Definition 6 If takes all value in (0,1), we can obtain a system of timeseriesforecastingmodel SDR ( ). The sum of system of timeseriesforecastingmodel SDR ( ) is called set of fuzzytimeseriesforecastingmodel based on difference rate, for short, it is written as SFTSFMBDR (The Set of FuzzyTimeSeriesForecasting Models Based on the Difference Rate). Its general element is SDR( ). SDR ( ) denotes a forecasting formula of fuzzytimeseries and a fuzzytimeseriesforecastingmodel.
Singh (2008, 2009) presented the computational method of forecasting based on fuzzytimeseries have been developed to provide improved forecasting results to deal with the situation containing higher uncertainty due to large fluctuations in consecutive year’s values in the timeseries data and having no visualization of trend or periodicity. He has developed the order three and higher order difference to use a time variant difference parameter on current state to forecast the next state. Ruey-Chyn Tsaur,(2012) developed the fuzzytimeseries-Markov chain approach for analyzing the linguistic or a small sample timeseries data is proposed to further enhance the predictive accuracy. In this method to transferred fuzzytimeseries data into fuzzy logic group, and using the obtained fuzzy logic group to derive a Markov chain transition matrix. Wangren Qiu et al., (2011) presented ensemble technique an effective method for improving the classification accuracy in data mining area. The ensemble technique was applied to fuzzytimeseries and improves that Song’s and Chissom (1993a, b), Chen (1996) and Lee et al. (2009) models can be approximated by the proposed model via the limitation of the fuzzy weights. The impact on the performance of the proposal model is discussed. Both university enrollment and Shanghai stock index are chosen as the forecasting targets. The empirical results not only testify the above assertion, but also show that the proposed model can provide better overall forecasting results than the previous models with appropriate parameters. Enjian Bai (2011) presented a simple heuristic time-invariant fuzzytimeseriesforecastingmodel, which was used to prediction accuracy of model observations to train the trend predictor in the training phase, and uses this trend predictor to generate forecasting values in the testing phase.
Abstract. In order to solve the question which the existing fuzzytimeseriesforecastingmodel prediction accuracy is not high, this paper proposes the fuzzytimeseriesforecastingmodel based on differential collection of SD. The general elements expressed in SD( ). Prove: IF a timeseriesforecasting method requires AFER<B (an arbitrary small positive number) and MSE<C (a positive number), then there exist independent variables 0 (0,1), when it uses forecastingmodel DR for timeseries prediction model to simulate the problem of history data prediction research, to ensure that the average prediction error rate AFER B and mean square error MSE C was established at the same time.
Abstract Fuzzytimeseriesforecasting models are used to overcome traditional timeseries methods when the historical data of traditional timeseries approaches contain uncertainty or need to be represented by linguistic values. Besides, fuzzytimeseriesforecasting methods do not require any assumption valid. Generally, fuzzytimeseriesforecasting methods consist of three major stages such as fuzzification, determination of fuzzy logic relationships or fuzzy relationship matrix, and defuzzification. All these stages of fuzzytimeseries are very important on the forecasting performance of the model. In this paper, a new hybrid fuzzytimeseriesforecastingmodel is proposed based on three computational approaches such as: the new concept of time-variant fuzzy relationship group is used to establish time-variant fuzzy relationship group in the determination of fuzzy logical relationships stage, named called the time - variant fuzzy logical relationship groups (TV-FLRGs), the proposed forecasting rules is applied to calculate the forecasting value for the TV-FLRGs and particle swarm optimization technique (PSO) is aggregated with TV-FLRGs to adjust interval lengths and find proper intervals in the universe of discourse with the objective of increasing forecasting accuracy. To verify the effectiveness of the proposed model, two numerical data sets are selected to illustrate the proposed method and compare the forecasting accuracy with existing methods. The results show that the proposed model gets a higher average forecasting accuracy rate to forecast the Taiwan futures exchange (TAIFEX) and enrolments of the University of Alabama than the existing methods based on the first – order and high-order fuzzytimeseries.
During last few decades, various approaches have been developed for timeseriesforecasting. Among them ARMA models and Box-Jenkins model building approaches are highly famous. In recent years, many researchers used fuzzytimeseries to handle prediction problems. Song and Chissom  presented the concept of fuzzytimeseries based on the concepts of fuzzy set theory to forecast the historical enrollments of the University of Alabama. Huarng  presented the definition of two kinds of intervals in the universe of discourse to forecast the TAIFEX. Chen  presented a method for forecasting based on high-order fuzzytimeseries. Lee  presented a method for temperature prediction based on two-factor high-order fuzzytimeseries. Melike  proposed forecasting method using first order fuzzytimeseries for forecasting enrollments in University of Alabama. Lee  Presented handling of forecasting problems using two-factor high order fuzzytimeseries for TAIFEX and daily temperature in Taipei, Taiwan. In (, , , ), Jilani and Burney presented a number of new fuzzy metrics for multivariate fuzzytimeseriesforecasting for car road casualties data, Taipei temperature data and enrollments data problem of University of Alabama.
