academic_74634_6760986_34783

UNCORRECTEDPROOF

PHYSA: 11049 +Model pp. 113 (col. fig: NIL)

ARTICLE IN PRESS

Physica A xx (xxxx) xxxxxxwww.elsevier.com/locate/physa

High-order fuzzy time-series based on multi-period adaptationmodel for forecasting stock markets

Tai-Liang Chena,, Ching-Hsue Chenga, Hia-Jong Teoha,b

aDepartment of Information Management, National Yunlin University of Science and Technology, 123, Section 3, University Road, Touliu,Yunlin 640, Taiwan, ROC

bDepartment of Accounting Information, Ling Tung University, 1, Ling Tung Road, Nantun, Taichung 408, Taiwan, ROC

Received 30 May 2007; received in revised form 5 September 2007

12

Abstract 3

Stock investors usually make their short-term investment decisions according to recent stock information such as late market 4news, yesterdays technical analysis reports, and the price fluctuations in these two days. To reflect these short-term factors which 5impact stock price, this paper proposes a comprehensive fuzzy time-series, which factors linear relationships between recent 6periods of stock prices and fuzzy logical relationships (nonlinear relationships) mined from time-series into forecasting processes. 7In empirical analysis, the TAIEX (Taiwan Stock Exchange Capitalization Weighted Stock Index) and HSI (Heng Seng Index) 8are employed as experimental datasets, and four recent fuzzy time-series models, Chens (1996), Yus (2005), Chengs (2006) 9and Chens (2007), are used as comparison models. Besides, to compare with conventional statistic method, the method of least 10squares is utilized to estimate the auto-regressive models of the testing periods within the databases. From analysis results, the 11performance comparisons indicate that the multi-period adaptation model, proposed in this paper, can effectively improve the 12forecasting performance of conventional fuzzy time-series models which only factor fuzzy logical relationships in forecasting 13processes. From the empirical study, the traditional statistic method and the proposed model both reveal that stock price patterns in 14the Taiwan stock and Hong Kong stock markets are short-term. 15c 2007 Published by Elsevier B.V. 16Keywords: High-order fuzzy time-series; Multi-period adaptation model; Stock index forecasting

17

1. Introduction 18

Time-series models have utilized the fuzzy theory to solve various domain forecasting problems, such as university 19enrollment forecasting [14], stock price forecasting [514] and temperature forecasting [15]. In the area of stock 20price forecasting, Huarng (2001) provided heuristic models [5] from stock price time-series to improve forecasting 21performance by integrating problem-specific heuristic knowledge with Chens (1996) model, which was proposed 22to forecasting university enrollment. In the following research, an N th-order heuristic fuzzy time-series model was 23proposed by Huarng (2003) to forecasting the TAIEX [7]. Additionally, the researcher has found that the length of 24

Corresponding author. Tel.: +886 920975168.E-mail address: [email protected] (T.-L. Chen).

0378-4371/$ - see front matter c 2007 Published by Elsevier B.V.doi:10.1016/j.physa.2007.10.004

Please cite this article in press as: T.-L. Chen, et al., High-order fuzzy time-series based on multi-period adaptation model for forecasting stockmarkets, Physica A (2007), doi:10.1016/j.physa.2007.10.004

UNCORRECTEDPROOF

PHYSA: 11049

ARTICLE IN PRESS2 T.-L. Chen et al. / Physica A xx (xxxx) xxxxxx

intervals for the universe of discourse, will affect forecasting results and proposed two linguistic interval partitioning1approaches, distribution-based and average-based length, to approach this issue [6].2

Another researcher, Yu (2005), proposed a weighted method to forecasting the TAIEX to tackle two issues,3recurrence and weighting, in fuzzy time-series forecasting [10]. Yu argued that recurrent fuzzy relationships, which4were simply ignored in Chens (1996) studies, should be considered in forecasting and recommended that different5weights should be assigned to various fuzzy relationships. In the following research, a trend-weighted model [12] was6proposed by Cheng (2006) to echo Yus research.7

From the literature above, mining fuzzy logical relationships (FLR) from time-series is considered as one of the8important factors influencing the forecasting accuracy of fuzzy time-series models. Therefore, in recent research,9some advanced algorithms such as genetic algorithms [4] and neural networks [9] are applied to improve this process.10Chen (2006) proposed a new method, by using genetic algorithms, to deal with the university enrollment forecasting11problems based on high-order fuzzy time-series, where the length of each interval in the universe of discourse is12tuned [4]. Besides, Yu (2006) applied a backpropagation neural network to handle nonlinear forecasting problems in13stock price forecasting [9].Two models, a basic model and a hybrid model, using a neural network approach were14proposed to forecast the TAIEX.15

Although these fuzzy time-series models using advanced algorithms have made great improvements on forecasting16performance, only nonlinear relationships such as fuzzy logical relationships are most concerned in forecasting and17the process of mining fuzzy logical relationships is not easily understandable just like a black box. Besides, stock18market investors usually make their short-term decisions based on the latest stock information such as yesterdays19market news and price fluctuations in these two days. Therefore, in stock price forecasting, we argue that the linear20relationships between recent periods of stock prices should be factored in forecasting models besides fuzzy logical21relationships [1214].22

