Bulanık Zaman Serisi Modellerinde Farklı Üyelik Fonksiyonlarının Başarıma Etkisi

Abstract

Üretim planlanması, hava veya sıcaklık tahmini, borsa endeksleri, bir okula kayıt yaptıracak öğrenci sayısının belirlenmesi gibi konularda var olan verilerden gelecek ile ilgili akla uygun tahminler yapılmaya çalışılır. Bu tahminler sonucunda, insanların karşılaşabileceği bazı zararlar önlenebilir veya çeşitli kazanımlar elde edilebilir. Bu nedenle, zaman serisi kavramı ve analizi önem taşımaktadır. Zaman serisi, bir büyüklüğün belirli aralıklar ile ölçülmesiyle oluşmuş bir veri kümesidir. Zaman serisi analizinde modelleme yapmak ve öngörüde bulunmak amaçlanır. Bulanık zaman serisi ise verileri dilsel terimler olarak ifade edilen zaman serileridir. Literatürde, bulanık zaman serisi kavramı ortaya atıldıktan sonra, verileri iyi modellemek ve öngörüde bulunabilmek için çeşitli bulanık zaman serisi modelleri önerilmiştir. İlk yıllarda önerilen modeller, daha sonradan referans alınarak sezgisel modelleme, yüksek dereceden modelleme gibi farklı çalışmalar yapılmıştır. Tahmin başarımında, seçilecek üyelik fonksiyonunun biçimi önemli bir etkiye sahip olmakla birlikte, yapılan çalışmalarda sadece parçalı keskin tipte üyelik fonksiyonu kullanılmıştır. Bu sebepten, bulanık çıkarım yerine aritmetik işlemler kullanılarak tahmin yapılmıştır. Bu çalışmada, yeni bir bulanık zaman serisi modeli önerilmiştir. Bu modelde, klasik parçalı fonksiyon yerine, üçgen ve yamuk üyelik fonksiyonları kullanılmıştır. Bulanık çıkarım mekanizması Mamdani tipindedir ve durulayıcı olarak ağırlık merkezi yöntemi kullanılmıştır. Oluşturulan üyelik fonksiyonlarının göbekleri orta nokta ve ağırlık merkezi olarak iki farklı tipte seçilerek sonuca etkisi tüm benzetim çalışmalarında incelenmiştir. Önerilen modelin geçerliliği ve etkinliği, literatürde de yaygın olarak kullanılan Alabama Üniversitesi öğrenci kaydı üzerinde gösterilmiştir. Ardından literatürdeki bir başka örnek olan TAIFEX(Taiwan Futures Exchange) verileri için de önerilen model denenmiştir. Alabama Üniversitesi verileri yumuşak değişim gösteren, TAIFEX verileri ise sert değişim gösteren verilerdir. Bu anlamda, iki farklı veri tipine göre de modelin başarımı incelenmiştir. Bu benzetim çalışmalarına ek olarak en iyileme çalışması yapılarak önerilen modelin parçalı keskin fonksiyon kullanılan model karşısındaki başarımı da incelenmiştir. Ayrıca, her iki örnek için, farklı sayıda bölmelenme ile modelleme yapmanın ve yüksek derece modelleme yapmanın sonuca etkisi incelenmiştir. Sonuçlar, MSE kriterine göre karşılaştırılmış ve önerilen model ile yapılan tahminin parçalı keskin fonksiyon modeller ile karşılaştırıldığında daha az hataya sahip olduğunu göstermiştir. Ayrıca farklı kriterlere göre yapılan modellemelerde önerilen model ile yapılan modellemelerin daha tutarlı sonuçlar verdiği gözlemlenmiştir.

