Advances in Time Series Forecasting

The researchers from various fields have been studying on time series forecasting for approximately one century in order to get better forecasts for the future. To achieve high forecast accuracy level, various time series forecasting approaches have been improved in the literature. During 1980s, some crucial developments happened and time series researches changed. More sophisticated algorithms could be improved since properties of computers were enhanced. Therefore, new time series forecasting approaches such as artificial neural networks and fuzzy time series could be proposed. In the applications, these approaches have proved its success in forecasting real life time series. In addition, hybrid forecasting methods which combine these new approaches have also been improved to obtain more accurate forecasts. In recent years, these advanced time series forecasting methods have been used to forecast real life time series and satisfactory results have also been obtained.

In recent years, artificial neural networks (ANN) have been widely used in real life time series forecasting. Artificial neural networks can model both linear and curvilinear structure in time series. Most of the conventional methods used in the analysis of time series are linear structure and fail to analyze non-linear time series. In conventional time series methods such as threshold autoregressive, bilinear model, which are used in non-linear time series modeling, a particular curvilinear model pattern is needed. Artificial neural network is a method based on data and does not require a model pattern. With its activation function, it provides flexible non-linear modeling. Additionally, when compared with conventional methods, successful results are obtained in forecasting time series via artificial neural networks in the literature. In this study, feed forward and feedback artificial neural networks which are widely used for time series forecasting were applied to Istanbul Stock Exchange Market (IMKB) time series and forecasting performances were evaluated.

In recent years, studies including long memory time series are existed in the literature. Such time series in real life may have both linear and nonlinear structures. Linear models are inadequate for this kind of time series. An alternative method to forecast these time series is artificial neural networks which is data based and can model both linear and nonlinear structure in these time series. In order to determine the number of nodes in the layers of a network is an important decision. This decision has been made by using various architecture selection criteria. The performance of these criteria varies, depending on components of time series, such as trend and seasonality. In this study, some architecture selection criteria are compared on real time series when artificial neural networks are employed in forecasting. Some advices are given for using artificial neural networks to forecast long memory time series.

Forecasting the number of outpatient visits plays important role in strategic decisions for the expert of healthcare administration. In order to manage hospitals effectively, it is needed to forecast the number of outpatient visits accurately. In the literature, there have been some methods proposed to forecast these time series. One of these methods is artificial neural networks approach. Although, artificial neural networks have proved its success in forecasting, there are still some problems with using this method. Determining the elements of this method is an important issue. Activation function is a crucial element of artificial neural networks. Therefore, in this study, we examined different activation functions to obtain more accurate out sample predictions while the number of patients is being forecasted. It is found that using different activation function affects the forecasting accuracy of feed forward neural network models.

In the literature, different selection criteria are used for determining the best architecture when time series is analyzed by artificial neural networks. Criteria available in the literature measure different properties of forecasts. To obtain better forecasts, Eǧrioǧlu et al. [1] proposed a criterion which can measure all properties of forecasts. Aladag et al. [2] improved the criterion proposed by [1] by using optimization. In this study, both the weighted information criterion proposed by Eǧrioǧlu et al. [1] and the adaptive weighted information criterion proposed by Aladag et al. [2] are introduced. These criteria are used in the architecture selection to analyze time series which are the import values of Turkey and the air pollution records in Ankara. As a result of computations, obtained results are compared and discussed. As a result of the comparison, it is seen that adaptive weighted information criterion produce more consistent results.

The accurate forecast of public expenditure is crucial for the success of the new public financial management approach developed in Turkey since the financial crisis of 2001. The public institutions are now obliged to align their expenditure with the framework shaped by the Public Financial Management and Control Law (No: 5018), the Middle-Term Programme of 2010-2012, and recently the Fiscal Rule envisaged to apply in the next budgetary period. This necessitates a better forecasting method than the traditional way of budget forecasting, which is typically based on the expenditures of previous years adjusted by inflation. Particularly focusing on the expenditure side of the budget, this chapter applies various artificial neural networks models to the expenditures of 1973-2008 of two Turkish public institutions, namely, the State Planning Organization and the Court of Accounts to achieve accurate forecast levels. The artificial neural networks approach is rarely applied for the forecasting of public expenditures, and as far as we know this is the first of such attempts involving Turkish data. The artificial neural networks application provided very accurate public expenditure forecasts for these public institutions, suggesting that the artificial neural networks is a very useful method for the public expenditure forecasting, as well.

Most of the time series faced in real life are fuzzy time series and these time series have to be forecasted by fuzzy time series forecasting methods. Therefore, there have been many studies in the literature in which various fuzzy time series approaches are proposed. The fuzzy time series methods introduced in the literature have been generally proposed to analyze fuzzy time series whose observations are fuzzy sets. On the other hand, Song et al. firstly improved a fuzzy time series model to analyze fuzzy time series whose observations are triangular fuzzy numbers [1]. Their method requires complex arithmetic operations for triangular fuzzy numbers. We propose a novel fuzzy time series forecasting approach based on simulation and feed forward neural networks to forecast fuzzy time series including triangular fuzzy numbers. The proposed method is applied to gold prices in Turkey series to show the applicability of the method.

The idea of exploring fuzzy set theory to time series forecasting issues has been enormously attracted researcher’s attention in recent years. Several new approaches on fuzzy time series have been put forward. These approaches have got some advantages related to classical methods and are complementary of them. Two of these kinds of procedures are FARIMA and FSARIMA. FARIMA and FSARIMA do not require a restriction of at least 50 observations and linearity assumption. The methods of FARIMA and FSARIMA provide interval estimates of a time series. ARIMA and SARIMA also provide interval estimation but it has been put forward that estimated intervals are large, therefore not informative. The width of estimated intervals obtained from FARIMA and SARIMA may generally tend to be less than ones from ARIMA and SARIMA. In the literature, there has been no study which provides a criterion for the comparisons of time series with respect to interval estimates. In this study, two criteria for such comparisons are presented.