In this paper, we propose a high-order fuzzy time-series model based on a multi-period adaptation model [16],23which is derived from the adaptive expectation model [17], to promote forecasting accuracy. In empirical analysis, we24employ three stock databases, TAIEX (Taiwan Stock Exchange Capitalization Weighted Stock Index), and HSI (Heng25Seng Index), from 1991 to 1999, as experimental datasets. And four recent fuzzy time-series models, Chens (1996),26Yus (2005), Chengs (2006), and Chens (2007), are employed as comparison models. The comparisons show that27the proposed model outperforms these conventional models, Chens (1996), and Yus (2005) which only mine fuzzy28logical relationships from time-series.29

Additionally, to verify the proposed model, we use a statistic method, the method of least squares [18], to estimate30auto-regressive models for the nine year of stock index within each stock database. The estimated results are employed31to check the time lags against the order recommended by the proposed model. This verification shows that they are32highly consistent.33

Based on the empirical analysis, two major conclusions are given: (1) the multi-period adaptation model can34effectively improve the forecasting performance of conventional fuzzy time-series; and (2) the price change patterns35in the two stock markets, TAIEX and HSI, are short-term.36

The remaining of this paper is organized as follows: Section 2 introduces fuzzy time-series model; Section 337introduces the proposed model and algorithm; Section 4 is empirical analysis and model comparisons; and, in the last38section, findings and concluding remarks are given.39

2. Fuzzy time-series40

Fuzzy theory, the modern concept of uncertainty, was introduced by Zadeh (1975) to deal with linguistic terms [194121]. The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of degree. Nowadays42fuzzy theory is vigorously studied in expert system, approximate reasoning, controls, pattern recognition, database,43and information retrieval systems, etc. Time-series models had failed to consider the application of this theory until44fuzzy time-series was defined by Song and Chissom [22,23]. The definitions and processes of the fuzzy time-series45presented by Song and Chissom (1993) are described as follows.46

Definition 1. Y (t) (t = . . . 0, 1, 2, . . .) is a subset of a real number. Let Y (t) be the universe of discourse defined47by the fuzzy set fi (t). If F(t) consists of Fi (t) (i = 1, 2, . . .), F(t) is defined as a fuzzy time-series on Y (t)48(t = . . . , 0, 1, 2, . . .).49Please cite this article in press as: T.-L. Chen, et al., High-order fuzzy time-series based on multi-period adaptation model for forecasting stockmarkets, Physica A (2007), doi:10.1016/j.physa.2007.10.004

UNCORRECTEDPROOF

PHYSA: 11049

ARTICLE IN PRESST.-L. Chen et al. / Physica A xx (xxxx) xxxxxx 3

Definition 2. If there exists a fuzzy logical relationship R(t 1, t), such that F(t) = F(t 1) R(t 1, t), where 1 represents an operation, then F(t) is said to be caused by F(t 1). The logical relationship between F(t) and 2F(t 1) is F(t 1) F(t). 3Definition 3. Let F(t 1) = Ai and F(t) = A j . The relationship between two consecutive observations, F(t) and 4F(t 1), referred to as a fuzzy logical relationship (FLR), can be denoted by Ai A j , where Ai is called the 5Left-Hand Side (LHS) and A j the Right-Hand Side (RHS) of the FLR. 6

Definition 4. All fuzzy logical relationships in the training dataset can be further grouped together into different fuzzy 7logical relationship groups according to the same Left-Hand Sides of the fuzzy logical relationship. For example, there 8are two fuzzy logical relationships with the same Left-Hand Side (Ai ): Ai A j1 and Ai A j2. These two fuzzy 9logical relationships can be grouped into a fuzzy logical relationship group. 10

Definition 5. Suppose F(t) is caused by F(t 1) only, and F(t) = F(t 1) R(t 1, t). For any t , if R(t 1, t) 11is independent of t , then F(t) is named a time-invariant fuzzy time-series, otherwise a time-variant fuzzy time-series. 12

Definition 6. Assume that F(t) is a fuzzy time-series and F(t) is caused by F(t 1), F(t 2), . . . , and F(t n), 13then the fuzzy logical relationship can be represented as follows: F(t 1), F(t 2), . . . , F(t n) F(t). This 14expression is called the nth-order fuzzy time-series forecasting model, where n = 2 [2,4]. 15

Song and Chissom employed six main procedures in time-invariant fuzzy time-series and time-variant fuzzy time- 16series models as follows: (1) define and partition the universe of discourse; (2) define fuzzy sets for the observations; 17(3) partition the intervals; (4) fuzzify the observations; (5) establish the fuzzy relationship and forecast; and (6) 18defuzzify the forecasting results. 19

3. Proposed model and algorithm 20

In fuzzy time-series models, mining fuzzy logical relationships (FLR) from time-series is the most critical process 21to influence forecasting accuracy [9]. In recent scientific research, artificial intelligence algorithms such as genetic 22algorithms and neural networks play an important role. To improve forecasting accuracy, these algorithms are also 23applied in fuzzy time-series models [4,9]. Although artificial intelligence algorithms perform well in forecasting, they 24only handle nonlinear relationships, which mean fuzzy logical relationships, in time-series. 25

Besides, as mentioned in the introduction section, recent stock information such as late market news and 26yesterdays price fluctuations are usually referenced for short-term decisions. And reasonable investors will modify 27their predictions with recent forecasting errors. Hence, a high-order fuzzy time-series model should be provided to 28meet the price patterns in history data such as Hurangs model (2003) [7]. 29

Based on these facts, we argue that a thoughtful fuzzy time-series model should consider two price patterns 30together in forecasting processes: (1) nonlinear high-order fuzzy logical relationships in historical data; and (2) linear 31relationships between recent periods of stock prices. Therefore, this paper proposes a high-order fuzzy time-series 32model (see Fig. 1) to implement this notion. 33

