procrustes test

Generalised ProcrustesAnalysis

applications in sensory evaluation and instrumental analysis

Procrustes principles

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

Program

Todays program Introduction The Procrustes principles Different data, different analysis Some sensory examples Examples of FCP data A hands-on demonstration Discussion

Introduction

Introduction

The history of GPA Description of the Senstools package:

functionality and tools GPA routine in Senstools

Introduction: history GPA

History of GPA

1952 (Green), 1962 (Hurley & Cattel): one-sided orthogonal Procrustes rotation

1968 (Schnemann), 1970 (Schnemann & Carroll): two-sided orthogonal Procrustes rotation with scaling

1971 (Wingersky): more than two configurations


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 simpleproblem

20 subjects - 8 products - 20 attributes

1989 13 hours1993 25 minutes1995 55 seconds2000 0,6 seconds

Introduction: Senstools package

Description of Senstools package

Data analysis tool for sensory professionals Uni- and multivariate statistics descriptive statistics analysis of variance assessor statistics and concordance


Description continued.

PCA 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 transfor-mations to the datasets

these transormations respect the relative distances between objects


Modern Proctustes

consider K configurations of n objects in p-dimensional 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


The first transformation

translation

move the centroids of each configuration to a common origin


Set C

Set B

A1

A2

A3

C2

C1

C3

B1

B2

B3

Set A

3 sets (A,B,C)3 products (1,2,3)2 attributes


A1

A2

A3

C2

C1

C3

B1

B2

B3

sets translated to common origin


The second transformation

isotropic scaling

shrink or stretch each configuration isotropically to make them as similar as possible


A1

A2

A3

C2

C1C3

B1

B2

B3

sets isotropically scaled


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=TT'=I

X a 3-way or K-sets datamatrix of order KNM (3-way, K individual data sets of N rowsMcolums) or KNMk (K-sets)

X the group average matrix =

K

kkkK

1

1 TX

kX a group average matrix excluding Xk from the average: =

=

K

kiiiik K

,1

1)1( TXX

Y a 2-way matrix of order (NJ), 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 TenBerge (1977). We will follow the algorithm as basically provided by the latter. It is convenient tointerpret 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 foran 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 kX by PQT = , where P and Q come from the SVD

QPXX = kk Evaluate the loss function:

=

=

K

kkks

1XTX

PRE-PROCESSING

Translation

Initializing rotation matrices on IInitializing scaling factors on 1

PCA on long individual sets Scaling the total variation

ROTATION set loop k=1,,K

SCALING set loop k=1,...,K

CONVERGENCE TEST

POST-PROCESSING PCA on group average Rotate sets to group average

Partitioning loss Correlations

yes

no

s-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 (VT) consists of consensus

variance (VC) and within variance (VW) the consensus variance (VC) consists of two

parts: the part explained by the first Q dimensions of the consensus space (VI) and the part left unexplained (VO, the part associated with the higher dimensions)


The Procrustes-Anova (Gower)Zero padding assym. data and centering

Scaling and rotation

Averaging individual spaces

PCA to lower dim. spaceGroup

average

space

VT

Loss VW

VC=(VT-VW)

VI=(VT-VW -VO)

Loss VO


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


Different data

Two basic types of data 3-mode data

productsassessorcharacteristics (conventional) profiling

K-sets data products(assessorsidiosyncratic characteristics) free choice profiling

Different data

N

p

r

o

d

u

c

t

s

M attributes

K assessors

( N M ) datamatrix Xkfor one assessor

3-mode data structure representing Conventional Profiling data: Nproducts are judged by K assessors using M attributes.

3 Mode data

Different data

N

p

r

o

d

u

c

t

s

K assessors

1 X 2 X 3 X K X

M 3 attributesM 2 attributesM 1 attributes M K attributes

Data structure representing Free Choice Profiling data: N products are judged by K assessors each using Mk attributes.

K-sets data

Different data

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

.. results in averaged data setProducts

attributes 1-n

Average

Analyses:PCA/FactorMDS...

