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2016 International Conference on Mathematical, Computational and Statistical Sciences and Engineering (MCSSE 2016) ISBN: 978-1-60595-396-0

Preliminary Theory of Set SDR of Fuzzy Time Series Forecasting Model

Hong-xu WANG

1

and Zhen-xing WU

2

1College of China-Austria, Hainan Tropical Ocean University, Hainan, Sanya, China, 572022 2College of Computer Engineering, Hainan Tropical Ocean University, Hainan, Sanya, China,

572022

Keywords: Fuzzy time series forecasting method, SDR, Difference rate, Suitable prediction model, Forecasting function SDR().

Abstract. In the past 23 years, many fuzzy time series forecasting models have studied the forecasting problem for registration number of University of Alabama. But the prediction accuracy of existing forecasting models is not ideal. This paper proposes set SDR in the differential rate domain of Fuzzy time series forecasting model and proves Any Accuracy Theorem of SDR. It makes this problem get initial solution. Introducing appropriate definitions, SDR() (≤0.004) is the standard fuzzy time series forecasting model, and it can be used for general research of time series prediction.

Introduction

Due to the extremely extensive application, the complexity and diversity of objective things, it makes the forecast model of time series forecasting is in constant innovation, new methods are constantly emerging. Particularly in 1993, Song et al. [1-3] introduced fuzzy set theory into time series prediction problem and proposed the first fuzzy time series forecasting model[4] and firstly used it to study the enrollment prediction of Alabama University from 1971 to 1992. So far, the fuzzy time series forecasting model with different ideas has emerged as the times require. In 2007, Jilani et al. [5] proposed the inverse fuzzy number firstly and built a forecasting model based on it for the study of fuzzy time series model. In 2009, Stevenson et al [6] proposed an improved fuzzy time series forecasting model. Especially in 2012, Saxena et al [5] proposed an improved fuzzy time series forecasting model based on inverse fuzzy number, when it is used to study at the number of registered prediction problem in the university of Alabama, the mean square error and the average forecasting error rate of their model are MSE=9169 and AFER=0.3406%, they provided the smallest AFER and MSE than the existing methods (until 2012) [13].The biggest characteristic of the three prediction models is application of the concept of inverse fuzzy number. Feng et al[8] and Wang et al [9-14] of the presented model respectively, are also used inverse fuzzy number concept, are all improved model prediction model is proposed for literature[5-7], which makes respectively applied to the number of registered at the university of Alabama at most have certain to improve prediction accuracy of forecasting problems. For example Feng et al[8] proposed method gets higher prediction accuracy of AFER=0.0099% and MSE=7.In this paper, it further promotes the literature [5-14] prediction model, builds a set of elements in fuzzy time series prediction method, it’s called a set of fuzzy time series forecasting model based on difference rate. SFTSFMBDR (The Set of Fuzzy Time Series Forecasting Models Based on the Difference Rate), further simplifies the abbreviation for SDR and preliminary studies of the basic theory of SDR.

Basic Concepts

Use the following basic concepts and equations.

Definition 1 For time series prediction problem of history data A = {A1, A2, …, An}.we define

the difference rate as

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Definition 2 Use the following formula for inspection Prediction error: Bd –Ad;

Predictive variance: (Bd –Ad )2;

MSE (Mean Square Error): MSE = 2 1

1

( )

n

d d d B A n

  ; Prediction error rate: |Bd –Ad|/Ad;

AFER (Average Forecasting Error Rate):

1 1

/ n

d d d d

AFER B A A

n

 ,

where Bd is called the forecast number of d year, Ad is called historical data of d year.

Definition 3 For difference rate of history data E= {E2, E3, …, En}. We define inverse fuzzy

number as

1 1

, 1

d d

r X

r EE

 

constant r(0,1) is called membership number of element Ed-1, Also known as the membership

number of inverse fuzzy number X.

