ABSTRACT
Fuzzy time series (FTS) forecasting is a technique based on time series and fuzzy logic theory developed for the purpose of analysis and prediction of time series events. The proposed Fuzzified Trend Mapping and Identification (FTMI) model uses a Re-Partitioning Discretization (RPD) approach to optimize the partitioning of the interval lengths and high-order fuzzy relations to construct the trend values. In the proposed model, the mapped out trends are fuzzified into classes both in linguistic and numeric terms to capture both the uncertainty and fuzziness inherent in the trends. Each trend class is given distinct ordinal position for ease of identification during deffuzzification and forecasting. The proposed model is tested on three time series data of different structural and statistical characteristics using mean average percentage error (MAPE) as statistical performance measure. The adaptability of the proposed model to different time series events is also tested using statistical measure of dispersion (variance). Empirical result shows an increase of over 50% in forecast accuracy over pioneer and recent models. Also, the statistical variance of the forecast errors of the proposed model from the MAPE were 0.12, 0.488 and 1.267 compared to 0.58, 8.037 and 4.915 of Shah’s (2012) model for the three time series data respectively. These results demonstrate both the superiority of the proposed FTMI model in accuracy of prediction and its robustness in adaptation to time series of different structural and statistical characteristics when compared to existing models. The effect of increasing the order of difference on both the data trend and the accuracy of forecast are also investigated. Results obtained show that it does not necessarily increase the forecast accuracy regardless of the structure of the time series. The FTMI model is also applied to forecast the short term Internet traffic data of ABU, Zaria. The empirical result shows a MAPE of 0.27 for the Internet traffic, indicating a good accuracy of prediction considering the large size of these traffics.
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TABLE OF CONTENTS
Title Page
Declaration ……………………………………………………………………………………………………………………… i
Certification …………………………………………………………………………………………………………………… ii
Dedication …………………………………………………………………………………………………………………….. iii
Acknowledgement …………………………………………………………………………………………………………. iv
List of Figures ……………………………………………………………………………………………………………… viii
List of Tables ………………………………………………………………………………………………………………… ix
List of Abbreviations ……………………………………………………………………………………………………… xi
List of Symbols ……………………………………………………………………………………………………………. xiii
Abstract ………………………………………………………………………………………………………………………. xiv
CHAPTER ONE: INTRODUCTION …………………………………………………………………………….. 1
1.1 Preamble …………………………………………………………………………………………………………………. 1
1.2 Motivation ……………………………………………………………………………………………………………….. 2
1.3 Aim and Objectives ………………………………………………………………………………………………….. 3
1.4 Statement of the Problem …………………………………………………………………………………………. 4
1.5 Methodology ……………………………………………………………………………………………………………. 5
1.6 Significant Contribution of the Developed FTMI Model ……………………………………………. 6
1.7 Scope of the Research ………………………………………………………………………………………………. 7
1.8 Thesis Organisation …………………………………………………………………………………………………. 7
CHAPTER TWO: LITERATURE REVIEW …………………………………………………………………. 9
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2.1 Introduction …………………………………………………………………………………………………………….. 9
2.2 Review of Fundamental Concepts …………………………………………………………………………….. 9
2.2.1 Time Series Models and Analyses …………………………………………………………………………….. 9
2.2.1.1 Statistical conventional methods…………………………………………………………………………… 11
2.2.1.2 Non conventional methods ………………………………………………………………………………….. 13
2.2.2 Fuzzy Set Theory ………………………………………………………………………………………………….. 13
2.2.2.1 Fuzzy time series………………………………………………………………………………………………… 17
2.2.3 Re-Partitioning Discretization (RPD) Approach ……………………………………………………….. 20
2.2.4 The Order of Difference (OD) ………………………………………………………………………………… 22
2.3 Review of Similar Works………………………………………………………………………………………… 23
