Compositional Data Analysis: a business case application to … · 2019-09-23 · Sommario In...

POLITECNICO DI MILANO

Scuola di Ingegneria dei Sistemi

Corso di Laurea Specialistica in Ingegneria Matematica

Dipartimento di Matematica

Compositional Data Analysis: a business caseapplication to Cross Selling

Relatore: Prof. Simone VANTINICorrelatrice: Dott.ssa Alessandra MENAFOGLIO

Tesi di Laurea di:

Luca FACCHETTI

Matr. 755105

Anno Accademico 2015–2016

Sommario

In questo lavoro si presentano le principali caratteristiche della Compositio-

nal Data Analysis, soffermandosi sul problema dell’applicazione di tecniche

classiche di statistica multivariata su dati composizionali.

Successivamente i metodi di analisi e trasformazione di dati composizionali

vengono applicati ad un dataset inerente a un caso aziendale di Cross Sel-

ling relativo alle quantità di macro famiglie di prodotti acquistate dai clienti

della società di automazione industriale Festo S.p.A. (ed alle relazioni che in-

tercorrono tra le varie famiglie), cercando di capire, tramite l’applicazione di

analisi di statistica multivariata come PCA, Clustering, K-means e Classifica-

zione Supervisionata, se le strategie aziendali per lo sviluppo di campagne di

marketing relative al Cross Selling abbiano evidenza statistica.

PAROLE CHIAVE: Compositional Data Analysis, Cross Selling, PCA, Cluste-

ring, K-means, Festo.

Abstract

In this work main characteristics of Compositional Data Analysis are pre-

sented, specially focusing on issue of application of classical multivariate sta-

tistical techniques to compositional data.

Afterwards methods of compositional data transformation are applied to a

dataset inherent to a Cross Selling business case related to product macro-

families purchased by customers of Industrial Automation company Festo Ltd

(and related to relations between product families) with the aim, through the

application of multivariate statistical methods such PCA, Clustering, K-means

and Supervised Classification, to understand if company strategies regarding

the development of marketing campaigns related to Cross Selling are sup-

ported by statistical results.

KEYWORDS: Compositional Data Analysis, Cross Selling, PCA, Clustering,

K-means, Festo

Contents

1 Introduction 2

1.1 Framework and motivation . . . . . . . . . . . . . . . . . . . . . 2

1.2 Case study: application of Composition Data Analysis to Cross

Selling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Cross Selling in a nutshell . . . . . . . . . . . . . . . . . . 3

1.2.2 Cross Selling: from heuristic to quantitative approach . . 4

1.3 Chapters description . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Business case: Festo and Cross Selling analysis 6

2.1 Introduction of Festo . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Festo Italy and Industry Automation Italian market . . . . . . . 7

2.2.1 Festo Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 F-IT market: Festo products . . . . . . . . . . . . . . . . . 8

2.2.3 F-IT customer segmentation . . . . . . . . . . . . . . . . . 9

2.3 Cross Selling as marketing and statistical approach . . . . . . . . 12

2.3.1 Cross and Up Selling . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Why Cross Selling in Festo? . . . . . . . . . . . . . . . . . 14

2.3.3 Festo first steps in Cross Selling . . . . . . . . . . . . . . . 15

2.4 Festo analytics and Status Quo . . . . . . . . . . . . . . . . . . . 16

2.4.1 Level 1: total amounts . . . . . . . . . . . . . . . . . . . . 16

2.4.2 Pros and Cons of Level 1 analysis . . . . . . . . . . . . . . 19

2.4.3 Level 2 analysis: Ratios between product families . . . . 21

2.4.4 Level 2 Ratios logic: Pneumatic Drives as family basis . . 22

2.4.5 Level 2 problems: the dependence from Drives trend . . 24

3 Compositional Data Analysis and the Aitchison Simplex 26

3.1 D-compositions and Simplex SD . . . . . . . . . . . . . . . . . . 26

3.2 Compositional analysis principles . . . . . . . . . . . . . . . . . . 30

3.2.1 Scale invariance . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.2 Permutation invariance . . . . . . . . . . . . . . . . . . . 31

3.2.3 Subcompositional coherence . . . . . . . . . . . . . . . . . 32

3.3 The Aitchison geometry . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 Defining a vector space structure . . . . . . . . . . . . . . 34

3.3.2 Aitchison inner product, norm, and distance . . . . . . . 36

3.3.3 Geometry on S3: figures on ternary diagrams . . . . . . . 38

4 Business case: Compositional data analysis of Cross Selling data 42

4.1 F-IT Customer dataset as Compositional dataset . . . . . . . . . 42

4.1.1 Considering Customer as compositional observations . . 43

4.2 Simplification of Dataset . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.1 Variables taken into consideration . . . . . . . . . . . . . 43

4.3 Dataset representation . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Compositional Principal Components Analysis . . . . . . . . . . 47

4.4.1 Visualizing and interpreting Biplot of compositional PCA

scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4.2 Visualizing and interpreting compositional PCA loadings 51

4.5 Classification and grouping with Aitchison distance . . . . . . . 55

4.5.1 Hierarchical Cluster Analysis in S5 and S3 . . . . . . . . 55

4.6 Compositional K-means Cluster Analysis . . . . . . . . . . . . . 61

4.6.1 K-means with k=6 . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Compositional Discriminant Analysis . . . . . . . . . . . . . . . 66

4.7.1 LDA to classify customers by ISM . . . . . . . . . . . . . 66

4.7.2 QDA analysis . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Conclusions 72

5.1 Targets achieved . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Possible future developments . . . . . . . . . . . . . . . . . . . . 73

List of Figures

2.1 Festo TC in Esslingen . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Example of typical Festo products and the final result, an indus-

try machine where these products are applied . . . . . . . . . . . 9

2.3 Two typical Customer Solutions composed by standard prod-

ucts; in general these are not serial projects. . . . . . . . . . . . . 10

2.4 F-IT market, in NTO and number of customers, split by cus-

tomer typology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 F-IT market proportion, in NTO (blue) and number of customers

(grey), split by Industry Sector . . . . . . . . . . . . . . . . . . . . 11

2.6 Related items suggested by Amazon website . . . . . . . . . . . 14

2.7 Bundling on Amazon website . . . . . . . . . . . . . . . . . . . . 14

2.8 Proposal based on customer purchasing choices. . . . . . . . . . 15

2.9 Specific section that show related products: a first Cross Selling

approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.10 Example of how products have been aggregated into bigger fam-

ilies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.11 Final macro categories taken into account for Level 1 analysis

(Customer Solutions have been excluded). . . . . . . . . . . . . . 18

2.12 YTD Result, on NTO and Volumes, of all product families for F-IT 18

2.13 Template Excel Analysis for Level 1 in Food & Beverage sector . 19

2.14 Relations between Drives and other product families . . . . . . 23

2.15 After the analysis, the resume sheet shows us the ratios for each

F-IT customer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Simplex inR3+ and its representation as ternary diagram (Source:

[6]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Example of distance in S3 . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Previous example with customers perturbed by y = [5, 30, 80]

and y = [10, 100, 10] . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Previous example with powering applied to customers withα =

0.5 andα = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Template for ternary diagrams and importance of amount com-

ponents proportion in S3. . . . . . . . . . . . . . . . . . . . . . . 39

3.6 Similar customers (by shape or colour) lying on same proportion

lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7 Parallel lines, circles and ellipses on ternary diagram. Source: [6] 41

4.1 Final macro families taken into analysis . . . . . . . . . . . . . . 45

4.2 Example of dataset . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Representation of dataset as absolute amounts . . . . . . . . . . 47

4.4 Representation of dataset on simplex . . . . . . . . . . . . . . . . 48

4.5 Results summary applied to all 5 components . . . . . . . . . . . 49

4.6 Proportion of variance on 4 Principal Components . . . . . . . . 49

4.7 Biplot of first 2 principal components . . . . . . . . . . . . . . . . 50

4.8 Loadings on first 2 Principal Components . . . . . . . . . . . . . 52

4.9 Barplot representing loadings value on all 4 principal components 52

4.10 Dataset with first (solid line) and second (dashed line) PCs . . . 53

4.11 Representation of first (solid line) and second (dashed line) PCs

on subcompositions for Sensors, Pneumatic Drives and Cylinder

Mountings coloured by ISM . . . . . . . . . . . . . . . . . . . . . 54

4.12 Dendograms for Euclidean and Manhattan distances . . . . . . 56

4.13 Dendograms for Euclidean and Manhattan distances with Ward 57

4.14 Cophenetic correlations for Euclidean and Manhattan distances 57

4.15 Data grouped in 3 clusters . . . . . . . . . . . . . . . . . . . . . . 58

4.16 Data grouped in 6 clusters . . . . . . . . . . . . . . . . . . . . . . 59

4.17 Subcompositions data grouped in 3 clusters for average, com-

plete and Ward linkage: first row Euclidean distance, second

row Manhattan distance . . . . . . . . . . . . . . . . . . . . . . . 59

4.18 Subcompositions data grouped in 6 clusters for average, com-

plete and Ward linkage: first row Euclidean distance, second

row Manhattan distance . . . . . . . . . . . . . . . . . . . . . . . 60

4.19 Results for k-means algorithm with 6 clusters and starting cen-

tres from hierarchical clustering . . . . . . . . . . . . . . . . . . . 62

4.20 Ternary diagrams for k-means with 6 clusters, purple squares

represent centres of the clusters . . . . . . . . . . . . . . . . . . . 62

4.21 Subcompositions’ original data coloured by ISM and coloured

by k-means clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.22 Extreme centres applied to k-means algorithm . . . . . . . . . . 64

4.23 Comparing results of k-means with 5 clusters . . . . . . . . . . . 64

4.24 Variance explained percentage in function of k . . . . . . . . . . 65

4.25 Comparing with k=3 and different initial centres . . . . . . . . . 65

4.26 Values of final clusters’ centres in both cases . . . . . . . . . . . . 65

4.27 Results for lda analysis using proportional priors . . . . . . . . . 67

4.28 Scalings of lda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.29 Dataset with new classification for ISM . . . . . . . . . . . . . . . 70

4.30 Dataset with new classification computed with LDA and QDA

algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

List of Tables

2.1 Example table: which Ratio is the correct one? . . . . . . . . . . 21

2.2 Opposite effect on ratios generated by Drives . . . . . . . . . . . 24

3.1 Euclidean Distance between customers on S3 . . . . . . . . . . . 33

3.2 Aitchison Distance between customers on S3 . . . . . . . . . . . 37

4.1 Extreme centres applied to k-means algorithm . . . . . . . . . . 63

4.2 Number of customers per ISM . . . . . . . . . . . . . . . . . . . . 66

4.3 ISM included in analysis . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Confusion matrix of dataset: on rows original classes, on column

predicted classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Confusion matrix for LDA using uniform priors for groups . . . 68

4.6 Confusion matrix related to QDA with uniform priors . . . . . . 69

1

Chapter 1

Introduction

1.1 Framework and motivation

This work aims to present the basic theory behind the branch of statistics

called Compositional Data Analysis, that is the set of statistical techniques de-

veloped to analyse multivariate dataset as compositional data, so as they are

quantitative descriptions of the parts of some whole, conveying exclusively

relative information.

