2005 Fiscal Year Final Research Report Summary
New developments in penalized optimal scoring
| Project/Area Number |
16500180
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Statistical science
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| Research Institution | Osaka University (2005)
Ritsumeikan University (2004) |
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Principal Investigator |
ADACHI Kohei Osaka University, Graduate School of Human Sciences, Professor, 人間科学研究科, 教授 (60299055)
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| Project Period (FY) |
2004 – 2005
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| Keywords | Multivariate analysis / Optimal scaling / Penalty functions / Principal component analysis / Spline functions / Nonlinear correlation / Transition trends / Cross-validation |
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| Research Abstract |
In optimal scoring, also known as multiple correspondence analysis and homogeneity analysis, multivariate categorical data is analyzed and optimal scores are assigned to categories and individuals. In this study, we extend the optimal scoring and developed (A) a method for graphically representing inter-variable nonlinear relationships and (B) a method for representing transition trends of individuals by vectors. In both (A) and (B), an objective function to be minimized is defined as combing the loss function of the existing optimal scoring and a penalty function. Next, we detailed the studies on (A) and (B).
(A)We developed a method for representing variables by nonlinear trajectories in a low-dimensional configuration. In this method, the values on quantitative variables are regarded as nominal categories to be given optimal coordinates, and the trajectories connecting the coordinates are defined as natural cubic spline functions. The penalty function expresses the loss of the smoothness of trajectories, and the penalty weight is chosen by a cross-validation procedure. Simulation study and real data analysis shows that the above method represents inter-variable nonlinear relations better than the existing optimal scoring and principal component analysis.
(B)We proposed a method for analyzing transition frequency tables and representing transition trends by vectors in a low-dimensional configuration. This method finds scores of individuals, those of categories, and vectors for trends, in such a way that individuals' scores become homogeneous to the scores of chosen categories and trend vectors become homogeneous to the inter-occasion changes in individuals' scores. The applications to real data show that the resulting configurations of trend vectors allow us easily to grasp trends. Further, this method is extended to treat movers and stayers differently.
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