Cairo University Institute of Statistical Studies and Research Department of Computer and Information Sciences Fuzzy Time Series Model for Egypt Gold Reserves Forecasting Based On Fuzzy Clustering ICCES'2011 Ashraf K. Abd-Elaal Department of Computer and Information Sciences, High Institute of Computer Science, Al-Kawser City, Sohag, Egypt Hesham A. Hefny Department of Computer and Information Sciences, Institute of Statistical Studies and Research, Cairo University, Egypt Ashraf H. Abd-Elwahab Department of Computer Sciences, Electronics Research Institute, National Center for Research, Egypt

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 Outlines 1- Introduction 2- Fuzzy Time Series Vs Traditional Time Series forecasting 3- Fuzzy Approaches for Forecasting 4- Proposed Model 5- Forecasting Egypt Gold Reserves 6- Conclusion 7- References

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES' Introduction The existence of uncertainty in the historical data given for a certain forecasting problem represents a real challenge to the traditional time series forecasting models. Song and Chissom presented the first concept of fuzzy time series model. They presented the time-invariant fuzzy time series model and the time-variant fuzzy time series model based on the fuzzy set theory for forecasting the enrolments of the University of Alabama. Since such a time, many researchers have contributed to developing and improving the fuzzy time series models.

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES' Fuzzy Time Series Vs Traditional Time Series forecasting The traditional forecasting methods fail to forecast the data with linguistic facts. Time series analysis often requires to turn a non-stationary series into a stationary series. The traditional time series requires more historical data along with some assumptions like normality postulates. The fuzzy forecasting methods success to forecast the data with linguistic facts. Fuzzy time series do not need to turn a non-stationary series into a stationary series and do not require more historical data along with some assumptions like normality postulates

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES' Fuzzy Approaches of Forecasting Fuzzy time series deal with data without any assumptions like normality, and not requires data normalization, training set and the test set. It works with scant historical data Song and Chissom (1993) presented the concept of fuzzy time series based on the historical enrollments of the University of Alabama. Fuzzy time series used to handle forecasting problems.

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 Song and Chissom methodology 1- Define the universe of discourse U. U=[D min – D 1, D max + D 2 ] where, D min is the minimum value, D max is the maximum value, D 1, D 2 is the positive real numbers to divide the U into n equal length intervals. 2- Partition universal of discourse U into equal intervals. Song and Chissom (1993) presented a methodology for establish fuzzy time series, most of authors in fuzzy time series field took the same path, but they differ in some steps.

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 Song and Chissom methodology 4- Fuzzify the historical data. 5- Build fuzzy logic relationships. 3- Define the linguistic terms:-

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES' Proposed Model 1- Cluster data into c clusters. 2- Determine membership values for each cluster. 3- Define the Universe of Discourse U. U=[D min – D 1, D max + D 2 ] Where, D min is the minimum value, D max is the maximum value, D 1, D 2 is the positive real numbers to divide the U into n equal length intervals. 4- Partition universal of discourse U into equal intervals. 5- Fuzzify the historical data. 6- Calculating the crisp value for each linguistic. Where mg j is the membership grade, X j is the actual value. 7- Re-fuzzification of historical data. 8- Determining the forecasted values.

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES' Forecasting Egypt Gold reserves The proposed model is applied to forecast the Egypt Gold Reserves according to official data found in the International Monetary Fund's International Financial Statistics. The used data represent the reserve holdings in US$ Millions during the period starting from first quarter of 2002 up to the first quarter 0f The forecasting accuracy is compared by using Normalized Root Mean Square Error (NRMSE) as follows:

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 Song and Chissom Model

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 ARIMA Model The ARIMA Model is: This model is a good model that can be used for forecasting the Egypt Gold reserves since the PACF and ACF of residual have no lags out of range

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 Data of Egypt Gold Reserves and Membership Grades YearGoldA1A1 A2A2 A3A3 A4A4 A5A5 A6A6 A7A

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 The crisp value for A 5 according to step 6 as follows: According to step 7, proposed model divides u 1 into four partitions and u 7 into three partitions. But when proposed model divided u 1 then linguistic A 2 will be converted to be linguistic A 5 and linguistic A 3 to be linguistic A 6 and so on According to step 8, the forecasting value for the first quarter of year 2002 is while the actual value was and the forecasting value for the second quarter of year 2007 is as the same with the actual value which was 28563