3.1. Linear forecasting model 34

In time-series research, the adaptive expectation model [17] is a reasonable forecast model to represent the 35prediction approach of stock investors for the future stock price, where the forecast is generated by the last one 36period of stock price and the correction for last one period of forecasting error. 37

In stock markets, practical experiences tell that investors usually make their decisions based on recent periods of 38stock prices. Based on this assumption, we can utilize recent periods of forecasting errors to modify the forecast for 39the future stock price. Therefore, we extend the adaptive expectation model to derive a multi-period adaptation model 40(defined in Eq. (1), where forecast (t + 1) is the prediction for the future price; P(t) is the present price; i is the 41previous i th period of forecasting error; and hi is the adaptation parameter for i ) [24]. 42

forecast(t + 1) = P(t)+k

i=1hi i . (1) 43


UNCORRECTEDPROOF

PHYSA: 11049


Fig. 1. Research process of the proposed model.

With the multi-period adaptation model model, we propose a comprehensive fuzzy time-series model which is capable1of mining nonlinear and linear relationships among stock price time-series.2

3.2. Proposed algorithm3

In this section, we provide stepwise computations and a numerical example to introduce the proposed algorithm as4follows.5

Step 1: Reasonably define the universe of discourse, U .6U = [Dmin D1, Dmax+ D2], where D1 and D2 are two proper positive numbers [1]. For example, the minimum7

and maximum of the TAIEX, in a training period, are 6430 and 7750, respectively. Then the universe of discourse8


UNCORRECTEDPROOF

PHYSA: 11049


Table 1Seven linguistic intervals for TAIEX

Linguistic interval Interval range

I1 [6400, 6600]I2 [6600, 6800]I3 [6800, 7000]I4 [7000, 7200]I5 [7200, 7400]I6 [7400, 7600]I7 [7600, 7800]

Table 2Assign a related linguistic value

Time Stock price Linguistic value

t = 1 6436 L1t = 2 6537 L1t = 3 6662 L2t = 4 6550 L1t = 5 6666 L2t = 6 6890 L3t = 7 6810 L3t = 8 7020 L4

can be defined as U = [6400, 7800], where D1 is 30 and D2 is 50. As a result, the defined universe of discourse, 1U = [6400, 7800], can cover every occurred stock index price in the training period.

Q1

2

Step 2: Partition U into several equal length of linguistic intervals. 3The number of interval is related to the count of linguistic values which are considered for the universe of discourse. 4

In Chens research [2002, 2006], different linguistic values, from 7 to 14, are employed to test his models and 5different forecasting performances were produced. Chens models showed that more linguistic values result in better 6performance for the enrollment dataset (the enrollments of the University of Alabama) which is only used as training 7dataset not testing. In this paper, seven different linguistic values, from 7 to 13, are taken to evaluate the proposed 8model. Take seven linguistic values as example, the linguistic intervals for these linguistic values, which partition the 9universe of discourse, U = [6400, 7800], are listed in Table 1. 10

Step 3: Establish a related fuzzy set for each observation in the training dataset. 11Firstly, define the fuzzy set, L1, L2, . . . , Lk , based on the linguistic intervals by Eq. (2).The value of ai j indicates 12

the grade of membership of u j in fuzzy set L i , where u j is a triangle fuzzy number with a linguistic interval, Ik , and 13ai j [0, 1] (1 i k and 1 j m). Q2 14

L1 = a11/u1 + a12/u2 + + a1m/umL2 = a21/u1 + a22/u2 + + a2m/um...

Lk = ak1/u1 + ak2/u2 + + akm/um .

(2) 15

Secondly, find out the degree of each stock price belonging to each L i (i = 1, . . . , k). If the maximum membership 16of the stock price is under Lk , then the fuzzified stock price is labeled as Lk . 17

Lastly, convert each stock price in training dataset to corresponding linguistic values, Lk . In the next step, the fuzzy 18logical relationships are constructed based on the fuzzified stock price. 19

For example, seven fuzzy linguistic vales from Step 2 can be defined as follows: L1 = (very low price), 20L2 = (low price), L3 = (little low price), L4 = (normal price), L5 = (little high price), L6 = (high price) and 21L7 = (very high price) [1]. And Table 2 demonstrates how to classify eight periods of stock prices into corresponding 22linguistic vales. 23


UNCORRECTEDPROOF

PHYSA: 11049


Table 3First-order fuzzy logical relationship

Time Linguistic value First-order FLR Time Linguistic value

t = 1 L1 t = 2 L1t = 2 L1 t = 3 L2t = 3 L2 t = 4 L1t = 4 L1 t = 5 L2t = 5 L2 t = 6 L3t = 6 L3 t = 7 L3t = 7 L3 t = 8 L4t = 8 L4 N.A.

Table 4Second-order fuzzy logical relationship

Time Linguistic value Time Linguistic value Second-order FLR Time Linguistic value

t = 1 L1 t = 2 L1 t = 3 L2t = 2 L1 t = 3 L2 t = 4 L1t = 3 L2 t = 4 L1 t = 5 L2t = 4 L1 t = 5 L2 t = 6 L3t = 5 L2 t = 6 L3 t = 7 L3t = 6 L3 t = 7 L3 t = 8 L4t = 7 L3 t = 8 L4 N.A.t = 8 L4 N.A.