The basics concepts that lead to the formulation of principles beyond these

techniques has ancients origins since in a paper of Karl Pearson (1897) [7] that

begin with the words "On a form of spurious correlation ..." he wrote about the

awareness of problems related to the analysis of data from their relative point

of view instead of their standard quantitative approach.

After a long period where most of statisticians community focused on devel-

opment of multivariate statistical analysis (in the classic meaning of the term),

in last decades these techniques have been developed specially thanks to the

work of John Aitchison and his working team that first in 1986 defined princi-

ples beyond analysis of data treated as compositions and second developed a

dedicated geometry and data transformation in order to be able to apply most

of classics multivariate statistical techniques (such as Principal Components

2

1.2. CASE STUDY: APPLICATION OF COMPOSITION DATA ANALYSIS TOCROSS SELLING

Analysis, Cluster Analysis, Data sampling, Classification etc).

Nowadays Compositional Data Analysis found its deserved space in statisti-

cal techniques since literature is full of quantities and data where it is more

interesting to focus on the relations between components instead of absolute

quantities. Fields with most applications are such as biosciences, geosciences,

chemistry (think about the proportion of elements in a chemical compound or

in the relations between parts of different material in an earth sample).

1.2 Case study: application of Composition DataAnalysis to Cross Selling

1.2.1 Cross Selling in a nutshell

One of the most important fields in developing in marketing analyses for

industrial and sales companies is the so-called Cross Selling, its specific con-

cepts and applications will be described in depth in next chapter: we can de-

scribe Cross Selling in a nutshell as the set of practices related to the opti-

mization of the sale of products or accessories belonging to a specific basket

of products related each others. These analyses are of critical importance as

gives to the company the latent information about the missing potential mar-

ket relying on sales data of its customers’ basket of purchased products.

Let’s make a simple example of Cross Selling logic: company Festo sells

two product families, Pneumatic Drives and Sensors: it is well-known that in

almost all application cases it is necessary to mount a Sensor on a Pneumatic

Drive, so quantities sold from Festo of these two product families are corre-

lated: if Customer A is buying only Pneumatic Drives from Festo it means

that it is potentially buying Positioning Sensors from a Festo competitor, and

3

1.3. CHAPTERS DESCRIPTION

the amount of Sensors that it is buying from the competitor is also quantified

(in relation to Pneumatic Drives quantities sold to Customer A by Festo).

We can generalize the concept to more than two product families, in this case

we are interested in the relation between a sold product family quantity with

respect to the whole components’ quantity.

1.2.2 Cross Selling: from heuristic to quantitative approach

Unfortunately Cross Selling is generally used in a heuristic way: data are

analysed from a qualitative point of view and only absolute quantities of sin-

gle product families are taken into account.

The target of this work is to develop propose the proper mathematical en-

vironment and the set of quantitative (and formally correct) methods for the

Cross Selling analysis applied to a business case related to Cross Selling data

for customers of Italian subsidiary of Festo, a German company worldwide

leader in Industry Automation.

After the definition of the right analysis environment for dataset composed

by Festo customers, some classical multivariate techniques are applied to the

dataset in order to verify if company strategy related to Cross Selling cam-

paigns definition is supported by statistical analysis of data.

1.3 Chapters description

In first part of Chapter 2 Festo’s brief history and company structure is de-

scribed, after that there is the description of the market where the company

operates, the strategical attributes for customers defined by the company and

a general description of products sold by Festo. In the second part of the chap-

4

1.3. CHAPTERS DESCRIPTION

ter it is described the status quo of Cross Selling analysis developed in the

company: issues and crucial points of weakness of the analysis are defined.

Chapter 3 is the one concerning the description of Compositional Data Anal-

ysis theory: after the formal description of basic principles of Composition

Data Analysis enunciated by John Aitchison and after verifying on a part of

dataset of compositions the issues related to the application of standard statis-

tical techniques, a proper data transformation using Aitchison Geometry is

developed.

In last part, considering the difference geometry rules of compositional data on

the simplex with respect to standard Euclidean spaces geometry rules, geome-

try on the simplex is described with examples of parallel lines, curves, circles

and ellipses .

Chapter 4 Is the joint between second and third chapter: Festo Cross Selling

data is transformed into compositional data and some multivariate analysis

techniques (such Principal Components Analysis, Hierarchic Cluster Anal-

ysis, K-means Clustering and Linear and Quadratic Discriminant Analysis)

have been developed in order to understand if company’s strategies are sup-

ported by data analysis.

For each type of analysis conclusions on the nature of Festo customers’ pur-

chasing attitudes are drawn and differences between the conclusions coming

from a standard approach are described.

5

Chapter 2

Business case: Festo and CrossSelling analysis

2.1 Introduction of Festo

Festo is worldwide leader in Industrial Automation sector, in training and

updating of industrial systems for its customers. Its Head Quarter is placed in

Esslingen am Neckar, near Stuttgart 1; it is also present almost worldwide with

61 abroad subsidiaries and almost 18700 employees.

Figure 2.1 – Festo Technology Centre in Esslingen

1. one of the most important European areas for industry development, for this motivationits region, Baden-Württemberg, is called one of the Four Motors for Europe together withItalian region Lombardia, Spanish Catalunya and French Rhône-Alpes

6

2.2. FESTO ITALY AND INDUSTRY AUTOMATION ITALIAN MARKET

Founded in 1925 as a wood working machines’ builder, Festo is now both a

global player and an independent family-run company that covers its financial

needs by their own means.

For such a motivation the company is not dependent by any kind of capital

market but it’s linked to its relations with customers, employees and business

partners. This allowed the company to define a long term planning even in

highly dynamic and competitive markets, often different by their starting spe-

cific market. The company is synonymous with innovation in industrial and

process automation, from individual single products up to customer solutions

ready for installation.

Festo’s innovative strength is demonstrated by the introduction of about 100

new products every year, the investment of 9% of its total turnover in Research

and Development and by the deposition of over 2900 patents worldwide.

With a total turnover of 2.64 billion Euros in 2015, Festo supplies pneumatic

and electrical automation technology for 300,000 customers in over 35 different

industry fields.

2.2 Festo Italy and Industry Automation Italian mar-ket

2.2.1 Festo Italy

Festo SpA, from now on F-IT, was born in 1956 as the first Festo abroad sub-

sidiary, after only one year since Festo started to spread in Germany pneu-

matic technology applied to industrial sectors different from its specific one,

the wood working sector.

It is from Italy that its vocation of international company started to become a

reality. Contributing to the affirmation of industrial development, Festo Italy

7


witnessed the birth of small, medium and large enterprises that arose during

the post-war period, and using simple technology, progressively reach more

complex and efficient working models.

With an approximate number of 240 employees and an established net-

work of authorized distributors F-IT successfully underpins the group’s mis-

sion. The flagship of Assago company offices is the Application Center, an ex-

clusive operating environment with full availability for F-IT customers: a func-

tional technological space divided into four application areas and with training

classrooms where customers can test solutions and simulate real-world appli-

cations.

In the headquarters of Assago also operational Festo C.T.E. Srl can be found.

Festo C.T.E. is part of international Festo Didactic, one of the world-leading

providers of equipment and solutions for technical education. The product

and service portfolio offers customers holistic education solutions for all ar-

eas of technology in factory and process automation, such as pneumatics, hy-

draulics, electrical engineering, production technology, mechanical engineer-

ing, mechatronics, CNC, HVAC and telecommunications.

2.2.2 F-IT market: Festo products

As mentioned before, F-IT is a leading company on the Italian manufactur-

ing market in industrial automation. The core business of F-IT is characterized

by the sale of almost all components for material handling, mainly pneumatic

(i.e. all you need to build an industrial machine such as valves, valve termi-

nals, cylinders and related accessories, positioning sensors, flow regulators,

cables, wires, tubes, pipes and fittings etc) and to a lesser extent but in big de-

velopment, for electric movement (electric drives, brushless motors, etc.) and

Process Automation (products related to the process industry market for the

8


Figure 2.2 – Example of typical Festo products and the final result, an industrymachine where these products are applied

Pharma, the Oil & Gas and for wastewater treatment). Figure 2.2 depicts some

examples of Festo products and one typical final application of them.

The most important part of Festo market is composed by standard prod-

ucts, or single products that could be used as components for a more complex

system. A smaller part of its market is also composed by the so-called Cus-

tomer Solutions, that are single projects specifically assembled with both Festo

and external products and customized for our customers (see two examples in

Figure 2.3).

2.2.3 F-IT customer segmentation

Excluding Distributors channel, Italian market is mainly composed by OEMs

(Original Equipment Manufacturers) or industrial machine manufacturers for

handling, production and packaging of products of all manufacturing sectors.

These represent about 80% of F-IT market; the residual part is made by End

Users, i.e., final users that purchase products only related to their actual busi-

9


Figure 2.3 – Two typical Customer Solutions composed by standard products;in general these are not serial projects.

ness requirements (these companies, for example, are in most cases customers

of OEM companies from whom they buy one or more machines and which

require only single products for personal needs such as replacing a Festo dam-

aged component or a modification of a machine project).

Customer Structure

Market Sales [LC] /# Customer by OEM/EU/DEALER

Market Sales [LC] # Customer

100%

90%

80%

70%

60%

50%

40%

30%

20%

10%

0%

IT

16,1%

6,6%

77,3%

23,1%

72,6%

Color by

OEM_End U

Dealer

End us

OEM

Figure 2.4 – F-IT market, in NTO and number of customers, split by customertypology.

10


An important additional segmentation in F-IT (and in a similar way in Festo

total market) is the allocation of its customers to a specific industry sector.

