egularised LSE variant for the baseline estimation [Whittaker,

nderson, 1924; Taylor, 1992].

Whittaker-Henderson algorithm

ttaker-Henderson algorithm (WH) is a special case of the ridge

ression algorithm (RLR). The main concept and implementation

has been introduced in the last chapter. The error term used in

amed as the fidelity in WH. Suppose a spectrum is expressed as

of n entries, ࢙ൌሺݏଵ, ݏଶ, ⋯, ݏ௡ሻ. Each entry of this vector is called

ntensity or simply an intensity. As aforementioned, such an

may not represent a real signal or a peak of a chemical. It is

mixture between a baseline intensity and a signal intensity, i.e.,

݁௜, where ܾ௜ stands for the baseline intensity contained in the i^th

and ݁௜ stands for the distance between the i^th signal intensity ݏ௜

h baseline intensity ܾ௜. Such a distance is also called an error. A

atrix expression of this relationship is ܛൌ܊൅܍, where three sets

are expressed by three vectors. Both b and e are unknown. In

o the fidelity definition, which is ܍ൌܛെ܊, WH defines another

ed the smoothness, which is ݀௜ൌܾ௜െܾ௜ିଵ. A vector d is used to

all the smoothness values. WH has employed an objective

by introducing a regularisation constant ߣ to make a balance

the fidelity (regression error) and the smoothness. The WH

function is defined as below,

ܱൌ܍^௧܍൅ߣ܌^௧܌

(5.1)

SE optimisation of this objective function leads to the solution of

wn baseline shown below,

܊ൌሺ۷ ൅ߣ܌^௧܌ሻ^ିଵܛ

(5.2)

e 5.1 shows such an example based on a simulated spectrum,

e baseline (b) is expressed by a solid line and the spectrum (s) is

d by a dotted line. It can be seen that if the baseline has been very