Table 5A fluctuation-type matrix for FLR groups (first order)

P(t 1) P(t)L1 L2 L3 L4 L5 L6 L7

L1 1 2 0 0 0 0 0L2 1 0 1 0 0 0 0L3 0 0 1 1 0 0 0L4 0 0 0 0 0 0 0L5 0 0 0 0 0 0 0L6 0 0 0 0 0 0 0L7 0 0 0 0 0 0 0

Step 4: Establish i th-order fuzzy logic relationship (FLR) for time-series.1First-order fuzzy logical relationship is composed of two consecutive linguistic values; second-order FLR is2

composed of three consecutive linguistic values; and i th-order fuzzy logical relationship is composed of i + 13consecutive linguistic values. For example, Tables 3 and 4 demonstrate how to establish first-and second-order fuzzy4logical relationships based on Table 2.5

Step 5: Establish FLR groups and assign a frequency weight for each FLR.6The FLR with the same LHS (Left-Hand Side) linguistic value can be grouped into one FLR group. All FLR groups7

will construct a fluctuation-type matrix. Tables 5 and 6 show the fluctuation-type matrix produced by the FLR from8Tables 3 and 4. Each row of the matrix represents one FLR group and each cell represents the occurrence frequency9of each FLR. Each FLR within the same FLR group should be assigned a weight. For example, in Table 4, the FLR10group of L1 is L1 L1, L2. The FLR of L1 L1 occurs once and the weight is assigned 1. The FLR of L1 L211occurs twice and the weight is assigned 2.12

In this paper, a frequency-weighted method, in which each FLR weight is determined by its occurrence frequency,13is employed. The sum of the weight of each FLR will be normalized to obtain a frequency-weighted matrix,W (L i , t),14


UNCORRECTEDPROOF

PHYSA: 11049


Table 6A fluctuation-type matrix for FLR groups (second order)

P(t 2), P(t 1) P(t)L1 L2 L3 L4 L5 L6 L7

L1, L1 1 2 0 0 0 0 0L1, L2 1 0 1 0 0 0 0L1, L3 0 0 1 1 0 0 0 L7, L5 0 0 0 0 0 0 0L7, L6 0 0 0 0 0 0 0L7, L7 0 0 0 0 0 0 0

which is defined in Eq. (3). 1

W (L i , t) =[W 1,W 2, . . . ,W k

] = W1k

i=1Wi

,W2k

i=1Wi

, . . . ,Wkk

i=1Wi

. (3) 2For example, the frequency-weighted matrix for the FLR group of L1 at t is specified as follows: W1 = 1/3, 3

W2 = 2/3, W3 = 0, W4 = 0, W5 = 0, W6 = 0 and W7 = 0. 4Step 6: Compute initial linguistic forecasts and defuzzify the forecasts to a numeric forecast. Therefore there are 5

two sub-procedures in this step as follows. 6

Step 6-1: Produce linguistic forecasts based on the rules of FLR groups. 7Take Table 5 as example, if a linguistic stock price is L1, which fit for the rule in the first row of Table 5, then 8

the linguistic forecasts for the future stock price will be L1 and L2, and the frequency-weighted matrix for these two 9linguistic forecasts is [1/3, 2/3, 0, 0, 0, 0, 0]. If the linguistic stock price is not found in the rules of FLR groups, the 10numeric stock price of L1 is employed as the forecast for the future stock price. 11

Step 6-2: Defuzzify the linguistic forecasts to generate a numeric forecasted output [1]. 12This procedure is called defuzzification, defined in Eq. (4), where Ld f is the defuzzified matrix, which is 13

composed of each midpoint of linguistic interval and W T (L i , t) is the transformed matrix of the frequency-weighted 14matrix from Step 5. 15

D(t) = Ld f W T (L i , t). (4) 16For example, if the present stock price is L1, which fit for the rule in the first row of Table 5, then the transformed 17

matrix of the frequency-weighted matrix is [1/3, 2/3, 0, 0, 0, 0, 0]T . From Table 1, the midpoint of I1 is 6500, I2 is 186700, I3 is 6900, I4 is 7100, I5 is 7300, I6 is 7500, and I7 is 7700. And, therefore, the defuzzified matrix, Ld f , is 19[6500, 6700, 6900, 7100, 7300, 7500, 7700]. 20

From the defuzzification equation (4), [6500, 6700, 6900, 7100, 7300, 7500, 7700] [1/3, 2/3, 0, 0, 0, 0, 0]T is 21computed to get the defuzzified result, 6633. Table 7 shows some examples of defuzzification using the data from 22Tables 1 and 5. 23

Step 7: Use the multi-period adaptation model to produce forecasts. 24The multi-period adaptation model is defined in Eq. (5), where Forecast(t + 1) is the conclusive forecast for the 25

future stock price; P(t) is the present stock price on time t ; i is the i th forecasting error (where = D(t) P(t)) 26and; hi is a linear parameter for the i th period of forecasting error, i . 27

Forecast(t + 1) = P(t)+k

i=1hi i . (5) 28

The linear parameters, hi , range from1 to 1 but 0 with the stepped value, 0.001, to adapt the forecasts to reach the 29best forecasting performance in training datasets. And the determined parameters from the training datasets are taken 30


UNCORRECTEDPROOF

PHYSA: 11049


Table 7Examples of defuzzification (first order)

