Partial least squares (PLS) regression [13-15] is a multivariate linear projection method, which used to find the fundamental relations between the predictor matrix 
and the response matrix
. PLS decomposes the matrix of zero-mean variables 
and the matrix of zero-mean variables 
into the form:
where 
is the 
score matrix; 
is the 
loading matrix; 
is the 
residual matrix; 
is the 
score matrix; 
is the 
loading matrix; 
is the 
residual matrix. 
and 
represent information after removing most noise. Based on the correlation between them, the linear regression model can be given by:
In practical, the relationship between predictor matrix and response matrix coming from experimental data is often not linear. Lambert-Beer’s law [
absorbers behaving independently, absorbers being distributed homogenously and low concentrations. Apparent deviations from Lambert-Beer’s law may be caused by chemical and/or physical effects, instrumental effects or both. So non-linearity in NIR spectra may arise from factors such as highly absorbing samples, the multiplicative effect of differences in particle size among samples, nonlinear detector responses, interactions between analytes, etc. In our type of system, the spectral instruments optical scattering, detector responses and high concentration may cause non-linear behavior.
Kernel partial least squares (KPLS) is a novel kernel method developed by Rosipal et al. [17,18]. Briefly speaking, the kernel methods could be performed in two successive steps. The first step is to embed the original data via a nonlinear mapping 
in the input space into a much higher dimensional feature space. The second step is that a linear algorithm is designed to discover the linear relationship in that feature space (see
KPLS is a nonlinear extension of linear PLS in which the input data are transformed into a high-dimensional feature space via the nonlinear mapping
. For example, the mapping 
transforms the 2-D data points into a new 3-D space.
points in 2-D input space. From
The nonlinear transformation effect in KPLS can be completed only by dot product as described in Equation (4):
where 
denotes kernel function, which satisfies Mercer’s theorem [17,18]. There are several kernel functions in common use. In this study we used the Radial Basis Function [22-24] :
where 
is kernel parameter. After kernel function and kernel parameter are determined, 
is the kernel matrix of training set, which is computed and centered by using Equations (6)-(8):
where 
(1 ≤
≤ n, n: the number of training samples, m: the number of wavenumber variables) denotes training set, and 
is a n-dimensional square matrix, in which each element is obtained by computing kernel function between the two training samples.
The algorithm of KPLS can be summarized as follows:
Step 1:
, 
Step 2: Randomly initialize 
Step 3:
, 
Step 4: 
Step 5:
, 
Step 6: Repeat Steps 3-6 until the convergence;
Step 7: residual matrix 
and 
were computed, 
, 
, where 
is a n-dimensional identity matrix;
Step 8: turn to step 3 until the convergence of residual matrix 
and
.
The predicted data of training set are evaluated by using Equation (9):
where 
is formed by the columns of latent vector
. 
is formed by the columns of latent vector
. 
is the response matrix.
For test set, 
is the kernel matrix, which is computed and centered by using Equations (10)-(12).
where 
(1 ≤ v ≤ l, l: the number of test samples) denotes test set. 
is a 
-dimensional matrix, in which each element is obtained by computing kernel function between test samples and training samples.
The predicted data of test set are evaluated by using Equation (13).
If the mode uses 
to be an indicator vector coding two classes: −1 for members of Class A, 1 for members of Class B, a kernel partial least squares-discriminant analysis (KPLS-DA) model is developed. The KPLS-DA model is developed by regression of the predictor matrix 
against the response matrix
.The model based on experimental data is established in order to assign unknown samples to a previously defined sample class based on pattern of its measured features. The threshold is set to an assigned value, and a sample is considered to be categorized correctly if the predicted value lies on the same side of the threshold.
The purpose of this study is merely to establish a rapid detection method for examining malathion residues in water. It simply detects whether there are malathion residues in water, without the demand for the strict linear relationship between absorbance and concentration. So the KPLS-DA method is used to build the model in this study.
The KPLS-DA codes were written by the author according to the algorithm proposed above.