Making reasonably accurate estimates is important in many topics such as prediction of weather or temperature, stock index, futures exchange, enrollments, etc. Based on these forecasts, damages can be prevented or benefit can be gained. Therefore, the time series concept and analysis become important. A time series consists of a data points sequence ,which measured periodically at uniform time intervals. In time series analysis, model fitting and forecasting is aimed. The fuzzy time series is a time series, whose data are linguistic values instead of real numbers. In literature, the concept of fuzzy time series and the procedures to develop their models were introduced and then several fuzzy time series models are proposed to obtain better forecasting performance. In the first proposed model, the max-min composition operation is used. After that, a simple fuzzy time series model was presented where the forecasting results were obtained by using simple interval arithmetic operations instead of complex max-min composition operation. In addition to these, many researchers proposed fuzzy time series models for handling forecasting problems, such as temperature forecasting, forecasting the enrollments of the University of Alabama, the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) and Taiwan Futures Exchange (TAIFEX). Many methods are also presented such as, high-order fuzzy time series, heuristic function to present a method for forecasting, two-factors high-order fuzzy time series, average-based length method and distribution-based length method, etc. Although the shape of membership functions has an important effect on the forecasting performance, piecewise crisp membership functions are used in all of these models without showing any plausible reason. Therefore, the forecasting is performed by using interval arithmetic operations instead of fuzzy inference. In this study, a new fuzzy time series model is proposed. Firstly, the minimum datum value and the maximum datum value should be found. Then, the universe is defined accordingly. After that, the universe is partitioned into several areas with equal lengths. The linguistic terms, defined by the fuzzy sets on the universe, are defined. Thereafter, the knowledge of each datum value is determined in terms of the linguistic values. Then, the fuzzy logical relationship of the data values are defined between the two sequential data values. First datum value called as current state value and the following called as next state datum value. After that, the fuzzy logical relationships of the data values are grouped based on their current state fuzzy logical relationships. Thereafter, output fuzzy sets related with the each fuzzy logical relationship group are defined. The rules are determined considering all these groups. The type of the membership function of the input and output fuzzy sets is chosen. In this model, triangular and trapezoidal membership functions are used instead of classical piecewise crisp membership functions. While creating the membership functions, the cores of the formed membership functions can be chosen in two different types, which are midpoint of the related interval and centroid of the data within the related interval. The effect of this selection is also observed through the forecasting results. Lastly, the forecasting is performed by using Mamdani-type fuzzy inference with centroid defuzzification. The results are compared with the well-known piecewise crisp membership function models. The success performance is evaluated considering the mean squared error criterion. The validation and effectiveness of the proposed model is shown on the forecast of the enrollments of the University of Alabama. In addition to this example, another well-known example called Taiwan Futures Exchange (TAIFEX) is used. The data of the enrollments of the Alabama University represents a time series with smooth changing data and the data of the TAIFEX represents a time series with aggressive changing data. In this manner, the effectiveness of the model is examined with two different types of time series, which have different data dynamism from each other. Furthermore, an optimization example is performed. With this optimization example, the success of the proposed model against the well-known piecewise crisp membership function model is presented. Finally, the effect of different length of intervals and high-order fuzzy time series modeling is analysed for both examples. The length of intervals are chosen considering the original length of intervals, which used in the literature. For each time series example, one wider than the original length and one narrower than the original length are chosen. The narrow length of interval models are expected to show better forecasting than the wide length of interval models. Moreover, high order models are expected to show better forecasting performance than the first order models. In every analysis, both triangular and trapezoidal membership function models are used and results are compared with each other as well as piecewise crisp membership function models. The results are compared in terms of mean squared error. While using the original lenght of intervals in the enrollment of the University of Alabama example, the results show that the proposed model has smaller forecasting mean square error value than piecewise crisp membership function models. The results are also show that, when cores of the formed membership functions are chosen based on centroid of the data within the related interval, the model shows better forecasting performance than cores of the formed membership functions are chosen as midpoint of the related interval. While using the original lenght of intervals in the TAIFEX example, triangular and trapezoidal models performs a better forecast than the classical piecewise crisp function model. Using genetic algorithm optimization technique, the piecewise crisp membership function model improves its forecasting performance. On the other hand, the genetic algorithm optimization of the proposed method gives much better forecasting performance than the optimized piecewise crisp membership function model for the same search criteria and options. For wider length of intervals the formed models with proposed method shows poor performance as expected. On the other hand, for narrower length of intervals the formed models with proposed method shows better performance as expected. Although, the piecewise crisp membership function model shows better performans for some examples, it doesn't show a consistent forecasting performance unlike the proposed model. For second order modeling, the formed models with proposed method shows better performance as expected. Piecewise crisp membership function models shows poor performance in the enrollment of the University of Alabama example. However, in TAIFEX example its performance is better than some of the proposed method models. The reason is that, the TAIFEX data is an aggressive changing data. Generally, for different length of intervals and high order models, the proposed method shows better and consistent forecasting performance than piecewiese crisp membership function models. For smooth changing data, the model shows quite well performance. On the other hand, the model shows sometimes low performance for the aggressive changing data. The reason for the low performance is that, the intervals are determined considering equal lengths rather than intensity of the data. Nevertheless, the proposed model shows a consistent forecasting unlike the piecewise crisp membership function models for both examples.