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{{Short description|Statistical concept}}
{{About|correlation and dependence in statistical data||Correlation (disambiguation)}}
{{Merge from|Correlation coefficient|discuss=Talk:Correlation#Proposed merge of Correlation coefficient into Correlation|date=February 2025}}
[[File:Correlation examples2.svg|thumb|400px|right|Several sets of (''x'',&nbsp;''y'') points, with the [[Pearson correlation coefficient]] of ''x'' and ''y'' for each set. The correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case, the correlation coefficient is undefined because the variance of ''Y'' is zero.]]

In [[statistics]], '''correlation ''' or '''dependence ''' is any statistical relationship, whether [[causality|causal]] or not, between two [[random variable]]s or [[bivariate data]]. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are ''[[line (geometry)|linearly]]'' related.  
Familiar examples of dependent phenomena include the correlation between the [[human height|height]] of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the [[demand curve]].

Correlations are useful because they can indicate a predictive relationship that can be exploited in practice.  For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a [[causality|causal relationship]], because [[extreme weather]] causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., [[correlation does not imply causation]]).

Formally, random variables are ''dependent'' if they do not satisfy a mathematical property of [[independence (probability theory)|probabilistic independence]].  In informal parlance, ''correlation'' is synonymous with ''dependence''. However, when used in a technical sense, correlation refers to any of several specific types of mathematical relationship between [[Conditional expectation|the conditional expectation of one variable given the other is not constant as the conditioning variable changes]]; broadly correlation in this specific sense is used when <math>E(Y|X=x)</math> is related to <math>x</math> in some manner (such as linearly, monotonically, or perhaps according to some particular functional form such as logarithmic). Essentially, correlation is the measure of how two or more variables are related to one another.  There are several [[correlation coefficient]]s, often denoted <math>\rho</math> or <math>r</math>, measuring the degree of correlation.  The most common of these is the ''[[Pearson product-moment correlation coefficient|Pearson correlation coefficient]]'', which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other).  Other correlation coefficients&nbsp;– such as ''[[Spearman's rank correlation coefficient]]'' –&nbsp;have been developed to be more [[robust statistics|robust]] than Pearson's and to detect less structured relationships between variables.<ref>Croxton, Frederick Emory; Cowden, Dudley Johnstone; Klein, Sidney (1968) ''Applied General Statistics'', Pitman. {{ISBN|9780273403159}} (page 625)</ref><ref>Dietrich, Cornelius Frank (1991) ''Uncertainty, Calibration and Probability: The Statistics of Scientific and Industrial Measurement'' 2nd Edition, A. Higler. {{ISBN|9780750300605}} (Page 331)</ref><ref>Aitken, Alexander Craig (1957) ''Statistical Mathematics'' 8th Edition. Oliver & Boyd. {{ISBN|9780050013007}}  (Page 95)</ref> [[Mutual information]] can also be applied to measure dependence between two variables.

==Pearson's product-moment coefficient==
{{Main|Pearson product-moment correlation coefficient}}
[[File:Pearson Correlation Coefficient and associated scatterplots.png|thumb|Example scatterplots of various datasets with various correlation coefficients]]

The most familiar measure of dependence between two quantities is the [[Pearson product-moment correlation coefficient]] (PPMCC), or "Pearson's correlation coefficient", commonly called simply "the correlation coefficient". It is obtained by taking the ratio of the covariance of the two variables in question of our numerical dataset, normalized to the square root of their variances.   Mathematically, one simply divides the [[covariance]] of the two variables by the product of their [[standard deviation]]s. [[Karl Pearson]] developed the coefficient from a similar but slightly different idea by [[Francis Galton]].<ref name="thirteenways">{{cite journal | last1 = Rodgers | first1 = J. L. | last2 = Nicewander | first2 = W. A. | year = 1988 | title = Thirteen ways to look at the correlation coefficient | journal = The American Statistician | volume = 42 | issue = 1| pages = 59–66 | jstor=2685263 | doi=10.1080/00031305.1988.10475524}}</ref>

A Pearson product-moment correlation coefficient attempts to establish a line of best fit through a dataset of two variables by essentially laying out the expected values and the resulting Pearson's correlation coefficient indicates how far away the actual dataset is from the expected values.  Depending on the sign of our Pearson's correlation coefficient, we can end up with either a negative or positive correlation if there is any sort of relationship between the variables of our data set.{{cn|date=November 2023}}

The population correlation coefficient <math>\rho_{X,Y}</math> between two [[random variables]] <math>X</math> and <math>Y</math> with [[expected value]]s <math>\mu_X</math> and <math>\mu_Y</math> and [[standard deviation]]s <math>\sigma_X</math> and <math>\sigma_Y</math> is defined as:

