ATTITUDE ANALYSIS Sample Clauses

ATTITUDE ANALYSIS. 4.1 Principal Component Analysis PCA is a statistical approach that converts a set of observed correlated variables into a group of linearly uncorrelated variables called factors or principal components using orthogonal transformation (Xxxx Xx, 2010). This approach is particularly useful in dealing with attitude questions, as many of the attitudes may be correlated, especially when there is a large number of attitudinal questions. This approach has been widely used for travel behavior application to identify individuals’ attitudes using a large number of responses to the SP surveys (Xxxxx et al., 2018; Xxxx et al., 2019; Xxxxxxxx et al., 2018; Xxxxxxxx et al., 2017; Xxxxxxx et al., 2019; Xxxxx et al., 2014; Xxxxxx et al., 2019; xxx Xxxxxx et al., 2018; Xxxxxxxx et al., 2017; Xxx et al., 2016; Xx & Titheridge, 2017; Xxx et al.,1994). The next section will elaborate on PCA and the procedure to identify attitudes. Consider a data matrix X where each of the N rows represents a respondent’s opinion, and each of the p columns shows different questions in the SP survey. X can be decomposed as X = UDVT using singular value decomposition transformation where U is an N×p orthogonal matrix, the columns are the orthonormal basis of the column space of X. V is a p×p orthogonal matrix and called the principal component directions of X. Finally, D is a p×p diagonal matrix with diagonal entries called the singular values of X. The principal components of X, as shown with green lines in Figure 5, can be computed as zm = Xvm. Each principal component (zm) explains a portion of the j variance in data which is calculated as Var (zj)=d2/N. In the example illustrated in Figure 5, most of the variance is explained by the first principle component and the data can be described with the first principle component, zi,1 , instead of original attributes (xi,1, xi,2). Interestingly, diagonal entries of D has the ordered nature, d1 ≥ d2 ≥ . . . dp ≥ 0, and as a result variance explained by each principal will be smaller as components increases (Xxxx Xx, 2010; Xxxx et al., 2017). Figure 5 Principal component analysis illustration.