Time Stock price Linguistic value Ld f W T (L i , t) D(t)t = 1 6436 L1 [6500, 6700, 6900, 7100, 7300, 7500, 7700] [1/3, 2/3, 0, 0, 0, 0, 0]T 6633t = 2 6537 L1 [6500, 6700, 6900, 7100, 7300, 7500, 7700] [1/3, 2/3, 0, 0, 0, 0, 0]T 6633t = 3 6662 L2 [6500, 6700, 6900, 7100, 7300, 7500, 7700] [1/2, 0, 1/2, 0, 0, 0, 0]T 6700t = 4 6550 L1 [6500, 6700, 6900, 7100, 7300, 7500, 7700] [1/3, 2/3, 0, 0, 0, 0, 0]T 6633t = 5 6666 L2 [6500, 6700, 6900, 7100, 7300, 7500, 7700] [1/2, 0, 1/2, 0, 0, 0, 0]T 6700t = 6 6890 L3 [6500, 6700, 6900, 7100, 7300, 7500, 7700] [0, 0, 1/2, 1/2, 0, 0, 0]T 7000t = 7 6810 L3 [6500, 6700, 6900, 7100, 7300, 7500, 7700] [0, 0, 1/2, 1/2, 0, 0, 0]T 7000t = 8 7020 L4 No rules

Table 8Examples of forecasting processes for adaptive forecasts (first order)

Time P(t) Linguisticvalue

D(t) =D(t) P(t)

Forecast (t + 1)

1 adaptationperiod

2 adaptation periods 3 adaptation periods

t = 1 6436 L1 6633 197 N.A. N.A. N.A.t = 2 6537 L1 6633 96 6436+ h1 197 N.A. N.A.t = 3 6662 L2 6700 38 6537+ h1 96 6537+ h1 96+ h2 197 N.A.t = 4 6550 L1 6633 83 6662+ h1 38 6662+ h1 38+ h2 96 6662+ h1 38+ h2 96+ h3 197t = 5 6666 L2 6700 34 6550+ h1 83 6550+ h1 83+ h2 38 6550+ h1 83+ h2 38+ h3 96t = 6 6890 L3 7000 110 6666+ h1 34 6666+ h1 34+ h2 83 6666+ h1 34+ h2 83+ h3 38t = 7 6810 L3 7000 190 6890+ h1 110 6890+ h1 110+ h2 34 6890+ h1 110+ h2 34+ h3 93t = 8 7020 L4 N.A. N.A. 6436+ h1 190 6436+h1 190+h2 110 6436+ h1 190+ h2 110+ h3 34

to adapt the forecasts for testing datasets. Table 8 demonstrates how to produce multi-period adaptations of forecasts1using the data from Tables 2, 5 and 7.2

4. Empirical analysis3

This section provides performance evaluation, comparison and model. The empirical databases to verify the4proposed model are two stock indexes as follows: TAIEX (Taiwan Stock Exchange Capitalization Weighted Stock5Index) and HSI (Heng Seng Index).6

A nine year period of each stock index, from 1991 to 1999, is selected as experimental datasets to evaluate the7forecasting performance of the proposed model. Each year of the selected stock index is split into two subsets, a8training dataset and a testing dataset. The previous ten months of one year, from January to October, is used for9training, and the last two months, November to December, is for testing. Each stock index database contains over102500 objects with one attribute, closing price of stock index.11

To measure forecasting performance of the proposed model, we employ the RMSE (Root Mean Squared Error) as12a performance indicator (defined in Eq. (6), where actual (t) is the actual trading stock price on time t ; forecast(t) is13the forecasting value for actual (t); and n is the number of times for forecasts).14

RMSE =

nt=1(actual(t) forecast(t))2n

. (6)15

4.1. Performance evaluation16

To evaluate the proposed model, various forecasting performances using different orders, from 1st order to 4th17order, and adaptation periods, form 1 to 4 periods, are produced. Tables 9 and 10 list the forecasting performances18


UNCORRECTEDPROOF

PHYSA: 11049


Table 9Forecasting performance for the TAIEX (testing datasets)

Proposed model YearOrder Adaptation period 1991 1992 1993 1994 1995 1996 1997 1998 1999

0 99.08 81.13 126.55 110.69 105.57 76.56 158.89 154.08 1331 43.44 43.81 105.58a 75.71a 54.99a 51.05 134.41 115.14 103.86

1 2 44.24 43.32 108.41 75.79 55.55 50.93a 137.32 114.58 102.02a

3 38.18a 41.78 109.59 76.42 58.28 51.63 135 115.63 102.134 38.66 41.2a 110.9 78.15 59.17 52.67 132.48 116.54 103.33

0 100.69 86.41 125.31 112.39 102.96 70.16 160 149.66 141.911 44.56 45.72 108.12 79.34 55.84 51.1 136.31 113.65a 104.06

2 2 38.9 43.38 110.44 79.79 56.41 51.66 133.25 114.97 104.253 41.49 43.49 111.56 80.82 58.34 51.78 131.15 115.88 105.814 42.88 44.41 112.74 85.78 58.88 54.55 128.89a 116.51 105.92

0 130.97 89.3 136.47 119.65 103.64 70.19 184.56 153.22 149.341 56.92 45.07 113.01 87.69 59.1 53.36 135.66 115.8 112.16

3 2 58.03 44.05 114.19 88.64 59.68 53.09 135.54 116.58 114.593 61.97 44.22 115.33 88.48 60.93 54.01 131.66 117.64 115.494 68.65 44.86 117.49 91.19 62.37 56.05 130.62 119.08 117.01

0 130.58 103.16 132.09 128.09 117.53 71.42 206.64 167.99 190.771 66.12 52.38 113.61 97.14 62.98 54.31 143.24 121.31 136.27

4 2 66.96 52.59 114.67 96.79 63.04 54.9 138.35 122.63 135.263 63.28 52.97 115.97 97.95 63.78 56.46 139.65 124.16 137.634 64.88 53.87 117.8 100.27 67.53 57.94 141.18 122.94 139.14

a Minimum RMSE.

