Download Table 4.2 The sample memberships and kernel set

行政院國家科學委員會專題研究計畫
成果報告
模糊統計分析在資料採礦以及軟計算方法之理論探討及其
應用
計畫類別： 個別型計畫
計畫編號： NSC91-2213-E-004-007執行期間： 91 年 08 月 01 日至 92 年 07 月 31 日
執行單位： 國立政治大學應用數學學系
計畫主持人： 吳柏林
報告類型： 精簡報告
處理方式： 本計畫涉及專利或其他智慧財產權，1 年後可公開查詢
中 華
民 國 92 年 10 月 13 日
模糊統計分析在資料採礦以及軟計算方法之理
論探討及其應用
計畫編號: NSC91-2213-E-004-007
吳柏林
國立政治大學應用數學系
2003/8/2
Introduction
Data mining for knowledge discovery is a new and rapidly developing area of
research towards applications in this information age. Although this area falls into
the field of multivariate data analysis, there is a tendency to consider alternative
techniques to traditional multivariate statistical techniques, which are borrowed from
computer science and artificial intelligence, such as neural networks, and rough sets
theory(see, e.g., [5], [7]). This emerging technique are somewhat ad-hoc and lack
theoretical foundations.
As Elder [2] pointed out, there is a compelling reason to
look at statistical perspectives of knowledge discovery in databases(KDD).
This is
mainly due to the fact that recent statistical contributions to multivariate data analysis
did not fully spread out to the KDD community.
Needless to say, the advantage of
using statistical methodology is that it will provide a firm theoretical foundation for
deriving inference procedure in decision-making.
In this research project, we will show that a relatively new area of probability
theory and statistics, namely random sets theory(see, e.g., [4], [7], and [15]), is
suitable as a framework for data mining problems.
Theorem 2.1. Let X be a universal set and A be a fuzzy set on X . Then the
complement of fuzzy set, Ac , is denoted by Ac  ∨ c (1   )  ( Ac )(1 ) .
(1- ) ( A )
(2.3)
where ( Ac ) is the level set of Ac , ( Ac ) (1 ) is a (1   )  cut of Ac , and ∨
denotes the finite fuzzy union.
Theorem 2.2. Let X be a universal set, A and B be two fuzzy sets on X . For
any significant levels 1 , 2  [0,1] , the standard fuzzy union and intersection of A
and B are denoted by
(i) A F B  [ ∨   ( A) ] F [ ∨   ( B)  ]
 ( A)
  ( B )
 [ Ker ( A)
F
Ker ( B)]
F
∨
[(
 ( A)  
  ( A) )
F
(
1
(ii) A
F
B  [ ∨   ( A) ]
 ( A)
 [ Ker ( A)
F
[(
F
∨
 ( A)  
  ( B )
F
[ Ker ( A)
  ( A) )
F
Ker ( B)]
  ( A) )
F
(
F
(
  ( B)  )]
∨
  ( B )  
2
1
F
[(
∨
 ( A)  
1
(2.12)
2
[ ∨   ( B)  ]
Ker ( B)]
F
  ( B)  )] . (2.11)
∨
  ( B )  
  ( B)  )] .
∨
  ( B )  
2
where  ( A) is the level set of A ,  ( B ) is the level set of B , ( A) is a   cut of
A , (B )  is a   cut of B , F and F denote the standard fuzzy union and
intersection respectively
Applications with the RRI data
The data analyzed here comes from the ICU of Taipei Veterans General Hospital,
2001. The data records RRI of the dead patients and survival patients for the first four
days of ICU. The RRI data for each patient is measured with 30 minutes. By
discarding the first 100 observation, we analysis the 101 to 600 observations from
each patient which contains about 1800 ~ 3000 observations of RRI. The purpose of
this study is to extract features and identify nonlinear time series for the ICU patients.
Figure 4.1 (a) and (b) plot respectively the dead and survival patients’ RRI.
The 2nd Day for 1st Dead Patient
The 2nd Day for 1st Survival Patient
The 2nd Day for 2nd Dead Patient
The 2nd Day for 2nd Survival Patient
The 2nd Day for 3rd Dead Patient
The 2nd Day for 3rd Survival Patient
The 2nd Day for 4th Dead Patient
The 2nd Day for 4th Survival Patient
The 2nd Day for 5th Dead Patient
The 2nd Day for 5th Survival Patient
(a)
(b)
Figure 4.1 Plots of RRI for dead and survival patients
For the 500 observations, we can find the cluster centers for each data set. Now,
under the significant level  = 0.9, we can construct our kernel sets by the proposed
procedures in the Section 3. In following, the dead patients’ and the survival patients’
cluster centers, radiuses of kernel set and ratios are showed in Table 4.1.
Table 4.1 The dead and the survival patients’ cluster centers, radiuses and ratios
Patient
Cluster
center
Radius of
kernel set
Ker( Di 2 ) / Di 2
D12
623.734
0.734
0.348
S12
750.546
Radius
of
kernel
set
0.546
D22
D32
976.018
1.018
0.118
850.132
0.868
0.006
883.592
0.592
0.088
S 22
S 32
561.882
0.882
0.066
D42
651.442
0.558
0.046
S 42
667.570
0.570
0.060
Average of Ker(Di 2 ) / Di 2
Its ratio
 0.15
Patient
Cluster
center
Average of Ker(Si 2 ) / Si 2
Its ratio
Ker(Si 2 ) / Si 2
0.078
 0.05
4
The kernel set learned from the dead patients is Ker ( D)  F Ker ( Di 2 ) . Then, we
i 1
can give the following testing-hypothesis procedure:
H 0 : the data belongs to the dead patterns.