<math display=block>\rho_{X,Y} = \operatorname{corr}(X,Y) = {\operatorname{cov}(X,Y) \over \sigma_X \sigma_Y} = {\operatorname{E}[(X-\mu_X)(Y-\mu_Y)] \over \sigma_X\sigma_Y}, \quad \text{if}\ \sigma_{X}\sigma_{Y}>0.</math>

where <math>\operatorname{E}</math> is the [[expected value]] operator, <math>\operatorname{cov}</math> means [[covariance]], and <math>\operatorname{corr}</math> is a widely used alternative notation for the correlation coefficient. The Pearson correlation is defined only if both standard deviations are finite and positive. An alternative formula purely in terms of [[moment (mathematics)|moments]] is:

<math display=block>\rho_{X,Y} =  {\operatorname{E}(XY)-\operatorname{E}(X)\operatorname{E}(Y)\over \sqrt{\operatorname{E}(X^2)-\operatorname{E}(X)^2}\cdot \sqrt{\operatorname{E}(Y^2)-\operatorname{E}(Y)^2} }</math>

===Correlation and independence===
It is a corollary of the [[Cauchy–Schwarz inequality]] that the [[absolute value]] of the Pearson correlation coefficient is not bigger than 1. Therefore, the value of a correlation coefficient ranges between −1 and +1. The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect inverse (decreasing) linear relationship ('''anti-correlation'''),<ref>Dowdy, S. and Wearden, S. (1983). "Statistics for Research", Wiley. {{ISBN|0-471-08602-9}} pp 230</ref> and some value in the [[open interval]] <math>(-1,1)</math> in all other cases, indicating the degree of [[linear dependence]] between the variables. As it approaches zero there is less of a relationship (closer to uncorrelated). The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.

If the variables are [[statistical independence|independent]], Pearson's correlation coefficient is 0. However, because the correlation coefficient detects only linear dependencies between two variables, the converse is not necessarily true. A correlation coefficient of 0 does not imply that the variables are independent{{Cn|date=May 2024}}.

<math display=block>\begin{align}
X,Y \text{ independent} \quad & \Rightarrow \quad \rho_{X,Y} = 0 \quad (X,Y \text{ uncorrelated})\\
\rho_{X,Y} = 0 \quad (X,Y \text{ uncorrelated})\quad & \nRightarrow \quad X,Y \text{ independent}
\end{align}</math>

For example, suppose the random variable <math>X</math> is symmetrically distributed about zero, and <math>Y=X^2</math>. Then <math>Y</math> is completely determined by <math>X</math>, so that <math>X</math> and <math>Y</math> are perfectly dependent, but their correlation is zero; they are [[uncorrelated]]. However, in the special case when <math>X</math> and <math>Y</math> are [[Joint normality|jointly normal]], uncorrelatedness is equivalent to independence.

Even though uncorrelated data does not necessarily imply independence, one can check if random variables are independent if their [[mutual information]] is 0.


===Sample correlation coefficient===
Given a series of <math>n</math> measurements of the pair <math>(X_i,Y_i)</math> indexed by <math>i=1,\ldots,n</math>, the ''sample correlation coefficient'' can be used to estimate the population Pearson correlation <math>\rho_{X,Y}</math> between <math>X</math> and <math>Y</math>.  The sample correlation coefficient is defined as

:<math>
r_{xy} \quad \overset{\underset{\mathrm{def}}{}}{=} \quad \frac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{(n-1)s_x s_y}
      =\frac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}
            {\sqrt{\sum\limits_{i=1}^n (x_i-\bar{x})^2 \sum\limits_{i=1}^n (y_i-\bar{y})^2}},
</math>

where <math>\overline{x}</math> and <math>\overline{y}</math> are the sample [[arithmetic mean|means]] of <math>X</math> and <math>Y</math>, and <math>s_x</math> and <math>s_y</math> are the [[Standard deviation#Corrected sample standard deviation|corrected sample standard deviations]] of <math>X</math> and <math>Y</math>.

Equivalent expressions for <math>r_{xy}</math> are
:<math>
\begin{align}
r_{xy} &=\frac{\sum x_iy_i-n \bar{x} \bar{y}}{n s'_x s'_y} \\[5pt]
       &=\frac{n\sum x_iy_i-\sum x_i\sum y_i}{\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}.
\end{align}
</math>
where <math>s'_x</math> and <math>s'_y</math> are the [[Standard deviation#Uncorrected sample standard deviation|''uncorrected'' sample standard deviations]] of <math>X</math> and <math>Y</math>.