Different data

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 matrixPRED PGREEN MOIST DRYMAT ACID ITHICK KATAC

PRED 1.000PGREEN 0.921 1.000MOIST 0.841 0.846 1.000DRYMAT 0.196 0.098 0.027 1.000ACID 0.609 0.526 0.502 0.534 1.000ITHICK 0.224 0.182 0.361 0.327 0.390 1.000KATAC -0.609 -0.587 -0.669 -0.207 -0.497 -0.301 1.000

Instrumental measurements on 66 apples.

Different data

Scree graph

55.883

19.286

1.0241.7074.516.834

10.756

0

10

20

30

40

50

60

1 2 3 4 5 6 7Dimension

%

V

A

F

Biplot

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

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

d

i

m

e

n

s

i

o

n

2

dimension 1

+

PRED

PGREENMOIST

DRYMAT

ACID

ITHICK

KATAC

1

2

34

5

6 7

8

9

101112 13

14

15

16

17

1819

2021 22232425

26

27 2829

30

313233

343536

37

3839 40

41

4243

4445

46

47

48

49

5051

52

53

54

555657

585960 61

6263

64

65

66

676869

70

7172

GPA in Sensory

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

attributes set 5

Format of free choice data

Products

attributes set 1

Set 1Set 2

Set 3 Set 4Set K

attributes set 2attributes set 3 ributes set 4 utes set K

only products are similar

GPA in Sensory

Format of K Sets data

Products

chemical data

Chemical

InstrumentalSensory

instrumental data

attributes

only products are similar

GPA in 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 problems

Effects of level Effects of range 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 bitterness Very strong

Very weak sweetness 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 FCP

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 configu-

rations 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 attributesfrom each individual space are projected


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

Program

Todays program Introduction The Procrustes principles Different data - different analysis Some sensory examples Examples of FCP data A hands-on demonstration Discussion


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 StdDevJ2 38 16J1 51 18J3 53 28


Judge#1 by attributes and carcassesSpiderplot: attributes over objects (J1)

att1att2att3att4att5att6att7

carc1

carc2

carc3carc4

carc5

carc6

carc7carc8

carc9

mean rating: 51

St. dev.: 18


Judge#3 by attributes and carcassesSpiderplot: attributes over objects (J3)

att1att2att3att4att5att6att7

carc1

carc2

carc3carc4

carc5

carc6

carc7carc8

carc9

mean rating: 53

St. dev.: 28


Averaged rating and standard deviation by carcass (27 obs)

carc8 carc1 carc3 carc4 carc5 carc7 carc2 carc6 carc9Mean 31 40 41 43 45 52 53 60 64StdDev 28 8 21 20 22 18 20 20 19


Carcass #1 by judges and attributesSpiderplot: sets over attributes (carc1)

J1J2J3

att1

att2

att3

att4

att5

att6

att7

mean rating: 53

St. dev.: 28


Carcass #3 by judges and attributesSpiderplot: sets over attributes (carc3)

J1J2J3

att1

att2

att3

att4

att5

att6

att7

mean rating: 53

St. dev.: 28


Averaged rating and standard deviation by attribute (21 obs)

att7 att1 att6 att4 att5 att3 att2Mean 35 41 43 43 51 55 64StdDev 8 22 22 26 23 20 16


Attribute #7 by judges and carcassesSpiderplot: sets over objects (att7)

J1J2J3

carc1

carc2

carc3carc4

carc5

carc6

carc7carc8

carc9

mean rating: 35

St. dev.: 8


Attribute #4 by judges and carcassesSpiderplot: sets over objects (att4)

J1J2J3

carc1

carc2

carc3carc4

carc5

carc6

carc7carc8

carc9

mean rating: 43

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, LETS PROCRUSTES


Scree plot - Gower data


Panova tableReal Residual Total

Dim 1 60,9 13,2 74,1Dim 2 8,1 2,4 10,5Dim 3 6,4 2,5 8,9Dim 4 2,7 1,0 3,7Dim 5 1,2 0,4 1,6Dim 6 0,4 0,2 0,6Dim 7 0,3 0,3 0,6