This classification is crucial since the market in which F-IT operates is vast

and eclectic. Therefore the business is divided in different industry sectors,

among which the most significant for F-IT are:

— Food & Beverage

— Packaging

— Automotive

— Electronic-Light Assembly

— Printing & Plastic

— Machine Tool & Handling

— Process Automation

Customer Structure

Market Sales [LC] /# Customer by ISM

ISM Desc. [short]

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Market Sales [LC]

# Customer

Figure 2.5 – F-IT market proportion, in NTO (blue) and number of customers(grey), split by Industry Sector

11

2.3. CROSS SELLING AS MARKETING AND STATISTICAL APPROACH

This classification is essential as the market in which F-IT operates is so dif-

ferentiated that reducing everything to a single action strategy would not be

only simplistic than ever misleading. In fact, in the European market Italy is

second only to Germany for the manufacture of industry machines, and our

market is also recognized as the one with the most innovative and creative so-

lutions.

Every sector is different and has particular industrial applications of its mar-

ket. By way of example consider the machinery market for Food & Beverage

where there are much more stringent sanitary certifications than other mar-

kets. In a market like this there will be specific products, waterproof or totally

aseptic for the treatment of the food or beverage compared to a market such

as the Automotive one that has different applications and standards in force

(think about bending machines of metal sheets of vehicle bodies).

2.3 Cross Selling as marketing and statistical ap-proach

2.3.1 Cross and Up Selling

For the varied composition of its market, Festo implements and supports

analytically different marketing campaigns that are placed on different planes

of action: there are campaigns related to specific industry sectors and cam-

paigns related to individual specific products or to macro-families of products.

Here is where comes into play the Cross Selling concept.

The term Cross Selling defines the set of practices, marketing actions, sta-

tistical analyses and business KPIs 2 related to the optimization of the sale of

2. Key Performance Indicators: a type of performance measurements calculated to evalu-ate the success of an organization or an activity with respect to a particular business need.

12


products or accessories belonging to a specific basket of products related each

others. This tool is necessarily linked to a partially loyal customers that are

already buying from Festo a basket of consolidated products.

The practice of Cross Selling is widely used as part of B2C 3 markets, where

the buyer is characterized mainly by End Users and the logic of the market is

mainly characterized by direct marketing 4.

A classic and simple example of this practice is the one implemented by

major e-commerce companies like Amazon: once a buyer has completed a pur-

chase on their site, the same service provider shows to the buyer some deals

related to the products of his latest purchase: this practice not only allows

one to bypass all the logic related to indirect marketing campaigns or the field

studies to understand the purchasing logics of a particular customer segment,

but also enables one to directly use the purchasing information of a customer

and to use other sales techniques. A classic example is the formulation of a

spot price of a given bundle of different products with a further saving for the

customers in case they decide to buy the entire package instead of the single

individual products.

Figure 2.6 shows an example of Cross Selling implemented by Amazon,

the largest e-commerce site on the European market. In Figure 2.7 we can see

how Cross Selling could be applied to bundling campaigns.

The first image shows how, after a careful categorization of products and

3. Business to Consumer (B2C) is business or transactions conducted directly between acompany and consumers who are the end-users of its products or services

4. from Wikipedia: Direct Marketing is a form of advertising which allows businesses andnonprofit organizations to communicate directly to customers through a variety of media in-cluding cell phone text messaging, email, websites, online adverts, database marketing, fliers,catalogue distribution, promotional letters and targeted television, newspaper and magazineadvertisements as well as outdoor advertising. Among practitioners, it is also known as directresponse.

13


Figure 2.6 – Related items suggested by Amazon website

Figure 2.7 – Bundling on Amazon website

links between them, the management and proposal of related products is im-

plemented almost automatically. In the second image we have an example of

the next automatic step for a cumulated offer.

2.3.2 Why Cross Selling in Festo?

As we explained in the first section of this chapter, F-IT market is mainly

composed by OEMs (Original Equipment Manufacturers).

It is easily understood that a practice like Cross Selling in Festo market is

much more suited to the market constituted by the so-called industry machine

manufacturers (OEMs) as they require almost all of the products and compo-

14


nents sold by Festo for the manufacture of their final products, as opposed

to End Users that will purchase only the necessary products to the resolution

of his momentary need, or buy products from Festo because "forced" by the

composition of its industrial plant (e.g. buy pneumatic components for re-

placement).

The ability to examine the basket of goods purchased by an OEM customer,

therefore, gives important information about what he is purchasing from Festo

and what is necessarily buying from its direct competitors.

2.3.3 Festo first steps in Cross Selling

Festo has already activated strategies of Cross Selling as part of its shares

trading. An example of this is a service similar to that described in the first

part of this chapter from Amazon. In fact, when customers decide to use our

Online Shop, they have also the ability to view in few clicks a series of acces-

sories products related to the product that they have just bought.

Figure 2.8 – Proposal based on customer purchasing choices.

However, this happens only when the indirect acquisition channel is used

by customers, not when, as often happens, the customer is followed directly

by a Sales Engineer. The target was to identify and develope the right tools to

15

2.4. FESTO ANALYTICS AND STATUS QUO

Figure 2.9 – Specific section that show related products: a first Cross Sellingapproach.

support and monitoring Sale Force Cross Selling actions in the best way.

There was also the problem of a lack of specific statistical analysis of Cross Sell-

ing monitoring for direct customers and to develop specific campaigns related

to this topic.

2.4 Festo analytics and Status Quo

2.4.1 Level 1: total amounts

Festo decided to start a centralized analysis few years ago by forming a

global team that was in charge to set and develop some KPIs and some mea-

surements in order to identify and positioning customers from Cross Selling

point of view.

The first step has been to aggregate all Festo products’ basket into macro-

families of products that could be compared between them (e.g. all the Pneu-

matic Drives, all the Sensors etc.), so Cross Selling works on the comparison of

absolute amounts from these different product families between them.

16


Figure 2.10 – Example of how products have been aggregated into bigger fam-ilies

The result has been the possibility to compute, for each Festo customer,

sector, or country (depending on the analysis level) the amount for a certain

analysis period, generally a moving year, of all macro-families and to compute

the total amount for every single product family: this concept was already

present in the past for main product families (such Drives, Sensors or Valves)

but with the new analysis we have the full view of F-IT split by all product

categories and for every single customer (see Figure 2.11). This allowed us to

make specific and strategic campaigns based on the information coming from

this analysis and would allow us to make specific statical analysis on them.

We remind that we exclude from the analysis the product category called Cus-

tomer Solutions as they don’t represent a serial product (except in specific

cases) but single customization made by the aggregation of single products.

We now have the possibility to aggregate yearly Market Sales or Volumes

generated for every product family for F-IT customers and start making some

assumption that will explain some strategic choices in the Cross Selling Level

17


Figure 2.11 – Final macro categories taken into account for Level 1 analysis(Customer Solutions have been excluded).

2 analysis.

In Figure 2.12 F-IT yearly market sales and volumes in quantity are shown.

CHART Cross Selling Basic Analysis (LEV 1) - only OEM (I) (GREEN)

Proportion of Market Sales [LC] into defined cluster - only OEMs!

Air supply Pneumatic Drives

Valves Valve Terminals

Throttles Tubings Fittings Sensors Electric Drives

Connecting cables

Other products

Pneumatic Drives Acc.

Valve Accessoiries

Other Accessoiries

Customer Solutions (VCC 3-13)

not classified

EW

IT

Proportion of Quantities into defined cluster - only OEMs!

Air supply Pneumatic Drives

Valves Valve Terminals

Throttles Tubings Fittings Sensors Electric Drives

Connecting cables

Other products

Pneumatic Drives Acc.

Valve Accessoiries

Other Accessoiries

Customer Solutions (VCC 3-13)

not classified

EW

IT

Figure 2.12 – YTD Result, on NTO and Volumes, of all product families for F-IT

It’s easy to understand from this picture that Pneumatic Drives is the most

profitable family (from NTO point of view) and it is quite known as Festo

is worldwide leader in production of Pneumatic Drives, Valves and Valves

Terminals. This information will be important in the next paragraphs.

Another thing to be understood and that is not visible in Figure 2.4 is that there

18


is a lot of variability in the relation between total amounts of product families

in different customers. This could be due to a lack of sales action in a customer

for a certain product families or it could be a pathological situation, due to the

incompatibility of a customer’s final product with a particular product family.

2.4.2 Pros and Cons of Level 1 analysis

Level 1 of Cross Selling analysis is therefore useful to make a rough anal-

ysis on amounts of all macro categories for all customers and to find where

the amount of particular product family is completely null or very small. An

example of this comparison is made in the template of Figure 2.13 for a partic-

ular F-IT industry sector’s customers (Food & Beverage). We can immediately

see which customerrs are not buying a particular product family. Every Sales

Engineer of F-IT receives an Excel file with data for his own customers every

quarter.

Figure 2.13 – Template Excel Analysis for Level 1 in Food & Beverage sector

The main benefit from Level 1 analysis is to identify in a fast and easy way

the customers with which F-IT can choose to start a marketing campaign rely-

ing on the information of null (red cells on Excel Table) or very small amount

19


for a certain product family.

There are two big problems with Level 1 analysis:

1. It doesn’t take in account the relation between different product fami-

lies: for a particular customer (a row in the Excel file), we cannot work

on the distribution of the composition of a product basket for a cus-

tomer: we can just act on single product families (such Drives, Valves,

Sensors etc.) but not on the relation between this family with the other

ones.

2. The second one is even more crucial: since it works on absolute quan-

tities of each product family, it doesn’t allow us to compare in the right

way customers with different dimensions (both from Volumes or NTO

point of view). What is the benefit of comparing customers with several

differences in the absolute amount on the same product families?

As we already understood, the relation between different families allow us

to understand the customer fidelity level, because if we know that 2 product fam-

ilies are positively correlated (e.g. if I buy a Pneumatic Drive I will need some

accessories for this Drive such Positioning Sensors, a Valve or a Valve Termi-

nal to let it move, a Cylinder Mounting etc.) and we note that this relation

is not satisfied for some customers, it means the latter are necessarily buying

some products form our competitors. Hence if we were able to define a sta-

tistically significant relation between correlated product families, we would be

able to make exploratory analysis and to cluster customers with methods that

are supported by statistical instruments.

20


2.4.3 Level 2 analysis: Ratios between product families

The way to build specific analysis to compare amounts of different product

families for a customer has been developed in the most feasible and easy to be

implemented method: Festo decided to calculate ratios between the amounts

of related macro families taken in pairs and decide, after defining which fam-

ilies were related together, and to study the distance of this ratio from 1 in

order to understand if this distance was a motivation for marketing actions or

if it was related to technical motivation (e.g. ratio Sensors/Drives could not

be ideally 1 for each industry application, in some case we need 2 Sensors for

each pneumatic drive to be controlled, in other cases, less frequent, we won’t

need any Sensor for a Pneumatic Drive). For this kind of analysis Festo needed

to limit data taken into account, as we could set a ratio between products only

for the ones whose technical relation we were sure about.

