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

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Determinant of a N symmetric square matrix with diagonal 1

What is the determinant of a symmetric $n \times n$ matrix with all diagonals be 1 and all others are $\rho$ (yes correlation matrix)? Anyone can tell me a method to work it out elegantly? Thanks!
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486 views

Eigenvalue decomposition of block covariance matrix for Canonical Correlation Analysis (CCA)

Edited: My question is related to a tutorial I was reading. The covariance matrix is a block matrix where $C_{xx}$ and $C_{yy}$ are within-set covariance matrices and $C_{xy} = C_{yx}^T$ are ...
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1answer
110 views

Independence of quadratic forms

Let us consider the quadratic form $$q_1 = \mathbf{x}_1^\mathrm{H} \mathbf{A}\,\mathbf{x}_1 $$ and the quadratic form $$ q_2= \mathbf{x}_2^\mathrm{H} \mathbf{A}\, \mathbf{x}_2 $$ where ...
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3answers
148 views

Prove dot product will get maximum

If we have two identical sets $A_1 = A_2 $, and we were asked to get the maximum sum of multiplying one distinct element from $A_1$ by another distinct element of $A_2$, for all elements in $A_1$. ...
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0answers
144 views

Correlated diffusion processes and covariance matrix

I'm really noob in maths topics so I hope you will excuse me if I use terms which aren't correct. I would like to simulate $n$ dimensional diffusion processes with $n$ noises. Each process has its ...
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2answers
192 views

Correlation and squared variables

According to my textbooks if two variables are uncorrelated, they are not necessarily independent (unless they are normally distributed). My question is, are 2 variables still not independent if they ...
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1answer
62 views

PCA vs Correlation

What is the relationship between (first) principal component(s) and the correlation matrix or the average correlation of the data. For example, in an empirical application I observe that the average ...
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1answer
180 views

Difference between positively correlated graph and a linear relationship?

So I understand the differences between positive and negative correlation and so on. However what I don't understand are the similarities and differences between a positive correlation and a linear ...
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3answers
83 views

Product of standard deviations

If $u=cx+dy$ and $v=cx-dy$ and $R$ is the co-efficient of correlation between variables $x$ and $y$ and variables $u$ and $v$ have 0 correlation, then how can I prove that ...
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1answer
447 views

Calculate a whitening matrix without using inverses?

Consider a random column vector $\mathbf{x}$, of dimension $m$. That is, it is a random vector, composed of $m$ random variables. The PDF of the random vector $\mathbf{x}$ is thus the joint-PDF of its ...
2
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1answer
89 views

Nonlinear regression with correlated errors

it's my first post here and I'm a newbie in statistics, so please forgive me if I'm doing something wrong or explaining myself badly. Anyway, I have a problem similar to this: How to perform ...
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0answers
130 views

How to perform nonlinear regression with correlated errors?

I have a nonlinear least squares problem, but the errors are correlated. I could use R's nls function to do the regression if the errors were independent, but I don't know the right way to handle ...
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111 views

Windowed Linear Correlation

$\DeclareMathOperator \Cov {Cov}$ $\DeclareMathOperator \Var {Var}$ $\DeclareMathOperator \E {E}$ Consider the following experiment: For $N\geq1$, consider $N$ black balls. Let us paint each black ...
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1answer
324 views

Necessary and sufficient conditions for a matrix to be a valid correlation matrix.

It's not too hard to see that any correlation matrix must have certain properties, such as all entries in the range -1 to 1, symmetric, positive semi-definite (excluding pathological cases like ...
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2answers
133 views

Correlation in errors

I'm not good in statistics, so please excuse my noob question. We want to ask a question from people (say what is $2+2$). They might make mistake. We assume that they give the correct answer with the ...
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1answer
218 views

Pearson Product Moment Correlation Coefficient for linear relationship only?

Pearson Product Moment Correlation Coefficient method is used only if variables are linearly correlated. But if they are linearly correlated, then correlation coefficient $$r=\pm 1$$ only. Then why we ...
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1answer
7k views

Generating correlated random numbers: Why does Cholesky decomposition work?

