Generalised Procrustes Analysis - OP&P Product Research

Generalised Procrustes 

Analysis 

applications in sensory evaluation and 

instrumental analysis

GPA course, ©OP&P Product Research, 

Utrecht 

Senstools ® is an OP&P trademark 

It is not allowed to copy or use parts of this 

course without proper reference to OP&P 

Utrecht, 2004 

Procrustes principles

• Introduction 

Today’s program 

• The Procrustes principles 

• Different data, different analysis 

• Some ‘sensory’ examples 

• Examples of FCP data 

• A hands-on demonstration 

• Discussion 

Program

• The history of GPA 

Introduction 

• Description of the Senstools package: 

functionality and tools 

• GPA routine in Senstools 

Introduction

History of GPA 

• 1952 (Green), 1962 (Hurley & Cattel): onesided 

orthogonal Procrustes rotation 

• 1968 (Schönemann), 1970 (Schönemann & 

Carroll): two-sided orthogonal Procrustes 

rotation with scaling 

• 1971 (Wingersky): more than two 

configurations 

Introduction: history GPA

History: continued 

• 1975 (Gower): generalised Procrustes with 

scaling factor and Anova (Psychometrika) 

• 1982-1986: practical application in sensory 

by Tony Williams, Gillian Arnold, Steve 

Langron and others 

• until 1989 GPA was only available as macro 

in SAS or Genstat 


History: continued 

• 1989: OP&P wrote the first PC routine for 

GPA in APL (Procrustes-PC) 

• 1993: the program was written in C 

• 1995: Senstools-for-Windows v1.0 was 

released 

• 2000: Senstools v3.0 was released 


Time needed to solve a simple 

problem 

20 subjects - 8 products - 20 attributes 

1989 13 hours 

1993 25 minutes 

1995 55 seconds 

2000 0,6 seconds 


Description of Senstools package 

Data analysis tool for sensory professionals 

Uni- and multivariate statistics 

• descriptive statistics 

• analysis of variance 

• assessor statistics and concordance 

Introduction: Senstools package

• PCA 

Description continued…. 

• Generalized Procrustes Analysis 

• MDPref 

• Latent Variable Cluster analysis 

• Graphics 


GPA routine in Senstools 










Program

Procrustes 

• Procrustes was a character of Greek myth. 

An innkeeper who plied his trade in Attica, 

he put his victims on an iron bed. If they 

were longer than the bed, he cut off their 

feet. If they were shorter, he stretched 

them…….. 


More or less like this…. 


The Procrustean principles 

Make the configurations fit each other: 

• do this by moving them to a common origin 

• stretch or shrink each configuration in order 

to make it fit as good as possible 

• if needed, flip them around 


in summary:…….. 

• Procrustes only allows ‘rigid-body’ transformations 

to the datasets 

• these transormations respect the relative 

distances between objects 


Modern Proctustes 

• consider K configurations of n objects in pdimensional 

spaces 

• how can we represent the K configurations 

in a common space while minimizing the 

goodness of fit criterion? 

• we do this with the aid of 3 transformations 


translation 

The first transformation 

• move the centroids of each configuration to 

a common origin 


3 sets (A,B,C) 

3 products (1,2,3) 

2 attributes 

A2 

A1 

Set A 

B1 

Procrustes principles 

C1 

B2 

A3 

Set B 

B3 

Set C 

C2 

C3

sets translated to 

common origin 

C1 


A2 

B1 

A1 

C2 

B2 

B3 

A3 

C3

The second transformation 

isotropic scaling 

• shrink or stretch each configuration 

isotropically to make them as similar as 

possible 


sets isotropically 

scaled 

B1 

C1 

A2 


A1 

C2 

B2 

A3 

B3 

C3

The third transformation 

rotation/reflection 

• turn or flip the configurations 


The notation and algorithm 


1.1 Notation 

T Transformation matrix, in GPA context a rotation matrix: T'T’=T’T'=I 

X a 3-way or K-sets datamatrix of order K×N×M (3-way, K individual data sets of N rows×M 

colums) or K×N×Mk (K-sets) 

X the group average matrix K 

− K 

1 

k= 

1 

Xk a group average matrix excluding Xk from the average: 

X 

k 

T 

k 

X 

k 

−1 

K 

Xi 

i 

i= 

1, 

i≠k 

= ( K −1) 

T 

Y a 2-way matrix of order (N×J), of design variables and/or physical/chemical variables. 