The first issue to be managed with ratios was to set which product family

between the two taken into account for each ratios was the one leading the

ratio meaning: to make a simple example, let’s have dummy values for the 3

Customers in Table 2.1

Table 2.1 – Example table: which Ratio is the correct one?

Name Pneumatic Drives Sensors Sens./Drives Drives/Sens.CUST 1 3500 2800 0,8 1,25CUST 2 23459 27021 1,15 0,87CUST 3 (perfect) 5500 5500 1,00 1,00

Let’s look to Customer 1 ratio between Sensors Vs Drives equal to 0.8: it

means that this customer is buying 80 sensors every 100 purchased drives;

does it means that Festo is not good selling Sensors to this customer or that it

is very good in selling Drives to it?

21


So how do we manage ratios with value >1? Do we leave them out from cam-

paigns or do we have to deal with them in another way?

2.4.4 Level 2 Ratios logic: Pneumatic Drives as family basis

We understand that to use ratios between two product families we have to

fix one of the values that are composing the ratio, in this way we are able to

create a feasible positioning for our customers by relating them to their ratios

(and in this way we give meanings to ratios higher than 1). We also under-

stand that this decision is completely strategic and will draw the direction of

the future campaigns to be done related to Cross Selling analysis.

As we already explained in the second section of this chapter, F-IT (and

generally Festo worldwide) core business is historically constituted by Pneu-

matic Drives product family. This is due to historic technical competence of

the company and to the quality of these products. Pneumatic Drives are also

the products with the highest number of other product families correlated with

them: it means that, by fixing these product families, we have the possibility

to set a good number of ratios with other product families or with product that

are accessories for Pneumatic Drives (necessarily correlated with the quantity

of Drives sold).

The final result are 6 main ratios all based on drives quantities, 4 of these

with a 1 by 1 relation with the number of drives, 2 ratios composed by for-

mulas with more than 2 different components (see Figure 2.14). The result

is an analysis with information about drives that allows us to compare cus-

tomers with different volumes generated, but this time comparable between

them with ratios: this enables us to identify groups of customers with the same

ratios positioning and to define specific strategies for them.

22


Figure 2.14 – Relations between Drives and other product families

Figure 2.15 – After the analysis, the resume sheet shows us the ratios for eachF-IT customer

In the right side of Figure 2.15, the Analysis Area, we can note all 6 ratios

amount for some F-IT customer and the logic (or at least the simplest logic) to

define an action or not with a customer on the basis of ratios’ values.

The header of Analysis Area shows us also the fact that all ratios (with the

exception of the last one) are defined by taking fixed in the denominator the

amount of Pneumatic Drives, in order to treat all customers in the same way,

23


thinking of Pneumatic Drives as the leading product family and trying to study

differences between all the other macro families.

2.4.5 Level 2 problems: the dependence from Drives trend

As we explained, all Ratio Analysis is based on the amount of Pneumatic

Drives, this means that if this family suffers from some specific trend, this will

affect all Ratios in the opposite way of its trend (as for all ratios the amount of

Drives is set as denominator). To make a simple example look at Table 2.2 and

study the case where F-IT Pneumatic Drives total amount decreases instead of

being the leading product family and at the same time all the other families’

amount remains the same: what will happen to first 2 Ratios?

Name Pneumatic Drives Sensors Mountings Sens./Drives Mount./DrivesCUST 1 3500 2800 2600 0,80 0,74CUST 1b 2500 2800 2600 1,12 1,04

Table 2.2 – Opposite effect on ratios generated by Drives

For the nature of ratios, this would be taken as good effect on all ratios in-

stead of a loss of market share on Pneumatic Drives family. This is something

very misleading and counter-intuitive and makes us unable to set specific sta-

tistical analysis on our dataset with coherent logics.

The solution to this issue is a middle way between Level 1 and Level 2 anal-

yses, that is to reconsider the total amounts of macro-families (so retake in

account the trend of Pneumatic Drives) and to study all customers as a com-

position (in the statistic meaning of the term) of the amounts of single families.

In this way we are able to avoid the Pneumatic Drives trend problem and we

also have a complete view of how a customer is positioned with respect to its

ideal situation.

In order to develop the right statistic environment for our dataset we will use

24


Compositional Data Analysis. We will also see that we have to define the right

geometry (and so the right basis transformation) for our case study in order

to obtain results coherent with our scope, namely to study the relation be-

tween the single components of a customer and not just the size of the abso-

lute amounts of a particular product family.

25

Chapter 3

Compositional Data Analysis andthe Aitchison Simplex

3.1 D-compositions and Simplex SD

As we introduced at the end of the last chapter we are more interested in the

study, for each F-IT customer, of the relation between the amounts of product

families than their absolute size.

By treating customers on an absolute scale we can run into misleading situa-

tions as we can deal with spurious correlations between components of a com-

position and also because most of standard multivariate statistical methods are

not applicable to this type of data.

This is a problem that was known since the end of nineteenth century: Karl

Pearson in a paper in 1897 [7] pointed out the problems arising from the use

of standard statistical methods with proportions. Nowadays some different

schools of thought arose and there are different approach to Compositional

Data Analysis, some of them seem to meet our needs. Before introducing

these techniques first of all let’s define what a composition is from a statistical

point of view.

Definition 1. A vector x = [x1, x2, . . . , xD] is a D-part composition when all its

components are strictly positive real numbers and carry only relative information.

26

3.1. D-COMPOSITIONS AND SIMPLEX SD

In our study case we are more interested in the relation between the com-

ponents of customer product basket than on the amount of a single product

family. Thus it seems to be reasonable to treat our data as compositions instead

of considering them as a standard multivariate dataset (with every customer

as a single observation of the dataset) with the absolute amounts of product

families as a variable.

Generally a composition has the intrinsic propriety that all its components sum

to a constant k (in this case we are talking about closed data), with k = 1 if we are

dealing with compositions that represent proportions (i.e. every components

will describe the part per unit of a component) or k = 100 if we are dealing

with percentages. In our case most of our observations will sum up to a total

different every time since we are dealing with customers with different total

amounts. In this case a kind of normalization or a transformation of data is

essential in order to have the possibility to manage the data in the proper way.

We will see different approaches to obtain our purpose that involve different

schools of thought. First of all we have to fix the concept that no matters if we

are dealing with compositions with different total sum of their components as

what we need is to study the ratios between them respect to the total.

Definition 2. Two vectors of D positive real components x, y ∈ RD+ (xi, yi > 0, ∀i =

1, 2, . . . , D) are compositionally equivalent if exists a positive constant number

c ∈ R+ such that x = c · y.

This definition allows us to begin to think that this statistical approach

is feasible for our purpose as it treats two customers (vectors) with same ra-

tios between components but different total amount (so with the components

that differ two by two only for a constant c) as compositionally equivalent, so

they would represent the same composition even if they had different total

27


amounts. This fact allows to understand that, with the appropriate scaling fac-

tor for each composition, we can represent all our customers’ D− composition

as elements of the hyperplane containing the vectors whose components sum

up to a given constant κ. To make this we need to define the closure operation,

that allows to assign a costant sum to all our D-compostions.

Definition 3. For any vector of D strictly positive real components,

z = [z1, z2, . . . , zD] ∈ RD+, zi > 0 ∀i = 1, 2, . . . , D,

the closure of z to κ > 0 is defined as

C(z) =

[κ · z1

∑Di=1 zi

,κ · z2

∑Di=1 zi

, . . . ,κ · zD

∑Di=1 zi

]. (3.1)

So we can rewrite the compositional equivalence in the next way: two vec-

tors x, y ∈ RD+ are compositionally equivalent if C(x) = C(y) for all κ closure

constant chosen.

With Closure operation we represent all our D-compositions (as customers’

components of all product families taken in analysis) rescaled to sum up to the

same constant. From now on we will considerκ = 1, as we are familiar to treat

Cross Selling ratios that are ideally close to 1.

With this operation we will be able to define the sample space for our composi-

tions.

Definition 4. The sample space of compositional data is the simplex,

SD =

{x = [x1, x2, . . . , xD]

∣∣∣∣∣xi > 0, i = 1, 2, . . . , D;D

∑i=1

xi = κ

}(3.2)

In Figure 3.1 it is shown the simplex SD of constant sum k relative to R3+:

vectors P and P’ are treated as equivalent in the simplex.

28


Figure 3.1 – Simplex in R3+ and its representation as ternary diagram (Source:

[6])

Now that the sample space for compositions is defined, we can also intro-

duce the concept of subcomposition.

Definition 5. Given a composition x and a selection of indices S = i1, i2, . . . , is, a

subcomposition xS, with s parts, is obtained by applying the closure operation to the

subvector [xi1 , xi2 , . . . , xis ] of x.

The set of subscripts S indicates which parts are selected in the subcomposition, not

necessarily the first s ones.

It is essential to remark that in most situations we are just able to measure

only subcompositions (let’s think, for instance, to measurements of basic com-

ponents of a chemical compost where there will be components not measured

because of their real absence but there will be also components that are not

measured because of instrument errors or other external factors). This happens

generally in analyses in biologic fields. We will see in Chapter 4 the importance

of how to treat subcompositions in relation to the whole compositions of the

same dataset. In our specific case we will make some analysis on subcomposi-

tions of our initial customers to obtain results on 3 components, even because

it will be more intuitive and it will help us to draw some graphical conclusions.

Subcompositions are one way to reduce the dimensionality of a compositional

29

3.2. COMPOSITIONAL ANALYSIS PRINCIPLES

data set: another quite commonly used way is to amalgamate some compo-

nents, that is, to sum them into one new part.

Property 1. Amalgamation: Given a composition x ∈ SD, and a selection of indices

A = {i1, . . . , ia} (not necessarily the first ones), D− a ≥ 1, and the set of remaining

indices A the value

xA = ∑j∈A

xi

is called amalgamated part or amalgamated component. The vector x’ = [xA, xA],

containing the components with subscript in A grouped in xA and the amalgamated

component xA, is called amalgamated composition which is in SD−a+1. Note that

using a fill-up or residual value is equivalent to using an amalgamated composition.

3.2 Compositional analysis principles

As it’s nature based on the relation between components with the whole

unit, it’s clear that compositions (as statical tool) must meet certain needed

requirements (or principles). Aitchison in 1986 defined three principles that

induce geometric transformations for the dataset allowing for the use of classic

statistical tools on the transformed dataset.