2.3.1 Summary ……………………………………………………………………………………………………………… 32
CHAPTER THREE: DEVELOPMENT OF THE NEW FTMI MODEL ………………………. 34
3.1 Introduction …………………………………………………………………………………………………………… 34
3.2 Conventional Fuzzy Trend Mapping Models …………………………………………………………… 34
3.3 The New FTMI Model ……………………………………………………………………………………………. 36
3.3.1 Algorithm of the New FTMI Model ………………………………………………………………………… 37
3.3.2 Trend Values ( ) and Trend Fuzzification ……………………………………………………………… 40
3.4 Summary ……………………………………………………………………………………………………………….. 41
CHAPTER FOUR: VALIDATION AND APPLICATION OF THE NEW FTMI MODEL……………………………………………………………………………………………………………………………………. 42
4.1 Introduction …………………………………………………………………………………………………………… 42
4.2.1 Performance of FTMI Model Vs Pioneer and Recent Models in Accuracy …………………… 60
4.2.2 Result Discussion ………………………………………………………………………………………………….. 63
4.3 Increment of the Order of Difference (OoD) ……………………………………………………………. 64
4.3.1 Effect of Increment of the Order of Difference (OoD) ……………………………………………….. 65
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4.3.2 Result Discussion ………………………………………………………………………………………………….. 67
4.4 The Model Adaptability …………………………………………………………………………………………. 68
4.4.1 Result Discussion ………………………………………………………………………………………………….. 71
4.5 Application of the FTMI Model on Internet Traffic ………………………………………………… 72
4.5.2 Comparison of the Actual and Predicted Traffic ……………………………………………………….. 77
CHAPTER FIVE: CONCLUSION AND SUGGESTIONS FOR FURTHER WORK…….. 79
5.1 Introduction …………………………………………………………………………………………………………… 79
5.2 Summary of Findings …………………………………………………………………………………………….. 79
5.3 Conclusions ……………………………………………………………………………………………………………. 80
5.4 Limitations …………………………………………………………………………………………………………….. 82
5.5 Suggestions for Further Work ………………………………………………………………………………… 82
References …………………………………………………………………………………………………………………… 84
CHAPTER ONE
INTRODUCTION
1.1 Preamble
Time series is simply a collection of quantitative variables at regular intervals of time. Whether discrete or continuous, time series is always both non linear and non stationary since they are sample functions realized from processes that are always stochastic (Subanar and Abadi, 2011). Time series analyses and forecasting play a vital role in planning, equipment maintenance and optimization, efficient quality of service (OoS), and even anomaly detection in diverse fields such as: engineering, medicine, stock market, information and communication technology (ICT) (Sah and Konstantin, 2005; Klevecka, 2011; Zhani et al., 2011; Cortez et al., 2012). Time series forecasting has been widely studied and investigated for the past three decades or so (Box and Jenkins, 1976; Song and Chissom, 1993a; Huarng and Yu, 2003; Wang et al., 2008; Singh and Borah, 2013). In simple terms, time series forecasting involves the analyses of historical time series data and prediction of future variables from the analyzed data (Box and Jenkins, 1976; Hassan et al., 2012). Traditionally, time series forecasting problems are being solved using a class of statistical linear autoregressive (AR), moving average (MA) and their hybrid (ARMA) models. These models and their subsequent extensions such as auto-regressive integrated moving average (ARIMA) and other linear models assume that the time series are both linear and stationary. A viable alternative to these linear techniques are soft computing techniques which are capable of approximating any real continuous function without making assumptions about the structure of the data (Subanar and Abadi, 2011). Among these techniques such as neural network, evolutionary algorithm etc, fuzzy logic has received a much greater attention because of its over-riding advantages (Song and Chissom, 1993a; Chabaa and Zeroual, 2009; Shah,
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2012). Of many critical issues in fuzzy time series (FTS) models, trend mapping and identification has not been fully exploited from the research community. This research is focused on the development of a robust FTS forecasting model from trend mapping and identification approach.