1.2 Algorithm 

The GPA method was first described by Gower (1975), and some modifications are found in Ten 

Berge (1977). We will follow the algorithm as basically provided by the latter. It is convenient to 

interpret the algorithm to consist of three main parts (see Fout! Verwijzingsbron niet gevonden.): 

1. Pre-processing and initialisation 

2. Procrustean iterations, possibly including isotropic scaling 

3. Post processing and presentation of results 

All three parts are potentially subject to adaptation due to our inclusion of a PLSR step to allow for 

an extra Y matrix to exert its influence. For the moment we will envision a PLSR step to be 

included in step 2 above. The original GPA algorithm according to Ten Berge (1977) is as follows, 

ignoring for the moment pre- and post-processing steps, which are not part of the GPA process 

proper. 

Initialisation of the necessary parameters. 

For k=1, …, K rotate Xk to k X by T = QP′ 

, where P and Q come from the SVD 

′ k k 

X X = P Q′ 

Evaluate the loss function: σ = 

s 

K 

k= 

1 

X T − X 

k 

k 


PRE-PROCESSING 

Translation 

PCA on long individual sets 

Scaling the total variation 

INITIALISATION 

Initializing rotation matrices on I 

Initializing scaling factors on 1 

ROTATION 

set loop k=1,…,K 

SCALING 

set loop k=1,...,K 

CONVERGENCE TEST 

σ s -σ s-1

Basic principles of GPA 

• Use the 3 transformations to make the 

individual spaces as similar as possible 

• Compute a Group-Average-Space of these 

individual spaces 

• Compute the difference between the Group 

and Individual spaces (the residuals) 

• Minimizes the total residual by applying the 

3 transformations 


The computations 

• GPA performs the transformations on each set 

• Individual configurations are averaged when 

they are as alike as possible 

• the resulting high-dimensional space is reduced 

by means of PCA to a lower dimensionality 

• the total variance in the data is partitioned over 

sets, objects or dimensions 


Procrustes-Anova (Panova) 

• The total variance (V T ) consists of consensus 

variance (V C ) and within variance (V W ) 

• the consensus variance (V C ) consists of two 

parts: the part explained by the first Q 

dimensions of the consensus space (V I ) and the 

part left unexplained (V O , the part associated 

with the higher dimensions) 


The Procrustes-Anova (Gower) 

Zero padding assym. data 

and centering 

V T 

Scaling and rotation 

Averaging individual spaces 

V C=(V T-V W) 

PCA to lower dim. space 

V I=(V T-V W -V O) 


Loss V W 

Loss V O 

Group 

average 

space

Panova output in Senstools 

• The total, consensus and residual variance is 

shown as it is distributed over sets, objects and 

dimensions 

• high consensus variance for objects indicates 

agreement about the position of the objects by 

the assessors 

• high residual variance for assessors indicate 

that the assessor does not agree with the others 


Significance of the results ? 

• in contrast with PCA, the amount of variance 

explained in itself does not give an indication 

for the significance or fit of the final solution 

• a permutation test is used to estimate the odds 

that a ‘random’ dataset would give a similar 

percentage consensus variance 


The Procrustes permutation test 

• take the original dataset, permute the rows 

within each set and run an analysis 

• repeat this 50 times 

• the 90th and 98th percentile of the percentage 

of consensus variance from these permuted 

sets are compared to the percentage in the 

actual dataset 










Program

• 3-mode data 

Two basic types of data 

– products×assessor×characteristics 

– (conventional) profiling 

• K-sets data 

– products×(assessors×idiosyncratic characteristics) 

– ‘free choice’ profiling 

Different data

N products 

M attributes 

3 Mode data 

Different data 

K assessors 

( N × M ) datamatrix X k 

for one assessor 

3-mode data structure representing Conventional Profiling data: N 

products are judged by K assessors using M attributes.

