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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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Perfect correlation between two random variables: some clarifications [on hold]

Consider two random variables $X,Y$. $X$ can take value $x_1$ with probability $p$ and $x_2$ with probability $1-p$. $Y$ can take value $y_1$ with probability $p$ and $y_2$ with probability $1-p$. ...
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15 views

Autocorrelation of Geometric Process

I have this scenario where $S[n] = 0$ if an email does not arrive and $S[n] = 1$ if an email arrives in the $n^{th}$ minute. Therefore, $S[n]$ follows a Bernoulli distribution. Now $U[m]$ is the ...
1
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1answer
14 views

Autocorrelation when $m = 0$

I am trying to understand something with respect to autocorrelation. If the process is iid, I can say: $R_{XX}[n, n + m] = E[x[n]] \times E[x[n + m]] \quad \textrm{if} \quad m \ne 0$ But if $m= 0$ ...
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Probability Practice Final Exam Question - University [closed]

Can anybody help me with this problem? You have data on: x~No(mean=2,variance=9) y~No(5,16) with correlation coefficient p=3/8, S= x+y, T=y-x What are the mean and variance of S? What is P[x>y] ...
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1answer
13 views

Why is a pole-polar relation preserved under mappings?

If $P$ and $p$ are pole and polar with respect to a polarity with matrix $C$; then $P’$ and $p’$, their images under a collineation, will be pole and polar with respect to the polarity with matrix $C’$...
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11 views

Auto-correlation function for a time signal

For a time signal $x(t)= \cos^2(2\pi t)$, solve for the Auto-correlation Function. Basically I have been following a guide on how to compute the auto-correlation function and am getting hung-up on ...
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10 views

cross correlation of monotonic signal

I have two monotonic signals and I want to calculate the time delay between them. Say for example my signals are: s1 = [1 2 3 4 5] s2 = [2 3 4 5 6] To me these appear to be monotonic signals, with ...
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5 views

Spearman rank correlation- level of ranks

I have a question concerning the calculation of the Spearman rank correlation. When calculating the formula you need to subtract the different ranks one from another. Why can we do that? What is ...
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1answer
25 views

How to show Sub-independent Random Variables are uncorrelated.

I want to prove the following: If two RVs $X, Y$ are sub-independent, i.e., $\phi_{X+Y}(t) = \phi_X(t)\phi_Y(t), t\in\mathbb{R}$ then $X, Y$are uncorrelated. Keep $Cov(X,Y) = E(XY)-E(X)E(Y) = 0$ in ...
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0answers
18 views

A research paper presents an equation for 'correlation distance between between two vectors'. But I cannot find information its derivation.

Hallo Mathematics StackExchange, I currently trying to pick apart a research paper titled 'Accounting for Label Unscertainty in Machine Learning for Detection of Acute Respiratory Distress Syndrome' (...
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0answers
11 views

stochastic Brownian Motion

Consider the stochastic process enter image description here enter image description here and N(x,y,p)denotes the bivariate standard Gaussian cumulative distribution function with correlation ...
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1answer
23 views

Correlation and Covariance on Standardized X

I am stuck on the following problem: Let $Z_X$ be the standardized $X$, $Z_X=(X-\mu_X)/\sigma_X$, and let $Z_Y$ be the standardized $Y$, $Z_Y=(Y-\mu_Y)/\sigma_Y$. Show that $Corr(X,Y)=Cov(Z_X,Z_Y)=E(...
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1answer
32 views

Calculate covariance and correlation

$X$ is to $Ber(1/2)$ and $Y$ is to $N(0,1)$. Assume $X$ is indep to $Y$. We define $Z=X+Y, W=X-Y$. Find $Cov(Z,W)$ and $Corr(Z,W)$. Firstly, we know some information: $E(X)=1/2, E(Y)=0$ $Var(X)=1/...
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13 views

Finding best match using correlation matrix

Peoplematching provides a data set with pairs of friends answering a list of questions. They have made a correlation matrix that describes the "friendship correlation" between the answers for each ...
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0answers
17 views

A negative correlation property in a random matrix

I am trying to prove the following negative correlation property. (where neither FKG or the BK inequality apply) Any input/idea is much appreciated: Suppose each row of an $n\times n$ matrix is ...
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0answers
9 views

Correlation between movement trajectories

I have the data from an object moving in a 2-D plane. The object does back and forth motion towards a choice from its initial position. I would like to see if the there is some sort of coorelation ...
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0answers
6 views

Addition of correlated probabilities?

Consider a set of random variables $X,Y,Z$. The random variables represent different measurements of an observable $O$ which can e.g., take on three different values $O\in\{1,0,-1\}$. The measurements ...
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27 views

Does negative correlation survive monotone transformation?

Let $X$ and $Y$ be two non-negative random variables and be negative correlated, i.e., $$\mathbb{E}[XY] \leq \mathbb{E}[X]\mathbb{E}[Y].$$ Let $h(\cdot)$ and $g(\cdot)$ be two non-negative, monotone ...
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92 views

Calculate boundries within which $r$ remains above a critical value given correlation coefficient $(x,f(x)) = r$

I'm experimenting with a trading indicator based on the correlation of the closing price of a day and the simple moving average of P periods. When the correlation is above certain threshold, I need to ...
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0answers
9 views

How to find Cross Correlation of two series over time containing periodic trends?

