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

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What is the expectation of the product of dependent, normal random variables

Question: Let's say I have $X \sim N(\mu_1, \sigma) $ and $Y \sim N(\mu_2, \sigma) $. I know that $ cor(X,Y) = \rho $. What is $E(XY)$? What I've tried Based on a similar question where X and Y are ...
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40 views

relationship between multiplication and correlation

is there a deep interpretation of multiplication as correlation? is this in some sense the most fundamental way to "combine" objects (eg numbers) into products? my reasons for asking are that the ...
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1answer
30 views

Correlation between 3d images and their slices

I work in the field of the image processing and I need to compare results of my algorithm with a gold standart results. For this purpose I calculate the Pearson correlation coefficient between the ...
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2answers
46 views

Show $E\left(\mathbf{X}_i \otimes \mathbf{u}_i\right)=\mathbf{0}$ implies $E\left(\mathbf{X}_i^{\top}\mathbf{G}\mathbf{u}_i\right)=\mathbf{0}$

Let $\mathbf{X}_i$ be a $G \times K$ random matrix, and let $\mathbf{u}_i$ be a $G \times 1$ random vector, and suppose we have a sample of $i=1,\ldots,N$ of each. Suppose the following condition ...
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51 views

Definition of Autocorrelation Function (ACF)

For a weakly stationary time series $\left\{r_{t}\right\}$, the definition of ACF is (from Ruey Tsay's "Analysis of Financial Time Series") $$ ...
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42 views

Finding a relative error measure on a data set proportional to another

I have a set of exact data points $\mathcal{X}=\{X_i\}$ and another approximate one $\mathcal{Y}=\{Y_i\}$ where there is a correspondence between $X_i$ and $Y_i$ for all $i$. If $\mathcal{Y}$ was ...
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12 views

Proving $Corr(\hat{e}_{ij}, \hat{e}_{jk}) = \frac{-1}{n_i-1}$ for $ j \neq k$

For the model of a single factor experiment: $y_{ij}= \mu + \alpha_i + e_{ij}$, $(1 \leq i \leq a, 1 \leq j \leq n_i)$, where a = the number of treatments, $n_i$ = the number of experimental units ...
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2answers
65 views

The inverse of AR structure correlation matrix / Kac-Murdock-Szeg ̈o matrix

I want to find the inverse of the following matrix: $$ R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots ...
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12 views

Matrix with highly correlated adjacency entries

I am learning about SVD from this book. One of the exercise questions asks me to create matrix with highly correlated adjacency entries and then conduct some experiments to discover the nature of the ...
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1answer
367 views

What does the multiplication of standard deviation of two variables gives?

If we need to find the correlation between two variables it is given by the formula - co variance of two variables divided by the multiplication of Standard deviation of the two variables. My ...
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41 views

Isotropic correlation function for a vector valued random field

I'm having trouble with some of the implications of the following theorem. Let $\mathbf{T} (\mathbf{x})$ be a mean-square continuous vector valued random field on $\mathbb{R}^3$ satisfying conditions ...
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1answer
23 views

Question on the correlation between two dependant variables

I'm working on this question and it's stumping me. Let Sn = X1 + ... + Xn (with n>=1) be a random walk with X1,...,Xn be iid RV's. E(Xk)=mu Var(Xk)=sigma^2. Find the covariance of Sn and Sm Can ...
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0answers
87 views

Correlated variables from Latin Hypercube

Say I have a vector $\mathbf{Y}$ of $n$ normally distributed random variables. I have its mean vector $\mu$ and covariance matrix $\Sigma$. Normally if I were to generate a sample, I would decompose ...
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39 views

Generate two sets of (nonlinearly) dependent random numbers

I would like to find a method to generate two sets of (nonlinearly) dependent random numbers. Solution for linear dependence (that is, correlation). Generate two sets of uncorrelated random numbers ...
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1answer
56 views

Combination problem: random selection in a group

A scientific committee of 4 persons is to be randomly selected from a group consisting of 3 biologists, 3 physicists and 4 mathematicians. Let X denote the number of biologists, Y the number of ...
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35 views

Best line fit for correlated points

Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = ...
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14 views

Is auto-correlation a particular and simple form of pattern in time series?

