p

p

p

filter out peaks to estimate a baseline [Hastie and Tibshirani,

also employs a regularised constant. If a wavenumber variable

analyser) is denoted by w and a baseline function s(w) is at least

ferentiable, the roughness of a function or a curve is defined as

here ݓଵ and ݓଶ are used to define the boundary of a spectrum,

ܴൌන

൫ݏ^"ሺݓሻ൯

ଶ݀ݓ

௪మ

௪భ

(5.4)

order of a spline function s(w) is greater than one, s(w) is thus

ferentiable. Suppose a spline function is shown below,

ݏሺݓሻൌܽݓ^ଷ൅ܾݓ^ଶ൅ܿݓ൅݀

cond derivative is still a continuous function of w,

ݏ^"ሺݓሻൌ6ܽݓ൅2ܾ

ar to WH, the error between a baseline and observed

etric data (intensity) is defined as below, where ݓ௜ is the i^th

nalyser value of a spectrum and ܾ௜ is the baseline intensity

nding to ݓ௜,

ܧൌ෍ሺܾ௜െݏሺݓ௜ሻሻ^ଶ

௡

௜ୀଵ

(5.5)

ularised spline is defined as below, where ߣ is the regularisation

ܱൌ෍ሺܾ௜െݏሺݓ௜ሻሻ^ଶ

௡

௜ୀଵ

൅ߣන

൫ݏ^"ሺݓሻ൯

ଶ݀ݓ

௪మ

௪భ

(5.6)

alue of the regularisation constant requires to be optimised for an

stimation of a baseline for a spectrum