Correlation

::This article is about the correlation coefficient between two random variables. The term correlation can also mean the cross-correlation of two functions or electron correlation in molecular systems.

Pearson's product-moment coefficient

Mathematical properties

The correlation ρxy between two random variables X and Y with expected values μX and μY and standard deviations σX and σY is defined as:

Related Topics:
Random variables - Expected value - Standard deviation

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:

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ho_{xy}={mathrm{cov}(X,Y) over sigma_X sigma_Y} ={E((X-mu_X)(Y-mu_Y)) over sigma_Xsigma_Y}.

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Since μX=E(X),

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σX2=E(X2)-E2(X) and

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likewise for Y, we may also write:

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: ho_{xy}=rac{E(XY)-E(X)E(Y)}{sqrt{E(X^2)-E^2(X)}~sqrt{E(Y^2)-E^2(Y)}}

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The correlation is defined only if both standard deviations are finite and at least one of them is nonzero. It is a corollary of the Cauchy-Schwarz inequality that the correlation cannot exceed 1 in absolute value.

Related Topics:
Cauchy-Schwarz inequality - Absolute value

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The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables. The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.

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If the variables are independent then the correlation is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables. Here is an example: Suppose the random variable X is uniformly distributed on the interval from −1 to 1, and Y = X2. Then Y is completely determined by X, so that X and Y are as far from being independent as two random variables can be, but their correlation is zero; they are uncorrelated. However, in the special case when X and Y are jointly normal, independence is equivalent to uncorrelatedness.

Related Topics:
Independent - Jointly normal

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The sample correlation

If we have a series of n  measurements of X  and Y  written as xi  and yi  where i = 1, 2, ..., n, then the Pearson product-moment correlation coefficient can be used to estimate the correlation of X  and Y . The Pearson coefficient is

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also known as the "sample correlation coefficient". It is especially important if X  and Y  are both normally distributed. The Pearson correlation coefficient is then the best estimate of the correlation of X  and Y . The Pearson correlation coefficient is written:

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:

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r_{xy}=rac{sum (x_i-ar{x})(y_i-ar{y})}{(n-1) s_x s_y}

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where ar{x} and ar{y} are the sample means of xi  and yi , sx  and sy  are the sample standard deviations of xi  and yi  and the sum is from i = 1 to n. As with the population correlation, we may rewrite this as

Related Topics:
Mean - Standard deviation

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:

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r_{xy}=rac{nsum x_iy_i-sum x_isum y_i}

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Again, as is true with the population correlation, the absolute value of the sample correlation must be less than or equal to 1.

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The sample correlation coefficient is the fraction of the variance in yi  that is accounted for by a linear fit of xi  to yi . This is written

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:r_{xy}^2=1-rac{sigma_{y|x}^2}{sigma_y^2}

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where σy|x2  is the square of the error of a linear fit of yi  to xi  by the equation y = a + bx.

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:sigma_{y|x}^2=sum_{i=1}^n (y_i-a-bx_i)^2

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and σy2  is just the variance of y

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:sigma_y^2=sum_{i=1}^n (y_i-ar{y})^2

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Note that since the sample correlation coefficient is symmetric in xi  and yi , we will get the same value for a fit of xi  to yi :

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:r_{xy}^2=1-rac{sigma_{x|y}^2}{sigma_x^2}

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This equation also gives an intuitive idea of the correlation coefficient for higher dimensions. Just as the above described sample correlation coefficient is the fraction of variance accounted for by the fit of a 1-dimensional linear submanifold to a set of 2-dimensional vectors (xi , yi ), so we can define a correlation coefficient for a fit of an m-dimensional linear submanifold to a set of n-dimensional vectors. For example, if we fit a plane z = a + bx + cy  to a set of data (xi , yi , zi ) then the correlation coefficient of z  to x  and y  is

Related Topics:
Dimension - Linear submanifold

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:r^2=1-rac{sigma_{z|xy}^2}{sigma_z^2}.,

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