If <math>x</math> and <math>y</math> are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range.<ref>{{cite journal|last=Francis|first=DP|author2=Coats AJ|author3=Gibson D|title=How high can a correlation coefficient be?|journal=Int J Cardiol|year=1999|volume=69|pages=185–199|doi=10.1016/S0167-5273(99)00028-5|issue=2|pmid=10549842}}</ref> For the case of a linear model with a single independent variable, the [[Coefficient of determination|coefficient of determination (R squared)]] is the square of <math>r_{xy}</math>, Pearson's product-moment coefficient.

===Example===
Consider the [[joint probability distribution]] of {{mvar|X}} and {{mvar|Y}} given in the table below.

:{| class="wikitable" style="text-align:center;"
|+ <math>\mathrm{P}(X=x,Y=y)</math>
! {{diagonal split header|{{mvar|x}}|{{mvar|y}}}}
!&minus;1
!0
!1
|-
!0
|0
|{{sfrac|1|3}}
|0
|-
!1
|{{sfrac|1|3}}
|0
|{{sfrac|1|3}}
|}

For this joint distribution, the [[marginal distribution]]s are:
:<math>\mathrm{P}(X=x)=
\begin{cases}
\frac 1 3 & \quad \text{for } x=0 \\
\frac 2 3 & \quad \text{for } x=1 
\end{cases}
</math>

:<math>\mathrm{P}(Y=y)=
\begin{cases}
\frac 1 3 & \quad \text{for } y=-1 \\
\frac 1 3 & \quad \text{for } y=0 \\
\frac 1 3 & \quad \text{for } y=1 
\end{cases}
</math>

This yields the following expectations and variances:
:<math>\mu_X = \frac 2 3</math>
:<math>\mu_Y = 0</math>
:<math>\sigma_X^2 = \frac 2 9</math>
:<math>\sigma_Y^2 = \frac 2 3</math>

Therefore:

: <math>
\begin{align}
\rho_{X,Y} & = \frac{1}{\sigma_X \sigma_Y} \mathrm{E}[(X-\mu_X)(Y-\mu_Y)] \\[5pt]
& = \frac{1}{\sigma_X \sigma_Y} \sum_{x,y}{(x-\mu_X)(y-\mu_Y) \mathrm{P}(X=x,Y=y)} \\[5pt]
& = \frac{3\sqrt{3}}{2}\left(\left(1-\frac 2 3\right)(-1-0)\frac{1}{3} + \left(0-\frac 2 3\right)(0-0)\frac{1}{3} + \left(1-\frac 2 3\right)(1-0)\frac{1}{3}\right) = 0.
\end{align}
</math>

==Rank correlation coefficients==
{{Main|Spearman's rank correlation coefficient|Kendall tau rank correlation coefficient}}

[[Rank correlation]] coefficients, such as [[Spearman's rank correlation coefficient]] and [[Kendall's tau|Kendall's rank correlation coefficient (τ)]] measure the extent to which, as one variable increases, the other variable tends to increase, without requiring that increase to be represented by a linear relationship. If, as the one variable increases, the other ''decreases'', the rank correlation coefficients will be negative. It is common to regard these rank correlation coefficients as alternatives to Pearson's coefficient, used either to reduce the amount of calculation or to make the coefficient less sensitive to non-normality in distributions. However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the [[Pearson product-moment correlation coefficient]], and are best seen as measures of a different type of association, rather than as an alternative measure of the population correlation coefficient.<ref name="Yule and Kendall">Yule, G.U and Kendall, M.G. (1950), "An Introduction to the Theory of Statistics", 14th Edition (5th Impression 1968). Charles Griffin & Co. pp 258–270</ref><ref name="Kendall Rank Correlation Methods">Kendall, M. G. (1955) "Rank Correlation Methods", Charles Griffin & Co.</ref>

To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers <math>(x,y)</math>:

:(0,&nbsp;1), (10,&nbsp;100), (101,&nbsp;500), (102,&nbsp;2000).

As we go from each pair to the next pair <math>x</math> increases, and so does <math>y</math>. This relationship is perfect, in the sense that an increase in <math>x</math> is ''always'' accompanied by an increase in <math>y</math>. This means that we have a perfect rank correlation, and both Spearman's and Kendall's correlation coefficients are 1, whereas in this example Pearson product-moment correlation coefficient is 0.7544, indicating that the points are far from lying on a straight line. In the same way if <math>y</math> always ''decreases'' when <math>x</math> ''increases'', the rank correlation coefficients will be −1, while the Pearson product-moment correlation coefficient may or may not be close to −1, depending on how close the points are to a straight line.  Although in the extreme cases of perfect rank correlation the two coefficients are both equal (being both +1 or both −1), this is not generally the case, and so values of the two coefficients cannot meaningfully be compared.<ref name="Yule and Kendall"/> For example, for the three pairs (1,&nbsp;1) (2,&nbsp;3) (3,&nbsp;2) Spearman's coefficient is 1/2, while Kendall's coefficient is&nbsp;1/3.

