PLS회귀를 이용한 포지셔닝맵의 구축

Abstract

Partial Least Squares regression(PLSR) proposed by Herman Wold in 1966 has been used as very valuable method to predict a set of response variables by a set of explanatory variables. PLSR is very useful for building a predictive model when variables are many and highly correlated. Multiple regression analysis also useful tool for building a prediction model. But it has much limitation when variables are many and highly correlated. In such cases, even though we can build a prediction model, it will fail to predict new data well (Tobias, 2007). Especially when the number of variables is much larger than the number of observations, the phenomenon of so-called 'overfitting' occurs. When the explanatory variables are highly correlated, one approach to overcome the problem is to remove the some of highly correlated explanatory variables. Another approach is to reduce the explanatory variables into small number of variables which have no correlations. The concept of PLS is to extract small numbers of latent variables which explain for highly correlated many variables. In that sense, PLS is a indirect modelling. But the way of extracting latent variables is different from the traditional method,The superiority of PLSR to PCR(Principal Component Regression) is very well known. Ryan et. al (1999) showed empirically that PLSR is better than PCR in prediction the response variable. They compared three models with mediators and collinearity among the response variables, for example, regression, PCR, and PLSR. As the hypothesized conceptual model had moderators and collinearity in their study, the regression model was not germane to the research objective. Hence their focus was on the comparison of PCR, with PLSR. Even though the fact that there was a difference in estimating the coefficients between PCR and PLSR was very confusing, But prediction of PLSR was better than PCR. Even though PLSR began in social sciences, it's uses are extended to the various fields like chemometrics (Westerhuis 1998; Wagon & Kowalski 1988; Geladi & Kowalski 1986) or sensory evaluation (Martens & Naes 1989), marketing (Abdi 2003; Chin et al. 2003; Graver, et al 2002; Ryan et al 1999; Fornell and Bookstein 1982; Japal 1982) and design (Han and Yang 2004). Interestingly, similarly to this research, Husson & Pages (2005) proposed the way of corresponding additional variables by the use of PLSR coefficients instead of the linear regression coefficients in Prefmap technique. Huh and colleagues proposed several quantification methods using traditional multivariate data analysis techniques (Kim, 2000; Yang, 1998; Park and Huh 1996a, b; Han, 1995). The quantification methods proposed by them are endeavors to reduce the multivariate data with interrelationship and to represent or to plot them onto the low dimensional space. Projection pursuit stands for those methods. It aims to analyze the characteristics and structure of data through projecting the multivariate data onto the lower dimensional space and through analyzing the projection. In that sense, quantification method means a technique for building map in marketing. The purpose of this research is to propose the algorithm for building positioning map by PLSR. The basis of the algorithm is a singular value decomposition. To derive the form of singular value decomposition, Lagrange multiplier method function was adopted. After components are extracted via singular value decomposition, the relationships between components and variables can be gotten by regressing variables on the components. The regression coefficients are the coordinates of the variables. Additionally we can get score vectors of components for observations from the same process. They are the coordinates of the observations. That is, The variables and observations can be positioned on the simple space generated by PLSR. The quantification technique for PLS method gives us the better understanding of structure of variables and observations. The limitation of this study is the situation when there are more than 2 sets of data. In that case it is very to difficult to solve the Lagrange multiplier method function due to the many constraints in the equation. Thus we should consider another method of extracting the principal components due to the many constraints in the equation.