Considering the data in the series is real time in nature and there are periodic trends within the series , how to do a Cross-Correlation in real time so that each periodic trend can be identified?
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1answer
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How is the auto-correlation of vectors defined?

Suppose $v$ is an $n$-ary vector with entries from the set $\{0,1\}$ (i.e. a vector of ones and zeros). A paper I am reading defines the "auto-correlation sequences" $$v*v$$ where $*$ denotes the ...
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1answer
51 views

Find variable that is uncorrelated but not independent

I am given PMF of random variable X. P(X=0) = P(x=1) =0.5. Now there is another RV Y such that Y = XZ. I have to find Z independent of X such that X and Y are uncorrelated but not independent. My ...
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16 views

Relation between Vector Auto-regressive models and correlation matrix

I am generating a multivariate time series using Vector Autoregressive Models- $$X(t) = AX(t-1) + \epsilon$$ where $X \in R^{n \times 1}$, $A \in R^{n \times n}$ and $\epsilon \in R^{n \times 1}$ is a ...
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2answers
30 views

Confusion in Relationship between regression line slope and covariance

In simple linear regression model between RVs $(X,Y)$, the slope $\hat\beta_1$ is given as $$ \hat\beta_1 = \dfrac{\sum_i^N(x-\overline{x})(y - \overline{y})}{\sum_i^N(x - \overline{x})^2} \tag{1} $$ ...
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1answer
37 views

Show that $X$ and $Y$ are independent.

Let $X$ and $Y$ be two discrete random variables with respective probability distributions $\Bbb P(X=x_1)=p_1,\Bbb P(X=x_2) = 1 - p_1$ and $\Bbb P(Y=y_1) = p_2, \Bbb P(Y=y_2)=1-p_2$. Show that if they ...
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0answers
8 views

Visualizing Pearson's Correlation Coefficient via Cosine Similarity

This is part of me trying to understand correlation via vector approach because I am not convinced with just empirical proof of final ratio working out between -1 and 1 as we needed. So was exploring ...
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0answers
9 views

Calculating the auto correlation of sum of two uniformly distributed Stochastic Prosseses

I am trying to calculate the auto-correlation function of a stochastic process defined by: $X(t) = Y_1(t) + Y_2(t).$ Where: $Y_1$~$unif(a_1,b_1) , Y_2$~$unif(a_2,b_2) , Y_1,Y_2$ are independent. I ...
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1answer
34 views

If two random variables have a correlation of $0$, do they have to be independent?

If we have two random variables $X$ and $Y$ that have non-zero standard deviations and their correlation is $0$, are $X$ and $Y$ independent? Say we have two random variables $X$ and $Y$ with a ...
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52 views

Why divide by $N^2$ for Covariance?

I am trying to understand Covariance intuitively. Below is the general formula $$ \mathrm{Cov}(X,Y) = \dfrac{\sum\limits_{i=1}^N(x_i - \overline{x})(y_i - \overline{y})}{N} \tag{1} $$ There is ...
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21 views

What is the normalized product of two correlated Gaussian PDFs?

I want to simulate a sensor, which exhibits a Gaussian error distribution and temporal correlation $\rho$ between measurements. I want to have a formal description what fusing measurements with known ...
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1answer
17 views

How to properly define a space for correlation matrices?

One usually defines the space $$\{M\in\mathrm{Mat}_n(\mathbb R):\forall j, m_{jj}=1,M^\top=M\ge0\}$$ to be the space of correlation matrices. It is clear that every correlation matrix is inside this ...
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1answer
29 views

Finding correlation given variance-covariance matrix

I've been looking all over the internet and have been having trouble finding good uses of a covariance matrix to find the correlation coefficient. I know that, from a simple 2x2 variance-covariance ...
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20 views

Covariance intuition to formula

How we prove below transition mathematically? Double Summation from $$ \dfrac{1}{N^2}\sum_{i=1}^{N}\sum_{j=i+1}^{N}(x_i - x_j)(y_i - y_j) \tag{1} $$ to $$ \dfrac{1}{2N^2}\sum_{i=1}^{N}\sum_{j=1}^...
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How to assess the convergence of a time series towards a periodic signal and assess the sufficiency of the sample size?

I have a sampled temporal signal, result of a transient fluid simulation. At the beginning the flow is being established, so the first few seconds of signal should be ignored (question : is there a ...
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0answers
27 views

Bounds on $R^2$ (or correlation coefficient) for sum of linear models

I am using simple linear models to explore the relationship between a predictor variable $p$ and two target variables $i$ and $r$. I built two models: Model 1 ($M_1$) uses $p$ to predict $i$ and has ...
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3answers
23 views

How to use Bayesian Inference for a large set of data?

I have a set of large data and need to come up with a way to quantify correlation. I am thinking that I should use Bayesian Inference to tackle the problem. The question in mind is to see how the ...
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0answers
10 views

partial Autocorrelation in terms of matrix

today i woud like to know how to compute partial autocorrelation when correlation values are known ?for instance i have following values for autocorrelation ...
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0answers
28 views

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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1answer
57 views

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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12 views

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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0answers
27 views

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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0answers
16 views

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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1answer
69 views

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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2answers
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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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0answers
13 views

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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1answer
24 views

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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55 views

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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0answers
39 views

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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1answer
8 views

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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0answers
23 views

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 $...