3.2.1 Scale invariance

As we mentioned, compositions carry only relative information. In our

business case no matter if are comparing a customer with generated volumes

much larger than another one, we want to analyse them for the relative rela-

tions between the different product families that they are buying, so we need

to define the concept of scale invariance.

Definition 6. A function f (·) defined on RD+ is scale invariant if, for any positive

real value λ ∈ R and for any composition x ∈ SD, it satisfies f (λ x) = f (x), so if it

30

3.2. COMPOSITIONAL ANALYSIS PRINCIPLES

returns the same result for all compositionally equivalent compositions.

There are a lot of functions that satisfy this principle, an example is the

simple ratio function as, given compositions x = [x1, x2, . . . , xD] and y = λ x is

easy to understand that f (x) = x1/x2 = (λ x1)/(λ x2) = f (y) = f (λx). In our

business case it means that if ratio between the first 2 product families was the

same (even with different amounts) we would treat them as compositionally

equivalent.

However, ratios on compositions, as we define them, are strictly positive and

depend on the ordering of parts, in fact x1/x2 and x2/x1 would return us two

completely different results. A convenient transformation of ratios is the cor-

responding logratio, f (x) = ln(x1/x2).

In this way the inversion of the ratio only produces a change of sign and de-

fines a symmetry in the function with respect to the ordering of composition’s

parts.

3.2.2 Permutation invariance

Another crucial point is that results of the analysis must not depend on the

sequence of the components our dataset.

Definition 7. A function f (·) of a vector argument x = [x1, x2, . . . , xD] is permu-

tation invariant if the value of f (·) do not change if we permute the components of

x

If we think to the log-ratio approach, this is a very important principle

when we ask which methods can be meaningfully applied to coordinates of

compositional data. A naive Euclidean distance of alr 1 transformed data is

not permutation invariant and is not the proper tool for cluster analysis. We

1. Additive Log-Ratio Transformation: alr(x) = (ln(x1/xD), . . . , ln(xD−1/xD))

31

3.3. THE AITCHISON GEOMETRY

would risk to have different clusterings depending on which was the last vari-

able in the dataset. An alternative transformation will be discussed below.

3.2.3 Subcompositional coherence

The final principle we need is subcompositional coherence: subcomposi-

tions behave as the equivalent in real analysis of orthogonal projections of their

original compositions. This led us to two important consequence:

— The distance (whatever the way in which we define it) between two

compositions must be greater or equal to the distance between them

when we are considering any subcompositions. This is called subcom-

positional dominance. It could be easily demonstrated that Euclidean

distance between compositional vectors does not fulfil this condition.

— If a non-informative part is removed, results should not change: it means

that measures of association or measures of dissimilarity between com-

ponents, for example correlations or distances are unaffected by consid-

ering subcompositions instead of the whole composition.

Scale invariance of the results is preserved within arbitrary subcompositions,

that is, the ratios between any parts in the subcomposition are equal to the

corresponding ratios in the original composition.

3.3 The Aitchison geometry

To work with our dataset and to treat it as a set of observations we need to

build a proper environment for it and, in the same time, we have to preserve

the principles defined above.

If we were working in the real space, we could add vectors, multiply them

by scalar values and look for properties such as orthogonality, or compute the

distance between two points.

32


Drives Sensors MountingsCust1 115 25 5Cust2 90 45 10Cust3 60 15 15Cust4 35 35 10

Drives Sensors

Mountings

●●

●●

Figure 3.2 – Example of representation of 4 Customers on S3

Cust1 Cust2 Cust3Cust2 32.40370Cust3 56.78908 42.72002Cust4 80.77747 55.90170 32.40370

Table 3.1 – Euclidean Distance between customers on S3

All of this, and much more, is possible because the real space is a linear vec-

tor space with a metric structure. We are familiar with its geometric structure,

the Euclidean geometry, and we are used to represent observations within this

geometry. But this geometry is generally not a proper geometry for composi-

tional data, because it is not able to capture relations between the components

of a composition. Let’s take as an example the simple case of 4 customers rep-

resentation in Figure 3.2.

If we treat the simplex as a standard Euclidean space we would come to

the (graphical) conclusion that Cust1 and Cust2 have different distance than

Cust3 and Cust4. Results given by Euclidean Distance are in Table 3.1.

The Euclidean distance between them is certainly the same, as there is a dif-

ference of 25, 20 and 5 units between the three respective components. But in

the second case, the proportion in the first component is almost doubled, while

in the second case the difference is lower; we have a similar situation also in

the third component. An approach that takes into account relative differences

seems more adequate to describe compositional variability.

33


This is not the only reason for discarding the usual Euclidean geometry as a

proper tool for analysing compositional data. Problems might appear in other

situations, such as those where results end up outside the sample space, for ex-

ample, when translating compositional vectors or computing joint confidence

regions for random compositions under assumptions of normality, ellipses and

lines. So we need to build a geometry to work with compositional data in the

simplex, as there things appear not as simple as they are in real space.

To make this we have to define two operations equip the simplex with a vector

space structure. The first one is perturbation, which is analogous to addition

in real space; the second one is powering, the analogous to multiplication by

a scalar in real space. Moreover, it is possible to obtain an Euclidean vector

space structure on the simplex adding an inner product, a norm, and a dis-

tance to the previous definitions. With all these definitions we can operate in

the simplex in the same way as one operates in real space.

3.3.1 Defining a vector space structure

We thus have to define the analogous operations to addition and multipli-

cation by a scalar in real space, using closure operation defined in Definition

3:

Definition 8. Perturbation of x ∈ SD by y ∈ SD,

x⊕

x = C[x1y1, x2y2, . . . , xD yD] ∈ SD

Definition 9. Power transformation or powering of x ∈ SD by a constantα ∈ R,

α⊙

x = C[xα1 , xα2 , . . . , xαD] ∈ SD

Now we can say that the triple (S ,⊕

,⊙) with perturbation and powering

is a vector space. This means that these properties are analogous to translation

and scalar multiplication in real space.

34


Drives Sensors

Mountings

●●

●●

Drives Sensors

Mountings

●●

●

●

Drives Sensors

Mountings

● ●●

●

Figure 3.3 – Previous example with customers perturbed by y = [5, 30, 80] andy = [10, 100, 10]

Drives Sensors

Mountings

●●

●●

Drives Sensors

Mountings

●●

●●

Drives Sensors

Mountings

● ●● ●

Figure 3.4 – Previous example with powering applied to customers with α =0.5 andα = 2

Property 2. (SD,⊕) is a commutative group structure; that is, for x, y, z ∈ SD, it

holds:

1. commutative property: x⊕

y = y⊕

x;

2. associative property: (x⊕

y)⊕

z = x⊕(y⊕

z);

3. neutral element:

n = C[1, 1, . . . , 1] =

[1D

,1D

, . . . ,1D

];

n is the barycentre of the simplex and is unique;

4. inverse of x : x-1 = C[x−11 , x−1

2 , . . . , x−1D ]: thus, x

⊕x-1 = n

By analogy with standard operations in real space, for the perturbation differ-

ence, we will write: x⊕

y-1 = x y

35


Property 3. Powering satisfies the properties of an external product. For x, y ∈

SD,α,β ∈ R, it holds:

1. associative property: α⊙(β⊙

x) = (α ·β)⊙ x;

2. distributive property 1: α⊙(x⊕

y) = (α⊙

x)⊕

(α⊙

y);

3. distributive property 2: (α +β)⊙

x = (α⊙

x)⊕

(β⊙

x);

4. neutral element: 1⊙

x = x; the neutral element is unique.

Note that the closure operation cancels out any constant and, thus, the closure

constant itself is not important from a mathematical point of view. This fact

allows us to omit the closure in intermediate steps of any computation without

problem.

It has also significant implications for practical reasons, as shall be seen during

simplicial principal component analysis. We can express this property for z ∈

RD+ and x ∈ SD as

x ⊕ (α� z) = x ⊕ (α� C(z))

Nevertheless, one should be always aware that the closure constant is very

important for the interpretation of the units of the problem at hand. Therefore,

controlling for the right units should be the last step in any analysis.

3.3.2 Aitchison inner product, norm, and distance

To obtain a Euclidean vector space structure, we take the following inner

product, with associated norm and distance (the subindex A stands for Aitchi-

son).

Definition 10. Aitchison scalar product for compositions x, y ∈ SD,

〈x, y〉A = 1D

D

∑i> j

lnxi

x jln

yi

y j

36


Definition 11. Aitchison norm for composition x ∈ SD,

‖x‖A =

√√√√ 12D

D

∑i=1

D

∑j=1

(ln

xi

x j

)2

Now we are also able to define a proper distance that will allow us to work

on our compositional dataset as in real space.

Definition 12. Aitchison distance: the distance between compositions x, y ∈ SD is

defined in the next way,

dA(x, y) = ‖x y‖A =

√√√√ 12D

D

∑i=1

D

∑j=1

(ln

xi

x j− ln

yi

y j

)2

With this new definition of distance let’s consider again the example of

4 customers in Figure 3.2 and now compute the Aitchison distance between

Customers 1 and 2 and between Customers 3 and 4, considering that these

distance were identical using Euclidean distance (although this conclusion was

misleading from a logic point of view):

Cust1 Cust2 Cust3Cust2 0.7269086Cust3 1.3747150 1.0646298Cust4 1.4143010 0.6917658 1.0815202

Table 3.2 – Aitchison Distance between customers on S3

We can see that now the two distances are different, as in this definition we

are considering also the relation between components on our customers. We

thus understand that geometry on the simplex has logics that differs from our

standard geometric conceptions: even in ternary diagram the two distances

seem to be different (contradicting the results of Euclidean distance between

customers).

37


Transformations

Different log-ratio transformations could be applied to compositional data,

each one that gives different results: most used are Centered log-ratio transfor-

mation (clr) and Isometric log-ratio transformation (ilr):

— Centered log-ratio transform (g(x) = D√

x1 · · · xD)

clr(x) = z =

[ln

x1

g(x); . . . ; ln

xD

g(x)

]

=ln (x)

D·

D− 1 −1 . . . −1−1 D− 1 . . . −1

...... . . . ...

−1 −1 . . . D− 1

clr−1(z) = C[exp(z)]

— Isometric log-ratio transform

ilrV(x) = clr(x) ·V = ln (x) ·V

For a given matrix V of D rows and (D-1) columns such that V · V t =

ID−1 (Identity matrix of D-1 elements) and V · V t = ID + a1, where a

may be any value and 1 is a matrix full of ones.

