Journal of Data Science 12(2014), 143-155
High Order Lambda Measure Based Choquet Integral Composition
Forecasting Model
Hsiang-Chuan Liu1, Wei-Sung Chen2, Ben-Chang Shia3, Chia-Chen Lee4,
Shang-Ling Ou5, Yih-Chang Ou6, Chih-Hsiung Su7
1
Department of Biomedical Informatics, Asia University
Department of Computer Science and Information Engineering, Asia University
3
Department of Statistics and Information Science, Fu Jen Catholic University
4
Department of Statistics, Xiamen University
5
Department of Agronomy, National Chung Hsing University
6
Department of Finance, Ling Tung University, Taichung
7
Department of Accounting Informatiot, ChihLee Institute of Technology
2
Abstract: In this paper, a novel fuzzy measure, high order lambda measure, was
proposed, based on the Choquet integral with respect to this new measure, a novel
composition forecasting model which composed the GM(1,1) forecasting model,
the time series model and the exponential smoothing model was also proposed. For
evaluating the efficiency of this improved composition forecasting model, an
experiment with a real data by using the 5 fold cross validation mean square error
was conducted. The performances of Choquet integral composition forecasting
model with the P-measure, Lambda-measure, L-measure and high order lambda
measure, respectively, a ridge regression composition forecasting model and a
multiple linear regression composition forecasting model and the traditional linear
weighted composition forecasting model were compared. The experimental results
showed that the Choquet integral composition forecasting model with respect to the
high order lambda measure has the best performance.
Key words: Composition forecasting model, fuzzy measure, Choquet integral,
extensional lambda measure, high order extensional lambda measure
1. Introduction and motivation
The well-known composition functions of forecasting model are linear, in our previous
works, the non-linear composition forecasting model were considered by using Choquet integral
with respect to some fuzzy measures 4. For any given fuzzy density, both Zadeh’s P-measure and
Sugeno’s λ-measure are univalent fuzzy measures with only one formulaic fuzzy measure
solution, a multivalent fuzzy measure with infinite formulaic fuzzy ……
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High Order Lambda Measure Based Choquet Integral Composition Forecasting Model
Fuzzy measures
Definition 1. Fuzzy measure, A measure  on a finite set X  x1 , x2 ,..., xn  is called a
fuzzy measure, denoted  - measure, if its measure function g : 2 X  0,1 satisfies following
conditions:
(i) g    0, g  X   1
(ii) A, B  X , A  B  g  A  g  B 
(1)
2. Experiment and Results
A real data was obtained from the paper of Zhang, Wang and Gao. For evaluating
the proposed new density based composition forecasting model, an experiment by using
the 5 fold cross validation mean square error was conducted. The performances of
Choquet integral composition forecasting model with high order λ-measure, L-measure,
λ-measure and P-measure, respectively, a ridge regression composition forecasting
model and a multiple linear regression composition forecasting model and the traditional
linear weighted composition forecasting model were compared. The result is listed in
Table 1, which shows that the composition forecasting model based on Choquet integral
with respect to L -measure and O-density has the best performance
Table 1: MSEs of 7 Composition forecasting models
Composition forecasting models
Choquet Integral
Composition forecasting
model
with O-density
5-fold CV MSE
L –measure
43012.03
L –measure
43319.99
λ-measure
44504.19
P –measure
43727.90
Ridge forecasting model
48109.21
Multiple forecasting model
57095.67
H. C. Liu, W. S. Chen, B. C. Shia, C. C. Lee, S. L. Ou, Y. C. Ou, C. H. Su
Figure 1: CRISP-DM
3. Summary
In this paper, based on high order λ-measure and O-measure, a novel Choquet
integral composition forecasting model is proposed. The experiment with a real data by
using the 5 fold cross validation mean square error shows that the new model is better
than others.
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High Order Lambda Measure Based Choquet Integral Composition Forecasting Model
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Received December 12, 2013; accepted July 26, 2014.
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High Order Lambda Measure Based Choquet Integral Composition Forecasting Model
Hsiang-Chuan Liu
Department of Biomedical Informatics
Asia University
Taichung 41354, Taiwan, ROC.
Wei-Sung Chen
Department of Computer Science and Information Engineering
Asia University
Taichung, 41354, Taiwan.
Ben-Chang Shia
Department of Statistics and Information Science
Fu Jen Catholic University
New Taipei City, 24205, Taiwan.
Chia-Chen Lee
Department of Statistics
Xiamen University
China.
Shang-Ling Ou
Department of Agronomy
National Chung Hsing University
Taichung, 40227, Taiwan.
Yih-Chang Ou
Department of Finance
Ling Tung University
Taichung, 40852, Taiwan.
Chih-Hsiung Su
Department of Accounting Informatiot
ChihLee Institute of Technology
New Taipei City, 22050, Taiwan.