H 1 : the data doesn’t belong to the dead patterns.
Decision rule: for the new sample Ker( Dnew ) , under the significant level  D , if
there exist i, such that Ker ( Dnew ) Ker ( Di 2 )  1  a D then we accept H 0 .
Otherwise, we reject H 0 .
4
Similarly, the kernel set learned from the survival patients is Ker ( S )  F Ker ( S i 2 ) ,
i 1
and the testing-hypothesis procedures:
H 0 : the data belongs to the survival patterns.
H 1 : the data doesn’t belong to the survival patterns.
Decision rule: for the new sample Ker ( S new ) , under the significant level  D , if
there exist i, such that Ker ( S new ) Ker ( S i 2 )  1   D then we accept H 0 .
Otherwise, we reject H 0 .
Now, we examine two new RRI samples of patients from ICU. First, we can find
the cluster centers for each data set and construct their kernel sets under the
significant level  = 0.9. Then we can find radiuses of kernel sets for these samples.
Finally, we can get their patterns according to their features by above the
testing-hypothesis procedures.
For these two samples, we get the cluster centers 592.624 and 761.658
respectively. We can calculate the membership for each observation via the distance
between observation and its cluster center. Under the significant level  = 0.9, if the
membership of observation is lager than 0.9, then this observation is a member of the
kernel set. Therefore, the results of two new samples are showed in Table 4.2.
Table 4.2 The sample memberships and kernel set for RRI of new samples
The sample memberships and kernel set
for RRI of a new sample (I)(Cluster The sample memberships and kernel set
for RRI of a new sample (II)
center: 592.624,  = 0.9)
(Cluster center: 761.658,  = 0.9)
Data
Memberships
Is a member of
the kernel set ?
Data
Memberships
Is a member of
the kernel set?
1
2
3
4
5
6
7
8
9
10
569
573
571
529
622
598
609
614
608
605
0.042
0.051
0.046
0.016
0.034
0.186
0.061
0.047
0.065
0.081
no
no
no
no
no
no
no
no
no
no
751
740
760
734
730
729
718
713
708
741
0.094
0.046
0.603
0.036
0.032
0.031
0.023
0.021
0.019
0.048
no
no
no
no
no
no
no
no
no
no
491
492
493
494
495
496
591
591
591
595
598
598
0.616
0.616
0.616
0.421
0.186
0.186
no
no
no
no
no
no
737
725
722
717
721
721
0.041
0.027
0.025
0.022
0.025
0.025
no
no
no
no
no
no
497
498
499
500
Total
595
592
598
596
0.421
1.000
0.186
0.296
no
yes
no
no
47 (0.094 > 0.05)
717
725
736
728
0.022
0.027
0.039
0.030
no
no
no
no
17 (0.034  0.05)
From Table 4.2 we can find that:
(1) For these two new samples, the radiuses of sample kernel sets are 0.624 and 0.658
respectively. This information seems insufficient if we want to classify them by
using the traditional identification method.
(2) For the new sample (I), the observations which belongs to the sample kernel set
with respect to total observations is 0.094 (=47/500), which is larger than 0.05.
By the radius of sample kernel set and the ratio 0.094, we can say that the patient
has the same features of dead patients.
From Table 4.1, we obtain the significant level  D = 0.05. Under this
condition, we find that the number of observations which belongs to the sample
kernel set with respect to the number of Ker ( Di 2 ) is lager than 0.95, i  3,4 .
By the decision rule, the RRI of this patient will be identified into the learned dead
pattern of RRI.
(3) Similarly, for new sample (II), the observations which belongs to the sample
kernel set with respect to total observations is 0.034(=17/500), which is smaller
than 0.05. By the radius of sample kernel set and the ratio 0.034, we can say that
the patient has the same features of survival patients.
Under the significant level  D = 0.05, we find that the number of observations
which belongs to the sample kernel set with respect to the number of Ker(S 22 ) is
lager than 0.95. By the decision rule, the RRI of this patient will be identified into
the learned survival pattern of RRI.
Conclusion
In the medical science analysis discussed above the time series data have the
uncertain property. If we use the conventional clustering methods to analyze these
data, it will not solve the orientation problem. The contribution of this paper is that it
provides a new method to cluster and identify time series. In this paper, we presented
a procedure that can effectively cluster nonlinear time series into several patterns
based on kernel set. The proposed algorithm also combines with the concept of a
fuzzy set. We have demonstrated how to find a kernel set to help to cluster nonlinear
time series into several patterns.
Our algorithm is highly recommended practically for clustering nonlinear time
series and is supported by the empirical results. A major advantage of this framework
is that our procedure does not require any initial knowledge about the structure in the
data and can take full advantage of much more detailed information for some
ambiguity.
However, certain challenging problems still remain open, such as:
(1) Since hardly ever any disturbance or noise in the data set can be completely
eliminated, therefore, for the case of interacting noise, the complexity of
multivariate filtering problems still remains to be solved.
(2) The convergence of the algorithm for clustering and the proposed statistics have
not been well proved, although the algorithms and the proposed statistics are
known as fuzzy criteria. This needs further investigation.
Although there remain many problems to be overcome, we think fuzzy statistical
methods will be a worthwhile approach and will stimulate more future empirical work
in time series analysis.