Financial Modelling in Python

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44 Financial Modelling in Python

for j in range(n):

s = 0.0

for jj in range(n):

s += v[jj, j]*tmp[jj]

x[j] = s

return x

4.6 GENERALISED LINEAR LEAST SQUARES

Generalised linear least squares is a method for fitting a set of data points (x

i

, y

i

)

i = 1,..,N

to a

linear combination of basis functions. The general form of this kind of model is

y(x) =

M



k=1

a

k

f

k

(x) (4.9)

where f

1

(x), ..., f

M

(x) are the basis functions. The central idea behind the method is to find

the fitting coefficients a

1

, ..., a

M

by minimising the merit function

χ

2

=

N



i=1



y

i

−



M

k=1

a

k

f

k

(x

i

)

σ

i



2

. (4.10)

The σ

i

represent the measurement error, or equivalently the standard deviation, of the ith data

point. In [5] it is shown that the solution of the above equation can be calculated by solving

the normal equations

M



j=1

α

kj

a

j

= β

k

(4.11)

where

α

kj

=

N



i=1

f

j

(x

i

) f

k

(x

i

)

σ

2

i

(4.12)

and

β

k

=

N



i=1

y

i

f

k

(x

i

)

σ

2

i

. (4.13)

The normal equations can be solved using LU decomposition and backsubstitution but the solu-

tion is susceptible to roundoff error. It is common practice to use singular value decomposition

to solve this problem, and this is the route we have taken.

The implementation of the generalised linear least squares algorithm can be found in the

ppf.math.generalised

least squares module and the details of the implementa-

tion are shown below.

def generalised least squares fit(y, x, sig, fit fos):

tol = ...