Due to the vagueness that they contain in their observations, fuzzy time series models worked in two main categories such as first order and high order models, has an ever expending field of study. Fuzzy time series analysis method is highly effective in uncovering the relations of this type of time series structure. In the implementation of fuzzy time series methods, it is crucial to determine the model order in terms of forecasting performance. Besides, regardless of the model order, the length of interval determined in the partition phase of the universe of discourse, greatly affects forecasting performance. Therefore, there have been numerous studies focusing on determining the length of interval in the literature. This study aims to introduce the significance of interval length determination in fuzzy time series analysis method on forecasting performance. For this purpose, related methods are introduced, implementation of two real time series is shown and some comparisons between methods are made and finally obtained results are discussed.

Various theoretical assumptions in conventional time series methods do not need to be checked in fuzzy time series approach. Therefore fuzzy time series are preferred in many applications. The identification of the length of intervals is an important issue and affects the forecasting performance. But in many studies in the literature, the length of intervals is determined randomly. Starting from this point, Huarng [1] has proposed two novel approaches which are based on the distribution and the average to choose a more effective length. Huarng and Yu [2] used a dynamic approach for adjusting lengths of interval. Huarng [3] suggested a different method which is called ratio based lengths of intervals. Cheng et al. [4] have proposed a new approach by using entropy. Eǧrioǧlu et al. [5] and Yolcu et al. [6] have determined the lengths of intervals by using optimization. At the first stage of the method proposed by Cheng et al. [4], a specific method has not been used and classes have been assigned intuitively while classes to which data belong were generating. In this study, the approach proposed by Degirmenci et al. [7] is applied to the enrollment data at the University of Alabama and the yearly data of the quantities of clean water used in Istanbul. Then obtained forecasts are compared with those obtained from other methods available in the literature.

In recent years, the most preferred forecasting method in time series forecasting has been artificial neural networks. In many applications, artificial neural networks have been successfully employed to obtain accurate forecasts in the literature. This approach has been preferred to conventional time series forecasting models because of its easy usage and providing accurate results. On the other hand, there are still some problems with using this method. Fining a good artificial neural network architecture which gives the most accurate forecasts is an important issue when the method is used for forecasting. Although, there are some systematical methods proposed to determine the best architecture, the most preferred method is trial and error method [1]. To solve the architecture selection problem, Aladag [2] also proposed an approach based on tabu search algorithm. In this study, the air pollution in Ankara time series is forecasted by utilizing artificial neural networks and the architecture selection algorithm proposed by Aladag [2] is used to determine the best architecture. The obtained results show that high accuracy level is reached when Aladag’s [2] algorithm is employed.

Fuzzy time series, subjected to many scientific studies, have been used in forecasting in recent years. Due to their uncertainty, time series encountered in daily life should be perceived as fuzzy time series and analyzed by fuzzy time series methods. Instead of representing time series, which may have different values during the time they measured, by instantaneous value of each observation, representing a fuzzy set which may contain several values provides more information and thus more realistic analyses. In such a situation, forecasting problem of time series whose observations are fuzzy sets emerges. In the literature, there are several methods and algorithms proposed for forecasting these types of fuzzy time series. However, one can say that most of the observed fuzzy time series contain seasonal structures. From this stand point, using seasonal fuzzy time series forecasting methods in analyzing fuzzy time series containing seasonal relations would be effective in terms of both forecasting performance and explanation of the relation of the data contained in. This study aims to introduce a partial high order bivariate fuzzy time series forecasting method hybridized with Box-Jenkins method seasonal autoregressive integrated moving average model (SARIMA), one of the conventional time series analysis methods used in forecasting seasonal time series, and its advantages. For this purpose, two real data are analyzed using this seasonal fuzzy time series forecasting method and results are evaluated with certain fuzzy and conventional seasonal time series methods.

Forecast combination is a method used for obtaining more accurate forecasts. Forecast combination consists of the combination of forecasts obtained from different models with various methods. There are several types of forecast combination in the forecasting literature. In this study, various fuzzy time series approaches are applied to Turkey’s daily highest gold prices series and forecasts obtained from these approaches are combined with variance covariance method (VCM), mean square forecast error method (MSFE) and artificial neural networks (ANN) approach. Results obtained from all of these methods are analyzed and the optimal forecast technique for Turkey’s daily highest gold prices series is determined.

Fuzzy time series approaches have been recently used for forecasting in many studies [1]. These approaches can be categorized into two subclasses that are univariate and multivariate approaches. It is a fact that many factors can actually affect real time series data. Therefore, using a multivariate fuzzy time series forecasting model can be more reasonable in order to get more accurate forecasts. The most preferred method is using tables of fuzzy relations for determining fuzzy relations in multivariate fuzzy time series approaches in the literature. However, employing this method is a computationally though task. In this study, we propose a new method based on utilizing artificial neural networks in determining fuzzy logic relations and using the formula defined by Jilani and Burney [2] in calculating defuzzyfied forecasts. Hence, it is aimed to produce more accurate forecasts and avoid intense computations. The proposed method is applied to the time series data of the total number of annual car road accidents casualties in Belgium from 1974 to 2004 and a comparison is made between our proposed method and the methods proposed by Jilani and Burney [2] and Lee et al. [3].