==Other measures of dependence among random variables==
{{See also|Pearson product-moment correlation coefficient#Variants}}

The information given by a correlation coefficient is not enough to define the dependence structure between random variables. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is a [[multivariate normal distribution]]. (See diagram above.) In the case of [[elliptical distribution]]s it characterizes the (hyper-)ellipses of equal density; however, it does not completely characterize the dependence structure (for example, a [[multivariate t-distribution]]'s degrees of freedom determine the level of tail dependence). 

For continuous variables, multiple alternative measures of dependence were introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables (see <ref name = karch>{{cite journal | last1 = Karch | first1 = Julian D. | last2 = Perez-Alonso | first2 = Andres F. | last3 = Bergsma | first3 = Wicher P. | title = Beyond Pearson's Correlation: Modern Nonparametric Independence Tests for Psychological Research | journal = Multivariate Behavioral Research | doi = 10.1080/00273171.2024.2347960 | date = 2024-08-04 | volume = 59 | issue = 5 | pages = 957–977 | pmid = 39097830 | hdl = 1887/4108931 | hdl-access = free }}</ref> and reference references therein for an overview). They all share the important property that a value of zero implies independence. This led some authors <ref name = karch/><ref>{{cite arXiv | last1 = Simon | first1 = Noah | last2 = Tibshirani | first2 = Robert | title = Comment on "Detecting Novel Associations In Large Data Sets" by Reshef Et Al, Science Dec 16, 2011 | year = 2014 | eprint = 1401.7645 | class = stat.ME | pages = 3 }}</ref> to recommend their routine usage, particularly of [[distance correlation]].<ref>{{cite journal | last1 = Székely | first1 = G. J. Rizzo | last2 = Bakirov | first2 = N. K. | year = 2007 | title = Measuring and testing independence by correlation of distances | journal = [[Annals of Statistics]] | volume = 35 | issue = 6| pages = 2769–2794 | doi = 10.1214/009053607000000505 | arxiv = 0803.4101 | s2cid = 5661488 }}</ref><ref>{{cite journal | last1 = Székely | first1 = G. J. | last2 = Rizzo | first2 = M. L. | year = 2009 | title = Brownian distance covariance | journal = Annals of Applied Statistics | volume = 3 | issue = 4| pages = 1233–1303 | doi = 10.1214/09-AOAS312 | pmid = 20574547 | pmc = 2889501 | arxiv = 1010.0297 }}</ref> Another alternative measure is the Randomized Dependence Coefficient.<ref>Lopez-Paz D. and Hennig P. and Schölkopf B. (2013). "The Randomized Dependence Coefficient", "[[Conference on Neural Information Processing Systems]]" [http://papers.nips.cc/paper/5138-the-randomized-dependence-coefficient.pdf Reprint]</ref> The RDC is a computationally efficient, [[Copula (probability theory)|copula]]-based measure of dependence between multivariate random variables and is invariant with respect to non-linear scalings of random variables.

One important disadvantage of the alternative, more general measures is that, when used to test whether two variables are associated, they tend to have lower power compared to Pearson's correlation when the data follow a multivariate normal distribution.<ref name=karch/> This is an implication of the [[No free lunch theorem]]. To detect all kinds of relationships, these measures have to sacrifice power on other relationships, particularly for the important special case of a linear relationship with Gaussian marginals, for which Pearson's correlation is optimal. Another problem concerns interpretation. While Person's correlation can be interpreted for all values, the alternative measures can generally only be interpreted meaningfully at the extremes.<ref>{{cite journal | last1 = Reimherr | first1 = Matthew | last2 = Nicolae | first2 = Dan L. | title = On Quantifying Dependence: A Framework for Developing Interpretable Measures | journal = Statistical Science | volume = 28 | issue = 1 | pages = 116–130 | year = 2013 | doi = 10.1214/12-STS405 | arxiv = 1302.5233 }}</ref>

For two [[binary data|binary variables]], the [[odds ratio]] measures their dependence, and takes range non-negative numbers, possibly infinity: {{tmath|[0, +\infty]}}. Related statistics such as [[Yule's Y|Yule's ''Y'']] and [[Yule's Q|Yule's ''Q'']] normalize this to the correlation-like range {{tmath|[-1, 1]}}. The odds ratio is generalized by the [[logistic regression|logistic model]] to model cases where the dependent variables are discrete and there may be one or more independent variables.

