in common? For example, if I have three ...
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0answers
27 views

Concentration bounds on Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let $\rho(...
0
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1answer
61 views

If $X$ and $Y$ are Normally distributed with correlation $\rho$, can we say anything about $E[Y \mid X]?$

Let $X \sim N(0, 1)$ and $Y \sim N(0, 1)$ and $\mathbb E[XY]=\rho$. Can one say anything about the conditional expectation $\mathbb E[X \mid Y]$? In general, this clearly does not seem to work, ...
0
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1answer
29 views

inference of causality for binary variables

Let's say that a data set has N random binary variables Xi and we want to infer which of these variables have a causal relationship with X1. The following table would describe the data, where each ...
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1answer
114 views

How should I calculate a rolling autocorrelation?

I have an array of data $ \mathbf{y} \in \mathbb{R}^n $, and I need to calculate the lag-1 autocorrelation between sections of this array 7 elements long. For all intents and purposes, we can imagine ...
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1answer
42 views

Do 'X' and "y' have 'zero' correlation , or can be anything between '-1' and '+1'?

let , we have bi-variate data on X and Y . Now corresponding to the value $x_0$ , y can take any value.but for all other values of x , y takes a constant value. what will be the correlation between ...
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40 views

Non-orthogonal space

What does the angle between two non-orthogonal basis denote? Is it correlation or some measure of dependence. Does that mean that coordinates of a point if moved in the direction of one axis also ...
2
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45 views

The concept of correlation in functional analysis

I am currently reading a book "measure, integral and probability" by Capinski and Kopp. The correlation between random variables $X$ and $Y$ is defined as the cosine of the angle between $X_c$ and $...
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2answers
50 views

Pearson's R and Correlation formula

I'm trying to make sense out of Pearson's $R$ and Pearson's correlation coefficient. I'm not sure I really see a difference. Let me just clear out any confusion, for me Pearson's $R$ is: $$ R = \...
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1answer
26 views

Interpretation of $N$ and $p$ in Pearson’s correlation test?

In this paper, the authors report an estimate of $r = 0.86,\, N = 28,\, p < 0.001,$ using Pearson’s correlation test. The parameter $r$ (or $\rho$) is clear to me, but how are $N$ and $p$ derived ...
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61 views

Determining statistical correlation between XYZ points

I have a set of two 3D points, both 3D points (two points in each set) representing the same object - just in different states (State A, State B). I'd like to see if it's possible to predict the ...
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0answers
50 views

partical correlation in mixed case binomial and gaussian

For Gaussian mutlivariate distributions it is known, that zero partial correlation corresponds to conditional independence. Is there a same result if one of the variables has a binomial distribution? ...
0
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1answer
114 views

sufficient conditions for a stochastic process to be wide sense stationary

From the page Stationary process, I have the following definition: WSS random processes only require that 1st moment and autocovariance do not vary with respect to time and from the page ...
3
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1answer
121 views

Covariance of 1-D random process is $n\times n$!!!!

I'm reading a tutorial on stochastic processes. There is an example in the tutorial as follows: General Moving Average random process given as $X[n]=\frac{(U[n]+U[n-1])}{2}$ where $E[U[n]]=\mu$ ...
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25 views

DFT of subdomain of periodic domain

$f(t_i,x_j)$ is a solution of stochastic differential equation on grid. $j=[0,N+1]$, $i=[0,\infty]$ and boundary conditions are periodic: $f(t_i,x_0) = f(t_i,x_N)$ and $f(t_i,x_{N+1}) = f(t_i,x_1)$ ...
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0answers
42 views

Multiple variable correlation

I have three data variables (let's call them $A$,$B$, and $C$) that each consist of $14$ samples. What I know is that the combination of $A$ and $B$ is related to $C$. I don't know what kind of ...
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1answer
52 views

does uncorrelation extend to product of complex random variables?

Give two uncorrelated complex variables, $X$ and $Y$. Are $XX^{*}$ and $YY^{*}$ also uncorrelated, where $*$ means complex conjugation?
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15 views

combining correlation from two different time periods

Suppose I have two time-series $X_t$ and $Y_t$ and I measure their correlations over two different time-periods $\rho_1 = corr(X_i, Y_i)$ for $i \in (t_{1a}, t_{1b})$ $\rho_2 = corr(X_i, Y_i)$ for $...
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2answers
52 views

Finding Linear independent vectors

Thanks for clarifications. Now i am posting the question in a different way. Suppose a vector $V$ is orthogonal to vectors $X1$ and $X2$. $X1$ and $X2$ are linearly independent. Now if $V$ is also ...
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1answer
28 views

In a simple regression model estimated using OLS, the covariance between the estimated errors and regressors is zero by construction

Is this statement true or false? I seem to remember that this relationship does not hold when the regression has no intercept, however my teacher said that this was true regardless of whether we ...
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1answer
12 views

How represent correlation of $(f_i - f_j) $ and some $ y$ by $cov(f_i, f_j)$, $cov(f_i, y)$ and $cov(f_j, y)$?

I am reading this paper: Face Alignment by Explicit Shape Regression. One of the significant step of algorithm which proposed in these paper connected with correlation. But my knowledge about ...
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1answer
44 views

Correlation coefficients of X and Y [closed]

I was wandering if anybody could help me with the following question. I am fairly new to correlation coefficients and was attempting to tackle this question but was unsure how to do so? Thanks.
3
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2answers
231 views

Generating correlated random variables with discrete distribution

I would like to find a simple way to generate two correlated random variables under the condition that each r.v has a same discrete distribution (for example Bernoulli distribution) This link provides ...
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2answers
41 views

Calculating the correlation coefficient between least square estimates

PROBLEM STATEMENT: Consider the following 2-variable linear regression where the error $e_i$ 's are independently and identically distributed with mean $0$ and variance $1$; $$y_i = α + β(x_i − \bar ...
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1answer
36 views

Cross-correlation of identical sets: not getting expected result

I'm trying to work out the correlation coefficient of two sets using a given formula, but I'm not getting a perfect correlation when using identical sets. The correlation between a client’s ...