Correlation

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Several sets of (x, 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. Correlation examples2.svg
Several sets of (x, 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 causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it normally refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the 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 so-called demand curve.

Contents

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 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 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 operations between the tested variables and their respective expected values. Essentially, correlation is the measure of how two or more variables are related to one another. There are several correlation coefficients, often denoted or , measuring the degree of correlation. The most common of these is the 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 – such as Spearman's rank correlation  have been developed to be more robust than Pearson's, that is, more sensitive to nonlinear relationships. [1] [2] [3] Mutual information can also be applied to measure dependence between two variables.

Pearson's product-moment coefficient

Example scatterplots of various datasets with various correlation coefficients. Pearson Correlation Coefficient and associated scatterplots.png
Example scatterplots of various datasets with various correlation coefficients.

Definition

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 deviations. Karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton. [4]

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.

The population correlation coefficient between two random variables and with expected values and and standard deviations and is defined as:

where is the expected value operator, means covariance, and 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 moments is:

Symmetry property

The correlation coefficient is symmetric: . This is verified by the commutative property of multiplication..

Correlation as product

Let random variables and have standard deviations and . Then:

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), [5] and some value in the open interval 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 independent, Pearson's correlation coefficient is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables.

For example, suppose the random variable is symmetrically distributed about zero, and . Then is completely determined by , so that and are perfectly dependent, but their correlation is zero; they are uncorrelated. However, in the special case when and are 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 measurements of the pair indexed by , the sample correlation coefficient can be used to estimate the population Pearson correlation between and . The sample correlation coefficient is defined as

where and are the sample means of and , and and are the corrected sample standard deviations of and .

Equivalent expressions for are

where and are the uncorrected sample standard deviations of and .

If and are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range. [6] For the case of a linear model with a single independent variable, the coefficient of determination (R squared) is the square of , Pearson's product-moment coefficient.

Example

Consider the joint probability distribution of X and Y given in the table below.

y
x
101
001/30
11/301/3

For this joint distribution, the marginal distributions are:

This yields the following expectations and variances:

Therefore:

Rank correlation coefficients

Rank correlation coefficients, such as Spearman's rank correlation coefficient and 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. [7] [8]

To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers :

(0, 1), (10, 100), (101, 500), (102, 2000).

As we go from each pair to the next pair increases, and so does . This relationship is perfect, in the sense that an increase in is always accompanied by an increase in . 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 always decreases when 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. [7] For example, for the three pairs (1, 1) (2, 3) (3, 2) Spearman's coefficient is 1/2, while Kendall's coefficient is 1/3.

Other measures of dependence among random variables

The information given by a correlation coefficient is not enough to define the dependence structure between random variables. [9] 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 distributions 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).

Distance correlation [10] [11] was introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables; zero distance correlation implies independence.

The Randomized Dependence Coefficient [12] is a computationally efficient, copula-based measure of dependence between multivariate random variables. RDC is invariant with respect to non-linear scalings of random variables, is capable of discovering a wide range of functional association patterns and takes value zero at independence.

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

The correlation ratio, 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 between them, while the coefficient of determination generalizes the correlation coefficient to multiple regression.

Sensitivity to the data distribution

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

Pearson/Spearman correlation coefficients between X and Y are shown when the two variables' ranges are unrestricted, and when the range of X is restricted to the interval (0,1). Correlation range dependence.svg
Pearson/Spearman correlation coefficients between X and Y are shown when the two variables' ranges are unrestricted, and when the range of X is restricted to the interval (0,1).

Most correlation measures are sensitive to the manner in which X and 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. [13]

Various correlation measures in use may be undefined for certain joint distributions of X and Y. For example, the Pearson correlation coefficient is defined in terms of moments, and hence will be undefined if the moments are undefined. Measures of dependence based on quantiles are always defined. Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being unbiased, or 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. [14] 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 random variables is the matrix whose entry is . Thus the diagonal entries are all identically unity. If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables for . This applies both to the matrix of population correlations (in which case is the population standard deviation), and to the matrix of sample correlations (in which case denotes the sample standard deviation). Consequently, each is necessarily a positive-semidefinite matrix. Moreover, the correlation matrix is strictly 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 and is the same as the correlation between and .

A correlation matrix appears, for example, in one formula for the 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 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 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.

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 [15] 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. [16]

This sparked interest in the subject, with new theoretical (e.g., computing the nearest correlation matrix with factor structure [17] ) and numerical (e.g. usage the Newton's method for computing the nearest correlation matrix [18] ) results obtained in the subsequent years.

Uncorrelatedness and independence of stochastic processes

Similarly for two stochastic processes and : If they are independent, then they are uncorrelated. [19] :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

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. [20] 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 relations (tautologies), where no causal process exists. 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

Anscombe's quartet: four sets of data with the same correlation of 0.816 Anscombe's quartet 3.svg
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. [21] In particular, if the conditional mean of given , denoted , is not linear in , the correlation coefficient will not fully determine the form of .