Table 10Forecasting performance for the HSI (testing datasets)

Proposed model YearOrder Adaptation period 1991 1992 1993 1994 1995 1996 1997 1998 1999

0 53.04 182.56 266.12 202.16 127.52 147.30 534.82 296.65 295.591 39.75 131.49 208.43 146.47a 76.31 144.32 267.97a 204.43 238.73

1 2 39.01 131.32a 206.05 152.94 76.21 147.06 274.11 205.83 244.703 39.57 132.38 209.41 154.51 76.97 139.60a 277.26 203.70a 239.754 40.01 138.72 211.52 162.94 74.58a 141.62 278.70 205.67 242.90

0 52.21 190.71 219.74 205.85 129.62 148.20 465.22 310.00 296.111 38.63a 133.78 205.07a 158.82 79.82 146.82 278.67 216.57 245.88

2 2 39.45 135.24 206.75 160.51 81.83 139.65 282.32 215.40 238.50a

3 40.66 136.67 207.38 162.59 80.03 141.52 280.94 217.39 243.664 40.30 143.96 208.36 170.60 80.88 143.45 282.12 211.39 247.60

0 48.28 216.62 217.71 227.71 128.21 149.87 431.08 333.08 295.501 39.18 144.28 206.88 174.34 82.68 139.69 292.96 247.29 240.75

3 2 40.62 145.88 207.22 176.74 79.92 141.46 291.75 249.23 243.913 40.96 147.72 208.38 179.25 80.42 143.44 292.21 243.55 248.824 41.56 154.14 210.91 189.12 79.74 145.49 294.19 243.09 254.46

0 48.49 178.25 220.05 219.22 146.20 139.69 399.76 328.65 292.561 40.61 142.30 207.54 184.34 93.36 141.46 299.83 259.35 248.25

4 2 40.73 144.19 208.83 187.01 92.57 143.44 299.84 249.57 251.353 41.68 146.10 211.28 189.72 89.68 145.49 301.48 252.39 254.954 42.13 147.70 211.06 195.07 89.10 147.56 284.68 248.83 254.21

a Minimum RMSE.

for the empirical databases (Table 9 for the TAIEX and Table 10 for the HSI) by using 9 linguistic values to partition 1the universe of discourse defined in training datasets. It shows that, in the TAIEX, the forecasting models using the 2


UNCORRECTEDPROOF

PHYSA: 11049


Table 11The statistics of best performance for the TAIEX

Year 1991 1992 1993 1994 1995 1996 1997 1998 1999

Forecasting performance 38.18 41.2 105.58 75.71 54.99 50.93 128.89 113.65 102.02Order 1 1 1 1 1 1 2 2 1Adaptation periods 3 4 1 1 1 2 4 1 2

Table 12The statistics of best performance for the HSI

Year 1991 1992 1993 1994 1995 1996 1997 1998 1999

Forecasting performance 38.63 131.32 205.07 146.47 74.58 139.60 267.97 203.70 238.50Order 2 1 2 1 1 1 1 1 2Adaptation periods 1 2 1 1 4 3 1 3 2

Table 13Performance comparison table (TAIEX)

Models 1991 1992 1993 1994 1995 1996 1997 1998 1999 Sum of RMSE

Chens model [1] 80 60 110 112 79 54 148 167 149 959Yus model [10] 61 67 105a 135 70 54 133 151 142 918Chens model [13] 47 44 107 78 55a 50a 136 115 118 750Chengs model [12,14] 43 44 105a 76a 55a 50a 134 115 104 726Proposed model (optimal order & adaptation periods) 38a 41a 106 76a 55a 51 129a 114a 102a 712a

a The best performance among 5 approaches (Minimum RMSE).

adaptation model (from 1 to 4) perform better than the one without using (0). This result is also discovered in the HSI1database.2

The performance tables for the stock databases are summarized in Tables 11 and 12 (Table 11 for the TAIEX,3and Table 12 for the HSI), which list the different orders and adaptation period under the best performance for the4nine experimental datasets. Take Table 11 as explanation example, among 4 forecasting models with different order,51st-order model takes the first place of forecasting performance (performs best in 7 testing periods, 19911996 and61999). While among 4 forecasting models using different adaptation periods, one adaptation period model take the7first place (performs best in 5 testing periods, 1993, 1994, 1995, 1998, and 1999). From these summarization tables,8Tables 11 and 12, it is found that higher-order models are not sequential to perform better lower-order models.9

4.2. Performance comparison10

To examine the improvement in performance, four fuzzy time-series models, Chens (1996), Yus (2005), Chengs11(2006) and Chens (2007) models, are employed as comparison models. Because the optimal linguistic values for12the proposed model are 9, we reduplicate the algorithms from the literature [1,10,1214] using 9 linguistic values to13produce performance comparison tables for the empirical databases shown in Tables 13 and 14.14

The comparison table for the TAIEX shows that the proposed model (using 9 linguistic values) performs best in154 testing datasets (1991, 1992, 1997 and 1999) and bears the smallest value of the sum of RMSE. However, in the16HSI, the proposed model is not the only one best model among five models, although the proposed model bears the17smallest value of the sum of RMSE.18

4.3. Model verification19

To compare the proposed model with conventional time-series models, a statistic method, the method of least20squares [18], is taken to estimate the stock price patterns of the three stock markets. By using the software (E-Views)21to estimate the time-series models for different periods of stock index (from 1991 to 1999), nine sets of statistics of22hypotheses testing for each stock market are generated.23


UNCORRECTEDPROOF

PHYSA: 11049


Table 14Performance comparison table (HSI)

Models 1991 1992 1993 1994 1995 1996 1997 1998 1999 Sum of RMSE

Chens model [1] 105 197 336 313 214 169 512 282 306 2434Yus model [10] 40 183 203 198 105 141 375 258 273 1776Chens model [13] 38a 130a 206 147 75a 147 274 205 241 1465Chengs model [12,14] 38a 130a 208 140a 76 144 274 202a 234a 1446a

Proposed model (optimal order & adaptationperiods)

39 131 205a 146 75a 140a 268a 204 239 1446a

a The best performance among 5 approaches (Minimum RMSE).