Let's say I want to generate correlated random variables. I understand that I can use Cholesky decomposition of the correlation matrix to obtain the correlated values. If $C$ is the correlation ...
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2answers
92 views

Proving that the magnitude of the sample correlation coefficient is at most $1$

How can you show that the magnitude of the sample correlation coefficient is at most $1$? The formula is huge, I'm not even sure how to approach this. Can anyone point me in the right direction? ...
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0answers
33 views

Fast approximate construction of orthogonal system

Assuming I have $d+1\in\mathbb{R}^d$ points that are not unfortunately chosen (in which case I can just resample, correct me if I $d$ points are enough), then these should span the whole space. What ...
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27 views

Dynamic Light Scattering

In DLS does the combination of the Time Average and Ensemble Average give a better statistical average than the results shown by each case considered separately ?
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1answer
650 views

Probability that one normal distribution is greater than the other when they are correlated i.e (Not Independent)

Would like to know how to approach this question: Probability that one normal distribution is greater than the other when they are correlated i.e (Not Independent). Seen the solution for the ...
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1answer
53 views

What is a common way to measure the “goodness of fit” of an individual data point to a correlation?

Let's say I have a collection of data points (X & Y values) that show some correlation when, eg, Pearson's correlation formula is applied. What is a good measure for determining which data points ...
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1answer
290 views

Correlation Coefficients with Unequal Data Sets

I would like to determine the correlation coefficients of two timesets of data with unequal entries to detemine how often they move together. I am performing the following computation but would like ...
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2answers
2k views

Correlations between 3 random variables

I have a question as follows: The correlation coefficients between three random variables are x, y, and z respectively. What relation do x, y, and z have to satisfy? Can someone help me? Thanks very ...
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1answer
64 views

Given series $A$ and a correlation, is it possible to randomly calculate a fitting series $B$?

With reference to the original thread on Stackexchange, my question is as follows. Usually, one would enter two value-series and a script or program calculates the correlation. For instance, with $x ...
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0answers
244 views

How to find a correlation and whether it is statistically significant

I have 2 columns with numers in them, x and y. I'd like to find out what y could be if x = n and how strong the correlation is, if it's statistically significant or not. How do I do that? Pretty ...
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1answer
301 views

MATLAB's implementation of cross correlation

Wikipedia gives the cross-correlation as $$ \begin{align*} (f \star g)[n] = \sum^{\infty}_{m = -\infty} f^{*}[m] g[n+m] \end{align*} $$ MATLAB's documentation gives ...
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2k views

Correlation Coefficient and Determination Coefficient

I'm really new to linear regression and am trying to teach myself. In my textbook there's a problem that asks why R^2 in the regression of Y on X = the sample correlation between X and Y the whole ...
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1answer
80 views

constructing “pseudonoise” sequences other than (2^n)-1? (low cyclical autocorrelation)

Pseudonoise LFSR sequences of length $N = 2^k-1$ have the nice property that their cyclical autocorrelation is $N$ when the sequence is lined up with itself, and $-1$ elsewhere. Is there a way to ...
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2answers
86 views

Independence of Random Variables (kernel ICA)

In the paper Bach, F. R., & Jordan, M. I. (2002). Kernel Independent Component Analysis. Journal of Machine Learning Research, 3(1), 1-48. doi:10.1162/153244303768966085 I stumpled upon ...
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0answers
739 views

Pseudo-random binary sequence generated by shift register

Binary sequence generated by shift register with feedback have periodic properties. A simple 4-bit shift register shown in Fig (a). For the initial condition shown, it can be verified that the ...
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3answers
2k views

Correlation between two linear sums of random variables

I understand how to create random variables with a prespecified correlational structure using a Cholsesky decomposition. But I would like to be able to solve the inverse problem: Given random ...
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0answers
49 views

Finding out whether or not two graphs are “close”, given 20 points on each graph whose Xs do not match

Let's assume we have two graphs where: Each graph has 600 points per minute. We're allowed to get only one point per minute. We do not get the same point per minute in both graphs. So for example, ...
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2answers
344 views

Error of Pearson Correlation Coefficient

I have a data set expressed as in the figure here. 'y' is some measured quantity with known error and 'fit' is some attempt to fit a function with zero error. In order to evaluate the quality of the ...
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0answers
153 views

Cross Correlation

The cross-correlation function is defined as follows if $\bar{f}$ is the complex conjugate of $f$ and we assume that $f$ is real, such that $\bar{f} = f$. $$ \begin{align} f \star g &= ...
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1answer
121 views

