IMMM 2014 : The Fourth International Conference on Advances in Information Mining and Management A Multi-Factor HMM-based Forecasting Model for Fuzzy Time Series Hui-Chi Chuang, Wen-Shin Chang, Sheng-Tun Li Department of Industrial and Information Management Institute of Information Management National Cheng Kung University No.1, University Road, Tainan City, Taiwan (R.O.C.) e-mails:{r78021017, r36024037, stli}@mail.ncku.edu.tw Abstract—In our daily life, people are often using forecasting in deciding the high-order. Later, Li and Cheng [13] techniques to predict weather, stock, economy and even some proposed the deterministic fuzzy time series model that uses important Key Performance Indicator (KPI), and so forth. the concept of state transition diagram to backtrack to the Therefore, forecasting methods have recently received most stable state. Wang and Chen [8] proposed two-factor increasing attention. In the last years, many researchers used fuzzy time series model and used automatic clustering fuzzy time series methods for forecasting because of their techniques to partition intervals. Joshi and Kumar [1] first capability of dealing with vague data. The followers enhanced extended the idea of fuzzy sets into Intuitionistic Fuzzy Sets their study and proposed a stochastic hidden Markov model, (IFS) and used it to construction a method for determining which considers two factors. However, in forecasting problems, the membership degree. an event can be affected by many factors; if we consider more Although fuzzy time series has been developed for factors for prediction, we usually can get better forecasting results. In this paper, we present a multi-factor HMM-based decades, there are still some problems. The matrix forecasting model, which is enhanced by a stochastic hidden computing for two-factor fuzzy time series models are too Markov model, and utilizes more factors to predict the future complex. Furthermore, the process of fuzzy time series trend of data and get better forecasting accuracy rate. forecasting is only concerned with whether or not the fuzzy rule is adopted, while ignoring the importance of frequency. Keywords-fuzzy time series; forecasting; hidden Markov Therefore, Sullivan and Woodall [4] used the Markov model model (HMM). to improve the traditional fuzzy time series model. But traditional HMM is unable to solve the two-factor problem. I. INTRODUCTION If we can analyze more factors, we can get higher accuracy and fully utilize all the available information. In our study, In the information explosion era, forecasting is a useful we enhance the study of Li and Cheng [12] and expand the methodology for enterprises or governments to predict future model for forecasting based on multiple factors. trends. The more precise the forecasting result, the more The fuzzy time series methods are good methodologies, appropriate the behavior conducted by managers. In general, which are suited for data composed of linguistic values, but many data are present with crisp value, but others are vague the fuzzy relationship ignores the importance of the and ambiguous instead, such as stock monitoring indicators, frequency. Hidden Markov models are powerful probability signals, and so on. With the purpose of forecasting with models which use categorical data include linguistic labels, vague data, Song and Chissom [9] first proposed the fuzzy and allow handling of two-factor forecasting problem. But in time series and suggested seven steps for forecasting. real world situations, multiple factor data tends to appear, Therefore, numerous fuzzy time series forecasting models rendering this model ineffective in making accurate have been proposed after them. assessments. The seven steps proposed by Song and Chissom [9] are: In this study, we want to combine the benefit of both (1) define the universe of discourse; (2) partition the universe methodologies, so that we can deal with the problem with into several intervals; (3) define fuzzy sets on the universe; many factors and fully utilize the feature of fuzzy theory to (4) fuzzify the historical data; (5) establish fuzzy relations; get higher accuracy result. There are five sections in this (6) calculate the forecasted outputs; (7) defuzzification. The study, which is organized as follows. In Section I, we literature used Alabama University enrollment data to carry introduce the background and motivation. In Section Ⅱ, we out one factor time relational, invariant fuzzy time series introduce the basic definition of fuzzy time series and hidden forecast model. Sullivan and Woodall [4] improved the Markov model. Section Ⅲ presents the model development. results by using Markov’s