Total 80,1 19,9 100,0


Explained (real) variance by objectReal Variance by Object

Dim 4

Dim 3

Dim 2

Dim 1

0.00

0.05

0.10

0.15

0.20

0.25

carc1 carc2 carc3 carc4 carc5 carc6 carc7 carc8 carc9


Residual by judge Residual Variance by Set

Data Gower, Psychometrica 1975

0.00

0.02

0.04

0.06

0.08

J1 J2 J3


Group average space and individual sets GPA Group Average : dimension 1 versus 2

-3,59 3,59

-3,59

3,59

J1J2

J3carc9J1

J2J3

carc8

J1

J2J3

carc7J1

J2

J3carc6J1J2

J3carc5

J1

J2J3 carc4 J1

J2

J3carc3 J1

J2 J3carc2

J1 J2

J3carc1


Group space with averaged attributesGPA Group Average : dimension 1 versus 2

-3,59 3,59

-3,59

3,59

att1

att2att3 att4

att5att6att7

carc9carc8carc7

carc6carc5

carc4carc3 carc2

carc1


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

Upper 10 % of the TVA in the permutateddata Sets : 69,2



PCA on averaged datasetPCA Results (Correlation) : dimension 1 versus 2

-2,97 2,97

-2,97

2,97

att1

att2att3

att4att5att6att7

carc1

carc2 carc3

carc4carc5carc6

carc7 carc8carc9


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 datasetsGPA Group Average : dimension 1 versus 2

-2,95 2,95

-2,95

2,95

PCA

GPAcarc9

PCA

GPAcarc8

PCAGPAcarc7

PCAGPA

carc6PCA

GPAcarc5

PCAGPAcarc4PCAGPAcarc3

PCA GPAcarc2

PCA

GPA

carc1


Is this result significant?

Permutation Results:Total VAF in Real Data Set : 82,8 at 4 %




Another sensory example

the data are collected by Michael Bom Frst, Ph.D. student Sensory Science Group -Department of Dairy and Food Science, KVL, Denmark


The dataset

7 judges 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?ANOVA : F Ratios by Attribute

0

20

40

60

80

Cream-smell Whiteness Blueness Glass coating Cream-flavour Sweet Creaminess-oral Overal fattinessBoiled milk-smell Yellowness Transparency Thickness-visual Boiled milk-fla Thickness-oral Residual mouth feel


Are there differences between the products?

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)

1

2

3

4

5

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


Are there judges consistent?

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)


Are there judges consistent?RANOVA Assessor Statistics F Ratios by Attribute

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

0

10

20

30

40

50

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


A different lookRANOVA Assessor Statistics F Ratios by Assessor

set 1

set 2

set 3

set 4

set 5

set 6

set 7

0

10

20

30

40

50

Cream-smell Whiteness Blueness Glass coating Cream-flavour Sweet Creaminess-oral Overal fattinessBoiled milk-smell Yellowness Transparency Thickness-visual Boiled milk-fla Thickness-oral Residual mouth feel


GPA group space with products and attributesGPA Group Average : dimension 1 versus 2

-1,63 1,63

-1,63

1,63

Cream-smell

Whiteness

Yellowness

BluenessTransparency

Glass coating

Cream-flavour

Boiled milk-fla

Sweet

Overal fattiness

M16M15 M14

M13

M12

M11

M10

M9

M8

M7M6

M5

M4M3

M2

M1


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 transpa-rancy


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)


Are there differences between the products?ANOVA F Ratios by Attribute

0

50

100

150

saltt sweet taste int broth creamy thick mf sticky mf aftert color thickness coarseness nat. odor broth odorsour bitter fullness spicy mealy fatty mf crunchy metal taste tapioca filling odor full odor


Are there differences between the products?

yes, very clear differences for each attribute the most outspoken difference is for tapioca


Expert data - sensory profilesAgreement Between Assessors (Correlations)

2

4

6

8

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


Are there judges consistent?