In search of memorable example of “(Pearson-)uncorrelated $\not\Rightarrow$ independent”

I am looking for an easy-to-remember (and non-trivial) example that vividly illustrates that the "uncorrelatedness" (in the sense of Pearson) of two random variables $X, Y$ does not imply that $X$ and ...
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1answer
97 views

Find 3 normal variables which are linear combinations based on 3 ind std normal variable given a correlation matrix

I am given $3$ normal random variables $X_1$,$X_2$,$X_3$ which are linear combinations of $Z_1$,$Z_2$,$Z_3$. $Z_1$,$Z_2$,$Z_3$ are mutually independent standard normal variables. I am given a ...
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1answer
124 views

Given $Z_1$ and $Z_2$ independent normal variables, find a pair $X_1,X_2$ with correlation $p$

I am given $Z_1$ and $Z_2$ independent standard normal variables, I have to find two random variables $X_1$ and $X_2$ with correlation $\operatorname{corr}(X_1,X_2) = p$, where $p\in (−1, 1)$. Any ...
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1answer
239 views

Calculate Upper Lower bound in $4\times 4$ correlation matrix

Let $X_i$, $i = 1,2,3 4$, be random variables on the same probability space such that $$\begin{align*} \mathrm{corr}(X_1,X_3) &= 0.3;\\ \mathrm{corr}(X_2,X_3) &= 0.1;\\ ...
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2answers
161 views

Autocovariances and Autocorrelations of Processes

I have two questions that are related to calculating autocovariances and autocorrelations of processes. Let's say we have a simple process: $y_t = ε_t + ε_{t-1}$ $ε_{t} =$ standard normal random ...
2
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0answers
214 views

Geometric interpretation of element by element division of one vector by another

This is my first post here, and I'm not a mathematician, so please go easy on me :) In statistics there is a geometric interpretation of correlation that uses basic vector geometry. This is fairly ...
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2answers
16k views

Determining variance from sum of two random correlated variables

I understand that the variance of the sum of two independent normally distributed random variables is the sum of the variances, but how does this change when the two random variables are correlated? ...
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3answers
465 views

Correlation between variables

I asked this question on stats SE but did not find a suitable answer so far. Maybe someone can help. Given n random variables x1,...,xn (one-dimensional). The following is known (corr() = Pearson ...
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2answers
60 views

Relationship between x and y

Studying a financial model from a third party we see: 36month - 1.2 multipler 48month - 1.3333 multipler 60month - 1.5 multipler I am trying to determine if ...
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1answer
284 views

Use Pearson's correlation coefficient on a matrix

I have a problem to interpret the following formula which is said to be the Pearson's correlation coefficient: $$r = \frac{N \left(\sum XY\right) - \left(\sum X\right) \left(\sum ...
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2answers
2k views

If two Gaussian random variables are uncorrelated, they are statistically independent

I read in a textbook that when two gaussian variables are uncorrelated, then they are statistically independent? How can I prove that?
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0answers
37 views

Sampling a distribution with restrictions: eliminating the correlation between two variables

I have a collection of 400.000+ word-pairs. Each word-pair has an association strength, which is a measure of how related the two words are to each other (as in cow-milk). Each word-pair also has a ...
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1answer
137 views

Correlations between neighboring Voronoi cells

For a sequence $X_1,X_2,X_3,\ldots$ of random variables, what it means to say $X_1$ is correlated with $X_2$ is unambiguous. It may be that the bigger $X_1$ is, the bigger $X_2$ is likely to be. If, ...
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1answer
357 views

Expectation product of pairwise uncorrelated variables

Suppose I have three uncorrelated random variables $X, Y$ and $Z$ (discrete or continuous) such that $$\newcommand{\Cov}{\mathrm{Cov}}\Cov(X,Y)=0;\quad \Cov(Y,Z)=0;\quad \Cov(X,Z)=0 \tag{$\ast$}$$ I ...
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3answers
133 views

correlated or independent

Let $(X_1, X_2)$ be a randomly chosen pair out of $\{1,2, \ldots, 20\}$ (draw without repetition). Are both events $$E_1:=\{X_1 \geq 8\}$$ and $$E_2:=\{X_2 \geq 12\}$$ positive or negative correlated. ...