matrix- based probability statistics We propose a fuzzy time series model to deal with the multi- method [9] to establish one-factor one-order time invariant factor forecasting problem and we use HMM to build the forecast model. The above introduced methods implemented one factor fuzzy relationship matrix. In Section Ⅳ, we present the to forecast, but in realistic circumstances, there is more than experiment result. We demonstrate the proposed model step one factor that affects the forecasted data, such as the by step and make a comparison with other existing methods. temperature and cloud density. Therefore, based on this idea, In Section Ⅴ, we present the conclusions by discussing the Lee, Wang, Chen, and Leu [6] proposed a two-factor high- results as explained in Section Ⅳ and proposes future work order forecast model, which many people debate problematic expecting to improve the proposed method. Copyright (c) IARIA, 2014. ISBN: 978-1-61208-364-3 17 IMMM 2014 : The Fourth International Conference on Advances in Information Mining and Management II. LITERATURE REVIEW relationship group: AAAAi1→ j 1, j 2 ,..., jn A. Fuzzy Time Series AAAAi2→ j 1, j 2 ,..., jn Although there are many statistical methods that can be (5) used to solve the problem of time series, it is impossible to M resolve this problem when the historical data is linguistic AAAAik→ j1, j 2 ,..., jn value. It was not until 1993, when Song and Chissom [9] first proposed the fuzzy time series, that this problem has a B. Forecasting Model of Fuzzy Time Series solution. Because of its easiness to use and comprehend, Song and Chissom [9] first proposed a complete fuzzy there are many scholars who have dedicated their time and time series forecasting model and divided it into seven steps. energy to the field of fuzzy time series making it more Later, many scholars have continuously revised this complete and accurate. The following is the basic definition framework in order to get better forecasting accuracy. of fuzzy time series. 1) Define Universe of Discourse U & Partition the ： 1) Definition 1 Fuzzy Time Series universe Let be a fuzzy time series, U is the {xt ∈ R, t= 1, 2,... n} Song and Chissom [9] defined the universe of discourse U as follows: universe of discourse. Let {Li ( t ) , i= 1,2,..., l } be the ordered UDDDD= − , + (6) linguistic variables. Each Xt in the universe of discourse can [ min 1 max 2 ] be the denoted as follows: where Dmin and Dmax are the minimum and the maximum in µ ( XXX) µ ( ) µ ( ) (1) the training date set and D1 and D2 are the two proper F( t )= 1t+ 2 t+ ...+ l t LLL positive integers decided by the analyst. After defining U, 1 2 l we partition the universe into several intervals. One is equal l length interval, and another is unequal length interval. where µ ∈ 0,1 and . µ ()X is k [ ] ∑µk(x t )= 1,∀ t= 1,2,..., n k t 2) Define Fuzzy Set and Linguistic Values k After fuzzyfying historical data we then begin to the membership value of the linguistic variable L . After k decide linguistic values like rare, few, plenty and so on. transforming, we can say F(t) is the fuzzy time series of X . t. Each interval A (i=1,…,n) will have its own membership to 2) Definition 2：One-order Fuzzy Relation Equation i get linguistic value u (k=1,…,n), which represents the Let F(t) be a fuzzy time series, the relationship between k degree of belonging to each interval of each. They used their and F(t-1)is denoted as below: F() t own experience to formulate the membership degree of each F( t )= F( t− 1) o R( t , t − 1) (2) element in each fuzzy set; the example is shown in (7). A1={ u 1/1, u 2 / 0.5, u 3 / 0, u 4 / 0, u 5 / 0, u 6 / 0, u 7 / 0} “ o ” is a composition operator, and where R(t,t-1) which A2={ u 1/ 0.5, u 2 /1, u 3 / 0.5, u 4 / 0, u 5 / 0, u 6 / 0, u 7 / 0} is composed of Rij is a one-order fuzzy relation. The A= u/ 0, u / 0.5, u /1, u / 0.5, u / 0, u / 0, u / 0 relationship shows below: 3{ 1 2 3 4 5 6 7 } (3) A4={ u 1/ 0, u 2 / 0, u 3 / 0.5, u 4 /1, u 5 / 0.5, u 6 / 0, u 7 / 0} (7) R( t , t− 1)= ∪ij R ij ( t , t − 1) A5={ u 1/ 0, u 2 / 0, u3/ 0, u 4 / 0.5, u 5 /1, u 6 / 0.5, u 7 / 0} where R( t , t − 1) is the fuzzy relation between Fj (t) and ij A6={ u 1/ 0, u 2 / 0, u 3 / 0, u 4 / 0, u 5 / 0.5, u 6 /1, u 7 / 0.5} F (t-1). i A7={ u 1/ 0, u 2 / 0, u 3 / 0, u 4 / 0, u 5 / 0, u 6 / 0.5, u 7 /1} 3) Definition 3：One-order Fuzzy Logic Relationship In order to reduce the complexity of matrix computation, 3) Fuzzify Historical Data Chen [11] proposed Fuzzy Logical Relationship (FLR) If the historical data is not ambiguous, we need to combined with simple arithmetic operation to replace matrix fuzzify it first, for example hot, cold and linguistic values.