Page 121 - Standard Handbook Of Petroleum & Natural Gas Engineering

106 Mathematics

A second definition of R is

where n = number of observations on Y
Sx,Sy = biased (n degrees of freedom) estimates of the standard deviations
of X and Y
Note: For small n, even high correlation may not indicate a significant relation-
ship between the variables.

Regression
The relationship between a criterion variable and two or more predictor
variables is given by a linear multivariate model:

Y = bo + b,x, + b,x, + . . . + bpxp
where p = number of predictor variables
x, = ith predictor variable
b, = iLh slope coefficient
bo = intercept coefficient
i = 1,2, . . .,p

The coefficients b, are the partial regression coefficients.
The principle of least squares is used to correlate Y with the XI values. The
error e (or residual) is defined as

e, = ?, - Y,
where ?, = ith predicted value of the criterion variable
Y, = ith measured value of the criterion variable
e = ith error
The purpose of the principle of least squares is to minimize the sum of the
squares of the errors so that

E = minf:(qi - Y)'
i=l
where n = number of observations of the criterion variable (i.e., sample size)

E = c(b, + b,Xi -Yi)*
i=l
By differentiating with respect to bo and b, and setting the equations equal
to zero, two equations in two unknowns are obtained (the summations are for
i = 1, . . .,n)