In a time serie context and considering the sign of the variation of the variable auto-correlation means that for couple of measures at (time t, time t+1) the ...
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37 views

OLS standard error that corrects for autocorrelation but not heteroskedasticity

Question: By mapping the OLS regression into the GMM framework, write the formula for the standard error of the OLS regression coefficients that corrects for autocorrelation but not ...
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1answer
26 views

Is $d(i,j) = 1-\textrm{corr}(i,j)$ a metric?

I need to make sure that this function is a metric: $d(i,j) = 1-\textrm{corr}(i,j)$ where $\textrm{corr}(x,y)$ is the Pearson correlation coefficient which ranges from $[-1,1]$. With this scaling I ...
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27 views

Bounding the Correlation Coefficient

Let us assume we have two random variables $X$ and $Y$ where $X = f(A, B, C)$ and $Y = g(A, B, C)$. $A, B, C$ are 3 independent random variables and the functions $f, g$ are known but rather expensive ...
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58 views

Difference between identity and diagonal covariance matrices

thanks in advance for the help. Suppose I am training a linear model. What are the conceptual differences between using a diagonal covariance matrix and the identity? It is clear to me that the ...
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1answer
19 views

Covariance matrix computed based on a covariance function

I am reading Chapter 4 of Gaussian Processes for Machine Learning. It says that a matrix $K$ whose entries are computed as $k_{ij} = k(x_i, x_j)$ where $k$ is a covariance function is a positive ...
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1answer
68 views

Constraints on correlation coefficients of multiple random variables

I am looking for a generalization of Correlation between three variables question for more than three variables. It is stated in one of the answers there that, for three variables with pairwise ...
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24 views

Test indipendence and stationarity

I have to apply a model on a dataset of $I$ variables, each one with $n$ observations, but I need that every variable is stationary and indipendent from the other ones for the model to work. My ...
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2answers
47 views

How to find relation between 2 numbers

I have been practicing programming for many months now and what I found difficult is not about solving problem. But it is how to find the "how to solve problem" to make computer solves that for me! ...
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0answers
41 views

Error propagation in pearson correlation

I have two data-sets $X$ and $Y$ with errors $\Delta X$ and $\Delta Y$. I calculated the Pearson Sample Correlation $r$. Is it possible to calculate the error of $r$ using propagation of uncertainty: ...
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13 views

Methods for Uncorrelating data - Comparison

I see that both PCA and Cholesky Decomposition could be used for uncorrelating correlated data. When should one be used? What are the assumptions made by each model. When do the methods fail? Are ...
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2answers
46 views

Correlation of random variables with joint PDF proportional to $x^{a-1}y^{b-1}(1-x-y)^{c-1} $

The random variables $X$ and $Y$ have joint PDF $$f(x,y)= \frac{\Gamma(a+b+c)}{\Gamma(a)\Gamma(b)\Gamma(c)}x^{a-1}y^{b-1}(1-x-y)^{c-1} $$ where $0 \leq x \leq 1 , 0 \leq y \leq 1, x+y < 1 $ where ...
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0answers
42 views

Checking hand manipulations of matrices

Beginning with a 4*3 matrix: 5 4 -1 2 3 -3 3 4 -4 1 3 -2 I have to perform four manipulations on it, which I did by hand. I wanted to ask if my thinking and/or ...
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2answers
59 views

What is the intuitive meaning of uncorrelated?

