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Published in:
Statistics and Empirical Research Read chapter Read first chapter

2019 | OriginalPaper | Chapter

12. Cluster Analysis

Author : Thomas Cleff

Published in: Applied Statistics and Multivariate Data Analysis for Business and Economics

Publisher: Springer International Publishing

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Abstract

Before we turn to the subject of cluster analysis, think for a moment about the meaning of the word cluster. The term refers to a group of individuals or objects that converge around a certain point and are thus closely related in their position. In astronomy there are clusters of stars; in chemistry, clusters of atoms. Economic research often relies on techniques that consider groups within a total population. For instance, firms that engage in target group marketing must first divide consumers into segments, or clusters of potential customers. Indeed, in many contexts researchers and economists need accurate methods for delineating homogenous groups within a set of observations. Groups may contain individuals (such as people or their behaviours) or objects (such as firms, products, or patents). This chapter thus takes a cue from Goethe’s Faust (1987, Line 1943–45): “You soon will [understand]; just carry on as planned/You’ll learn reductive demonstrations/And all the proper classifications”.

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previous chapter Time Series and Indices
next chapter Factor Analysis
Footnotes
1
By contrast, divisive clustering methods start by collecting all observations as one cluster. They proceed by splitting the initial cluster into two groups and continue by splitting the subgroups, repeating this process down the line. The main disadvantage of divisive methods is their high level of computational complexity. With agglomerative methods, the most complicated set of calculations comes in the first step: for n observations, a total of n(n−1)/2 distance measurements must be performed. With divisive methods containing two non-empty clusters, there are a total of 2(n−1)−1 possible calculations. The greater time required for calculating divisive hierarchical clusters explains why this method is used infrequently by researchers and not included in standard statistics software.
 
2
In the case of two dimensions, the Euclidean distance and the Pythagorean theorem provide the same results.
 
3
In standardization – sometimes also called z-transform – the mean of x is subtracted from each x variable value and the result divided by the standard deviation (S) of the x variable: \( {z}_i=\frac{x_i-\overline{x}}{S} \).
 
4
Say we wanted to dichotomize calories per fl. oz. using three calorie variables. Calorie variable 1 assumes the value of one when the calories in a beer lie between 60 and 99.99 calories, otherwise it is equal to zero. Calorie variable 2 assumes the value one when the calories in a beer lie between 100 and 139.99 calories, otherwise it is equal to zero. Calorie variable 3 assumes the value one when the calories in a beer lie between 140 and 200 calories; otherwise it is equal to zero.
 
5
The centroid is determined by calculating the mean for every variable for all observations of each cluster separately.
 
6
Euclidean distance of #9 to centroid CLU#1: \( \sqrt{{\left(-0.571-\left(-0.401\right)\right)}^2+{\left(0.486-\left(-0.563\right)\right)}^2}=1.06 \).
 
7
Euclidean distance of #9 to centroid CLU#2: \( \sqrt{{\left(1.643-\left(-0.401\right)\right)}^2+{\left(0.719-\left(-0.563\right)\right)}^2}=2.41 \).
 
8
Euclidean distance of #9 to centroid CLU#3: \( \sqrt{{\left(-0.401-\left(-0.401\right)\right)}^2+{\left(-1.353-\left(-0.563\right)\right)}^2}=0.79 \).
 
Literature
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Everitt, B.S., Rabe-Hesketh, S. (2004). A Handbook of Statistical Analyses Using Stata, 3rd Edition. Chapman & Hall: Boca Raton.
Goethe, J.W. (1987). Faust Part One. Translated with an Introduction and Notes by David Luke. New York: Oxford University Press.
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Kaufman, L., Rousseeuw, P.J. (1990). Finding Groups in Data. New York: Wiley.CrossRef
Mooi, E., Sarstedt, M. (2019). A Concise Guide to Market Research. The Process, Data, and Methods Using IBM SPSS Statistics, 3rd Edition. Berlin, Heidelberg: Springer.
Ward, J. H., Jr. (1963). Hierarchical grouping to optimize an objective function. Journal of the American Statistical Association, 58, 236–244.CrossRef
Metadata
Title
Cluster Analysis
Author
Thomas Cleff
Copyright Year
2019
Publisher
Springer International Publishing
Book
Applied Statistics and Multivariate Data Analysis for Business and Economics

Print ISBN: 978-3-030-17766-9

Electronic ISBN: 978-3-030-17767-6

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-17767-6_12