which simplifies to

[

n

∑n

∑n i=1 xi

i=1 xi

∑n

i=1 x

(^2) i
][
β 0
β 1
]

[ ∑n
∑ni=1 yi
i=1 xiyi
]
(11 .92)

which is the matrix form of the equations (11.90). This presents an alternative way

to solve for the coefficients β 0 and β 1

Linear Regression

The previous least squares method applied to a straight line fit of data. The ideas

presented can be generalized to fitting data to any linear combination of functions.

Given a set of data points (xi, yi), for i= 1, 2 ,.. ., n , assume a curve fit function of the

form

y=y(x) = β 0 f 0 (x) + β 1 f 1 (x) + β 2 f 2 (x) + ···+βkfk(x) (11 .93)

where β 0 , β 1 ,... , β kare unknown coefficients and f 0 (x), f 1 (x), f 2 (x),... , fk(x) represent

linearly independent functions, called the basis of the representation. Note that for

the previous straight line fit the independent functions f 0 (x) = 1 and f 1 (x) = xwere

used. In general, select any set of independent functions and select the βcoefficients

such that the sum of squares error

E=E(β 0 , β 1 ,... , β k) =

∑n

i=1

(y(xi)−yi)^2

E=E(β 0 , β 1 ,... , β k) =

∑n

i=1

[β 0 f 0 (xi) + β 1 f 1 (xi) + β 2 f 2 (xi) + ···+βkfk(xi)−yi]^2

(11 .94)

is a minimum. The determination of the β-values requires a solution be found from

the set of simultaneous least square equations

∂E

∂β 0 = 0,

∂E

∂β 1 = 0, ···,

∂E

∂β k= 0. (11 .95)

Another way to obtain the system of equations (11.95) is to first represent the data

in the matrix form Aβ=y







f 0 (x 1 ) f 1 (x 1 ) f 2 (x 1 ) ··· fk(x 1 )

f 0 (x 2 ) f 1 (x 2 ) f 2 (x 2 ) ··· fk(x 2 )

..

.

..

.

..

.

... ..

.

f 0 (xn) f 1 (xn) f 2 (xn) ··· fk(xn)















β 0

β 1

β 2

..

.

βk









=







y 1

y 2

..

.

yn







(11 .96)