二阶矩阵公式相关系数计算公式

2020年9月19日发(作者：黄凯芹)
相关系数计算公式

相关系数计算公式
Statistical correlation coefficient
Due to the statistical correlation coefficient used more frequently, so here
is the use of a few articles introduce these coefficients.
The correlation coefficient: a study of two things (in the data we call the
degree of correlation between the variables).
If there are two variables: X, Y, correlation coefficient obtained by the
meaning can be understood as follows:
(1), when the correlation coefficient is 0, X and Y two variable relationship.
(2), when the value of X increases (decreases), Y value increases (decreases),
the two variables are positive correlation, correlation coefficient between 0
and 1.
(3), when the value of X increases (decreases), the value of Y decreases
(increases), two variables are negatively correlated, the correlation
coefficient between -1.00 and 0.
The absolute value of the correlation coefficient is bigger, stronger
correlations, the correlation coefficient is close to 1 or -1, the higher
degree of correlation, the correlation coefficient is close to 0 and the
correlation is weak.
The related strength normally through the following range of judgment
variables:
The correlation coefficient 0.8-1.0 strong correlation
0.6-0.8 strong correlation
0.4-0.6 medium degree.
0.2-0.4 weak correlation
0.0-0.2 very weakly correlated or not correlated
Pearson (Pearson) correlation coefficient
1, introduction
Pearson is also known as the correlation (or correlation) is a kind of
calculation method of the linear correlation of British statistician Pearson in
twentieth Century.
Suppose there are two variables X, Y, then the Pearson correlation
coefficient between the two variables can be calculated by the following
formula:
A formula:
Formula two:
Formula three:
Formula four:
Four equivalent formulas listed above, where E is the mathematical
expectation, cov said the covariance, N represents the number of variables.
2, scope of application
When the two variables of the standard deviation is not zero, the
correlation coefficient is defined, the correlation coefficient for Pearson:
(1), is the linear relationship between the two variables, are continuous
data.
(2) overall, two variables are normally distributed, or near normal unimodal
distribution.
(3) and the observation values of two variables is in pairs, each pair of
observations are independent of each other.
3, Matlab
Pearson correlation coefficient Matlab (according to the formula four):
[cpp] view plaincopy
Function coeff = myPearson (X, Y)
% of the function of the realization of the Pearson correlation coefficient
calculating operation
%
% input:
% X: numerical sequence input
% Y: numerical sequence input
%
% output:
% coeff: two input numerical sequence X, the correlation coefficient of Y
%
If length (X) ~ = length (Y)
Error (two 'numerical sequence dimension is not equal to');
Return;
End
Fenzi = sum (X * Y) - (sum (X) * sum (Y)) length (X);
(fenmu = sqrt (sum (X.^2) - sum (X) ^2 length (X)) * (sum (Y.^2) - sum (Y)
^2 length (X)));
Coeff = fenzi fenmu;
End% myPearson end function
Calculate the Pearson correlation coefficient function can also be used in
existing Matlab:
[cpp] view plaincopy
Coeff = corr (X, Y);
4, reference content
Spearman Rank (Spielman rank correlation coefficient)
1, introduction
In statistics, Spielman correlation coefficient is named for Charles
Spearman, and often use the Greek symbol (rho) said its value. Spielman
rank correlation coefficient is used to estimate the correlation between the
two variables X and Y, the correlation between variables can be used to
describe the monotone function.
If the two sets of two variable does not have the same two elements, so,
when one of the variables can be expressed as a monotone function well
when another variable (i.e. changes in two variables of the same trend),
between the two variables can reach +1 or -1.
Suppose that two random variables were X, Y (also can be seen as a set of
two), the number of their elements are N, two I
(1<=i<=N) random variables take values respectively with Xi, Yi said. Sort
of X, Y (at the same time as ascending or descending), two ranking
elements set X, y, Xi, Yi elements which are Xi in X and Yi ranking in the Y
ranking. The collection of X, y elements in the corresponding subtraction
to get a list of difference set D, di=xi-yi, 1<=i<=N. Spielman rank
correlation coefficient between random variables X and Y can be obtained
by X, y or D calculation, the calculation methods are as follows:
By ranking difference calculated from D diversity (formula one):
From the top set X, calculated from Y (Spielman rank correlation coefficient
were also considered after ranking two random variables Pearson
correlation coefficient, the following is the actual Pearson calculated the
correlation coefficient X, y) (formula two):
The following is a set of elements in the list of examples of calculation
(calculated only for Spielman rank correlation coefficient)
Note: when the two variables of the same, their ranking is obtained by the
average of their positions.
2, scope of application
Spielman rank correlation coefficient of the data conditions without
Pearson correlation coefficient is strict, as long as the observed values of
two variables is the rating data pairs,
or transformed by continuous variable data level data, regardless of the
overall distribution of the two variables of the form, the size of the sample,
we can use Spielman correlation the coefficient of.
3, Matlab
A source program:
Spielman rank correlation coefficient Matlab (based on ranking difference
diversity D calculated using the above formula)
[cpp] view plaincopy
Function coeff = mySpearman (X, Y)
% of the function used to achieve computing Spielman rank correlation
coefficient
%
% input:
% X: numerical sequence input
% Y: numerical sequence input
%
% output:
% coeff: two input numerical sequence X, the correlation coefficient of Y
If length (X) ~ = length (Y)
Error (two 'numerical sequence dimension is not equal to');
Return;
End
N = length (X);% by the length of the sequence
Xrank = zeros (1, N);% of elements stored in the X list
Yrank = zeros (1, N);% of elements stored in the Y list
% calculated value in Xrank
For I: N = 1
Cont1 = 1; the number of records is higher than the specified element%
Cont2 = -1;% records with specific elements of the same number of
elements
For J: N = 1
If X (I) < X (J)
Cont1 = cont1 + 1;
Elseif X (I) = X (J)
Cont2 = cont2 + 1;
End
End
Xrank (I) = cont1 + mean ([0: cont2]);
End
% calculated value in Yrank
For I: N = 1
Cont1 = 1; the number of records is higher than the specified
element%
Cont2 = -1;% records with specific elements of the same number
of elements
For J: N = 1
If Y (I) < Y (J)
Cont1 = cont1 + 1;
Elseif Y (I) = Y (J)
Cont2 = cont2 + 1;
End
End