The [[correlation ratio]], [[Entropy (information theory)|entropy]]-based [[mutual information]], [[total correlation]], [[dual total correlation]] and [[polychoric correlation]] are all also capable of detecting more general dependencies, as is consideration of the [[copula (statistics)|copula]] between them, while the [[coefficient of determination]] generalizes the correlation coefficient to [[multiple regression]].

==Sensitivity to the data distribution==
{{Further|Pearson product-moment correlation coefficient#Sensitivity to the data distribution}}

The degree of dependence between variables {{mvar|X}} and {{mvar|Y}} does not depend on the scale on which the variables are expressed.   That is, if we are analyzing the relationship between {{mvar|X}} and {{mvar|Y}}, most correlation measures are unaffected by transforming {{mvar|X}} to {{math|''a'' + ''bX''}} and {{mvar|Y}} to {{mvar|''c'' + ''dY''}}, where ''a'', ''b'', ''c'', and ''d'' are constants (''b'' and ''d'' being positive).  This is true of some correlation [[statistic]]s as well as their [[Population (statistics)|population]] analogues. Some correlation statistics, such as the rank correlation coefficient, are also invariant to [[monotone function|monotone transformations]] of the marginal distributions of {{mvar|X}} and/or {{mvar|Y}}.

[[File:correlation range dependence.svg|300px|right|thumb|[[Pearson product moment correlation coefficient|Pearson]]/[[Spearman's rank correlation coefficient|Spearman]] correlation coefficients between {{mvar|X}} and {{mvar|Y}} are shown when the two variables' ranges are unrestricted, and when the range of {{mvar|X}} is restricted to the interval (0,1).]]Most correlation measures are sensitive to the manner in which {{mvar|X}} and {{mvar|Y}} are sampled.  Dependencies tend to be stronger if viewed over a wider range of values. Thus, if we consider the correlation coefficient between the heights of fathers and their sons over all adult males, and compare it to the same correlation coefficient calculated when the fathers are selected to be between 165 cm and 170 cm in height, the correlation will be weaker in the latter case. Several techniques have been developed that attempt to correct for range restriction in one or both variables, and are commonly used in meta-analysis; the most common are Thorndike's case II and case III equations.<ref>{{cite book|last=Thorndike|first=Robert Ladd|title=Research problems and techniques (Report No. 3)|year=1947|publisher=US Govt. print. off.|location=Washington DC}}</ref>

Various correlation measures in use may be undefined for certain joint distributions of {{mvar|X}} and {{mvar|Y}}.  For example, the Pearson correlation coefficient is defined in terms of [[moment (mathematics)|moments]], and hence will be undefined if the moments are undefined.  Measures of dependence based on [[quantile]]s are always defined.  Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being [[bias of an estimator|unbiased]], or [[consistent estimator|asymptotically consistent]], based on the spatial structure of the population from which the data were sampled.

Sensitivity to the data distribution can be used to an advantage. For example, [[scaled correlation]] is designed to use the sensitivity to the range in order to pick out correlations between fast components of [[time series]].<ref name = "Nikolicetal">{{cite journal | last1 = Nikolić | first1 = D | last2 = Muresan | first2 = RC | last3 = Feng | first3 = W | last4 = Singer | first4 = W | year = 2012 | title = Scaled correlation analysis: a better way to compute a cross-correlogram | journal = European Journal of Neuroscience | volume = 35| issue = 5| pages = 1–21 | doi = 10.1111/j.1460-9568.2011.07987.x | pmid = 22324876 | s2cid = 4694570 }}</ref> By reducing the range of values in a controlled manner, the correlations on long time scale are filtered out and only the correlations on short time scales are revealed.

==Correlation matrices==

The correlation matrix of <math>n</math> random variables <math>X_1,\ldots,X_n</math> is the <math>n \times n</math> matrix <math>C</math> whose <math>(i,j)</math> entry is
:<math>c_{ij}:=\operatorname{corr}(X_i,X_j)=\frac{\operatorname{cov}(X_i,X_j)}{\sigma_{X_i}\sigma_{X_j}},\quad \text{if}\ \sigma_{X_i}\sigma_{X_j}>0.</math>
Thus the diagonal entries are all identically [[unity (number)|one]].  If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the [[covariance matrix]] of the [[standardized variable|standardized random variables]] <math>X_i / \sigma(X_i)</math> for <math>i = 1, \dots, n</math>.  This applies both to the matrix of population correlations (in which case <math>\sigma</math> is the population standard deviation), and to the matrix of sample correlations (in which case <math>\sigma</math> denotes the sample standard deviation). Consequently, each is necessarily a [[positive-semidefinite matrix]]. Moreover, the correlation matrix is strictly [[positive definite matrix|positive definite]] if no variable can have all its values exactly generated as a linear function of the values of the others.