The inverse is:

ilr−1V (x) = C[exp (x · V t)]

3.3.3 Geometry on S3: figures on ternary diagrams

Ternary diagrams interpretation

Now that we have a well-defined geometry, it is useful to show how some

geometric figures are represented on the simplex S3 using ternary diagrams.

This short explication is due as we will need to know how these geometric

38


elements have to be interpreted in order to understand results of our statical

analyses, for instance for PCA analysis to understand the meaning of Principal

Components directions, for the interpretation of the position on ternary dia-

gram of our customers and understand the meaning of clusters derived from

classification.

Figure 3.5 – Template for ternary diagrams and importance of amount compo-nents proportion in S3.

In Figure 3.6 some customers are represented on ternary diagrams in order

to show how they act on ternary diagrams: customers with same shape are

similar. The more a customer is near a vertex, the more the proportion of the

components of this vertex is important in the compositions (Figure 3.5): this

means that if a point is lying near the opposite side of a certain vertex, the

component related the vertex is very weak.

Shapes on ternary diagrams

As we are working on simplex, in ternary diagrams geometric shapes will

behave in different way from real space. In left ternary diagram of Figure 3.7

we show how parallel lines are represented and in right ternary diagram of

39


Drives Sensors

Mountings

●●

●

●

Figure 3.6 – Similar customers (by shape or colour) lying on same proportionlines.

circles and ellipses change respect to real space.

For some characteristics of ternary diagram interpretation is intuitive, for in-

stance the positioning of a point on ternary diagram and the relation with its

distance from the vertex is intuitive, for others is not so common: lines’ direc-

tion and shape together with translation of elements on ternary diagram has

to be treated with proper attention to not mislead data interpretation.

Now we have a proper mathematical environment to work on simplex

with F-IT customers’ data and we have also a proper representation tool such

ternary diagrams to support analysis results.

40


Figure 3.7 – Parallel lines, circles and ellipses on ternary diagram. Source: [6]

41

Chapter 4

Business case: Compositional dataanalysis of Cross Selling data

Now with Aithcison geometry we have the sample space for our compo-

sitional data that is the 5-part simplex. The latter is the 4-dimensional subset

of RD−1 that contains all 5-part compositions that sum up to a prescribed con-

stant, in our case to 1.

For all the analyses of this chapter it has been used a specific R package named

compositions [8], that already provides some operations on compositional

data (such Aitchison distance and Compositional PCA). For analyses like K-

means clustering, Linear and Quadratic Discriminant Analysis specific opera-

tions have been defined using ilr and clr transformations.

4.1 F-IT Customer dataset as Compositional dataset

We already introduced the Festo approach to Cross Selling with the defini-

tion of its Ratios and also with its weaknesses and we also denoted importance

in analysis of Industry Sector variable, we want to define our data in order to

be able to be coherent with Company’s strategies and also in order to be able

to make statistical analysis on it.

42

4.2. SIMPLIFICATION OF DATASET

4.1.1 Considering Customer as compositional observations

As explained in Chapter 1, regarding Cross Selling, by now customers have

been analysed on Level 1 as observations of absolute amounts for each product

family (but from the relation between these macro-families) and on Level 2 by

ratios for some product families (the one we are have relations with Drives).

We can indeed take all macro-families that we are sure have relations between

them (i.e. all the macro families used on Level 2 ratios and also the amount of

Pneumatic Drives).

Our aim is to be able to use classic statistical procedures such as PCA,

Cluster Analysis, Linear and Quadratic Discriminant Analysis in order to con-

firm or not the direction that Festo wants to tread for next marketing cam-

paigns, and which variable is better to take as guide. For the theory describ-

ing the above mentioned classical statistical methods we refer to (ref Johnson-

Wichern).

Our first target is to reduce the variables of our dataset.

4.2 Simplification of Dataset

4.2.1 Variables taken into consideration

The focus is to reduce dataset variables in order to keep only the ones that

we want to use for our analyses. We split the data reduction based on their

nature.

In our final dataset every observation will corresponds to a single customer

with some categorical and some numerical variables.

43

4.2. SIMPLIFICATION OF DATASET

Categorical variables

To be able to interpret our final results we will maintain Customer Num-

ber and Customer Name variables, even if they show unique values for each

observation. Another variable we will keep is the one referring to the code

of the F-IT Sales Engineer of each customer. As we explained in first chapter,

our aim is to make analysis on the Industry Sector of F-IT customers, it is cru-

cial as we want to understand if the idea of F-IT to make specific Cross Selling

campaigns differentiated by ISM, is reflected by results that come out from our

classification analyses.

Numerical variables: macro-families amounts taken into account

Not all macro families are perfectly related to each other. Ratios of Level

2 analysis are just six and two of them (last two) are stand-alone ratios not

related to the other one. Thus we decided to take into account only strictly re-

lated macro families: Pneumatic Drives, Proximity Sensors, Throttles, Cylin-

der Mountings and Piston Rod Attachment. We stated in Chapter 3 that neu-

tral element n of a perturbation is the one with uniform amounts in each part,

i.e., for our dataset it would be

n = C[k, k, k, k, k] =

[15

,15

,15

,15

,15

]We will also find some issues in interpreting graphical results for some analy-

ses applied to the complete dataset with compositions of 5 components, since

we are not able to represent data in S5 but we will need to represent data on

the S3 subspace. In these cases we will repeat some analyses on subcomposi-

tions of three components where results are more clearly explicable.

In Figure 4.1 we have some examples of products composing the selected

macro families.

44

4.3. DATASET REPRESENTATION

Figure 4.1 – Final macro families taken into analysis

4.3 Dataset representation

The final dataset is composed by 595 observations of 10 variables (six of

which are numerical, one for total Market Sales of customers and five for the

product quantities which form our compositions).

In Figure 4.2 a template of a part of final dataset is shown: we can immediately

see that there is a lot of variability between the amounts of the five components

of observations (last five columns).

Graphical representation: RD Vs SD

A first graphical and exploratory analysis approach reveals issues related

to analysing compositions as standard data. In Figure 4.3 observations are

shown with absolute amounts and coloured by ISM: the analysis is misleading

and shows how customers dimension affects the view.

As we are more interested in the relation between components than in di-

mension of families’ amounts, we apply Closure to 1 for our compositions and

represent them on the simplex S5: Figure 4.4 shows data projected on S3 sub-

45

4.3. DATASET REPRESENTATION

Figure 4.2 – Example of dataset

space by placing on the vertex couples of components and on the opposite

vertex the amalgamation of remaining components.

In this representation it becomes evident something that was already no-

ticed in Chapter 2: Pneumatic Drives (first row or column of ternary diagrams)

is the predominant component, as the observations are generally concentrated

toward the Drives vertex. The simplex representation shows that proportions

of Sensors and Throttles seem to be balanced (see on specific ternary diagram

for these two components) while Cylinder Mountings and Piston Rod Attach-

ments are less relevant components. From a "chromatic" point of view it seems

not evident a relation between colors of ISM and positioning of customers in

the simplex.

46

4.4. COMPOSITIONAL PRINCIPAL COMPONENTS ANALYSIS

Pneumatic_Drives

0 2000 5000

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0 1000 2500

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0 1000 2000

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Pist_Rod

Figure 4.3 – Representation of dataset as absolute amounts

4.4 Compositional Principal Components Analysis

After the definition of dataset, data reduction and a first graphical analysis,

let us start with first statistical exploratory analysis: PCA.

We have to remind that Aitchison geometry, as it is defined, performs a data

reduction in the equivalent working space, so as we pass from SD to a RD−1

47


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Pist_Rod

Figure 4.4 – Representation of dataset on simplex

equivalent space, we expect to have informations on four principal compo-

nents, fifth component is present but it is not shown because it is obviously

null (due to the constraint of definition of composition).

First of all, let us give a look to scree plot in Figure 4.5, where we can see the

first results of this first exploratory method in terms of amount of variability

explained by the different Principal Components.

Thanks to Figure 4.6 we can clearly see how the first principal component

already explains a great part of the phenomenon variability, that is almost

42%. With the second principal component, accounting for 33% of the vari-

ability, we reach an acceptable level of approximation in explaining the model

(74.6%).

48


●

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a_princ

Var

ianc

es

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Comp.1 Comp.2 Comp.3 Comp.4

Figure 4.5 – Results summary applied to all 5 components

Figure 4.6 – Proportion of variance on 4 Principal Components

4.4.1 Visualizing and interpreting Biplot of compositional PCAscores

We can then represent our Scores, taking in consideration Biplot of the first

two principal components. Origin of plot represents the centre of the compo-

sitional dataset (in Aitchison mean sense).

It must be clear that the fundamental elements of a compositional biplot are

the links, not the rays as in the case of variation diagrams for unconstrained

multivariate data. Links specify all the relative variances, informs us about

the compositional covariance structure and provides hints about subcomposi-

49


−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

−0.

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050.

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050.

100.

15

Comp.1

Com

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Pneumatic_Drives

Sensors

ThrottlesCil_Mountings

Pist_Rod

Figure 4.7 – Biplot of first 2 principal components

tional variability and independence. If distance between two vertex of com-

ponents’ vectors is high, it means that variance of ratio between these two

components is high in our dataset. In our case it would mean that we have a

very varied percentage of purchasing for these components in our customers.

For instance some customers are buying more component 1 and almost noth-

ing of component 2, some customers the opposite and some other customers

are buying similar quantities of two components.

On the contrary when link between two vertex is very low it means that varia-

tion in ratio of these two components is low, so inside the dataset ratio between

proportion of the two components remain almost constant.

50


In our case we have that length of segment linking vertex on Cylinder Mount-

ings and Piston Rod Attachments is very low. We can thus assume that more

or less all customers have similar ratios between these two product families.

On the contrary length of segment linking Throttles and Sensors (same expla-

nation for Throttles and Mountings) is high so in our customers there is more

variability in the ratios of these two product families. This in fact contradicts

our first simple graphical interpretation of data.

Results on Biplot reveal different information with respect to graphical con-

clusions given in the first exploratory analysis: for example the variability on

ratios between Throttles and Sensors was not easy to be identified directly

from ternary diagrams. This is due to the fact that analysing and interpret-

ing ternary diagrams could sometimes be not so easy and statistical analyses

results could help us.

4.4.2 Visualizing and interpreting compositional PCA loadings

Unlike a classical Principal Components Analysis applied to a standard

dataset lying in a Euclidean space, in this case, given that we are dealing with

compositional data, we must pay attention in the interpretation phase of the

results of such analysis, mostly in the meaning that the final loadings have.