N products 

X 

1 

M 1 attributes 

K-sets data 

X 

2 


K assessors 

X 

3 


Data structure representing Free Choice Profiling data: N products are judged 

by K assessors each using M k attributes. 


X 

K 

M K attributes

Averaging sensory data 

• Conventional profiling: we can average and 

use PCA to summarize 

• FCP: we cannot average, we need GPA 

• in case of individual differences, can we 

average at all? 

Different data

ΣΣ.. results in averaged data set 

Products 

attributes 1-n 

Average 


Analyses: 

PCA/Factor 

MDS 

…...

PCA 

• Fit a low dimensional structure that captures 

the most variance of the high dimensional 

structure 

• shows (cor)relations between variables 

• shows similarities between objects 

• gives fit of dimensions 

see Jolliffe, I.T. (1986). Principal Component Analysis. Springer-Verlag. 

Different data

Correlation matrix 

PRED PGREEN MOIST DRYMAT ACID ITHICK KATAC 

PRED 1.000 

PGREEN 0.921 1.000 

MOIST 0.841 0.846 1.000 

DRYMAT 0.196 0.098 0.027 1.000 

ACID 0.609 0.526 0.502 0.534 1.000 

ITHICK 0.224 0.182 0.361 0.327 0.390 1.000 

KATAC -0.609 -0.587 -0.669 -0.207 -0.497 -0.301 1.000 

Instrumental measurements on 66 apples. 

Different data

%VAF 

60 

50 

40 

30 

20 

10 

0 

55.883 

Scree graph 

19.286 

10.756 


6.834 

4.51 

1.707 

1 2 3 4 5 6 7 

Dimension 

1.024

dimension 2 

0.8 

0.6 

0.4 

0.2 

0 

-0.2 

-0.4 

-0.6 

Biplot 

DRYMAT 

ITHICK 

ACID 

15 

18 

KATAC 

6 30 

9 

19 7 

226 

+ 

35 29 2021 23 

22 

14 17 

24 

31 

3275 

8 

2834 

12 

1 10 

25 

33 

32 

13 

11 4 

PRED 

MOIST 

16 PGREEN 

36 

54 

63 

62 

6647 

57 5641 

64 

71 

44 58 

65 

72 

69 6042 38 

55 

5345 

59 

61 

68 

48 39 50 51 

43 40 67 

46 

49 

70 

52 

37 

-0.8 

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 

dimension 1

Applications of GPA -1 

Why use GPA instead of PCA for conventional 

profiling? 

• GPA will preserve the individual information: 

- information about individual scale-usage 

- the relative contribution of individuals to the 

group result 

- the consensus of individuals with the group 

GPA in Sensory

Applications of GPA -2 

The special case: non-matching attributes 

• GPA is the only way to analyze: 

- data from different countries/languages 

- free choice profiling data 

- in general: K Sets data 

GPA in Sensory

Format of free choice data 


attributes set 1 

Set 1 

Set 2 

GPA in Sensory 

only products are similar 

attributes set 2 attributes set 3 ributes set 4 

Set 3 

Set 4 

utes set K 

attributes set 5 

Set K


Format of K Sets data 

chemical data 

Chemical 

GPA in Sensory 

attributes 

instrumental data 

Instrumental 

only products are similar 

Sensory

GPA in sensory & consumer 

science - 1 

GPA in sensory science: 

• shows the relationships between products and 

attributes 

• monitoring panelist performance 

• relating sensory data to instrumental or 

chemical data 

GPA in Sensory

GPA in sensory & consumer 

science - 2 

GPA in consumer science: 

• shows the relationships between products and 

attributes 

• takes individual differences into account 

• corrects for lack of training 

GPA in Sensory

Basic assumptions in classical 

sensory profiling 

• assessors know meaning of attributes 

• assessors know meaning of scale 

• assessors use scale in consistent manner 

Training will provide the necessary skills ! 