The correlation matrix is symmetric because the correlation between <math>X_i</math> and <math>X_j</math> is the same as the correlation between <math>X_j</math> and <math>X_i</math>.

A correlation matrix appears, for example, in one formula for the [[coefficient of multiple determination#Computation|coefficient of multiple determination]], a measure of goodness of fit in [[multiple regression]].

In [[statistical modelling]], correlation matrices representing the relationships between variables are categorized into different correlation structures, which are distinguished by factors such as the number of parameters required to estimate them. For example, in an [[Exchangeability|exchangeable]] correlation matrix, all pairs of variables are modeled as having the same correlation, so all non-diagonal elements of the matrix are equal to each other. On the other hand, an [[Autoregressive model|autoregressive]] matrix is often used when variables represent a time series, since correlations are likely to be greater when measurements are closer in time. Other examples include independent, unstructured, M-dependent, and [[Toeplitz matrix|Toeplitz]].

In [[exploratory data analysis]], the [[iconography of correlations]] consists in replacing a correlation matrix by a diagram where the "remarkable" correlations are represented by a solid line (positive correlation), or a dotted line (negative correlation).

===Nearest valid correlation matrix===
In some applications (e.g., building data models from only partially observed data) one wants to find the "nearest" correlation matrix to an "approximate" correlation matrix (e.g., a matrix which typically lacks semi-definite positiveness due to the way it has been computed).

In 2002, Higham<ref>{{cite journal|title=Computing the nearest correlation matrix—a problem from finance|journal=IMA Journal of Numerical Analysis|date=2002|first=Nicholas J.|last=Higham|volume=22|issue=3|pages=329–343|doi=10.1093/imanum/22.3.329|citeseerx=10.1.1.661.2180}}</ref> formalized the notion of nearness using the [[Frobenius norm]] and provided a method for computing the nearest correlation matrix using the [[Dykstra's projection algorithm]], of which an implementation is available as an online Web API.<ref>{{Cite web|url=https://portfoliooptimizer.io/|title=Portfolio Optimizer |website=portfoliooptimizer.io|access-date=2021-01-30}}</ref>

This sparked interest in the subject, with new theoretical (e.g., computing the nearest correlation matrix with factor structure<ref>{{cite journal|title=Computing a Nearest Correlation Matrix with Factor Structure.|journal= SIAM J. Matrix Anal. Appl.|date=2010|first1=Rudiger|last1=Borsdorf|first2=Nicholas J.|last2=Higham|first3=Marcos|last3=Raydan|volume=31|issue=5|pages=2603–2622|doi=10.1137/090776718|url= http://eprints.maths.manchester.ac.uk/1523/1/SML002603.pdf}}</ref>) and numerical (e.g. usage the [[Newton's method]] for computing the nearest correlation matrix<ref>{{cite journal|title=A quadratically convergent Newton method for computing the nearest correlation matrix.|journal= SIAM J. Matrix Anal. Appl.|date=2006|first1=HOUDUO|last1=Qi|first2=DEFENG|last2=Sun|volume=28|issue=2|pages=360–385|doi=10.1137/050624509}}</ref>) results obtained in the subsequent years.

==Uncorrelatedness and independence of stochastic processes==
Similarly for two stochastic processes <math>\left\{ X_t \right\}_{t\in\mathcal{T}}</math> and <math>\left\{ Y_t \right\}_{t\in\mathcal{T}}</math>: If they are independent, then they are uncorrelated.<ref name=KunIlPark>{{cite book | author=Park, Kun Il| title=Fundamentals of Probability and Stochastic Processes with Applications to Communications| publisher=Springer | year=2018 | isbn=978-3-319-68074-3}}</ref>{{rp|p. 151}} The opposite of this statement might not be true. Even if two variables are uncorrelated, they might not be independent to each other.

==Common misconceptions==

===Correlation and causality===
{{Main|Correlation does not imply causation}} {{See also|Normally distributed and uncorrelated does not imply independent}}
The conventional dictum that "[[correlation does not imply causation]]" means that correlation cannot be used by itself to infer a causal relationship between the variables.<ref>{{cite journal | last=Aldrich | first=John | journal=Statistical Science | volume=10 | issue=4 | year=1995 | pages=364–376 | title=Correlations Genuine and Spurious in Pearson and Yule | jstor=2246135 | doi=10.1214/ss/1177009870| doi-access=free }}</ref> This dictum should not be taken to mean that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with [[identity (mathematics)|identity]] relations ([[tautology (logic)|tautologies]]), where no causal process exists (e.g., between two variables measuring the same construct). Consequently, a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction).

