p

g

g

stimated peak spectrum was better. The middle panel of Figure

ws this estimation. However, this estimation still had some

at the right side, where a noise curve made signal identification

Lots of the intensities may not be true signals, but artifacts. As

tioned, the spline smoother algorithm requires some update when

with a spectra data set with multiple replicates.

adaptive iterative reweighted penalised least square smoother

ptive iterative reweighted penalised least square algorithm

employs another smoothing mechanism [Zheng, et al., 2010;

t al., 2020]. First, a spectrum is denoted by s and a baseline is

by b. A penalised LSE function is defined as below,

ܱൌሺܛെ܊ሻ^௧ሺܛെ܊ሻ൅ߣሺ۲ܢሻ^௧ሺ۲ܢሻ

(5.8)

is the second-order difference operator and ߣ is the regularisation

The regularisation constant is again used to trade-off between the

r the fidelity (or the regression error) and the smoothness of a

curve. If the length of a spectrum is 6, D is defined as below,

۲ ൌቌ

1

െ2

1

0

0

0

0

1

െ2

1

0

0

0

0

0

0

1

0

െ2

1

1

െ2

0

1

ቍ

(5.9)

olution to the Equation (5.8) is shown below,

܊ൌሺ۷ ൅ߣ۲^࢚۲ሻ^ି૚ܛ

(5.10)

troducing a weight w into the system, the above equation is

as below, where W is a diagonal matrix of w,

܊ൌሺ܅൅ߣ۲^࢚۲ሻ^ି૚܅ܛ

(5.11)

d on this equation, an iterative update process is used in airPLS.

age can be downloaded from GitHub, which was contributed by