International Journal of Analytical Chemistry

Volume 2019, Article ID 7314916, 12 pages

https://doi.org/10.1155/2019/7314916

## A Comparison of Sparse Partial Least Squares and Elastic Net in Wavelength Selection on NIR Spectroscopy Data

Correspondence should be addressed to Lun-Zhao Yi; nc.ude.tsumk@oahznuliy

Received 29 April 2019; Revised 23 June 2019; Accepted 2 July 2019; Published 1 August 2019

Academic Editor: Jiu-Ju Feng

Copyright © 2019 Guang-Hui Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Elastic net (Enet) and sparse partial least squares (SPLS) are frequently employed for wavelength selection and model calibration in analysis of near infrared spectroscopy data. Enet and SPLS can perform variable selection and model calibration simultaneously. And they also tend to select wavelength intervals rather than individual wavelengths when the predictors are multicollinear. In this paper, we focus on comparison of Enet and SPLS in interval wavelength selection and model calibration for near infrared spectroscopy data. The results from both simulation and real spectroscopy data show that Enet method tends to select less predictors as key variables than SPLS; thus it gets more parsimony model and brings advantages for model interpretation. SPLS can obtain much lower mean square of prediction error (MSE) than Enet. So SPLS is more suitable when the attention is to get better model fitting accuracy. The above conclusion is still held when coming to performing the strongly correlated NIR spectroscopy data whose predictors present group structures, Enet exhibits more sparse property than SPLS, and the selected predictors (wavelengths) are segmentally successive.

#### 1. Introduction

One of characteristics of near infrared spectroscopy (NIR) data is that the number of predictors is much more than the size of observations. Taking corn data [1] as an example, the number of predictors is up to 700 but the sample size is just 80. Thus a problem in building calibration model for NIR is how to select a set of important predictors among a large number of candidate covariates. Wavelength selection for spectroscopy is a classic topic [2] and many methods have been proposed, such as VIP [3], MWPLS [4, 5], and MC-UVE [6]. A drawback of the above algorithms is that model calibration and wavelength selection are separated into two steps: the calibration model is firstly established and then the variable selection procedures are performed based on the model from the first step. Recently, sparse variable selection methods [7–16] have gained much attention for dealing with high-dimensional data from various fields. One of advantages of sparse methods is that they can perform the model calibration and variable selection simultaneously. In addition, sparse algorithm can shrink some estimation coefficients to exactly zero, thus the predictors corresponding to zero-valued coefficients are eliminated from the original calibration model. This is extremely useful when coming to model interpretation. Nowadays, there are many useful sparse methods for addressing the NIR spectroscopy data [17–23]. In this paper, we focus on two of them: elastic net [17] and sparse partial least squares (SPLS) [18]. Both Enet and SPLS can obtain sparse coefficients by choosing appropriate parameters.

Another feature of NIR spectroscopy is multicollinearity among the predictors. The neighboring predictors are continuous wavelength intervals and they are highly correlated. In this situation, the problem is that which strategy should be accepted when doing the model calibration and wavelength selection? In other words, to select a single wavelength each time or an entire interval of strongly correlated and adjacent wavelengths? On one hand, selecting the entire variable group can obtain better calibration and prediction accuracy compared with selecting single predictor from the group when multicollinearity or high correlation is present in the group variables [24–26]. On the other hand, the interval of wavelengths among which the pairwise correlations are strongly correlated should be regarded as a natural group when this wavelength interval is associated with a particular type of chemical bonding. So those predictors in the same group should be in or out of the calibration model simultaneously. For the above two considerations, the sparse methods for NIR spectroscopy data should be able to handle group variables (wavelength intervals) selection, which is called group effect in [17]. Fortunately, both Enet and SPLS can automatically group the multicollinear predictors and select (or eliminate) the entire predictor group simultaneously from the model. Therefore, Enet and SPLS are two potential powerful methods which are suitable for addressing the NIR spectroscopy data. In fact, many references [27–38] have introduced Enet or SPLS to analysis of NIR spectroscopy data. The purpose of this article is to compare the performance of them when dealing with the NIR spectroscopy data.

The remainder of this paper is organized as follows: Section 2 offers the basic theory of Enet and SPLS. Sections 3 and 4 give the experimental results on simulation data and real data sets, respectively. In Section 5, we give the conclusion and make a brief discussion.

#### 2. Theory of Enet and SPLS

##### 2.1. Sparsity of Enet and SPLS

We consider the following linear model for variable selection and estimation:where is the regression coefficient vector. is usually the Gauss noise, namely, . is the response and is the predictor matrix, where is the predictors. For the simplicity, we also assume that the response variable is centered and the predictors are standardized to have zero mean and unit length, namely, Traditional methods to obtain the regression coefficients in the linear model (1) are ordinary least squares (OLS). The solution of OLS generally has not sparsity (the term “sparsity”, as used here, refers to the linear model (1) having many zero-valued regression coefficients). The OLS is often overfitting and has poor predictive performance when applied to those highly correlated data. To date, there are many ways to deal with this issue. The OLS with the norm constraint, which is called LASSO [7], may be the most important one [39], as LASSO can perform variable selection and estimation simultaneously.

Enet [17] is an improved version of the LASSO by using doubly regularized parameters and can be expressed by the following constrained OLS optimization problem:where and are two nonnegative regularization parameters; is the -norm; and is the -norm. If , Enet is exactly equivalent to LASSO. The scale factor “” should be “” when the predictors are not standardized to have mean zero and -norm one. Enet penalty “” is the combination of -norm and -norm. The -norm constraint induces sparsity; namely, it can shrink those small coefficients being exactly zero. -norm constraint addresses the potential singularity and produces lower prediction error. The Enet constraint can be seen as a mix norm, which is like a fish net (that is why it is called elastic net) (see Figure 1). The Enet ball is a (hyper)cube with corners on the coordinate axes where all but one parameter is exactly zero. It is geometrically easy to see that the loss contours always touches the hypercube in a corner with some of the parameters being exactly zero. So, Enet shrinks some coefficients being exactly zero when the Enet constraint is active.