The first following line returns their coefficients (Figure 4.8), whose sum is

checked to be zero in the second line.

These coefficients are not free to vary in a five-dimensional space, but linked

to each other through this null sum. We cannot interpret a low value as an ab-

sence of dependence. For instance, in our case, we should not think that the

first PC is not affected by Pneumatic Drives, in spite of having a very low load-

ing (0.09): we can nevertheless say that the Cylinder Mountings vs. Sensors log

ratio increases along the first PC with a slope of 0.417 - 0.030 = 0.387.

51


Figure 4.8 – Loadings on first 2 Principal Components

We may also say that along the first PC the ratio Sensors/Pneumatic Drives is

almost unchanged, because their loadings difference is almost zero, whereas

along the second PC, this log ratio increases with a slope of 0.765.

We can represent our loadings as in Figure 4.9.

Comp.1 Comp.2 Comp.3 Comp.4

Pist_RodCil_MountingsThrottlesSensorsPneumatic_Drives

Loadings

01

23

45

Figure 4.9 – Barplot representing loadings value on all 4 principal components

A correct way to interpret the values of the loadings is the following: if a bar

has a unitary height for a given PC, along that PC the ratio of the two parts

52


involved is not modified; otherwise, if the bar is significantly lower (higher)

than 1, along that specific PC the ratio is reduced (increased).

Based on this way of thinking, we can see, for example, that Throttles variable

has a bar significantly lower than 1 for the first component. Along the 1st pc

the ratio is reduced.

Let us represent now the whole dataset together with the direction of the

direction of the first two principal components in Figure 4.10.

Pneumatic_Drives


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Pist_Rod

Figure 4.10 – Dataset with first (solid line) and second (dashed line) PCs

In order to better understand this type of picture, let us simplify the repre-

sentation, using only three parts and drawing Cylinders, Mountings and Sen-

sors together with the direction of the first two principal components. The red

53


line is an ellipse representing a 2σ predictive region of the simplex.


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Figure 4.11 – Representation of first (solid line) and second (dashed line) PCson subcompositions for Sensors, Pneumatic Drives and Cylinder Mountingscoloured by ISM

On this basis we can try to interpret the first two principal components.

As we see from Figure 4.11 the first PC (solid line), having its extreme points

in Cylinder Mountings and Sensors, suggests that the quantity of Pneumatic

Drives represents an influential part of the composition. The second PC, (dashed

line), being similar to a straight line, shows that the ratio between Cylinder

Mountings and Sensors is not far from being constant in the dataset, no matter

the amount of Pneumatic Drives.

54

4.5. CLASSIFICATION AND GROUPING WITH AITCHISON DISTANCE

4.5 Classification and grouping with Aitchison dis-tance

Next analyses are crucial to know if strategies of Festo are going in the right

direction: the use of Industry Sector as a driving attribute to develop specific

marketing campaigns (because of the differentiation of products between dif-

ferent ISMs) instead of using groups of customers with similar purchasing be-

haviours (regarding positioning related to components).

We will develop a series of analysis in order to see if ISM is a significant at-

tribute to drive a marketing campaign based on compositions dataset.

4.5.1 Hierarchical Cluster Analysis in S5 and S3

Hierarchical clustering has been applied on the dataset to see if it is possi-

ble to identify groups related on ISM variable: hierarchical clustering will be

applied after a preliminary part where different metrics (Euclidean and Man-

hattan) and linkage methods (single, average, complete and then Ward) effec-

tiveness will be evaluated.

Dendogram analysis

We remark that when we talk about Euclidean and Manhattan distances

we refer to their equivalence in Aitchison Geometry transformed data.

First of all in Figure 4.12 dendograms for the two distances and for first two

methods are compared.

Average linkage shows us the presence of two clusters with both distances

considered, one of them composed by only one customer.

Complete linkage shows us two or at maximum three clusters in both dis-

tances. We can easily remove cases where Single Linkage is applied as it

55

4.5. CLASSIFICATION AND GROUPING WITH AITCHISON DISTANCE0

12

34

56

7

Average linkage

Euc

lidea

n di

stan

ce

01

23

Single linkage

Hei

ght

02

46

810

Complete linkage

Hei

ght

02

46

810

12

Average linkage

Man

hatta

n di

stan

ce

01

23

45

67

Single linkageH

eigh

t

05

1015

20

Complete linkage

Hei

ght

Figure 4.12 – Dendograms for Euclidean and Manhattan distances

doesn’t seem to be appropriate to agglomerate data in significant clusters and

also distance between clusters’ levels has small values.

In Figure 4.13 Ward linkage is considered in the analysis: it seems to be the

best way to cluster data as it has clearly both separate cluster and high value

for distance between them. The right choice for number of cluster seems to be

3 or at maximum 6 clusters.

Before choosing which method is the best to cluster our compositions a

check on correlation between distances and their own cophenetic indexes has

to be done to measure how faithfully the methods preserve the pairwise dis-

tances between the original compositions.

For both distance methods highest values are related to average linkage,

other linkage methods have lower values and no cophenetic correlation gives

us the certainty to be the best (as all correlation indexes have values < 0.8).

Dendograms suggests to use Manhattan distance with Ward linkage.

56

4.5. CLASSIFICATION AND GROUPING WITH AITCHISON DISTANCE0

12

34

56

7

Average linkage

Euc

lidea

n di

stan

ce

02

46

810

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Hei

ght

050

100

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Hei

ght

02

46

810

12

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Man

hatta

n di

stan

ce

05

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eigh

t

010

020

030

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Hei

ght

Figure 4.13 – Dendograms for Euclidean and Manhattan distances with Ward

Figure 4.14 – Cophenetic correlations for Euclidean and Manhattan distances

Graphical representation for clusters

In Figure 4.15 data are shown coloured by 3 clusters calculated with Eu-

clidean Distance and complete linkage.

In Figure 4.16 data are shown coloured by 6 clusters calculated with Man-

hattan Distance and Ward linkage. In both cases we are able to identify quite

well differentiated clusters with a clear meaning: customers are grouped with

57


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Manhattan, Ward linkage with 3 clusters

Figure 4.15 – Data grouped in 3 clusters

respect to their attitude to buy product families.

It is noticed a problem of overlapping between clusters but this is due to a

technical motivation: representing our 5-compositions as projection in the sub-

space S3 doesn’t allow us to see the real separation between clusters. To have

a better view of different clusters let’s repeat the analysis but on a subcompo-

sition that take into account only 3 components.

Data reduction

Let’s take in analysis only subcomposition of Pneumatic Drives, Proximity

Sensors and Throttles and check if customers are clustered with the same logics

of the complete compositions case. First row of each picture represents data

clustered using euclidean distance, second row using manhattan distance.

58


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Manhattan, Ward linkage with 6 clusters

Figure 4.16 – Data grouped in 6 clusters


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Figure 4.17 – Subcompositions data grouped in 3 clusters for average, com-plete and Ward linkage: first row Euclidean distance, second row Manhattandistance

59



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Figure 4.18 – Subcompositions data grouped in 6 clusters for average, com-plete and Ward linkage: first row Euclidean distance, second row Manhattandistance

Operating on 3 components’ subcompositions, cluster are perfectly distin-

guishable and there is no sign of overlapping.

We repeat the interpretation already given on complete case: in case of choice

of 3 clusters, we can distinguish a central cluster composed by customers that

have proportions balanced (so they are buying homogeneously all product

families) and two marginal clusters composed by customers that are not buy-

ing respectively Sensors (cluster blue) and Throttles (cluster green).

In case of 6 clusters with Ward linkage three other groups are identified, one

regarding customers with a strong values of proportion of Pneumatic Drives

component, one in the opposite position of the simplex and another one posi-

tioned in the middle between cluster of customers near the center of simplex

and cluster of customers with a weak proportion on Throttles component.

It seems that hierarchical cluster analysis gives us the information that cus-

tomer should have to be treated in the way of how they approach market and

60

4.6. COMPOSITIONAL K-MEANS CLUSTER ANALYSIS

not only about their reference Industry Sector.

Last doubt that remains is the correct number of clusters to be chosen between

6 and 3: K-means clustering could help us discover it.

4.6 Compositional K-means Cluster Analysis

We understand from hierarchical cluster analysis that the customers group

together in terms of their purchasing behaviours but, since cophenetic corre-

lation values are too low to have certainty dendograms describe well our data,

we are not sure of which number of clusters is optimal to group our customers

in therms of their purchasing attitudes on our components.

4.6.1 K-means with k=6

K-means clustering is a method of vector quantization that can help us dis-

covering the optimal number of clusters for our compositional dataset. It aims

to partition n observations into k clusters in which each observation belongs to

the cluster with the nearest mean, serving as a prototype of the cluster.

As we studied in Section 4.5.1, it seems from hierarchical clustering that

a proper number of clusters for our dataset would be 3 or 6: we start analy-

sis with 6 clusters and using as starting centres for k-means algorithm values

coming from hierarchical clustering using Manhattan distance (in Aitchison

geometry) with Ward linkage.

In Figure 4.19 we can see results for k-means algorithm applied to transformed

data for 6 clusters with the final centres of algorithm computed in 3 iterations.

Motivation to the overlapping in single ternary diagrams has been explained

in previous section, we can repeat the simplified analysis as in the previous

61


Figure 4.19 – Results for k-means algorithm with 6 clusters and starting centresfrom hierarchical clustering

Pneumatic_Drives


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Figure 4.20 – Ternary diagrams for k-means with 6 clusters, purple squaresrepresent centres of the clusters

case to see at the result on the subcomposition of Pneumatic Drives, Proximity

Sensors and Throttles, results are shown in Figure 4.21: now data are visibly

62


separated into 6 clusters that seem to respond to a similar interpretation of hi-

erarchical clustering.


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Figure 4.21 – Subcompositions’ original data coloured by ISM and coloured byk-means clusters

Before drawing conclusions it is necessary to understand if 6, 5 or 3 is the

right number of clusters as this number is not so easy to be identified: for in-

stance if we choose rerun the algorithm with 5 cluster with same initial centres

and also with new centres positioned in the extreme cases (5 near simplex ver-

texes and 1 near the neutral element, the centre of simplex), we obtain results

in in Figure 4.22.

Centres Pneumatic_Drives Sensors Throttles

C1 800 50 200

C2 40 5000 40

C3 45 25 5000

C4 35 200 700

C5 501 499 502

Table 4.1 – Extreme centres applied to k-means algorithm

Results between first case and second one are graphically represented in

Figure 4.23: it is clearly evident that algorithm generates very different clusters

from the previous one, this allow us to understand that 5 could not be the

appropriate number of clusters.