1.2 Motivation
Time series analyses and forecasting is one of the ways humans try to exert some form of control in the future in order to avoid catastrophes, build capacities and to efficiently maintain and maximize scarce resources. Although, time series analyses and forecasting has arguably received the greatest attention in engineering, it spans in almost all the fields of human endeavor (Sah and Konstantin, 2005). The complexities involved in the analyses and training of large amount of historical time series data is of great concern in time series forecasting (Hassan et al., 2012). Another major challenge in addressing time series forecasting problems is the non linear nature of time series in addition to its attendant causative factors that are highly unpredictable. Hence, linear techniques and models have often fallen short of the basic requirements needed for effective time series analyses and forecasting, such as prediction accuracy. More so, because linear models assume that the underlying generation process of time series is time invariant, not all time series can be analyzed by these conventional models (Subanar and Abadi, 2011). A major way to overcome these shortcomings in linear techniques is by the use of non linear techniques which do not require large amount of training data and which can analyze time series events in ways humans think, namely linguistic terms. Fuzzy logic, more than any other non linear techniques, has proved robust and efficient in dealing with time series forecasting issues (Shahida et al., 2003; Chabaa and Zeroual, 2009; Shah, 2012). Being a universal approximator, fuzzy systems have advantages that the developed models are characterized by both numeric and
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linguistic interpretability and the generated rules can be understood, verified, extended and incorporate expert knowledge (Shah, 2012; Subanar and Abadi, 2011). Hence, fuzzy time series (FTS) developed in the past two decades (Song and Chissom, 1993b; Song and Chissom, 1993a), has commanded unprecedented attention from the research community to date. However, the issue of accuracy of forecasting is still a major challenge in FTS since prediction accuracy reported in the open literature is not enough.
Another issue to contend with in time series analyses and prediction is the fact that times series has varying structural and statistical characteristics depending on both the data source and time of collection. Although the inherent characteristics of some time series data vary negligibly with source and time, such as internet traffic (Zhani et al., 2011), others vary significantly. As a result, long-term prediction is often associated with variability of errors (Papadopouli et al., 2006). These have led researchers to favor short-term prediction in place of long-term prediction. Also, to improve accuracy of forecasting, researchers have tried to adapt time series models to the structure of time series under analyses. Nevertheless, this has led to the problem of over fitting of models (Shah, 2012) – another challenge bedeviling time series forecasting. Consequently, models developed may be effective in predicting the time series under study, but erratic in prediction of other time series of different or even similar structure. It has become imperative therefore, to develop robust FTS model that will adapt to the varying dynamics and characteristics of time series data while simultaneously improving the forecasting accuracy.
1.3 Aim and Objectives
The aim of the research is the development of a fuzzified-trend mapping and identification (FTMI) based fuzzy time series model.
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This aim is achieved with the following objectives:
I. Development of a Fuzzified-Trend Mapping and identification (FTMI) fuzzy time series forecasting model using the Re-Partitioning Discretization (RPD) approach (Singh and Borah, 2013) for optimizing the partitioning of the interval lengths and using the Order of Difference (OOD) for trend mapping and identification.
II. Validation of the developed model using the standard data of the students’ enrolment of the University of Alabama and two other time series data with different structural characteristics; and comparing its performance with pioneer and existing models.
III. Determination of effect of increment of the Order of Difference (OoD) on the forecast accuracy.
IV. Application of the model on the Internet traffic data of Ahmadu Bello University, Zaria.
1.4 Statement of the Problem
Many FTS models have been developed and used to forecast time series data in various domains. Regardless of the targeted domains, the main focus of these variant models is to improve the accuracy of forecasting relative to the pioneer model (Song and Chissom, 1993a; Song and Chissom, 1994), and any prevailing model(s) as the case may be. However, a fundamental limitation of the existing fuzzy time series models in the open literature has been the problem of over fitting of models (Shah, 2012). That is, the models are adapted to a particular time series data only and therefore, not effective in predicting other time series of different structure or even similar structure, in some cases. Many researchers have used first or second-order difference to see how the accuracy of forecasting may be improved. But the problem of how increasing the orders of difference affect both the data trend of a time series and the accuracy of forecasting has not been investigated. It is crucial, therefore, to develop robust fuzzy forecast model which will
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adapt to the varying dynamics (non-linearity, non-stationarity) of different time series data while simultaneously improving the forecast accuracy.