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 Forecasting Egypt Gold reserves by Proposed Model

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 Forecasting Egypt Gold reserves from Different Models YearGold Jilani (2008) Cheng (2008) Chen (2008) ARIMA Abd-Elaal (2010) Proposed …………………… …………………… NRMSE

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 Forecasting Results Curve of Egypt Gold Reserves

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 NRMSE Comparisons for Different Models

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES' Conclusion This paper presents an efficient fuzzy time series model based on fuzzy clustering. The proposed model has been used successfully to forecast the Egypt Gold reserves based on official data during the period starting from 2002 up to the first quarter of The proposed model has been evaluated through a comparison with other fuzzy time series forecasting models as well as ARIMA model. The result of comparison ensures the higher performance of the proposed model.

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES' References 1.A. K. Abd-Elaal, H. A. Hefny, and A. H. Abd-Elwahab, "An Improved Fuzzy Time Series Model For Forecasting ", IJCSIS, vol. 8, pp , A. K. Abd-Elaal, H.A. Hefny, and A. H. Abd-Elwahab, "Constructing Fuzzy Time Series Model Based on Fuzzy Clustering for a Forecasting", J. Computer Sci., vol. 7, pp , T.-L. Chen, C.-H. Cheng, and H.-J. Teoh, "High-order fuzzy time-series based on multi-period adaptation model for forecasting stock markets", Physica A, vol.387, pp. 876–888, C.-H. Cheng, J.-W. Wang, and G.-W. Cheng, "Multi-attribute fuzzy time series method based on fuzzy clustering", Expert Systems with Applications, Vol.34, pp. 1235–1242, M. Friedman and A. Kandel, "Introduction to pattern recognition statistical, structural, neural and fuzzy logic approaches", Imperial college press, London, 1999, p K. Huarng, "Effective lengths of intervals to improve forecasting in fuzzy time series", Fuzzy Sets and Systems, vol.123, pp. 387–394, T.A. Jilani and S. Burney, "Multivariate stochastic fuzzy forecasting models", Expert Systems with Applications, vol.35, pp. 691–700, F. Kai, Fang-Ping, and C. Wen-Gang, "A novel forecasting model of fuzzy time series based on K- means clustering", IWETCS, IEEE, 2010, pp.223– G. Kirchgässner and J. Wolters, "Introduction to modern time series analysis", Springer- Verlag.Berlin, Germany, 2007, p A. Konar, "Computational intelligence principles, techniques and applications", Springer Berlin Heidelberg, Netherlands, pp , 2005.

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES' H.-T. Liu, "An improved fuzzy time series forecasting method using trapezoidal fuzzy numbers". Fuzzy Optimization and Decision Making, vol. 6, pp.63-80, A.K. Palit and D. Popovic, "Computational intelligence in time series forecasting theory and engineering applications", Springer-Verlag.London, UK, 2005, p W. Qiu, X. Liu, and A. H. Li "generalized method for forecasting based on fuzzy time series", Expert Systems with Applications, vol. 38, pp , Q. Song and B.S. Chissom, "Forecasting enrollments with fuzzy time series. I", Fuzzy sets and systems, vol. 54, pp. 1-9, Q. Song and B.S. Chissom, "New models for forecasting enrollments: fuzzy time series and neural network approaches", ERIC, p. 27, 1993, 16.R.-C. Tsaur, J.-C. Yang, and H.-F. Wang, "Fuzzy relation analysis in fuzzy time series model", Computers and Mathematics with Applications, vol.49, pp , R.-C. Tsaur, and T.-C. Kuo, "The adaptive fuzzy time series model with application to Taiwan's tourism demand", Expert Systems with Applications, vol.38, pp , C.C.Wang, "A comparison study between fuzzy time series model and ARIMA model for forecasting Taiwan export", Expert Systems with Applications, vol.38, , H.-K. Yu, "Weighted fuzzy time series models for TAIEX forecasting", Physica A, vol.349, pp.609–624, 2005.

A.K.Abd-Elaal, H.A Hefny, A.H.Abd-ElwahabICCES'2011 Thank you