## DEVELOPMENT OF A FUZZIFIED-TREND MAPPING AND IDENTIFICATION (FTMI) MODEL FOR FUZZY TIME SERIES FORECASTING

Title Page
List of Abbreviations
Abstract

CHAPTER ONE: INTRODUCTION
1.1 Preamble
1.2 Motivation
1.3 Aim and Objectives
1.4 Statement of the Problem
1.5 Methodology
1.6 Significant Contribution of the Developed FTMI Model
1.7 Scope of the Research
1.8 Thesis Organisation

CHAPTER TWO: LITERATURE REVIEW
2.1 Introduction
2.2 Review of Fundamental Concepts
2.2.1 Time Series Models and Analyses
2.2.1.1 Statistical conventional methods
2.2.1.2 Non conventional methods
2.2.2 Fuzzy Set Theory
2.2.2.1 Fuzzy time series
2.2.3 Re-Partitioning Discretization (RPD) Approach
2.2.4 The Order of Difference (OD)
2.3 Review of Similar Works
2.3.1 Summary

CHAPTER THREE: DEVELOPMENT OF THE NEW FTMI MODEL
3.1 Introduction
3.2 Conventional Fuzzy Trend Mapping Models
3.3 The New FTMI Model
3.3.1 Algorithm of the New FTMI Model
3.3.2 Trend Values (   ) and Trend Fuzzification
3.4 Summary

CHAPTER FOUR: VALIDATION AND APPLICATION OF THE NEW FTMI MODEL
4.1 Introduction
4.2.1 Performance of FTMI Model Vs Pioneer and Recent Models in Accuracy
4.2.2 Result Discussion
4.3 Increment of the Order of Difference (OoD)
4.3.1 Effect of Increment of the Order of Difference (OoD)
4.3.2 Result Discussion
4.4.1 Result Discussion
4.5 Application of the FTMI Model on Internet Traffic
4.5.2 Comparison of the Actual and Predicted Traffic

CHAPTER FIVE: CONCLUSION AND SUGGESTIONS FOR FURTHER WORK
5.1 Introduction
5.2 Summary of Findings
5.3 Conclusions
5.4 Limitations
5.5 Suggestions for Further Work
References

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.

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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