In view of this, a novel FTS model is developed in this research using an adaptive optimizing approach namely, Re-Partitioning Discretization (RPD) to optimize the determination of its interval length and high-order trend relation to properly identify the general trend(s). The general trends identified are fuzzified to capture the uncertainty or fuzziness inherent in them. Also, this research extended the order of difference to fourth-order and investigated the effect of increasing the order of difference on both the data trend and accuracy of forecasting.
1.5 Methodology
Like most fuzzy time series systems, the Trend Mapping and Identification technique has two main components: Fuzzification and defuzzification. These components have many parts and an inference system which is rule based. The general methodology adopted in this model is as follows:
I. Apply the Re-Partitioning Discretization (RPD) approach presented in Section 2.5 based on the collected data to optimize both the partitioning of data sets and the universe of discourse.
II. Define linguistic terms for each of the intervals.
III. Establish the Fuzzy Membership Functions and the fuzzy sets.
IV. Fuzzify the time series data set and establish the high-order fuzzy relations (FLRs).
V. Find the Order of Difference (OoD) between successive data to map out the trends.
VI. Fuzzification of the trends. The trends are fuzzified based on the number of classes of the historical data.
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VII. Development of a fuzzy Inference System of rules based on the fuzzified trends for defuzzification and forecasting.
VIII. Validation of the developed model using the standard data of the students’ enrolments of the University of Alabama and two time series data of different structural characteristics by comparison with the results obtained using the Song and Chissom’s and Shah’s methods on the bases of MAPE and variance statistic.
IX. Determination of the effect of increasing the Order of Difference on the forecast accuracy.
X. Application of the model on the Internet traffic data obtained from Ahmadu Bello University, Zaria.
1.6 Significant Contribution of the Developed FTMI Model
The contributions of the developed FTMI Model in time series forecasting are as follows:
1. The developed FTMI model is able to capture the trend’s vagueness and has significantly improved the forecast accuracy (over 50% when compared to recent models such as Shah (2012) model.
2. The developed FTMI model can trap both the fuzziness and uncertainty inherent in the trend, and has significantly eliminated noise in all the time series tested.
3. The developed FTMI model is robust and adapts well to time series of different statistical and structural characteristics.The statistical measure of dispersion (variance) is introduced in the thesis to validate the proposed model and those of previous models considered in this paper. The variance of the proposed FTMI model for the three time series data tested are 0.12, 0.488 and 1.267 and are all less than their respective MAPE of
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0.37, 1.054 and 2.400 unlike other models. Hence, the proposed FTMI model does not have high variability in forecasting time series of different structural and statistical characteristics. Therefore, it is suitable for different time series events
4. Using three different time series of different structural characteristics, this research extended the order of difference to fourth-order and demonstrated that though, increasing the order of difference affected the data trend, it did not increase the accuracy of forecasting.
1.7 Scope of the Research
The research borders on the development of a fuzzy time series (FTS) model based on the trend mapping and identification method for short-term prediction of time series data based purely on a univariate approach. The scope of this research did not cover the following:
Effect of multi-variables in time series.
Seasonal time series.
Long-term prediction.
Residual and stationarity control.
Post-sample prediction
1.8 Thesis Organisation
The summary of how the rest of the chapters are organised is presented thus:
Having presented the introduction with the general background of the thesis in chapter one, chapter two presents the literature review consisting of systematic review of some fundamental concepts theories relevant to FTS and the proposed FTMI Model. Also in this chapter is a comprehensive review of works similar to this investigation. The development of the proposed
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FTMI Model is clearly analysed in chapter three. Prior to this, the general structure of the conventional fuzzy trend mapping models is briefly highlighted in the chapter. Chapter four presents the validation and application of the proposed FTMI Model. Three time series of varying structural characteristics are all used in the validation. Subsequently, the proposed FTMI Model is applied to forecast the internet traffic of Ahmadu Bello University, Zaria. Chapter five presents the conclusions of the study. Also in this chapter are the summary of findings and the suggestions for further work. Finally, all the cited references are presented at the end of this thesis work.
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