GPA in Sensory

Nevertheless: three possible 

• Effects of level 

• Effects of range 

problems 

• Effects of meaning/interpretation 

GPA in Sensory

Effects of level 

Very weak Very strong 

assessor 1 


assessor 2 

GPA in Sensory

Effects of range 



GPA in Sensory

Effects of meaning/interpretation 

Very weak sweetness Very strong 

Very weak bitterness Very strong 

GPA in Sensory

GPA removes the effects of level, range and 

interpretation from each individual dataset by 

applying 3 transformations: 

- translation to common mean 

- isotropic scaling (stretch or shrink) 

- rotation/reflection 

In summary: 

GPA in Sensory

FCP or Sensory-Instrumental 

relations 

• in these situations, we can not average 

• ‘attributes’ have different meanings for each 

set and each set can have different numbers 

of attributes 

• still, we want to find a common, underlying 

structure 

GPA in FCP

The ‘FCP’ principles: 

• GPA allows us to match different configurations 

without assumptions about the axis 

• these configurations are made as similar as 

possible by using the 3 transformations 

• on the basis of the individual spaces, a group 

space is computed in which the ‘attributes’ 

from each individual space are projected 

GPA in FCP

Other methods to relate Sensory - 

Instrumental data 

• assymetric methods (try to predict one set 

from another) 

for example: PLS, PCR, MulReg 

• symmetric methods (only relations between 

sets are studied) 

for example: CCA, GPA 





• Different data - different analysis 





Program

The classical example: beef data 

from Gower 

• 3 judges rated 9 beef carcasses with respect 

to 7 different attributes (k=3, m=7 and n=9) 

• this results in 3 matrices of 9 x 7 data points 

• first: descriptives 


Averaged rating and standard 

deviation by judge (63 obs) 

Mean StdDev 

J2 38 16 

J1 51 18 

J3 53 28 


Judge#1 by attributes and carcasses 

Spiderplot: attributes over objects (J1) 

carc8 

carc6 

carc5 

carc7 

carc4 

carc3 


carc9 

carc2 

carc1 

att1 

att2 

att3 

att4 

att5 

att6 

att7 

mean rating: 51 

St. dev.: 18

Judge#3 by attributes and carcasses 

Spiderplot: attributes over objects (J3) 

carc8 

carc6 

carc5 

carc7 

carc4 

carc3 


carc9 

carc2 

carc1 

att1 

att2 

att3 

att4 

att5 

att6 

att7 


St. dev.: 28


deviation by carcass (27 obs) 

carc8 carc1 carc3 carc4 carc5 carc7 carc2 carc6 carc9 

Mean 31 40 41 43 45 52 53 60 64 

StdDev 28 8 21 20 22 18 20 20 19 


Carcass #1 by judges and attributes 

Spiderplot: sets over attributes (carc1) 

att6 

att5 

att4 

att3 


att7 

att2 

att1 

J1 

J2 

J3 


St. dev.: 28

Carcass #3 by judges and attributes 

Spiderplot: sets over attributes (carc3) 

att6 

att5 

att4 

att3 


att7 

att2 

att1 

J1 

J2 

J3 


St. dev.: 28


deviation by attribute (21 obs) 

att7 att1 att6 att4 att5 att3 att2 

Mean 35 41 43 43 51 55 64 

StdDev 8 22 22 26 23 20 16 


Attribute #7 by judges and carcasses 

Spiderplot: sets over objects (att7) 

carc6 

carc5 

carc7 

carc4 

carc8 

carc3 


carc9 

carc2 

carc1 

J1 

J2 

J3 


St. dev.: 8

Attribute #4 by judges and carcasses 

Spiderplot: sets over objects (att4) 

