### Abstract

This study examines the methodology of "individual scaling method" in order to value each person's vocabulary and viewpoints. "Individual scales" refers to evaluation items made by subjects' own terms. Thus, these scales are different from person to person. In the previous report Part1, we proposed principal component analysis (PCA) method for evaluation data measured by individual scales. In this PCA method, evaluation object is regarded as observations, and individual scales of all subjects is regarded as variables. And, individual difference in vocabulary and viewpoint is described as difference in distribution of factor loadings vectors in principal component space. However, suitable method for analyzing individual differences that are described as such has not yet been proposed because of so-called "rotational indeterminacy" related problem. In this paper, first, when analyzing individual differences based on proposed PCA method, we discuss the problems caused by so-called "rotational indeterminacy". Next, as a solution to this problem, the new method of orthogonal rotation is proposed. This rotation method has the following features. • By the criterion in this method, R-squares matrix of [each person × rotated axes] is directed to simple structure. • By this rotation, individual differences in semantic space is maximized. R-squares is measured by personal-PCR (principal component regression analysis), i.e. PCA is performed by each subject, and then by using their each PCA scores as explanatory variables, multiple regression analysis is performed by each subject. Thus, the number of personal PCA scores is an important option in the analysis. The criteria to be maximized, we propose the following three.These are another important option. a) Sum of variances of columns of R-squares matrix b) Sum of variances of columns of Normalized R-squares matrix divided by row sum. c) Simply, variance of R-squares When replacing R-squares matrix to factor loadings squares matrix, each of the criteria is equivalent to the following well-known criteria. a) Row varimax criterion, b) Normal varimax criterion, c) Quartimax criterion As a case study, the proposed rotation method was applied on evaluation data of townscape in Kanda area. The results obtained and discussion are as follows. 1) In this case of data, the results by criterion c) seems to be most useful. 2) It's recommended to use personal PCA scores of slightly many numbers. In this case, six score is necessary. 3) Rotated axes are interpreted as follows; R1: History and taste, R2: Disorderly-Orderly, R3: Harmony and familiar, R4: Enjoying 4) Among these axes, R3 and R4 are large individual differences of R-squares. About these individual differences, the following tendencies were found. (1) Correlation between R-squares of R3 and R-squares of R4 is negative. Thus, terms to express R3 and terms to express R4, there seem to be not so many people with the both of vocabulary. (2) Subjects majoring in architecture have larger R-squares of R3 than other majors, and have smaller R-squares of R4 than other majors. Furthermore, from the viewpoint of validity of R-square as indicator of explanatory power, diagnostics of the number of personal PCA scores was discussed. Finally, it was noted that there are cases not to require rotation proposed in this paper. And some cases of analysis on such data were presented.

Original language | English |
---|---|

Pages (from-to) | 1-9 |

Number of pages | 9 |

Journal | Journal of Environmental Engineering (Japan) |

Volume | 82 |

Issue number | 731 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

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### Keywords

- Individual differences
- Individual scales method
- Orthogonal rotation
- Principal component analysis
- Quartimax criterion
- Semantic space

### ASJC Scopus subject areas

- Environmental Engineering

### Cite this

**Axes rotation for maximization of individual differences in individual scales method : - Studies on the environmental evaluation by subjects. Own term part 2.** / Kojima, Takaya.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Axes rotation for maximization of individual differences in individual scales method