I was going through the derivation of the Kalman filter and it mentions that since noise (v) is uncorrelated to the state (x) and the state estimate (xbar), the following quantity is zero: E((x - ...
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24 views

Exponentially weighted rank ordered correlation matrix

Is there any well-known method to apply exponential weighting (similar to EWMA) to rank ordered correlation matrices such as Kendall tau or Spearman?
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53 views

Generate Correlated Normal and Log-Normal Random Variable

The standard approach for generating two normally distributed random variables some with correlation $\rho$ is explained here: Generate Correlated Normal Random Variables. Now let $X,Y$ be normally ...
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2answers
43 views

Probability of observing a false correlation and confidence limits

In oil and gas exploration/development it is common to use acustic impedance derived from reflection seismic surveys to predict the porosity measured in wells drilled in the reservoir. I often use ...
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1answer
20 views

Re-calculating Value of $100 in Each State by Specific State

I'm using this Tax Foundations graphic for data. How would I re-calculate each state based on a specific state? For example, what if I wanted to base the control state on Missouri, which is $113.51. ...
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89 views

Can Pearson's correlation coefficient be computed for 3d surfaces

I have two functions $f(x,y)=z$ and $g(x,y)=z$ given on discrete intervals $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$... In other words, I have 2 matrices of dimension $3\times n$ representing discrete ...
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33 views

similarities between two binary matrices

I want to measure the similarities between two matrices A and B. Both A and B contains the feature vectors of sounds and are in binary format. i want to see what is the similarities between these two ...
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25 views

Calculate an ensemble average on computer

In a book I am reading, they have the expression $<j_1(u(0)) \, j_1(u(t))>$ where $j_1$ is a first order spherical bessel function. Now I have an expression for $u(t)$ and would like to ...
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1answer
56 views

Determine Patterns

I have some weather data that I would like to analyze. I have about a millions rows of data, and each row has about 100 attribute values. Each attribute value represents some measurement (i.e., ...
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2answers
158 views

Autocorrelation function and power spectral density

I want to get the autocorrelation function of the power spectral density of the wind. This function is defined by: $$s(\omega)=\frac{c_1}{(1+1.5 \times c_2\omega)^{5/3}}$$ $c_1$ and $c_2$ are ...
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1answer
72 views

Regression with Mean, Standard Deviation, Range and Correlation

A research team collected data on students in a statistics course. Their dependent variable was the student’s score on the final examination, which ranged from 200 to 800 points. The observed average ...
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1answer
143 views

Weighing correlation by sample size

I'm a scholar in the humanities trying to not be a complete idiot about statistics. I have a problem relevant to some philological articles I'm writing. To avoid introducing the obscure technicalities ...
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1answer
55 views

Sample Size for Correlation Testing

A research team wishes to test the null hypothesis: $H_0, r=0$ at $\alpha = 0.025$ against the alternative: $H_1, r>0$ using Fisher’s transformation of the Pearson product moment correlation ...
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1answer
25 views

Work out if the relationship between 2 datasets is constant

I have 2 one-dimensional datasets, let's call them a and b. I want to know the correlation between ...
0
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1answer
97 views

Correlated Rayleigh random variable generation

How to generate 4 correlated random variables Rayleigh distributed with parameter "a" and the co-variance matrix is toeplitz (1 rho rho^2 rho^3) where rho is the correlation coefficient
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1answer
76 views

What test should I use to statistically compare two intraclass correlation coefficients (ICC)?

I need to compare two generalizability (G) coefficients for data that are from two separate populations. G coefficients are a type of intraclass correlation coefficient (ICC). The literature on ...
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36 views

Correlation Statistic with a neutral score

I'm trying to develop a correlation statistic to compare lines/random variables (represented as a series of points). I want the correlation value to convey little information if one the lines is a ...
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1answer
94 views

Summing dependent random variables with unknown joint cdf

Suppose that X_1, X_2,... X_5000 are discrete and dependent non-identically distributed random variables, whose marginal distributions are known, but whose joint distribution is not known. Is there ...
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0answers
89 views

Difference between Kendall tau distance and Kendall tau rank correlation coefficient

What is the difference between Kendall tau distance and Kendall tau rank correlation coefficient? Which one I should use to calculate how similar two arbitrary permutations of ranks are? Suppose, we ...
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0answers
31 views

Correlating random numbers seems to skew the data

I am trying to generate a series of correlated random numbers that represent currency exchange rates for a Monte-Carlo simulation. I am attempting to do this via a Cholesky decomposition of the ...
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1answer
38 views

correlation estimator variance

Consider I have realisations of two random variables $X$ and $Y$ and I estimate their correlation thanks to the classic formula : ...