An example of sorting task
J.1 J.2 J.3 J.4 J.5Angel exotic;sweet;pleasant flowery;soft fruity;strong vanilla;spicy;spirit of islands to eat;sweetAromatics Elixir strong;alcoholized strong;man heady rough;strong oldChanel5 soap;clean G4 heady toilet soapCinema flowery;lilac flowery;artificial;grass fruity;middle sweet softy; y; ;g y;Coco Mademoiselle soap;clean flowery;soft fruity;middle softness;flowery softJadoreEP soap;clean flowery;soft sweet;light softness;flowery floweryJadoreET soap;clean flowery;artificial;grass sweet;light softness;flowery floweryLinstant G3 flowery;soft fruity;strong sweet oldLolita Lempicka exotic;sweet;pleasant flowery;soft fruity;middle vanilla;spicy;spirit of islands to eat;sweetPleasures G3 strong;man fruity;strong sweet flowery
12/07/2012 Sensometrics 2012 2
Pleasures G3 strong;man fruity;strong sweet floweryPure Poison flowery;lilac flowery;soft tangy;deodorant softness;flowery softShalimar strong;alcoholized flowery;artificial;grass strong;lavander mustiness;aggressive old
An example of sorting taskConfidence ellipses for the mean points
2.0
p p
Angel
1.0
1.5
g
Lolita Lempicka
0.5
1
Dim
2 (1
3.83
%)
Aromatics ElixirCinema
-0.5
0.0
D
Chanel5Coco MademoiselleJadoreEP
Linstant
Pure Poison
Shalimar
-1.0
Coco Mademoiselle
JadoreETPleasures
12/07/2012 Sensometrics 2012 3
-1 0 1 2
Dim 1 (16.99%)
Bootstrap technique
Real jury Virtual juryj yJ1 J2 J3 … J96 J97 J98
P1P2
J1 J1 J19 … J98P1P2P3P3
…P10P11
P3…P10P11P11
P12P11P12
=> Random re-sampling with
l t
12/07/2012 Sensometrics 2012 4
replacement
Real jury Virtual jury
Bootstrap techniqueReal jury
J1 J2 J3 … J96 J97 J98P1P2
Virtual jury…
P1P2
Virtual jury…
P1Virtual jury
…Virtual juryVi t l j
P3…P10
P3…P10
P2P3…
P1P2P3
…P1P2P3
Virtual jury…
P1P2
Virtual jury…
P1P11P12
P11P12P10P11P12
…P10P11P12
P3…P10P11
P2P3…P10
P2P3…P12P11
P12P11P12P10P11P12
• 2 ways to use bootstrapped virtual juries :By projection (partial bootstrap)B t t ti (t t l b t t )
12/07/2012 Sensometrics 2012 5
By procrustean rotation (total bootstrap)
Partial bootstrap1. Principal Component methodon the data of the real jury
2. Projection of the judges of a virtual jury
3a Calculation of the virtual jury’s3a. Calculation of the virtual jury’sbarycenter
3b. Calculation of the barycentersy
4. Ellipse including 95% of the points
12/07/2012 Sensometrics 2012 6
Total bootstrap
Virtual jury 1 Virtual jury 2 Virtual jury 3 Virtual jury 200
2. Principal Component method on each virtual jury
Real jury
Virtual jury 1 Virtual jury 2 Virtual jury 3 j y
1. Principal Component method on the data of
the real jury
Expansion/Shrinkage
Translation 3. ProcrusteanTranslation
Rotation
3. Procrusteanrotation
12/07/2012 Sensometrics 2012 7
4. Confidence ellipse including 95 % of the points
Example of a sorting task dataset
Comparison of partial and total bootstrap
Partial bootstrap Total bootstrap
3
Confidence ellipses for sorting task
2.0
Confidence ellipses for the mean points
Angel
12
.83%
)
Angel
Lolita Lempicka
Cinema Aromatics ElixirLi t t5
1.0
1.5
3.83
%)
Angel
Lolita Lempicka
-10
Dim
2 (1
3
Chanel5Shalimar
Coco MademoiselleJadoreEP
JadoreET
Pleasures
Pure Poison
Linstant
0.5
0.0
0.5
Dim
2 (1
3
Aromatics Elixir
Chanel5
Cinema
JadoreEP
Linstant
Pure Poison
Shalimar
-3 -2 -1 0 1 2 3 4
-3-2
-1 0 1 2
-1.0
-0
Coco MademoiselleJadoreEP
JadoreET Pleasures
12/07/2012 Sensometrics 2012 8
Dim 1 (16.99%)Dim 1 (16.99%)
Example of a RANDOM sorting task dataset
Comparison of partial and total bootstrap
Partial bootstrap Total bootstrap
2
Confidence ellipses for the mean points
2
Confidence ellipses for sorting task
1
.02%
)
10
12 12
.02%
)
10
12
0
Dim
2 (1
5.
1
234
5
6
8
9
11
-10
Dim
2 (1
5.
1
11
234
5
6
7
8
9
1 0 1 2
-1
7
2 1 0 1 2 3
-210 judges3 groups/judge
12 products
12/07/2012 Sensometrics 2012 9
-1 0 1 2
Dim 1 (22.8%)
-2 -1 0 1 2 3
Dim 1 (22.8%)
pRandom data
Example of a RANDOM sorting task dataset
Comparison of partial and total bootstrap
Partial bootstrap Total bootstrap
1.0
Confidence ellipses for the mean points
7 1.5
Confidence ellipses for sorting task
0.5
.53%
)
1
2
9 0.5
1.0
.53%
)
1
2
7
9
0.0
Dim
2 (1
0. 1
3
4
5
8
10
11 -0.5
0.0
Dim
2 (1
0.
1
10
11
12
3
4
5
6
8
1 0 0 5 0 0 0 5 1 0
-0.5 3
6
12
2 1 0 1 2
-1.5
-1.0
300 judges3 groups/judge
12 products
12/07/2012 Sensometrics 2012 10
-1.0 -0.5 0.0 0.5 1.0
Dim 1 (10.88%)
-2 -1 0 1 2
Dim 1 (10.88%)
pRandom data
I R d lit ti d t tConfidence ellipses for the mean pointsConfidence ellipses for the mean points
I. Random qualitative dataset
0.0
0.5
1.0
32%
) 1
2
3
6
7
9 11
12
12
6.56
%)
2
35
6
78
-1.0
-0.5D
im 2
(12.
3
4
5
8
10
-2-1
0
Dim
2 (2
6
1
4
9
10
11
12 30 judges3 groups/judge12 productsRandom dataset
3 judges3 groups/judge12 productsRandom dataset
Confidence ellipses for the mean pointsConfidence ellipses for the mean points
-1.5 -1.0 -0.5 0.0 0.5 1.0
-1.5
Dim 1 (13.46%)
-3 -2 -1 0 1 2
Dim 1 (35.62%)
When the number of judges increases► The dimensionality increases
Th lli b ll
0.5
) 3
4
8 10
0.5
%) 5
6
8
9
The ellipses become smaller
-0.5
0.0
Dim
2 (9
.501
%)
1
2
3
5 7
9
12
-0.5
0.0
Dim
2 (1
0.14
%
12
3
4
7
10
11 300 judges3 groups/judge
3000 judges3 groups/judge
12/07/2012 Sensometrics 2012 12-0.5 0.0 0.5 1.0
-
Dim 1 (9.582%)
6 11
-1.0 -0.5 0.0 0.5 1.0
Dim 1 (10.85%)
312
g j g12 productsRandom dataset
g p j g12 productsRandom dataset
II Q tit ti d t tAn example of QDA (RANDOM quantitative dataset)
II. Quantitative dataset
Confidence ellipses for the mean pointsConfidence ellipses for the mean points
1020
2
11
4
2
0
Dim
2 (1
0.35
%) 1
3 5
6
8
9
02
Dim
2 (1
8.43
%)
1
3
4
5
7
9
11
12
-10
D
4
7
10
12
-2
D 4
6 8
9
10
-20 -10 0 10 20
-20
Dim 1 (10 7%)
7
-4 -2 0 2 4 6
-4
Dim 1 (25 5%)
30 judges900 desc12 products
30 judges10 desc12 products
12/07/2012 Sensometrics 2012 13
Dim 1 (10.7%)Dim 1 (25.5%)
II Q tit ti d t tA simple RANDOM dataset
II. Quantitative dataset
ndesc=3 ndesc=30000
PCA on the average tableProjection of individual judgements in illustrativeProjection of the average points of the virtual juries in illustrative
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PCA on the average table=> A point represents a product as seen by the jury
Projection of individual judgements in illustrative=> A point is a product as seen by a judgeProjection of the average points of the virtual juries in illustrative=> A point is a product as seen by a virtual jury
Th f l tl d d tThe case of completly random data
Confidence ellipses for the mean points
RANDOM data without structure
Confidence ellipses for the mean pointsConfidence ellipses for the mean points
20
05
6.16
%)
112
2
3 4
568
9
10
10.8
2%)
10
12
-5
Dim
2 (1
6
10
11
7
-10
0
Dim
2 (1
1 11
2 3
4
5
67
8
9
-10
30 judges12 products30 descriptors -2
0-
30 judges12 products300 descriptors
12/07/2012 Sensometrics 2012 16
-10 -5 0 5 10
Dim 1 (18.96%)
-20 -10 0 10 20 30
Dim 1 (11.49%)
D t i l ti dData simulation procedure
« Pure » configuration
11 ndes
c1
1ndesc
11
ndesc
configurationAverage table
np
« Pure » nj times duplication
« Real » datasetNoise ~ N(0, sigma)« Pure »
configurationAverage table
« Pure » configuration
Average table
nj*npnj*np
« Pure » configuration
Average tablenj*np
12/07/2012 Sensometrics 2012 17
j pnj npnj*np
D th lli i l d th l t t f Do the ellipses include the « real » structure of the data ?
Sd Frequence
ataset 0.1 92,09%
0.5 91,5%1 91,67%
Count the number of times that a point is included in an ellipse, given its coordinates on the factorial plan
Confidence ellipses for the mean points
ructured
da 1 91,67%1.5 91,09%2 89,5%3 91,75%4 91,92%2
46
1
11
3
57 9
Str 4 91,92%
7 94,92%
nj 12 30 90 210
-20
2
Dim
2 (2
3.08
%)
2
4
6
α=
5%
12 88,50 88,08 89,17 92,42
30 91,42 90,00 91,00 93,92
200 94,08 94,33 95,33 93,92
-6-4 10
128
200 94,08 94,33 95,33 93,92
The proportion tends to 95% when the number of judges increases(asymptotic behavior of bootstrap techniques)
-6 -4 -2 0 2 4 6 8
Dim 1 (45.36%)
12/07/2012 Sensometrics 2012 18
(asymptotic behavior of bootstrap techniques)
C l i• One parameter must be chosen:
Conclusion• One parameter must be chosen:
• Number of dimensions for the procrustean rotations=> n = 2 in many sensometry applications
• Dimensionality problem highlighted :Confidence ellipses are essential (but may be built according to total bootstrap)
• Total bootstrap can be applied to all holistic approaches:• Napping• Sorting taskSorting task• Hierarchical sorting• Free Choice Profiling
A il bl i t th R k S Mi R th h th b t f ti• Available into the R package SensoMineR through the boot function
12/07/2012 Sensometrics 2012 19
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