A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health, or does good health lead to good mood, or both? Or does some other factor underlie both? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.

=== Simple linear correlations ===

[[File:Anscombe's quartet 3.svg|thumb|325px|right|[[Anscombe's quartet]]: four sets of data with the same correlation of 0.816]]
The Pearson correlation coefficient indicates the strength of a ''linear'' relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the [[conditional expectation|conditional mean]] of <math>Y</math> given <math>X</math>, denoted <math>\operatorname{E}(Y \mid X)</math>, is not linear in <math>X</math>, the correlation coefficient will not fully determine the form of <math>\operatorname{E}(Y \mid X)</math>.

The adjacent image shows [[scatter plot]]s of [[Anscombe's quartet]], a set of four different pairs of variables created by [[Francis Anscombe]].<ref>{{cite journal | last=Anscombe | first=Francis J. | year=1973 | title=Graphs in statistical analysis | journal=The American Statistician | volume=27 | issue=1 | pages=17–21 | jstor=2682899 | doi=10.2307/2682899}}</ref> The four <math>y</math> variables have the same mean (7.5), variance (4.12), correlation (0.816) and regression line (<math display="inline">y=3+0.5x</math>). However, as can be seen on the plots, the distribution of the variables is very different. The first one (top left) seems to be distributed normally, and corresponds to what one would expect when considering two variables correlated and following the assumption of normality. The second one (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear. In this case the Pearson correlation coefficient does not indicate that there is an exact functional relationship: only the extent to which that relationship can be approximated by a linear relationship. In the third case (bottom left), the linear relationship is perfect, except for one [[outlier]] which exerts enough influence to lower the correlation coefficient from 1 to 0.816. Finally, the fourth example (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.

These examples indicate that the correlation coefficient, as a [[summary statistic]], cannot replace visual examination of the data. The examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a [[normal distribution]], but this is only partially correct.<ref name="thirteenways"/> The Pearson correlation can be accurately calculated for any distribution that has a finite [[covariance matrix]], which includes most distributions encountered in practice. However, the Pearson correlation coefficient (taken together with the sample mean and variance) is only a [[sufficient statistic]] if the data is drawn from a [[multivariate normal distribution]]. As a result, the Pearson correlation coefficient fully characterizes the relationship between variables if and only if the data are drawn from a multivariate normal distribution.

==Bivariate normal distribution==
If a pair <math>\ (X,Y)\ </math> of random variables follows a [[bivariate normal distribution]], the conditional mean <math>\operatorname{\boldsymbol\mathcal E}(X \mid Y)</math> is a linear function of <math>Y</math>, and the conditional mean <math>\operatorname{\boldsymbol\mathcal E}(Y \mid X)</math> is a linear function of <math>\ X ~.</math> The correlation coefficient <math>\ \rho_{X,Y}\ </math> between <math>\ X\ </math> and <math>\ Y\ ,</math> and the [[Marginal distribution|marginal]] means and variances of <math>\ X\ </math> and <math>\ Y\ </math> determine this linear relationship:

:<math>\operatorname{\boldsymbol\mathcal E}(Y \mid X ) = \operatorname{\boldsymbol\mathcal E}(Y) + \rho_{X,Y} \cdot \sigma_Y \cdot \frac{\ X-\operatorname{\boldsymbol\mathcal E}(X)\ }{ \sigma_X }\ ,</math>

where <math>\operatorname{\boldsymbol\mathcal E}(X)</math> and <math>\operatorname{\boldsymbol\mathcal E}(Y)</math> are the expected values of <math>\ X\ </math> and <math>\ Y\ ,</math> respectively, and <math>\ \sigma_X\ </math> and <math>\ \sigma_Y\ </math> are the standard deviations of <math>\ X\ </math> and <math>\ Y\ ,</math> respectively. 


The empirical correlation <math>r</math> is an [[Estimation|estimate]] of the correlation coefficient <math>\ \rho ~.</math> A distribution estimate for <math>\ \rho\ </math> is given by

: <math display="block">\pi ( \rho \mid r ) =
\frac{\ \Gamma(N)\ }{\ \sqrt{ 2\pi\ } \cdot
\Gamma( N - \tfrac{\ 1\ }{ 2 } )\ } \cdot
\bigl( 1 - r^2 \bigr)^{ \frac{\ N\ - 2\ }{ 2 } } \cdot
\bigl( 1 - \rho^2 \bigr)^{ \frac{\ N - 3\ }{ 2 } } \cdot
\bigl( 1 - r \rho \bigr)^{ - N + \frac{\ 3 \ }{ 2 } } \cdot F_\mathsf{Hyp} \left(\ \tfrac{\ 3\ }{ 2 } , -\tfrac{\ 1\ }{ 2 } ; N - \tfrac{\ 1\ }{ 2 } ; \frac{\ 1 + r \rho\ }{ 2 }\ \right)\ </math>

where <math>\ F_\mathsf{Hyp} \ </math> is the [[Gaussian hypergeometric function]].