carc6 

carc5 

carc7 

carc4 

carc8 

carc3 


carc9 

carc2 

carc1 

J1 

J2 

J3 


St. dev.: 26

Univariate results 

• the carcasses differ on 5 out of 7 attributes 

• the judges differ in level and range effect 

• there is very little variation in the rating of 

carcass 1 and in the use of attribute 7 

NOW, LET’S PROCRUSTES 


Scree plot - Gower data 


Panova table 

Real Residual Total 

Dim 1 60,9 13,2 74,1 

Dim 2 8,1 2,4 10,5 

Dim 3 6,4 2,5 8,9 

Dim 4 2,7 1,0 3,7 

Dim 5 1,2 0,4 1,6 

Dim 6 0,4 0,2 0,6 

Dim 7 0,3 0,3 0,6 

Total 80,1 19,9 100,0 


0.25 

0.20 

0.15 

0.10 

0.05 

0.00 

Explained (real) variance by object 

Real Variance by Object 

carc1 carc2 carc3 carc4 carc5 carc6 carc7 carc8 carc9 


Dim 4 

Dim 3 

Dim 2 

Dim 1

0.08 

0.06 

0.04 

0.02 

0.00 

Residual by judge 

Residual Variance by Set 

J1 J2 J3 

Data Gower, Psychometrica 1975 


Group average space and individual sets 

GPA Group Average : dimension 1 versus 2 

J3 

3,59 

-3,59 

J1 

carc8 J2 

J1 

carc1 

J2 J3 

J1 

carc5 

J2 

J2 

carc3 J2 J1 carc2 J1 

J1 

J3 carc4 carc7 

J3 

J2 

J2 J3 

J2 

carc6 

J3 J3 

J1 J1 

J1 carc9 

J3 

3,59 

J3 

J2 

-3,59 


Group space with averaged attributes 


carc1 

3,59 

carc5 

carc6 

att1 att7 

-3,59 carc3 

carc2 att5 att6 

3,59 

carc4 carc7 

att4 

carc8 

att3 

carc9 

att2 

-3,59 


Permutation Results: 

Permutation test 

Total VAF in Real Data Set : 80,1 at 0 % 

Upper 10 % of the TVA in the permutated 

data Sets : 69,2 




PCA on averaged dataset 

PCA Results (Correlation) : dimension 1 versus 2 

2,97 

carc1 

carc2 

carc3 

-2,97 

att1 

att6 att7 

att5 

carc6 att4 

carc5carc4 

2,97 

carc9 

carc7 carc8 

att2 att3 

-2,97 


How similar are the results? 

• compare n-dim PCA space with n-dim GPA 

space 

• this is a 2-set GPA problem (‘free choice’) 

can the two [m x n] sets be fitted into a 

common group space? 


Group space for the two datasets 


PCA 

carc1 

2,95 

PCA 

carc5 

GPA 

PCA carc6 

GPA 

-2,95 

GPA carc4 

PCA 

carc7 

GPA 

2,95 

GPA 

carc8 

PCA 

PCA 

PCA 

carc9 

GPA 

PCA 

GPA 

GPAcarc3 

PCA carc2 GPA 

-2,95 


Is this result significant? 








Another sensory example 

• the data are collected by Michael Bom Frøst, 

Ph.D. student Sensory Science Group - 

Department of Dairy and Food Science, KVL, 

Denmark 


• 7 judges 

The dataset 

• 16 different milk samples (triplicated) 

• 23 sensory attributes 


Questions to be answered 

• are there differences between the products 

• are the judges consistent 

• how can we characterize the products 


Are there differences between the products? 

80 

60 

40 

20 

ANOVA : F Ratios by Attribute 

0 

Cream-smell Whiteness Blueness Glass coating Cream-flavour Sweet Creaminess-oral Overal fattiness 

Boiled milk-smell Yellowness Transparency Thickness-visual Boiled milk-fla Thickness-oral Residual mouth feel 



• yes, very clear differences for each attribute 

• the most outspoken difference is for ‘glass 

coating’ 

• the least outstanding difference is for 

‘boiled milk’ and ‘sweet’ 


Are there judges consistent? 