Fig. 2. Statistics of hypotheses testing for the TAIEX of 1991.

In each year of stock index, five linear regression variables, from price (t 1) to price (t 5), are selected to be 1estimated and tested. The p-value given just below the t-statistic, denoted probability (t-statistic) [18], is the marginal 2significance level of the t-test. If the p-value is less than the significance level which is testing, given at 0.01 here, 3the null hypothesis that all slope coefficients are equal to zero is rejected. Take Fig. 2 as example, in the TAIEX of 41991, the time-series variable significantly related to present variable, price (t), is previous the variable, price (t 1), 5because only the p-value (0.0000) for price (t 1), is less than the significance level (0.01) among five variables 6(from price (t 1) to price (t 5)). 7

To summarize nine sets of statistics of hypotheses testing for the stock databases, Tables 15 and 16 list the 8summarizations of estimated variables and statistics under 1% significance level, which include the estimated time- 9series variables, coefficients, t-statistic, and p-value, for the nine year period of each stock database (Table 15 for the 10TAIEX, and Table 16 for the HSI). 11

From the statistics of hypotheses testing in Tables 15 and 16, nine estimated time lags, from 1991 to 1999, for each 12stock database are derived, and used to compare with the orders of the proposed model under the best performance. 13Tables 17 and 18 lists two types of estimated time lags produced by using least squares method and the proposed model 14(Table 17 for the TAIEX, and Table 18 for the HSI). From these tables, it is apparent that there is high consistency 15between the statistic model and the proposed model for the two stock markets (the percentage of consistency is 1613/18 = 72%). Based on the evidence, we can argue that the proposed model derive most the same forecasting 17patterns with conventional statistic method in stock markets. 18

5. Findings and conclusions 19

This paper provides a high-order fuzzy time-series model and employ the multi-period adaptation model to enhance 20forecasting accuracy. In empirical analysis, there are three findings as follows. 21


UNCORRECTEDPROOF

PHYSA: 11049


Table 15Summarizations of statistics under significance level for TAIEX

Year Variable Coefficient t-statistic Prob.

1991 price (t 1) 1.007620 17.03325 0.00001992 price (t 1) 1.013346 16.88899 0.00001993 price (t 1) 1.146745 18.80499 0.00001994 price (t 1) 1.029945 17.74115 0.00001995 price (t 1) 0.963102 16.14425 0.00001996 price (t 1) 1.002557 16.76250 0.00001997 price (t 1) 0.997510 16.78753 0.00001998

price (t 1) 1.072445 17.43625 0.0000price (t 4) 0.245564 2.756455 0.0062

1999 price (t 1) 1.107256 17.88475 0.0000

Table 16Summarizations of statistics under significance level for HSI

Year Variable Coefficient t-statistic Prob.

1991 price (t 1) 0.970594 15.13667 0.00001992

price (t 1) 1.080045 17.05132 0.0000price (t 4) 0.302411 3.267867 0.0012

1993 price (t 1) 1.120111 17.51516 0.00001994 price (t 1) 0.997210 15.51544 0.00001995 price (t 1) 1.078850 16.73271 0.00001996 price (t 1) 0.955913 14.93345 0.0000

price (t 1) 1.028787 16.04084 0.00001997 price (t 3) 0.320416 3.692714 0.0003

price (t 4) 0.391072 4.393940 0.00001998

price (t 1) 1.116729 17.44999 0.0000price (t 2) 0.254831 2.680829 0.0078

1999 price (t 1) 1.065948 16.49485 0.0000

Table 17The estimated time lags for the TAIEX

Methods Year1991 1992 1993 1994 1995 1996 1997 1998 1999

Least squares method (p = 0.01) t 1 t 1 t 1 t 1 t 1 t 1 t 1 t 1/t 4 t 1Proposed model (order) 1 1 1 1 1 1 2 2 1Consistency Yes Yes Yes Yes Yes Yes No No Yes

Table 18The estimated time lags for the HSI

Methods Year1991 1992 1993 1994 1995 1996 1997 1998 1999

Least squares method (p = 0.01) t 1 t 1/t 4 t 1 t 1 t 1 t 1 t 1/t 3/t 4 t 1/t 2 t 1Proposed model (order) 2 1 2 1 1 1 1 1 2Consistency No Yes No Yes Yes Yes Yes Yes No

(1) From Tables 9 and 10, it is discovered that the forecasting models using lower order (1st or 2nd order) and1fewer adaptation periods (1 and 2 adaptation periods) perform better than higher-order models. Besides, most of the2estimated time lags from the method of least squares are less than 2 (see Tables 17 and 18). These findings both3show that the stock price patterns in the stock markets are short-term. This implies that investment decisions of stock4investors are influenced by recent 1 or 2 periods of price fluctuations.5