Preliminary Theoretical Research of Set SDR

Prediction Model of Set SDR

Definition 4 For time series prediction problem of history data A = {A1, A2, …, An}. Given

d∈{3,4,…,n} and difference rate E= { E2, E3, …, En}, we define the inverse fuzzy number function

as

1 1 ( )

1 d

d d X

E E

 

, (1)

where Ed-1 is the difference rate of history data of d-1- year, Ed is the difference rate of history data

of d- year, ∈(0,1) is called membership number of Ed-1, also it is the independent variable and

known as fuzzy numbers of the inverse function Xd().When  given a specific argument value,

inverse fuzzy number function Xd() is an inverse fuzzy number.

Definition 5 For given d∈{3,4,…,n}, time series prediction problem of history data A = {A1,

A2, …, An} and difference rate of history date E= { E1, E2, E3, …, En}, we define the fuzzy function

as

1 1

1 1

( ) 1 ( ) 1

1

d d d d

d d

B A X A

E E

 

 

   

 

 

, (2)

where Bd() is the fuzzy function of history data of d- year, ∈(0,1) is a independent variable , it

also called membership number of the fuzzy function. Where Ad is the history data of d-1- year, and

Ed-1 is the difference rate of history data of d-1- year, Ed is the difference rate of history data of d-

year.When given the independent variables of a specific value, the function can be called a prediction formula.

Theorem 1 (Convergence Theorem Based on Fuzzy Function of set SDR) For time series prediction problem of history data A = {A1, A2, …, An}, difference rate of history date E= { E2,

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0

limBd( ) Ad

   .

Proof For d∈{2,3,…,n}, according to the difference formula (1) we have

Ed = (Ad –Ad-1)/Ad-1  Ad = Ed Ad-1 +Ad-1 = Ad-1 (1+ Ed).

Letting  0, we get

1 1

0 0 0

1 1

0

1 1

0 1

1 1

lim ( ) lim 1 1 lim

1 1

lim 1

1 = (1 )

1 lim

d d d

d d d d

d d d d

d d

B A A

E E E E

A A E A

E E

  

 

 

 

  

 

 

 

   

 

   

   

 

   

   

 

 

 

 

 

Set up the Set SDR of Fuzzy Time Series Forecasting Model

When determining the membership number (0,1) as a specific numerical value, we can apply SDR () to study time series analysis. The application procedure is as follows.

1). Establishing historical data table for time series prediction problem;

2). Establishing theory of domain system (include domain of history data A, difference rate of history date E);

3). Writing prediction formula SDR();

4). Applying SDR() to compute the prediction value of history data;

Theorem 2 When determining the membership number  (0,1) as a specific numerical value,

the prediction formula SDR() is also a forecasting model of time series. The prediction formula of the model is SDR().

Set of Forecasting Model of Fuzzy Time Series SDR

Definition 6 If  takes all value in (0,1), we can obtain a system of time series forecasting model SDR (). The sum of system of time series forecasting model SDR () is called set of fuzzy time series forecasting model based on difference rate, for short, it is written as SFTSFMBDR (The Set of Fuzzy Time Series Forecasting Models Based on the Difference Rate). Its general element is SDR(). SDR () denotes a forecasting formula of fuzzy time series and a fuzzy time series forecasting model.

Remark SDR is a collection of time series forecasting model, Since Theorem 1 is the theory foundation of time series forecasting model and Theorem 2 depend on inverse fuzzy function (1), so we call the time series forecasting model as fuzzy time series forecasting model.

Any Accuracy Theorem of SDR

Theorem 3 (Any Accuracy Theorem) Let A = {A1, A2, …, An} be time series prediction problem

of history data, E= { E2, E3, …, En} be difference rate of history date. For a given arbitrarily small

positive number p and any positive number q, then for every 0(0,1), when prediction model

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3

1

( ) / 2

n

d d d

d

AFER B A A p

n  

  

and

2 3

1

( ( ) ) 2

n

d d

d

MSE B A q

n  

  

Proof It is not difficult to apply the SDR prediction function convergence theorem to prove this theorem.

Reasoning 1 Forecasts of the registered number of historical data for theory domain A = {A1971=13055, A1972=13563, …, A1992=18876}[1-3] at the University of Alabama in 1971 ~ 1992,

difference rate of history date E= { E1972=0.0389, E1973=0.0224, …, E1992= -0.0238}. Then for every

d∈{2,3,…,n}, p=0.3406%,q=9169. There are independent variables 0(0,1), when prediction

model SDR for the time series prediction problem of simulations to predict the historical data,it can ensure that the average prediction error rate AFERp and mean square error MSEq was established at the same time.