Agreement Between Assessors (Correlations) 

5 

4 

3 

2 

1 

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 



• yes, there is a very good correlation 

between each judge and the group average 

without that judge 

• inspect also the assessor statistics of the 

repeated measures anova (ratio of variance 

between products and within replications 

for each assessor and attribute) 


50 

40 

30 

20 

10 

0 


RANOVA Assessor Statistics F Ratios by Attribute 

set 1 set 2 set 3 set 4 set 5 set 6 set 7 


Cream-smell 

Boiled milk-smell 

Whiteness 

Yellowness 

Blueness 

Transparency 

Glass coating 

Thickness-visual 

Cream-flavour 

Boiled milk-fla 

Sweet 

Thickness-oral 

Creaminess-oral 

Residual mouth feel 

Overal fattiness

50 

40 

30 

20 

10 

0 

A different look 

RANOVA Assessor Statistics F Ratios by Assessor 

Cream-smell Whiteness Blueness Glass coating Cream-flavour Sweet Creaminess-oral Overal fattiness 

Boiled milk-smell Yellowness Transparency Thickness-visual Boiled milk-fla Thickness-oral Residual mouth feel 


set 1 

set 2 

set 3 

set 4 

set 5 

set 6 

set 7

GPA group space with products and attributes 


1,63 

-1,63 

M1 

M3 

M2 M15 

Glass coating Yellowness M4 

M16 

M6 

Overal fattiness 

Whiteness 

M5 M7 

Cream-flavour 

Sweet 

Cream-smell 

M8 

M9 

M12 

Boiled milk-fla 

M11 Blueness 

Transparency M13 

M14 M10 

1,63 

-1,63 


Relation between attributes and products 

• the solution is almost uni-dimensional (dim 

1 explains 74% and dim 2 only 4%) 

• the major distinction is based on fattiness 

and creaminess versus color and transparancy 


A third example 

Expert data - sensory profiles 

• 15 different tomato soups are rated by 14 

experts on 25 attributes 

• soups vary in type (can, glass, instant and 

freshly made) 



150 

100 

50 

0 

ANOVA F Ratios by Attribute 

saltt sweet taste int broth creamy thick mf sticky mf aftert color thickness coarseness nat. odor broth odor 

sour bitter fullness spicy mealy fatty mf crunchy metal taste tapioca filling odor full odor 



• yes, very clear differences for each attribute 

• the most outspoken difference is for ‘tapioca’ 


Expert data - sensory profiles 

Agreement Between Assessors (Correlations) 

8 

6 

4 

2 

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 



• yes, there is a very good correlation between 

each judge and the group average without that 

judge (except for one judge) 

• assessor statistics 


200 

150 

100 

50 

0 


RANOVA Assessor F Ratios by Attribute 

set 1 set 2 set 3 set 4 set 5 set 6 set 7 set 8 set 9 set 10 set 11 set 12 set 13 set 14 