UNCORRECTEDPROOF

PHYSA: 11049


(2) The best performance is generated from different order forecasting models (from 1 to 2) and the models 1using different adaptation periods (from 1 to 4) (see Tables 15 and 16). This finding can be explaining by assuming 2that different years of the stock data are generated from non-equivalent and auto-regressive time-series models (see 3Tables 17 and 18). 4

(3) The experiment datum, from Tables 9 and 10, shows that the adaptation model can effectively reduce the RMSE 5within the proposed model in the all testing periods, and from performance comparison tables, Tables 13 and 14, it is 6obvious that the proposed model using different adaptation periods outperforms the three listing conventional fuzzy 7time-series models, Chens (1996) and Yus (2005), which only mine fuzzy logical relationship from time-series. In 8stock markets, investors usually make their decisions based on recent stock prices in these two days not long time ago. 9However, most of the conventional models only extract fuzzy logical relationship from a long-period of historical data 10to generate forecasting rules. Therefore, the listing models, Chens (1996) and Yus (2005) models, cannot adapt their 11forecasts to meet recent price fluctuations to reduce forecasting error. 12

On these findings, two conclusions are remarked: (1) the multi-period adaptation model, proposed in this paper, can 13effectively improve the forecasting performance of conventional fuzzy time-series models; and (2) the price patterns 14in the TAIEX and HSI are short-term. 15

In the future research, two suggestions are provided to improve this paper: (1) refine the evaluating approaches 16with various ratios of training to testing to split experimental datasets; and (2) develop a computer system to provide 17forecasts for stock markets and evaluate the profit. 18

References 19

[1] S.M. Chen, Forecasting enrollments based on fuzzy time-series, Fuzzy Sets and Systems 81 (1996) 311319. 20[2] S.M. Chen, Forecasting enrollments based on high-order fuzzy time series, Cybernetics and Systems 33 (2002) 116. 21[3] S.M. Chen, C.C. Hsu, A new method to forecast enrollments using fuzzy time series, Applied Science and Engineering 2 (2004) 234244. Q3 22[4] S.M. Chen, N.Y. Chung, Forecasting enrollments using high-order fuzzy time series and genetic algorithms, International Journal of Intelligent 23

Systems 21 (2006) 485501. 24[5] K. Huarng, Heuristic models of fuzzy time series for forecasting, Fuzzy Sets and System 123 (2001) 369386. 25[6] K. Huarng, Effective lengths of intervals to improve forecasting in fuzzy time series, Fuzzy Sets and Systems 123 (2001) 155162. 26[7] K. Huarng, H.K. Yu, An N -th order heuristic fuzzy time series model for TAIEX forecasting, International Journal of Fuzzy Systems 5 (4) 27

(2003) 247253. 28[8] K. Huarng, H.K. Yu, A type-2 fuzzy time-series model for stock index forecasting, Physica A 353 (2005) 445462. 29[9] K. Huarng, T.H.K. Yu, The application of neural networks to forecast fuzzy time series, Physica A 336 (2006) 481491. 30[10] H.K. Yu, Weighted fuzzy time-series models for TAIEX forecasting, Physica A 349 (2005) 609624. 31[11] H.K. Yu, A refined fuzzy time-series model for forecasting, Physica A 346 (2005) 657681. 32[12] C.H. Cheng, T.L. Chen, C.H. Chiang, Trend-weighted fuzzy time-series model for TAIEX forecasting, Lecture Notes in Computer Science 33

4234 (2006) 469477. 34[13] T.-L. Chen, C.-H. Cheng, J.-T. Hia, Fuzzy time-series based on fibonacci sequence for stock price forecasting, Physica A 380 (July) (2007) 35

377390. 36[14] C.H. Cheng, T.L. Chen, H.J. Teoh, C.H. Chiang, Fuzzy time-series based on adaptive expectation model for TAIEX forecasting, Expert 37

Systems with Applications (2006) (in press). Q4 38[15] S.M. Chen, Temperature prediction using fuzzy time-series, IEEE Transactions on Cybernetics 30 (2000) 263275. 39[16] C.H. Cheng, T.L. Chen, H.J. Teoh, Multiple-period Modified Fuzzy Time-series for Forecasting TAIEX, 2007. 40[17] J. Kmenta, Elements of Econometrics, MacMillan, 1986. 41[18] Douglas C. Montgomery, George C. Runger, Norma Faris Hubele, Engineering Statistics, 3rd ed., Wiley, New York, 2004. 42[19] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning I, Information Science 8 (1975) 199249. 43[20] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning II, Information Science 8 (1975) 301357. 44[21] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning III, Information Science 9 (1976) 4380. 45[22] Q. Song, B.S. Chissom, Forecasting enrollments with fuzzy time-series Part I, Fuzzy Sets and Systems 54 (1993) 110. 46[23] Q. Song, B.S. Chissom, Forecasting enrollments with fuzzy time-series Part II, Fuzzy Sets and Systems 62 (1994) 18. 47[24] C.-H. Cheng, T.-L. Chen, H.-J. Teoh, Multiple-period Modified Fuzzy Time-series for Forecasting TAIEX, 2007. 48


High-order fuzzy time-series based on multi-period adaptation model for forecasting stock marketsIntroductionFuzzy time-seriesProposed model and algorithmLinear forecasting modelProposed algorithm

Empirical analysisPerformance evaluationPerformance comparisonModel verification

Findings and conclusionsReferences

Date post:	16-Oct-2015
Category:	Documents
Upload:	aditya-baid
View:	4 times
Download:	0 times

academic_74634_6760986_34783

Documents