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Best Model for Swiss Banknote Data

Explanation 1 of Matroska Feature-Selection Method (Method 2)

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New Theory of Discriminant Analysis After R. Fisher

Abstract

When we discriminate Swiss banknote data by IP-OLDF, we find that these data are linearly separable data (LSD). Because we examine all possible combination models, we can find that a two-variable model, such as (X4, X6), is the minimum linearly separable model. A total of 16 models, including these two variables, are linearly separable by the monotonic decrease of MNM (MNM p ≥ MNM(p+1)), and other 47 models are not linearly separable. Therefore, we compare eight LDFs by the best models with the minimum error rate mean in the validation sample (M2) and obtain good results. Although we could not explain the useful meaning of the 95 % CI of discriminant coefficient until now, the pass/fail determination using examination scores provide a clear understanding by normalizing the coefficient in Chap. 5. Seven LDFs become trivial LDFs. Only Fisher’s LDF is not trivial. Seven LDFs are Revised IP-OLDF based on MNM, Revised LP-OLDF, Revised IPLP-OLDF, three SVMs, and logistic regression. We successfully explain the meaning of coefficient. Therefore, we discuss the relationship between the best model and coefficient more precisely by Swiss banknote data in Chap. 6. We study LSD discrimination by Swiss banknote data, Student linearly separable data in Chap. 4, six pass/fail determinations using examination scores in Chap. 5, and Japanese-automobile data in Chap. 7, precisely. When we discriminate six microarray datasets that are LSD in Chap. 8, only Revised IP-OLDF can naturally make feature-selection and reduce the high-dimensional gene space to the small gene space drastically. In gene analysis, we call all linearly separable models, “Matroska.” The full model is the largest Matroska that includes all smaller Matroskas in it. As we already knew, the smallest Matroska (BGS) can explain the Matroska structure completely through the monotonic decrease of MNM. We propose the Matroska feature-selection method for the microarray dataset (Method 2). Because LSD discrimination is no longer popular, we explain Method 2 through detailed examples of the Swiss banknote and Japanese-automobile data. On the other hand, LASSO attempts to make feature-selection. If it cannot find the small Matroska (SM) in the dataset, it cannot explain the Matroska structure. Swiss banknote data, Japanese-automobile data, and six microarray datasets are helpful for evaluating the usefulness of other feature-selection methods, including LASSO.

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Authors and Affiliations

  1. Faculty of Economics, Seikei University, Musashinoshi, Tokyo, Japan

    Shuichi Shinmura

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  1. Shuichi Shinmura
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Correspondence to Shuichi Shinmura .

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Shinmura, S. (2016). Best Model for Swiss Banknote Data. In: New Theory of Discriminant Analysis After R. Fisher. Springer, Singapore. https://doi.org/10.1007/978-981-10-2164-0_6

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