yes, there is a very good correlation between each judge and the group average without that judge (except for one judge)

assessor statistics


Are there judges consistent?RANOVA Assessor F Ratios by Attribute

salttsoursweetbittertaste intfullnessbrothspicycreamymealythick mffatty mfsticky mfcrunchyaftertmetal tastecolortapiocathicknessfillingcoarsenessodornat. odorfull odorbroth odor

0

50

100

150

200

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


Expert data - dimensionalityScreeplot

40,6%

57,4%70,4%

77,4% 82,6% 87,5% 90,9% 93,5%

0.0

0.2

0.4

0.6

0.8

1.0

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


Expert data - all attributesGPA Group Average : dimension 1 versus 2

-1,14 1,14

-1,14

1,14

saltt

sour

sweet bitter

taste int

fullness

broth

spicycreamy mealy

thick mf

fatty mf

sticky mf

crunchy

aftert

metal tastecolor

tapioca

thickness

fillingcoarsenessodor nat. odorfull odor

broth odor

Can5

Standard

Can/cream4

Instant1

Can4

Fresh

Glass1

Can3

Can/cream2

Froz/cream Can2Frozen1

HomeMade

Can/creamCan1


Expert data - explained variance Real Variance by Object

Dim 3

Dim 2

Dim 1

0.00

0.05

0.10

0.15

Can1 HomeMade Can2 Can/cream2 Glass1 Can4 Can/cream4 Can5Can/cream Frozen1 Froz/cream Can3 Fresh Instant1 Standard


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, lets see


Expert data - attribute tapiocaGPA Group Average : dimension 1 versus 2

-1,14 1,14

-1,14

1,14

set 1 tapioca

set 2 tapioca

set 3 tapiocaset 4 tapiocaset 5 tapiocaset 6 tapiocaset 7 tapioca

set 8 tapiocaset 9 tapiocaset 10 tapiocaset 11 tapiocaset 12 tapiocaset 13 tapiocaset 14 tapioca

Can5

Standard

Can/cream4

Instant1

Can4

Fresh

Glass1

Can3

Can/cream2


HomeMade

Can/cream1Can1


Expert data - attribute bitterGPA Group Average : dimension 1 versus 2

-1,14 1,14

-1,14

1,14

set 1 bitter

set 2 bitter

set 3 bitterset 4 bitter

set 5 bitter

set 6 bitter

set 7 bitter

set 8 bitter

set 9 bitter set 10 bitter



Can5

Standard

Can/cream4

Instant1

Can4

Fresh

Glass1

Can3

Can/cream2


HomeMade

Can/cream1Can1


Expert data - attribute colorGPA Group Average : dimension 1 versus 2

-1,14 1,14

-1,14

1,14

set 1 color set 2 color

set 3 colorset 4 color

set 5 color

set 6 color set 7 color

set 8 color

set 9 colorset 10 color

set 11 color

set 12 color

set 13 color

set 14 color

Can5

Standard

Can/cream4

Instant1

Can4

Fresh

Glass1

Can3

Can/cream2


HomeMade

Can/cream1Can1


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 mk 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 multidimen-sional 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 GPA Group Average : dimension 1 versus 2

-3,02 3,02

-3,02

3,02

set 2

set 5

set 6

set 7

object 7

set 2set 5 set 6set 7object 6

set 2set 5

set 6set 7object 5

set 2

set 5

set 6

set 7object 4

set 2set 5

set 6

set 7

object 3set 2set 5

set 6set 7object 2

set 2set 5set 6

set 7

object 1


Attributes of expert 2GPA Group Average : dimension 1 versus 2

-2,85 2,85

-2,85

2,85

set 2 sour

set 2 bitterset 2 fullness

set 2 broth

set 2 mealy

object 7

object 6

object 5

object 4

object 3object 2

object 1


Attributes of expert 7GPA Group Average : dimension 1 versus 2

-2,85 2,85

-2,85

2,85

set 7 sweet

set 7 fullness

set 7 broth set 7 creamy

set 7 mealy

object 7

object 6

object 5

object 4

object 3object 2

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


procrustes test

Documents