This density is both a Bayesian [[posterior probability|posterior]] density and an exact optimal [[confidence distribution]] density.<ref>{{cite journal |last=Taraldsen |first=Gunnar |date=2021 |title=The confidence density for correlation |journal=Sankhya A |volume=85 |pages=600–616 |lang=en |s2cid=244594067 |issn=0976-8378 |doi=10.1007/s13171-021-00267-y |doi-access=free|hdl=11250/3133125 |hdl-access=free }}</ref><ref>{{cite report |last=Taraldsen |first=Gunnar |date=2020 |title=Confidence in correlation |lang=en |type=preprint |doi=10.13140/RG.2.2.23673.49769 |website=researchgate.net |url=http://rgdoi.net/10.13140/RG.2.2.23673.49769}}</ref>

==See also==
{{Portal|Mathematics}}
{{Further|Correlation (disambiguation)}}
{{cols|colwidth=26em}}
* [[Autocorrelation]]
* [[Canonical correlation]]
* [[Coefficient of determination]]
* [[Cointegration]]
* [[Concordance correlation coefficient]]
* [[Cophenetic correlation]]
* [[Correlation disattenuation]]
* [[Correlation function]]
* [[Correlation gap]]
* [[Covariance]]
* [[Covariance and correlation]]
* [[Cross-correlation]]
* [[Ecological correlation]]
* [[Fraction of variance unexplained]]
* [[Genetic correlation]]
* [[Goodman and Kruskal's lambda]]
* [[Iconography of correlations]]
* [[Illusory correlation]]
* [[Interclass correlation]]
* [[Intraclass correlation]]
* [[Lift (data mining)]]
* [[Mean dependence]]
* [[Modifiable areal unit problem]]
* [[Multiple correlation]]
* [[Point-biserial correlation coefficient]]
* [[Quadrant count ratio]]
* [[Spurious correlation]]
* [[Correlation ratio|Statistical correlation ratio]]
* [[Subindependence]]
{{colend}}

==References==
{{Reflist|colwidth=35em}}

==Further reading==
* {{cite book |author=Cohen, J. |author2=Cohen P. |author3=West, S.G. |author4=Aiken, L.S.|author4-link= Leona S. Aiken |name-list-style=amp |year=2002 |title=Applied multiple regression/correlation analysis for the behavioral sciences |edition=3rd |publisher=Psychology Press |isbn= 978-0-8058-2223-6 }}
* {{springer|title=Correlation (in statistics)|id=p/c026560}}
* {{cite book|last1=Oestreicher|first1=J. & D. R.|title=Plague of Equals: A science thriller of international disease, politics and drug discovery|date=February 26, 2015|publisher=Omega Cat Press|location=California|isbn=978-0963175540|pages=408}}

== External links ==
{{Wiktionary|correlation|dependence}}
{{Commons category|Correlation}}
{{Wikiversity|Correlation}}
* [http://mathworld.wolfram.com/CorrelationCoefficient.html MathWorld page on the (cross-)correlation coefficient/s of a sample]
* [http://peaks.informatik.uni-erlangen.de/cgi-bin/usignificance.cgi Compute significance between two correlations], for the comparison of two correlation values.
* {{cite web|url=http://www.mathworks.com/matlabcentral/fileexchange/20846|title=A MATLAB Toolbox for computing Weighted Correlation Coefficients|archive-url=https://web.archive.org/web/20210424091029/https://www.mathworks.com/matlabcentral/fileexchange/20846-weighted-correlation-matrix|archive-date=24 April 2021}}
* [https://www.scribd.com/doc/299546673/Proof-that-the-Sample-Bivariate-Correlation-has-limits-plus-or-minus-1 Proof that the Sample Bivariate Correlation has limits plus or minus 1]
* [http://nagysandor.eu/AsimovTeka/correlation_en/index.html Interactive Flash simulation on the correlation of two normally distributed variables] by Juha Puranen.
* [https://web.archive.org/web/20150407112430/http://www.biostat.katerynakon.in.ua/en/association/correlation.html Correlation analysis. Biomedical Statistics]
* R-Psychologist [https://rpsychologist.com/correlation/ Correlation] visualization of correlation between two numeric variables 

{{Statistics |correlation}}
{{Authority control}}

{{DEFAULTSORT:Correlation And Dependence}}
[[Category:Covariance and correlation]]
[[Category:Dimensionless numbers]]