T2 - - Studies on the environmental evaluation by subjects. Own term part 2

AU - Kojima, Takaya

PY - 2017/1/1

Y1 - 2017/1/1

N2 - This study examines the methodology of "individual scaling method" in order to value each person's vocabulary and viewpoints. "Individual scales" refers to evaluation items made by subjects' own terms. Thus, these scales are different from person to person. In the previous report Part1, we proposed principal component analysis (PCA) method for evaluation data measured by individual scales. In this PCA method, evaluation object is regarded as observations, and individual scales of all subjects is regarded as variables. And, individual difference in vocabulary and viewpoint is described as difference in distribution of factor loadings vectors in principal component space. However, suitable method for analyzing individual differences that are described as such has not yet been proposed because of so-called "rotational indeterminacy" related problem. In this paper, first, when analyzing individual differences based on proposed PCA method, we discuss the problems caused by so-called "rotational indeterminacy". Next, as a solution to this problem, the new method of orthogonal rotation is proposed. This rotation method has the following features. • By the criterion in this method, R-squares matrix of [each person × rotated axes] is directed to simple structure. • By this rotation, individual differences in semantic space is maximized. R-squares is measured by personal-PCR (principal component regression analysis), i.e. PCA is performed by each subject, and then by using their each PCA scores as explanatory variables, multiple regression analysis is performed by each subject. Thus, the number of personal PCA scores is an important option in the analysis. The criteria to be maximized, we propose the following three.These are another important option. a) Sum of variances of columns of R-squares matrix b) Sum of variances of columns of Normalized R-squares matrix divided by row sum. c) Simply, variance of R-squares When replacing R-squares matrix to factor loadings squares matrix, each of the criteria is equivalent to the following well-known criteria. a) Row varimax criterion, b) Normal varimax criterion, c) Quartimax criterion As a case study, the proposed rotation method was applied on evaluation data of townscape in Kanda area. The results obtained and discussion are as follows. 1) In this case of data, the results by criterion c) seems to be most useful. 2) It's recommended to use personal PCA scores of slightly many numbers. In this case, six score is necessary. 3) Rotated axes are interpreted as follows; R1: History and taste, R2: Disorderly-Orderly, R3: Harmony and familiar, R4: Enjoying 4) Among these axes, R3 and R4 are large individual differences of R-squares. About these individual differences, the following tendencies were found. (1) Correlation between R-squares of R3 and R-squares of R4 is negative. Thus, terms to express R3 and terms to express R4, there seem to be not so many people with the both of vocabulary. (2) Subjects majoring in architecture have larger R-squares of R3 than other majors, and have smaller R-squares of R4 than other majors. Furthermore, from the viewpoint of validity of R-square as indicator of explanatory power, diagnostics of the number of personal PCA scores was discussed. Finally, it was noted that there are cases not to require rotation proposed in this paper. And some cases of analysis on such data were presented.

AB - This study examines the methodology of "individual scaling method" in order to value each person's vocabulary and viewpoints. "Individual scales" refers to evaluation items made by subjects' own terms. Thus, these scales are different from person to person. In the previous report Part1, we proposed principal component analysis (PCA) method for evaluation data measured by individual scales. In this PCA method, evaluation object is regarded as observations, and individual scales of all subjects is regarded as variables. And, individual difference in vocabulary and viewpoint is described as difference in distribution of factor loadings vectors in principal component space. However, suitable method for analyzing individual differences that are described as such has not yet been proposed because of so-called "rotational indeterminacy" related problem. In this paper, first, when analyzing individual differences based on proposed PCA method, we discuss the problems caused by so-called "rotational indeterminacy". Next, as a solution to this problem, the new method of orthogonal rotation is proposed. This rotation method has the following features. • By the criterion in this method, R-squares matrix of [each person × rotated axes] is directed to simple structure. • By this rotation, individual differences in semantic space is maximized. R-squares is measured by personal-PCR (principal component regression analysis), i.e. PCA is performed by each subject, and then by using their each PCA scores as explanatory variables, multiple regression analysis is performed by each subject. Thus, the number of personal PCA scores is an important option in the analysis. The criteria to be maximized, we propose the following three.These are another important option. a) Sum of variances of columns of R-squares matrix b) Sum of variances of columns of Normalized R-squares matrix divided by row sum. c) Simply, variance of R-squares When replacing R-squares matrix to factor loadings squares matrix, each of the criteria is equivalent to the following well-known criteria. a) Row varimax criterion, b) Normal varimax criterion, c) Quartimax criterion As a case study, the proposed rotation method was applied on evaluation data of townscape in Kanda area. The results obtained and discussion are as follows. 1) In this case of data, the results by criterion c) seems to be most useful. 2) It's recommended to use personal PCA scores of slightly many numbers. In this case, six score is necessary. 3) Rotated axes are interpreted as follows; R1: History and taste, R2: Disorderly-Orderly, R3: Harmony and familiar, R4: Enjoying 4) Among these axes, R3 and R4 are large individual differences of R-squares. About these individual differences, the following tendencies were found. (1) Correlation between R-squares of R3 and R-squares of R4 is negative. Thus, terms to express R3 and terms to express R4, there seem to be not so many people with the both of vocabulary. (2) Subjects majoring in architecture have larger R-squares of R3 than other majors, and have smaller R-squares of R4 than other majors. Furthermore, from the viewpoint of validity of R-square as indicator of explanatory power, diagnostics of the number of personal PCA scores was discussed. Finally, it was noted that there are cases not to require rotation proposed in this paper. And some cases of analysis on such data were presented.

KW - Individual differences

KW - Individual scales method

KW - Orthogonal rotation

KW - Principal component analysis

KW - Quartimax criterion

KW - Semantic space

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U2 - 10.3130/aije.82.1

DO - 10.3130/aije.82.1

M3 - Article

VL - 82

SP - 1

EP - 9

JO - Journal of Environmental Engineering, ASCE

JF - Journal of Environmental Engineering, ASCE

SN - 1348-0685

IS - 731

ER -