Proof This is the direct inference of Any Accuracy Theorem.

For example, in the study of the number of registered prediction problems at the University of Alabama in 1971 ~ 1992, if required AFER < 0.3406% and MSE < 9169 formed at the same time. For the variable0=0.004(0,1), when use the forecast model SDR(0.004) of set SDR to research

this problem, it would get AFER=0.0841%<0.3406% and MSE=1382<9169 at the same time. But the prediction accuracy of prediction model of fuzzy time series forecasting models of the set SDR is not all higher, some of the forecasting model is standard, accuracy is high; There are also many forecasting model is not standard, accuracy is not high. It needs specific conditions to measure if predictive model is standard. Saxena etc.[7] proposed a forecasting model of fuzzy time series in 2012 and get very good results, i.e., Mean Square Error MSE=9169 and AFER=0.3406% to predict the enrollment number of Alabama university in 1971~1992 year, so those data are used to measure the criterion of good forecasting model of fuzzy time series. With these data as a condition to distinguish if a fuzzy time series forecasting model is standard. Therefore, we establish the following definition.

Definition 7 A forecasting model of fuzzy time series is called standard, it was noted above that, if we use it to predict the enrollment number of Alabama University in 1971~1992 year, we can get AFER=0.0841% and MSE=1382, So the SDR (0.004) is also the standard fuzzy time series forecasting model.

Conclusion

The structure of each prediction formula of forecast model of set SDR of Fuzzy time series prediction model is simple and convenient. Convergence Theorem Based on Fuzzy Function of set SDR (Theorem 1) is the theory basis of the set SDR of fuzzy time series forecasting model. Any Accuracy Theorem (Theorem 3) preliminary solved the problem of that the existing fuzzy time series forecasting model prediction accuracy is not high. SDR() (≤0.004) is the standard fuzzy time series forecasting model, and it can be used for general research of time series prediction. The task in future will focus on studying how to develop the predicted value of the SDR() for the unknown data prediction method.

Acknowledgement

Fund projects: Sanya Institute of science and technology cooperation projects (2015YD36, 2014YD10) Natural science fund project of Hainan province (20166225).

References

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[2] Q Song, B S Chissom. Forecasting enrollments with fuzzy time series-Part I. Fuzzy Sets and Systems, Vol.54, pp. 1-9, 1993.

[3] Q Song, B S Chissom. Forecasting enrollments with fuzzy time series-Part II. Fuzzy Sets and Systems, Vol.62, pp. 1-8, 1994.

[4] L A Zadeh. Fuzzy set [J]. Fuzzy Sets and Systems, 8: 338-353, 1965.

[5] T A Jilani, S M A Burney, C Ardil. Fuzzy metric approach for fuzzy time series forecasting based on frequency density based partitioning. Proceedings of World Academy of Science, Engineering and Technology, Vol. 34, pp, 333-338, 2007.

[6] Meredith Stevenson and John E. Porter. Fuzzy time series forecasting using percentage change as the universe of discourse. Proceedings of World Academy of Science, Engineering and Technology, Vol. 55, pp, 154-157, 2009.

[7] Preetika Saxena, Kalyani Sharma, Santhosh Easo. Forecasting enrollments based on fuzzy time series with higher forecast accuracy rate. Int. J. Computer Technology& Applications, Vol.3 (3), pp, 957-961, 2012.

[8] Hao Feng, Jianchun Guo, Hongxu Wang*, Fujin Zhang. A modified method of foreecasting enrollments based on fuzzy time series [C]. 2014 2nd International Conference on Soft Computing in Information Communication Technology (SCICT2014), The Authors- Published by Atlantis Press, pp. 176-179, 2014.

[9] Hongxu Wang#, Jianchun Guo, Hui Wang, Hao Feng*. A fuzzy time series forecasting model based on yearly difference of the student enrollment number [C]. 2014 2nd International Conference on Soft Computing in Information Communication Technology (SCICT2014), The Authors- Published by Atlantis Press, pp. 172-175, 2014.

References

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