Information AboutCorrelation |
| CATEGORIES ABOUT CORRELATION | |
| covariance and correlation | |
| experimental design | |
|
In Probability Theory and Statistics , correlation, also called '''correlation coefficient''', indicates the strength and direction of a linear relationship between two Random Variables . In general statistical usage, ''correlation'' or co-relation refers to the departure of two variables from independence. In this broad sense there are several coefficients, measuring the degree of correlation, adapted to the nature of data. A number of different coefficients are used for different situations. The best known is the Pearson Product-moment Correlation Coefficient , which is obtained by dividing the Covariance of the two variables by the product of their Standard Deviation s. Despite its name, it was first introduced by Francis Galton . PEARSON'S PRODUCT-MOMENT COEFFICIENT See Also: Pearson product-moment correlation coefficient Mathematical properties The correlation coefficient ρ''X, Y'' between two Random Variables ''X'' and ''Y'' with Expected Value s μ''X'' and μ''Y'' and Standard Deviation s σ''X'' and σ''Y'' is defined as: : where ''E'' is the Expected Value operator and cov means Covariance . Since μ''X'' = E(''X''), σ''X''2 = E(''X''2) − E2(''X'') and likewise for ''Y'', we may also write : The correlation is defined only if both of the standard deviations are finite and both of them are nonzero. It is a corollary of the Cauchy-Schwarz Inequality that the correlation cannot exceed 1 in Absolute Value . 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. If the variables are . However, in the special case when ''X'' and ''Y'' are Jointly Normal , independence is equivalent to uncorrelatedness. A correlation between two variables is diluted in the presence of measurement error around estimates of one or both variables, in which case Disattenuation provides a more accurate coefficient . 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 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: : where and are the sample Mean s of ''X'' and ''Y'' , ''s''''x'' and ''s''''y'' are the sample Standard Deviation s of ''X'' and ''Y'' and the sum is from ''i'' = 1 to ''n''. As with the population correlation, we may rewrite this as : Again, as is true with the population correlation, the absolute value of the sample correlation must be less than or equal to 1. Though the above formula conveniently suggests a single-pass algorithm for calculating sample correlations, it is notorious for its numerical instability (see below for something more accurate). The square of the sample correlation coefficient, which is also known as the Coefficient Of Determination , is the fraction of the variance in ''yi'' that is accounted for by a linear fit of ''xi'' to ''yi'' . This is written | ||
| Where ''s''<sub>''y''''x''</sub><sup>2</sup>  Is The Square Of The Error Of A | "http://wwwinformationdelightinfo/information/entry/linear_regression" class="copylinks">Linear Regression of ''x<sub>i</sub>''  on ''y<sub>i</sub>''  by the Equation ''y = a + bx'': |
|
| :<math>s {yx}^2 | rac{1}{n-1}\sum_{i=1}^n (y_i-a-bx_i)^2,</math> |
|
| :<math>r {xy}^2 | 1- rac{s_{xy}^2}{s_x^2}</math> |
|
| :<math>r^2 | 1- rac{\sigma_{zxy}^2}{s_z^2}</math> |
|
| <!-- Cos Theta | (X dot Y) / X Y = 293 / sqrt(103 00983) = 0920814711 --> |
|
| :<math> \cos Heta | rac { \bold{x} \cdot \bold{y} } { \left\ \bold{x}
ight\ \left\ \bold{y}
ight\ } = rac { 293 } { \sqrt { 103 } \sqrt { 00983 } } = 0920814711 </math> |
|
| <!-- Cos Theta | (X dot Y) / X Y = 0308 / sqrt(308 000308) = 1 --> |
|
| :<math> \cos Heta | rac { \bold{x} \cdot \bold{y} } { \left\ \bold{x}
ight\ \left\ \bold{y}
ight\ } = rac { 0308 } { \sqrt { 308 } \sqrt { 000308 } } = 1, </math> |
|
| {class | "wikitable" align="right" |
|
|