The adjacent image shows scatter plots of Anscombe's quartet, a set of four different pairs of variables created by Francis Anscombe. [22] The four variables have the same mean (7.5), variance (4.12), correlation (0.816) and regression line (y = 3 + 0.5x). 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. [4] 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 of random variables follows a bivariate normal distribution, the conditional mean is a linear function of , and the conditional mean is a linear function of . The correlation coefficient between and , along with the marginal means and variances of and , determines this linear relationship:

where and are the expected values of and , respectively, and and are the standard deviations of and , respectively.


The empirical correlation is an estimate of the correlation coefficient . A distribution estimate for is given by

where is the Gaussian hypergeometric function and . This density is both a Bayesian posterior density and an exact optimal confidence distribution density. [23] [24]

See also

Related Research Articles

Autocorrelation Correlation of a signal with a time-shifted copy of itself, as a function of shift

Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.

Variance Statistical measure of how far values spread from their average

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or .

Multivariate normal distribution Generalization of the one-dimensional normal distribution to higher dimensions

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.

In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other,, the covariance is negative. The sign of the covariance therefore shows the tendency in the linear relationship between the variables. The magnitude of the covariance is not easy to interpret because it is not normalized and hence depends on the magnitudes of the variables. The normalized version of the covariance, the correlation coefficient, however, shows by its magnitude the strength of the linear relation.

Covariance matrix Measure of covariance of components of a random vector

In probability theory and statistics, a covariance matrix is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances.

Pearson correlation coefficient Measure of linear correlation

In statistics, the Pearson correlation coefficient ― also known as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ― is a measure of linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviations; thus it is essentially a normalized measurement of the covariance, such that the result always has a value between −1 and 1. As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationship or correlation. As a simple example, one would expect the age and height of a sample of teenagers from a high school to have a Pearson correlation coefficient significantly greater than 0, but less than 1.

Spearmans rank correlation coefficient Nonparametric measure of rank correlation

In statistics, Spearman's rank correlation coefficient or Spearman's ρ, named after Charles Spearman and often denoted by the Greek letter (rho) or as , is a nonparametric measure of rank correlation. It assesses how well the relationship between two variables can be described using a monotonic function.

Homoscedasticity Statistical property

In statistics, a sequence of random variables is homoscedastic if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The spellings homoskedasticity and heteroskedasticity are also frequently used.

Canonical correlation Way of inferring information from cross-covariance matrices

In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors X = (X1, ..., Xn) and Y = (Y1, ..., Ym) of random variables, and there are correlations among the variables, then canonical-correlation analysis will find linear combinations of X and Y which have maximum correlation with each other. T. R. Knapp notes that "virtually all of the commonly encountered parametric tests of significance can be treated as special cases of canonical-correlation analysis, which is the general procedure for investigating the relationships between two sets of variables." The method was first introduced by Harold Hotelling in 1936, although in the context of angles between flats the mathematical concept was published by Jordan in 1875.

In statistics, propagation of uncertainty is the effect of variables' uncertainties on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations which propagate due to the combination of variables in the function.

Cross-correlation Covariance and correlation

In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology. The cross-correlation is similar in nature to the convolution of two functions. In an autocorrelation, which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero, and its size will be the signal energy.

Fisher transformation Statistical transformation

In statistics, the Fisher transformation of a Pearson correlation coefficient is its inverse hyperbolic tangent (artanh). When the sample correlation coefficients r is significant, its distribution is highly skewed, which makes it difficult to estimate confidence intervals and apply tests of significance for the population correlation coefficient ρ. The Fisher transformation solves this problem by yielding a variable whose distribution is approximately normally distributed, with a variance that is stable over different values of r.

In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function of the independent variable.

Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the variance of observations is incorporated into the regression. WLS is also a specialization of generalized least squares.

Simple linear regression Linear regression model with a single explanatory variable

In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable and finds a linear function that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. The adjective simple refers to the fact that the outcome variable is related to a single predictor.

In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. If we are interested in finding to what extent there is a numerical relationship between two variables of interest, using their correlation coefficient will give misleading results if there is another, confounding, variable that is numerically related to both variables of interest. This misleading information can be avoided by controlling for the confounding variable, which is done by computing the partial correlation coefficient. This is precisely the motivation for including other right-side variables in a multiple regression; but while multiple regression gives unbiased results for the effect size, it does not give a numerical value of a measure of the strength of the relationship between the two variables of interest.

In statistics, the RV coefficient is a multivariate generalization of the squared Pearson correlation coefficient. It measures the closeness of two set of points that may each be represented in a matrix.

Financial correlations measure the relationship between the changes of two or more financial variables over time. For example, the prices of equity stocks and fixed interest bonds often move in opposite directions: when investors sell stocks, they often use the proceeds to buy bonds and vice versa. In this case, stock and bond prices are negatively correlated.

In probability theory and statistics, cokurtosis is a measure of how much two random variables change together. Cokurtosis is the fourth standardized cross central moment. If two random variables exhibit a high level of cokurtosis they will tend to undergo extreme positive and negative deviations at the same time.

In statistics, functional correlation is a dimensionality reduction technique used to quantify the correlation and dependence between two variables when the data is functional. Several approaches have been developed to quantify the relation between two functional variables.

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Further reading