Page 521 - Modelling in Transport Phenomena A Conceptual Approach

A.6. REGRESSION AND C0RRE;cATION 501

Now suppose we wish to minimize S with respect to the mean value x,, i.e.,
N N
dS
-- - 0 = - 2 (xi - x,) = 2 Nx, - xi) (A.6-3)
axrn i=l i=l
or.
(A.6-4)

Therefore, the mean value which minimizes the sum of the squares of the deviations
is the arithmetic mean, f.

A.6.3 The Method of Least Squares

The parameters a and b in Eq. (A.6-1) are estimated by the method of least
squares. These values have to be chosen such that the sum of the squares of the
deviations
N
S = [vi - (azi + b)12 (A.6-5)
i=l
is minimum. This is accomplished by differentiating the function S with respect
to a and b, and setting these derivatives equal to zero:
as
- = 0 = - 2 ( yi - u xi - b) xi (A.6-6)
dU
i
(A.6-7)

Equations (A.6-6) and (A.6-7) can be simplified as

(A.6-8)

axxi + Nb = (A.6-9)
i i
Simultaneous solution of Eqs. (A.6-8) and (A.6-9) gives

(A.6-10)

(A.6- 11)

Example A.2 Experimental measurements of the density of benzene vapor ut
563K are given as follows: