Introduction of Symbolic Data Analysis and its Applications
Huiwen Wang (
王惠文)
School of Economic Management,
Beijing Univ. of Aeronautics and Astronautics, Beijing 100083, China
E-mail: [email protected]
Outline
A Short Introduction on Symbolic Data
Concept of Interval Data
Descriptive Statistics of Interval Data
Principal Component Analysis of Interval Data
Application I:
Analysis on Speculation in China’s Stock Market
Factor Interval Data Analysis
**Application II:
Data Analysis on Features Market of China
Founder of SDA: Edwin Diday (Paris IX)
z In the first conference of the International Federation of Classification Societies (IFCS) , 1987.
z Many literatures on symbolic data has been published in different journals, proceedings and working reports.
z About 17 European research groups have cooperated in an European Esprit Project (No.20821) named Symbolic Official Data Analysis System. As the result of this project a software package SODAS has been developed.
z H.-H. Bock, E. Diday (Editors), Analysis of Symbolic Data, Springer, 2000
I. A Short Introduction on Symbolic Data
Data Mining:
How to treat the huge sets of data
(1) Classify the dataset into several groups (2) Summarize each group by a symbolic data
Knowledge mining:
Extend the classical statistical methods (1) Descriptive statistics of symbolic data:
Mean, Variance, Histogram (2) Multivariate analysis of symbolic data:
Factor analysis, Classification, Regression In many domains of human activities it is quite common to record huge sets of data.
Motivation of SDA
Example: Analysis on Stock Market of China
The total number of stocks is very large.
The number of individuals changes every year.
Some of individuals may be renamed.
Year 1990 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Number
of Stocks 10 306 509 714 819 912 1088 1137 1134 1215 1234
Total number of stocks in Market
Problem:
REGR factor score 1 for analysis 1
6 4
2 0
-2 -4
-6
REGR factor score 2 for analysis 1
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1178 117711731176 11751174 1172
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2 1
Variables: Negotiable Market Capitalization (NMC) , Return rate, Turnover rate, P/E ratio, Volatility
PCA on 1088 stocks of China (2001 )
REGR factor score 1 for analysis 1
6 4
2 0
-2 -4
-6
RE G R f ac tor s cor e 2 f or a na ly si s 1
8
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1163 11621161 1160 1158 1159
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84 8281 807879 83
7677 7475 73 72 71
70 69
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54 53
5152
50 49
48 47 46
44 45 424143 403938
3637
35 34
33 32
3029 31 272628 24 25
23 22
20 182119
17 16 14 15
13 12
11 10
9 7 8
6 5 4
3
2 1
PCA after deleting 15 outliers
REGR factor score 1 for analysis 1
4 3
2 1
0 -1
-2 -3
-4
R EG R f ac to r s co re 2 f or a na lys is 1
8
6
4
2
0
-2
-4
-6
PCA after deleting 113 outliers
Stock Style Classification Based On CITIC Style Indices
Tab.4-1 The classification of the six CITIC style indices Low B/P value High B/P value High market
capitalization
Large-cap
growth Large-cap value Mid-market
capitalization Mid-cap growth Mid-cap value Low market
capitalization
Small-cap
growth Small-cap value
Size ： represented by Negotiable Market Capitalization (NMC) B/P Ratio ： net assets / market value
CITIC: China International Trust Investment Company
Object
Market Capitalization
Turnover
Rate Return Rate P/E Ratio Volatility Large-Cap
Growth Large-Cap
Value Mid-Cap
Growth Mid-Cap
Value Small-Cap
Growth Small-Cap
Value
[ x 11 , x 11 ] [ x 2 1 , x 2 1 ] [ x 3 1 , x 3 1 ]
[ x 4 1 , x 4 1 ]
[ x 5 1 , x 5 1 ]
[ x 6 1 , x 6 1 ]
[ x 12 , x 1 2 ] [ x 22 , x 22 ]
[ x 32 , x 32 ]
[ x 42 , x 42 ]
[ x 52 , x 52 ] [ x 62 , x 62 ]
[ x 1 3 , x 1 3 ] [ x 23 , x 23 ] [ x 33 , x 33 ] [ x 43 , x 43 ] [ x 53 , x 53 ] [ x 63 , x 63 ]
[ x 1 4 , x 1 4 ] [ x 24 , x 24 ]
[ x 34 , x 34 ]
[ x 44 , x 44 ] [ x 54 , x 54 ]
[ x 64 , x 64 ]
[ x 1 5 , x 1 5 ]
[ x 25 , x 25 ] [ x 35 , x 35 ] [ x 45 , x 45 ] [ x 55 , x 55 ] [ x 65 , x 65 ]
Interval Data Table
Reflect: Central Tendency & Dispersion Tendency of each class
{ } { }
1, , 1, ,
min , max
i i
k k
ij ij ij ij
k m k m
x a x a
= =
= =
** Here we use Interquartile Range to prevent the influence of outliers.
Variable
The Scatter Plot of PCA
Quick Impression:
¾Size Style is the primary factor influencing activity of market deals.
¾Growth or Value Style are also important factors that can describe the behaviours of stock market in China.
LargeCapG.
LargeCapV.
SmallCapG.
SmallCapV. MidCapV.
MidCapG.
Definition of Symbolic Data
E. Diday, 1987,1996
(1) Traditional Data:
A single quantitative or Qualitative value
(2) A Set of Value or Categories (Multivalued Variable) EX: Height (w)={ 3.5, 2.1, 5 } means that :
the height of w can be ether 3.5 or 2.1 or 5 (3) An Interval
EX: Height (w)=[3,5] means that:
the height of w varies in the interval [3,5]
(4) A Set of Value with Associated Weights (Distribution Var.) histogram , membership function.
Cell
Data Table
Supplier Weekly working time
Customer Delivery’s Town Weight of delivery
S1 34 SG1,SG3
Paris(0.5), Lannion(0.5) [12,15]
S2 45 SG2
Toulouse (1) [20,25]
S3 40 SG4,SG5
Lille(0.4), Lyon(0.6) [34,40]
Example:
We consider three relations: Supplier, Delivery, and Customer
Multivalued Variable Distribution variable Interval variable
Note: Compositional data is a kind of distribution data.
Characteristics
Reduce complexity of data system, since SDA reduces
aggregation scale by pretreatment of classifying original dataset.
Powerful in summarizing global features of the data
set, because it can hold overall characteristics of the sample group, and extract internal rules of the data set.
Visualize analysis results, especially it is suitable for
knowledge mining and visualization research of large
scale data set.
Interval Data: the elements of matrix are interval of real data.
The interval data may result from three sources:
– Imprecision of the measurement:
– Expert’s knowledge: including a range of uncertainty – Summary a class of dataset (second-order-object)
The object i to be described is a class of elements.
Let be m i observations of the feature j for the object i:
ij , ij
a δ a δ
⎡ − + ⎤
⎣ ⎦
1 2
{ a a ij , ij , , a ij m i }
{ } { }
1, , 1, ,
,
min , max
i i
ij ij ij
k k
ij k m ij ij k m ij
x x
Where x a x a
ξ
= =
⎡ ⎤
= ⎣ ⎦
= =
Range of error
II. Concept of Interval Data
Interval Data Matrix
¾ Let denote n objects (classes) described by p features of interval type.
¾ Let be the interval which include all the possible values of feature j for object i , the resulting symbolic data matrix is given by:
1, 2, ,
i = n
1 , 2 , , p
Y Y Y
ij x x ij , ij
ξ = ⎣ ⎡ ⎤ ⎦
[ ]
[ ]
11 11 1 1
11 1
1 1 1
, ,
, ,
p p
p
n np n n np np
n p
x x x x
X
x x x x
ξ ξ
ξ ξ
×
⎛ ⎡ ⎤ ⎞
⎛ ⎞ ⎜ ⎣ ⎦ ⎟
⎜ ⎟
= ⎜ ⎟ = ⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎡ ⎤ ⎟
⎝ ⎠ ⎝ ⎣ ⎦ ⎠
III. Descriptive Statistics of Interval Data
Basic set: E={1,2,…,n}
X is an univariable interval-valued variable measured for each observed unit of the basic set E={1,2,…,n}
For each k∈ E, we denote
Where, is the lower bound of the interval; and is the upper bound of the interval.
]
, [
) (
X k = x k x k
x k x k
The Equi-Distribution Hypothesis:
(1) Each observed unit k∈ E is equally likely, i.e. each observed unit k∈ E has been selected with the same probability 1/n.
(2) The values of x are uniformly distributed in the interval.
[ ]
( )
⎪ ⎪
⎩
⎪⎪ ⎨
⎧
≥
<
− ≤
− <
=
∈
≤
if 1
if
if 0
, Pr
k k k
k k
k
k k k
x
x x x
x
x
x x
x x
x
ξ
ξ ξ ξ
ξ
∀ ξ ∈ R
Basic set
E={1,2,…,n}
Empirical Distribution Function of X
EDF is the distribution function of the uniform mixture of n uniform distributions defined on the interval , for all k∈ E.
[ ]
( )
{ } { }
{ }
∑
∑
∑
∈
∈
∈
≥ + ∈
−
= −
≥
⋅ ∈
< +
⋅ ∈
− +
= −
∈
≤
=
) ( ) (
1 #
#
# 1 1 0
) (
: Thus
, 1 Pr
) (
k X
j k
k
k k
X
i j k
k X k
E k
k k X
n
x E
j x
x
x n
n
x E
j n
x E
i x
x
x F n
x x x
n x F
ξ ξ
ξ ξ
ξ ξ ξ ξ
ξ ξ
] , [ ) (
X k = x k x k
{ j ∈ E ξ ≥ x j }
#
Where the notation of :
is the number of intervals with X ( j ) = [ x j , x j ] ξ ≥ x j
By taking the derivation of the empirical distribution function, we get the Empirical Density Function of X
∈ ∑ −
=
) (
1 ) 1
( f
k
X k k
X ξ n ξ x x
{ }
∈ ∑
≥ + ∈
−
= −
) (
1 # )
(
k X
j k k
X k
n
x E
j x
x
x F n
ξ
ξ ξ ξ
Here is an indicator function:
E k k
X ξ
k X ξ
k
X ∈
⎩ ⎨
⎧
∉
= ∈
) (
if 0
) (
if ) 1
) (
( ξ
1 )
) (
( k ξ 1 X
Empirical Density Function of X
From EDF
∑ ∈ −
=
E
k k k
k X
x x
n ( )
1 1 ( ) ξ
∈ R
ξ
Empirical Mean of X
( ) ξ ξ
ξ d
X : = ∫ − +∞ ∞ f X
) 1 (
: ∫ − + ∞ ∞ ∑ ( )
∈ −
⋅
= ξ ξ
ξ d
x x
X n
E
k k k
k
1 X
∑ ∈
= +
E k
k x k
x X n
2 1
∑ ∈ −
=
E
k k k
k X
X n 1 x x ( )
) (
f ( ) ξ
ξ 1
Empirical Density Function
] ,
[
if 0
] , [
if ) 1
) (
( ⎩ ⎨ ⎧
∉
= ∈
k k
k k k
X ξ x x
x x ξ ξ
1
indicator function
k k
k
k x x x
x
+ 2
[ ]
∑∫ ∈
∞ +
∞
− −
=
E
k k k
k
X d
x x
n ( ξ ) ξ ξ
1 1 ( )
1 ∑∫
∈ −
=
E k
x
x k k
k k
x d x
n ξ ξ
∑ ∈
= +
E k
k x k
x
n 2
1
Empirical Standard Deviation:
( 2 2 ) 2 ( ) 2
2
4 1 3
1 ⎥
⎦
⎢ ⎤
⎣
⎡ +
− +
+
= ∑ ∑
∈
∈ k E
k k E
k
k k k
k
Y x x
x n x
x n x
s
( ξ X ) ( ) ξ d ξ
s X : = ∫ − + ∞ ∞ − 2 f X
∫
∫
∞ +
∞
− +∞
∞
−
=
−
=
ξ ξ
ξ
ξ ξ
ξ
d M
X d
s
X X X
) ( f :
) ( f
2 2
2 2
2
ξ ξ x d x
n k E
x
x k k
k
∑∫ k
∈ −
= 1 2 ∑
∈ −
= −
E
k k k
k k
x x
x x
n 3 ( )
1 3 3
( )
∑ ∈
+ +
=
E k
k k k
k x x x
n x
2 2
3
1
Example : We observe an interval-valued variable X defined on a set E={1,2,…,8}. The observed values of X at set E are:
X(E)={[0,2]; [1,3]; [1.5,2.5]; [2,4]; [3.5,5]; [4.5,5.5]; [5,7]; [6.5,7.5]}
045 .
2 781
. 24 3
5 . 443
781 .
3 )
7 6 2 5
5 . 3 8
2 2 1 8 ( 1
2 ≅
−
=
≅ +
+ + +
+ + +
=
s X
X
Depends on the equi-distribution hypothesis:
(1) Each observed unit k∈ E has been selected with the same probability 1/n :
(2) The values of x are uniformly distributed in the interval.
Example: An interval-valued variable X defined on a set E={1,2,3}.
The observed values of X at set E are: X(E)={[0,2]; [2,4] ;[1,3]}
3 1
3 , 1
3 , 1
) 3 , 1 [ )
4 , 2 [ )
2 , 0
[ = p = p =
p
6 1 1
3 1 3
1 ~
6 1 2
4 1 3
1 ~
6 1 1
2 1 3
~ 1
) 1 3
~ (
), 2 4
~ (
), 0 2
~ (
) 3 , 1 [
) 4 , 2 [
) 2 , 0 [
) 3 , 1 [ )
3 , 1 [ )
4 , 2 [ )
4 , 2 [ )
2 , 0 [ )
2 , 0 [
− =
×
=
− =
×
=
− =
×
=
−
×
=
−
×
=
−
×
=
f f f
f p
f p
f p
0 0 0
1/7 1/5 1/4 2/7 1/3
1 2 3 4
0 1 2 3 4 1/3
1/6 f
Histogram:
(1) The interval that contains all the observed single values of X is given by I = [0, 4].
(2) There are 5 distinct observed boundaries that are ranked in increasing order:
(3) Consider the partition of I by the 4 intervals
4
, 3
, 2
, 1
,
0 1 2 3 4
0
) 4], , [3 ( ),
3), , ([2 ),
2), , 1 [ ( ), ), 1 , 0
([ f 1 f 2 f 3 f 4
=
=
=
=
= u u u u
u
X(E)={[0,2]; [1,3]; [2,4]}
0 0 0
1/7 1/5 1/4 2/7 1/3
1 2 3 4
0 1 2 3 4 1/3
1/6 f
6 1 2
4 1 3
~ 1
3 1 2
4 1 1
3 1 3
~ 1 ~
3 1 1
3 1 0
2 1 3
1
~ ~
6 1 1
2 1 3
1 ~
) 4 , 2 [ 4
) 4 , 2 [ )
3 , 1 [ 3
) 3 , 1 [ )
2 , 0 [ 2
) 2 , 0 [ 1
− =
×
=
=
⎟ =
⎠
⎜ ⎞
⎝
⎛
+ −
× −
= +
=
⎟ =
⎠
⎜ ⎞
⎝
⎛
+ −
× −
= +
=
− =
×
=
=
f f
f f
f
f f
f
f f
Calculation Process:
Example: We observe an interval-valued variable X defined on a set E={1,2,…,8}. The observed values of X at set E are:
X(E)={[0,2]; [1,3]; [1.5,2.5]; [2,4]; [3.5,5]; [4.5,5.5]; [5,7]; [6.5,7.5]}
(1) The interval that contains all the observed single values of X is given by I = [0, 7.5].
(2) The 14 distinct observed boundaries that are ranked in increasing order:
(3) Consider the partition of I by the 13 intervals 5 . 7
, 7 ,
, 5 . 1
, 1
,
0 1 2 13 14
0 = u = u = u = u =
u
13 , , 2 , 1
, ) ,
[ 1 =
= u − u j
I j j j
two repeated boundaries: 2 , 5
The corresponding Density Function is described by the list of pairs:
Let p j indicates the area of the bar of base I j in the histogram.
( [ u j − 1 , u j ) ， f j ) , j = 1 , 2 , , 13
( j j j ) j j j
j x u u f u u f
p = Pr( ∈ [ − 1 , ) ， = ( − − 1 ) ×
X(E)={[0,2]; [1,3]; [1.5,2.5]; [2,4]; [3.5,5]; [4.5,5.5]; [5,7]; [6.5,7.5]}
4 1 5
. 1 5 . 2
1 1
3 1 0
2 1 8
1 : ) 2 , 5 . 1 [
8 1 1
3 1 0
2 1 8
1 : ) 5 . 1 , 1 [
16 1 0
2 1 8
1 : ) 1 , 0 [
3 3
2 2
1 1
⎥⎦ =
⎢⎣ ⎤
⎡
+ − + −
× −
=
=
⎥⎦ =
⎢⎣ ⎤
⎡
+ −
× −
=
=
− =
×
=
=
f I
f I
f I
= 8 n
5 . 7
, 7 ,
, 2 ,
5 . 1
, 1
,
0 1 2 2 13 14
0 = u = u = u = u = u =
u
observed boundaries
Observed values:
Thus the corresponding list is given by:
( [ u j − 1 , u j ) ， f j ) , j = 1 , 2 , , 13
，
），
，
，
），
，
，
），
，
，
），
，
，
），
，
，
），
，
，
），
，
），
，
，
），
，
，
），
，
，
，
，
，
），
，
，
），
，