63



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Figure 4.22 – Extreme centres applied to k-means algorithm


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Figure 4.23 – Comparing results of k-means with 5 clusters

Let’s study the cumulative variance explained (in both between cluster

an within clusters) as a function of the number of clusters using the elbow

method:

Right value of k seems to be 3 as for k with higher values we lose slope in

cumulative chart lines.

If we rerun k-means algorithm with k=3 and check if results are similar with

different choice of initial centres we note a big improvement in results and now

algorithm gives us robust results: in Figures 4.25 and 4.26.

We can conclude that results from k-means clustering suggest us to choose

3 as number of clusters, but the most important result is that it returns us a sim-

64


2 4 6 8 10

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Figure 4.25 – Comparing with k=3 and different initial centres

Figure 4.26 – Values of final clusters’ centres in both cases

ilar result of hierarchical clustering: customers have to be treated in function

of their purchasing attitude (described by their positioning on the simplex)

instead of their industry sector natures.

65

4.7. COMPOSITIONAL DISCRIMINANT ANALYSIS

4.7 Compositional Discriminant Analysis

4.7.1 LDA to classify customers by ISM

In order to check statement produced by results in Section 4.6 it is appropri-

ate to compute a supervised classification analysis to know if Industry Sector

is a label that could drive classification of our customers or if it would be better

to consider it in other purposes different from Cross Selling campaigns.

It is important to remark that we are not interested in implementation and

creation of a classifier for Industry Sector as this information is strategically

defined by Festo sales department but this analysis could be useful to look

how classifier works on the generation of classes.

Before starting test we have to execute a further data reduction as Industry

Sectors with small amount of customers could drive analysis to misleading

conclusions, in Table 4.2 there is a resume of numerosity by ISM.

ISM AC AMI BIOPHA BUILD CHEM ELA ELP FOOD FOODPR FOOT GENPA GLCER HEAVY

freq 3 35 12 2 2 52 95 99 2 1 2 5 21

ISM HYPNEU MTOOL OTHER PLAST PRINT PULPAP SPCIAL STOLI TEST TEXTIL TYRE WELD WOOD

freq 1 51 8 30 47 2 93 8 3 4 5 1 11

Table 4.2 – Number of customers per ISM

We need to exclude from analysis all ISM with a numerosity too low to be

statistical significant as label, final result is shown in Table 4.3 where only 8 ISM

with total amount of customers higher than 25 are considered (95 customers

have been excluded from analysis):

ISM AMI ELA ELP FOOD MTOOL PLAST PRINT SPCIALfreq 35 52 95 99 51 30 47 93

Table 4.3 – ISM included in analysis

We take a part of our compositions (a sample of 70% of our dataset) as

training set and, by performing a Discriminant Linear Analysis, we want to

66


find a good predictor for the class ISM for our customers and look if we will be

able to use such classifier in the future. First we will assume that proportion of

customers in each ISM is relevant so we will use it as prior to group belonging

probability.

In Figure 4.27 We can check results for discriminant linear analysis.

Figure 4.27 – Results for lda analysis using proportional priors

Confusion matrix of our dataset is shown in Table 4.4: we can see how some

classes are cut off the analysis (we would expect a good amount of customers

to remain on diagonal of the matrix as it shown the combination between pre-

dicted and real Industry Sector of our customers) and that there are four pre-

dominant Industry Sectors in which customer are classified.

Looking to values of priors used to perform the analysis it seems that In-

dustry Sectors with higher values on prior are the ones where customers are

classified with more frequency, so there is a strong dependence between prior

and final classification for customers: we can perform a LDA using as prior for

Industry Sector affiliation a uniform distribution P(xi ∈ c j) =( 1

8

)∀c j classes

67


ISM AMI ELA ELP FOOD MTOOL PLAST PRINT SPCIALAMI 0 3 5 6 0 0 0 21ELA 0 18 8 10 0 0 0 16ELP 0 7 29 32 3 0 2 22FOOD 0 9 30 46 1 0 2 11MTOOL 0 5 14 15 5 0 1 11PLAST 0 3 8 5 3 0 0 11PRINT 0 2 9 21 1 0 2 12SPCIAL 0 9 16 20 1 0 5 42

Table 4.4 – Confusion matrix of dataset: on rows original classes, on columnpredicted classes

considered in the analysis and see if we obtain a better classifier for our data.

Results for confusion matrix in this case are shown in Table 4.5:


Table 4.5 – Confusion matrix for LDA using uniform priors for groups

Results obtained are very different with first LDA analysis but classification

remains real distant from our target, so that classifier mislead in classifying a

customer in its own Industry Sector.

We can visualize discrimination functions in term of their scalings on a barplot

in Figure 4.28: it’s really not that easy to interpret scalings in compositional

data but variables with large absolute values in scalings are more likely to

influence the classification process made by LDA algorithm. In our case, for

instance Cylinder Mountings macro family is affecting the definition of first

discriminant more than the other components.

We can draw our dataset coloured in function of new industry classification

as shown in Figure 4.29.

68


LD1 LD2 LD3 LD4

Pist_RodCil_MountingsThrottlesSensorsPneumatic_Drives

0.0

0.2

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Figure 4.28 – Scalings of lda

4.7.2 QDA analysis

Last analysis done is a Quadratic Discriminant Analysis in order to under-

stand if customers would be classified in a better way respect to LDA. In Table

4.6 is represented confusion matrix for Quadratic Discriminant Analysis using

uniform priors for our data:


Table 4.6 – Confusion matrix related to QDA with uniform priors

Even classifier built with QDA is not satisfactory to group our dataset cus-

69


Pneumatic_Drives


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Pist_Rod

Figure 4.29 – Dataset with new classification for ISM

tomers in the proper Industry Sector.

We can conclude that discriminant analysis doesn’t lead to desired results:

both LDA algorithms and QDA algorithm, with proportional and uniform pri-

ors, give different results but, in both cases, very misleading in term of data

classification.

If we observe data on the simplex with new classifications we denote that

even in LDA and QDA analysis graphically seems to tend on the definition

of classes based on the proportion on compositions, this would be a further

confirmation that nature of data for our customers is strictly related to their

purchasing attitudes and to their positioning on the Simplex instead of their

70


categorical characteristics such Industry Sector belonging.

Figure 4.30 shows us the comparison between data classified with LDA (first

column of charts) and QDA methods (second column) for sucbcompositions

already taken into account in previous analyses: first row results obtained us-

ing proportional priors second row with uniform priors. R package composi-

tions


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Figure 4.30 – Dataset with new classification computed with LDA and QDAalgorithms.

71

Chapter 5

Conclusions

5.1 Targets achieved

Cross Selling is a very interesting field in marketing but by now there are

not mathematical developments to support in the proper way information

coming from its data: now we defined the proper mathematical environment

to front a Cross Selling business case and Festo customers’ compositions have

been analysed. In this way we have been able to study not only which Cross

Selling product families our customer purchased by Festo but how this prod-

uct families are statistically related together.

This approach satisfies the original aim of Cross Selling analysis for Festo and

let us develop some multivariate analysis on our customers.

Results on these analyses drove us to the conclusion that some customers’

categorical attributes, such Industry Sector, should not be taken as leading

variable to define Cross Selling marketing campaigns, since all multivari-

ate analyses developed in this work showed that most important elements of

Cross Selling data are strictly related to customers’ positioning on the sim-

plex, that it means that it could be more profitable to analyse customers from

their purchasing attitudes point of view.

72

5.2. POSSIBLE FUTURE DEVELOPMENTS

Another good result has been the development of ternary diagrams as a new

visualization tool.

Though the analysis on ternary diagrams could be sometimes very hard to be

interpreted, after the definition of how data are positioned on the Simplex, it

provides the possibility to analyse customers’ positioning in a very intuitive

way by using a bidimensional tool with information about the proportion of

three product families.

Beyond the analysis of specific Festo business case, we have to remark that

this Thesis work provides a modeling framework for the analysis of Cross

Selling data, applicable on the most varied business and sales fields: no mat-

ters which type of products have to be analysed, what matters is to be able to

analyse product quantities sold from the point of view of their inner relations.

5.2 Possible future developments

Now that a proper geometric space for compositions of customers it has

been developed, it is possible to apply to Cross Selling dataset most of statisti-

cal techniques.

A further development would be to study customers on the simplex and to

identify Sales Engineer Barycentres (computed by the application of Aitchi-

son mean to customers assigned to the same Sales Engineer) and study mean

differences between groups in order to understand if there are Sales Engineers

with peculiar selling attitudes.

Another further interesting approach would be to develop a Composi-

tional Times Series Analysis on customers: studying the evolution of cus-

tomers’ positioning on the simplex and their compositions development, we

73

5.2. POSSIBLE FUTURE DEVELOPMENTS

would be able to identify how a marketing campaign could be effective by

studying how customers involved moves on the simplex before and after the

campaigns in different periods. By studying the evolution of customers posi-

tioning in different time steps we could also try to define customers’ specific

paths on the simplex and identify specific purchasing behaviours related to

these specific paths. (for more information see [5])

Several and fancy statistical analysis could be developed as now data have

been studied from compositional point of view: so almost all standard anal-

yses already developed for classic statistics could be applied, but this time

taking into count the relations between components of compositions.

74

Bibliography

[1] Aitchison, J.P. (1973). Principal component analysis of compositional data.

Biometrika 70

[2] Aitchison, J. (1986) The Statistical Analysis of Compositional Data. Mono-

graphs on Statistics and Applied Probability, Chapman & Hall Ltd.,

London, (Reprinted in 2003 with additional material by The Blackburn

Press)

[3] J. A. Martin-Fernandez, C.Barcelo-Vidal, V. Pawlowsky-Glahn Measures

Of Difference For Compositional Data And Hierarchical Clustering Methods

[4] R. A. Johnson, D. W. Wichern, Applied Multivariate Statistical Analysis,

6th Edition, 2008 Pearson

[5] P. Legendre, O. Gauthier, Statistical methods for temporal and spaceâtime

analysis of community composition data, The Royal Society 2015

[6] V. Pawlowsky-Glahn, J.J. Egozcue, R. Tolosana-Delgado, Modeling and

Analysis of Compositional Data, Wiley 2015

[7] Pearson, K. (1897) Mathematical contributions to the theory of evolution.

On a form of spurious correlation which may arise when indices are used in the

measurement of organs. Proceedings of the Royal Society of London, LX

[8] R Package: K. Gerald van den Boogaart, Raimon Tolosana and Mat-

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75

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