Data Analysis Machine Learning and Applications Episode 1 Part 2 potx

54 Kamila Migdađ Najman and Krzysztof Najman

itself6. Since the learning algorithm of the SOM network is not deterministic, in

subsequent iterations it is possible to obtain a network with very weak discriminating

properties. In such a situation the value of the Silhouette index in subsequent stages

of variable reduction may not be monotone, what would make the interpretation

of obtained results substantially more difficult. At the end it is worth to note that

for large databases the repetitive construction of the SOM networks may be time

consuming and may require a large computing capacity of the computer equipment

used.

In the opinion of the authors the presented method proved its utility in numerous

empirical studies and may be successfully applied in practice.

References

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variables for cluster analysis, Journal of Classification, vol. 12, p. 113-136.

GORDON A.D. (1999), Classification , Chapman and Hall / CRC, London, p.3

KOHONEN T. (1997), Self-Organizing Maps, Springer Series in Information Sciences,

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MILLIGAN G.W., COOPER M.C. (1985), An examination of procedures for determining the

number of clusters in data set. Psychometrika, 50(2), p. 159-179.

MILLIGAN G.W. (1994), Issues in Applied Classification: Selection of Variables to Cluster,

Classification Society of North America News Letter, November Issue 37.

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MIGDAĐ NAJMAN K., NAJMAN K. (2003), Zastosowanie sieci neuronowej typu SOM w

badaniu przestrzennego zróznicowania powiatów ˙ , Wiadomosci Statystyczne, 4/2003, p. ´

72-85.

ROUSSEEUW P.J. (1987), Silhouettes: a graphical aid to the interpretation and validation of

cluster analysis. J. Comput. Appl. Math. 20, p. 53-65.

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the degree of Master of Science in Engineering, Helsinki University of Technology.

6 The quality of the SOM network is assessed on the basis of the following coefficients:

topographic, distortion and quantisation.

Calibrating Margin–based Classifier Scores into

Polychotomous Probabilities

Martin Gebel1 and Claus Weihs2

1 Graduiertenkolleg Statistische Modellbildung,

Lehrstuhl für Computergestützte Statistik,

Universität Dortmund, D-44221 Dortmund, Germany

[email protected]

2 Lehrstuhl für Computergestützte Statistik,

Universität Dortmund, D-44221 Dortmund, Germany

[email protected]

Abstract. Margin–based classifiers like the SVM and ANN have two drawbacks. They are

only directly applicable for two–class problems and they only output scores which do not

reflect the assessment uncertainty. K–class assessment probabilities are usually generated by

using a reduction to binary tasks, univariate calibration and further application of the pairwise

coupling algorithm. This paper presents an alternative to coupling with usage of the Dirichlet

distribution.

1 Introduction

Although many classification problems cover more than two classes, the margin–

based classifiers such as the Support Vector Machine (SVM) and Artificial Neural

Networks (ANN), are only directly applicable to binary classification tasks. Thus,

tasks with number of classes K greater than 2 require a reduction to several binary

problems and a following combination of the produced binary assessment values to

just one assessment value per class.

Before this combination it is beneficial to generate comparable outcomes by cali￾brating them to probabilities which reflect the assessment uncertainty in the binary

decisions, see Section 2. Analyzes for calibration of dichotomous classifier scores

show that the calibrators using Mapping with Logistic Regression or the Assign￾ment Value idea are performing best and most robust, see Gebel and Weihs (2007).

Up to date, pairwise coupling by Hastie and Tibshirani (1998) is the standard ap￾proach for the subsequent combination of binary assessment values, see Section 3.

Section 4 presents a new multi–class calibration method for margin–based classifiers

which combines the binary outcomes to assessment probabilities for the K classes.

This method based on the Dirichlet distribution will be compared in Section 5 to the

coupling algorithm.