⎟ ⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎛
⎟ ⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎛
⎟ ⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎛
8 5 1
. 7 7 16 [
7 3 5 . 6 16 [
5 1 . 6 5 . 5 16 [
5 3 . 5 5 24 [
5 5 5 . 4 [
24 5 2
. 4 4 48 [
4 7 5 . 3 16 [
5 1 . 3 3 8 [
3 1 5 . 2 [
4 5 1
. 2 2 4 [
) 1 2 5 . 1 8 [
5 1 . 1 1 16 [
1 1 0 [
8 ) 1 5 . 1 2 4 (
1
16 ) 1
1 5 . 1 8 (
1
16 ) 1
0 1 16 (
1
3 2 1
…
…
=
−
×
=
=
−
×
=
=
−
×
=
p p p
0
1/ 20 1/ 10 3/ 20 1/ 5 1/ 4 3/ 10
1 1. 5 2 2. 5 3 3. 5 4 4. 5 5 5. 5 6. 5 7 7. 5
) ( − − 1
×
= j j j
j f u u
p
Now, consider a new partition of the interval with 5 subintervals:
] 8 , 0
= [ I ′
] 8 , 5 . 6 [ )
5 . 6 , 5 [ )
5 , 5 . 3 [ )
5 . 3 , 5 . 1 [ )
5 . 1 , 0
[ ∪ ∪ ∪ ∪
′ = I
5 , 4 , 3 , 2 , 1
,
=
′ = ∑
∈ ′
α
α
α
I I
j
j
p
Remark that p
For
( )
12 1 5
. 1
8 / 1
8 0 1
5 . 1
8 ) 1 1 5 . 1 8 ( ) 1 0 1 16 ( 1
: ) 5 . 1 , 1 [ ) 1 , 0 [ ) 5 . 1 , 0 [
1
1 1
1
=
=
∴
=
−
×
=
=
− +
−
=
∪
=
′ =
f
f p
p I
∵
⎟ ⎠
⎜ ⎞
⎝
⎛
12 1 ), 5 . 1 , 0
Thus, [
0
1/ 20 1/ 10 3/ 20 1/ 5
1 2 3 4 5 6 7 8
With the same methodology, we get the empirical frequency distribution:
⎟ ⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎛
48 ]; 5 8 , 5 . 6 [ 48 ;
); 5 5 . 6 , 5 [ 48 ;
); 7 5 , 5 . 3 64 [
); 11 5 . 3 , 5 . 1 [ 12 ;
]; 1
5
.
1
,
0
[
[ ]
[ ]
11 11 1 1
11 1
1 1 1
, ,
, ,
p p
p
n np n n np np
x x x x
X
x x x x
ξ ξ
ξ ξ
⎛ ⎡ ⎤ ⎞
⎛ ⎞ ⎜ ⎣ ⎦ ⎟
⎜ ⎟
= ⎜ ⎟ = ⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎡ ⎤ ⎟
⎝ ⎠ ⎝ ⎣ ⎦ ⎠
IV. PCA of Interval Data
(Cazes, Chouakria, Diday& Schektman,1997) Interval Data Matrix
Let denote n objects described by p features of interval type value.
1, 2, ,
i = n
1 , 2 , , p
Y Y Y
1. The VERTICES Method
z Let denote the interval data vector of object i . This data point can be visualized as a hyper-rectangle in space with 2 p vertices.
z For p = 2, the object vector is
z With 2 2 vertices as follows:
( 1 , , ) ( [ 1 , 1 ] , , , )
i i ip i i ip ip
x = ξ ξ ′ = x x ⎡ ⎣ x x ⎤ ⎦ ′
R p
[ ] [ ]
( 1 , 1 , 2 , 2 )
i i i i i
O = x x x x ′
1 2
1 2
1 2
1 2
i i
i i
i
i i
i i
x x
x x
M x x
x x
⎡ ⎤
⎢ ⎥
⎢ ⎥
= ⎢ ⎥
⎢ ⎥
⎣ ⎦
R i
X 2
X 1 M i
R i
Transformation to Numerical Matrix
¾ Describe each interval-valued object O i by a numerical data matrix M i with 2 p rows and p columns.
¾ Compile the matrices into the numerical matrix M with rows and p columns.
[ ]
[ ]
11 11 1 1
1
1 1
, ,
, ,
p p
n p
n n n np np
x x x x
O X
O x x x x
×
⎛ ⎡ ⎤ ⎞
⎛ ⎞ ⎜ ⎣ ⎦ ⎟
⎜ ⎟
= ⎜ ⎜ ⎝ ⎟ ⎜ ⎟ ⎜ ⎠ = ⎜ ⎝ ⎡ ⎣ ⎤ ⎦ ⎟ ⎟ ⎟ ⎠
2 p n ×
11 1
11 1
1 2
1
1
p