saltt 

sour 

sweet 

bitter 

taste int 

fullness 

broth 

spicy 

creamy 

mealy 

thick mf 

fatty mf 

sticky mf 

crunchy 

aftert 

metal taste 

color 

tapioca 

thickness 

filling 

coarseness 

odor 

nat. odor 

full odor 

broth odor









1.0 

0.8 

0.6 

40,6% 

0.4 

0.2 

Expert data - dimensionality 

57,4% 

70,4% 

Screeplot 

77,4% 


82,6% 87,5% 

90,9% 93,5% 

0.0 

dim 1 dim 2 dim 3 dim 4 dim 5 dim 6 dim 7 dim 8


Expert data - all attributes 

1,14 

-1,14 Can/cream 

Can5 

Froz/cream 

Frozen1 

Can2 broth 

Can4 

fatty mf saltt Can1 

broth odor 

Fresh 

Can3 

1,14 

Can/cream4 

Can/cream2 

creamy mealysticky 

mf sweet 

metal taste 

HomeMade 

bitter 

odor 

crunchy 

color spicy 

fullness full 

nat. Standard odor tapioca 

odor coarseness filling 

sour aftert taste int 

Glass1 

thick mf 

thickness 

Instant1 

-1,14 


0.15 

0.10 

0.05 

0.00 

Expert data - explained variance 

Real Variance by Object 

Can1 HomeMade Can2 Can/cream2 Glass1 Can4 Can/cream4 Can5 

Can/cream Frozen1 Froz/cream Can3 Fresh Instant1 Standard 


Dim 3 

Dim 2 

Dim 1

Expert data: individual performance 

• GPA allows us to inspect the performance 

of individuals in the group average space 

• in the case of experts or trained panels, the 

variability between individuals should be 

low 

so, let’s see 


Expert data - attribute tapioca 


1,14 

Froz/cream 

Can5 Frozen1 

Can2 

Can4 

Can1 

-1,14 Can/cream1 

Fresh 

Can3 

1,14 

Can/cream4 

Can/cream2 

Instant1 

-1,14 

HomeMade 


set 2 tapioca 

set set 

3 4 5 

tapioca 

Standard set 

set 10 

9 8 tapioca 

tapioca 

set set 12 14 11 13 tapioca 

tapioca 

set set 7 6 tapioca 

set 1 tapioca 

Glass1

Expert data - attribute bitter 


1,14 

Instant1 

set 11 Froz/cream bitter 

Can5 Frozen1 set 12 bitter 

Can2 

Can4 set 5 bitter 

Can1 


Fresh 

Can3 

1,14 

Can/cream4 

set 9 bitter 

Can/cream2 

set set 14 set bitter 13 3 bitter 

set 10 bitter 

HomeMade 

set 6 bitter set 4 bitter 

set 8 bitter 

set 1 bitter 

Standard 

set 7 bitter 

Glass1 

set 2 bitter 

-1,14 


Expert data - attribute color 


1,14 

Instant1 


set 1 color 

Froz/cream 

Fresh 

Frozen1 Can5 

set 2 color 

Can2 

Can4 

set 8 color 

Can1 

set 5 color set 13 color 

Can3 

1,14 

Can/cream4 

Can/cream2 

HomeMade 

set 6 color 

set 14 color 

set 12 color 

set 10 color 

Standard set set 9 color 7 color 

-1,14 


Glass1 set 

set 

4 

11 

color 

color 

set 3 color

Performance of individuals 

• for some attributes, there is excellent 

agreement 

• for other attributes there is much less 

agreement 

• the GPA results allow direct feedback to the 

tasters 










Program

Free Choice Profiling and GPA 

• N products are rated by K sets on m k attributes 

• each of the k sets uses different attributes 

• no descriptives possible 


Basic assumptions in FCP/GPA 

• the N products can be fitted in a K multidimensional 

spaces 

• the spatial structure of the K spaces can be 

defined in different ways 


Example 1: the dataset 

• experts from different countries rated the 

same products 

• 4 experts, 7 products, up to 6 attributes 


Four experts in space 


3,02 

set 6 

set 5 

set 7 

-3,02 

set 6 

set 2 

object set 7 5 

set 5 

object 3 

set object set set 7 

5 2 6 

set 5 

set 2 set 2 

set 7 

set 7 

set set 7object 5 6set 

2 

set 6 

object 1 

set 25 

set 6 

3,02 

-3,02 

set 5 

object 4 

set 2set 

7 

set 2 

set 6 

set 6 

object 7 










Attributes of expert 2 


2,85 

-2,85 

object 5 

set 2 mealy 

object 

object 

set 2 2 

3 

bitter 

2,85 

set 2 fullness 

object 6 

-2,85 

object 4 

set 2 sour 


object 7 

set 2 broth 

object 1

Attributes of expert 7 


2,85 

-2,85 

object 5 

set 7 sweet 

set 

object 

object 

7 broth 

2 

3 

set 7 creamy 

2,85 

-2,85 

object 4 

object 7 

set 7 fullness 

set 7 mealy 


object 6 

object 1

Conclusions: 

• the data from the different experts can be 

summarized in a common group space 

• by comparing the attribute vectors of the 

different experts, we can identify the nature 

of the dimensions 


The data sets used in this presentation: 

• fcp.sts 

• expert75.sts 

• gower.sts 

• twoset.sts 

can be downloaded from our website 










Program

Generalised Procrustes Analysis - OP&P Product Research

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