In stock market, it is well known that historic timeseries imply the fluctuation rules and can be used to forecast the future of its fluctuation trends. In 1993, Song and Chissom proposed the fuzzytimeseriesforecastingmodel [25-27]. Since then, researchers have proposed various fuzzytimeseriesforecasting models and employed them to predict stock market[3, 5-6, 21], electricity load demand[13, 22], project cost, and the enrollment at Alabama University[14, 24], etc. In order to improve the accuracy of the forecastingmodel, some researchers combine fuzzy and non-fuzzytimeseries heuristic optimization methods for stock market forecasting [1, 19-20, 30].
by using a simple fuzzytimeseriesforecastingmodel. In recent years, some researchers focused on the research topic of using fuzzytimeseries for handling forecasting problems, such as forecasting the enrollments  - , crop forecast , , stock index prediction , , the temperature prediction . Chen also extended his previous work  to present several forecast models based on the high- order FTS to deal with the enrollments forecasting problem , . Huarng  pointed out that the different lengths of intervals in the universe of discourse can affect the forecasting result and a proper choice of the length of each interval can greatly improve the forecasting accuracy rate. In other words, the choice of the length of intervals can improve the forecasting results. Ref.  presented a heuristic model for fuzzyforecasting by integrating Chen’s fuzzyforecasting method . In  the length of intervals for the FTS model was adjusted by the K- mean clustering algorithm to forecast the enrollments. Some other techniques for determining proper intervals and interval lengths is used automatic clustering technique ,  and particle swarm optimization algorithm , . Additionally, in  proposed a new method to forecast Temperature and TAIFEX based on automatic clustering algorithm and two – factors high – order fuzzytimeseries.
T his paper deals with the design of forecastingmodel for annual district outlay for district Nagpur using fuzzytimeseries with first order differencing. Paper suggested modification in FuzzyTimeSeries algorithm while defuzzification to forecast values of annual outlay. In fuzzytimeseries numerous modifications were suggested by various authors to improve forecasting accuracy or computation methodology. It is shown that proposed model achieves a significant improvement in forecasting accuracy as compared to fuzzytimeseriesforecasting with equal weight method while defuzzification. To illustrate the forecasting process, the historical annual district outlay for Nagpur district is used.
An ordered sequence of observed values is known as timeseries. If the observed values represent measured values, it is often not possible to assign precise numerical values to the observed data, they then possess data uncertainty. This paper concerns with the timeseries comprised of imprecise i.e., uncertain observed values. In the case of timeseries the uncertainty of the individual observed values as well as the interpretation of a sequence of uncertain observed values are of interest. The uncertain observed value is thus modeled as a fuzzy variable. Fuzzy sets represent concepts such as low etc. are called fuzzy variables. Modeling of the individual observed values as fuzzy variables results in fuzzytimeseries. Forecasting using fuzzytimeseries has been widely used in many activities. It arises in forecasting the weather, earthquakes, stock fluctuations and any phenomenon indexed by variables that change unpredictably in time The traditional timeseriesforecasting methods cannot be used for forecasting problems in which the historical data are linguistic values. Song and Chissom (1993, 1994) proposed time variant and time invariant fuzzytimeseries models and fuzzyforecasting to model and forecast processes whose observation are linguistic values. Instead of complicated maximum minimum composition operations Chen (1996) used a simple arithmetic operation for timeseriesforecasting. Thereafter a number of related research works have been reported that follow their framework and aim to improve forecasting accuracy and reduce the computational overhead. These works include enrollments, length of intervals, temperature prediction, weighted method, stock price, hidden Markov model, genetic algorithm, neural – fuzzy system, bulk shipping, seasonal and heuristic models.
forecasting techniques to forecast real problems, such as forecasting stock market, forecasting enrolments, temperature prediction, population growth prediction, etc. In recent years, many researchers used fuzzytimeseries to handle prediction problems. When forecasting these problems based on fuzzytimeseries, it is obvious that the length of intervals in the universe of discourse is important because it can affect the forecasting accuracy rate. However, some of the existing fuzzyforecasting methods based on fuzzytimeseries used the static length of intervals, i.e., the same length of intervals. The disadvantage of the static length of intervals is that the historical data are put into the intervals in a rough way, even if the change of the historical data is not large. Therefore, the forecasting accuracy rates of the existing fuzzyforecasting methods are not good enough. Consequently, we need to propose a new fuzzyforecasting method to overcome the drawbacks of the existing forecasting models to increase the forecasting accuracy rates. In this paper, a hybrid forecastingmodel based on two computational methods, the fuzzy logical relationship groups and clustering algorithm, is presented for forecasting enrolments and the Taiwan Futures Exchange (TAIFEX). Firstly, we use the automatic clustering algorithm to divide the historical data into clusters and adjust them into intervals with unequal lengths. Then, based on the new intervals, we fuzzify all the historical data of the enrolments of the University of Alabama and calculate the forecasted output by the proposed method. Compared to the other methods existing in literature, particularly to the first-order fuzzytimeseries and high – order fuzzytimeseries using two data sets: the historical data of the enrolments of the University of Alabama and the stock index data set of TAIFEX, our method gets a higher average forecasting accuracy rate than the existing methods.
This paper presents the development of fuzzytimeseries; time invariant model development and its implementation in forecasting the agricultural production. It contains a comparative study of three models and their testing for the forecasting of Lahi crop production of University farm. The timeseriesforecasting is based on the historical data of 21 years of University farm. The robustness of the models is examined on the basis of error estimates. The study reveals some interesting feature of fuzzytimeseriesforecasting for the production of Lahi crop and can be used for short term forecasting of agriculture crop production.
In the present work the time invariant model have been developed refined and applied on the forecasting of Lahi crop production. The objective of applying the fuzzytime invariant seriesforecasting models is to develop better forecasting models for the prediction of crop yield, a real non-deterministic process. Further, the area specific crop yield forecasting will help in better crop and agro based business planning and can be used in economics and business analysis.
 Wang, N.-Y, & Chen, S.-M. Temperature prediction and TAIFEX forecasting based on automatic clustering techniques and two-factors high-order fuzzytimeseries. Expert Systems with Applications, 36, 2143–2154, 2009.  Kuo, I. H., Horng, S.-J., Kao, T.-W., Lin, T.-L., Lee, C.- L., & Pan. An improved method for forecasting enrollments based on fuzzytimeseries and particle swarm optimization. Expert Systems with applications, 36, 6108–6117, 2009a.
HE traditional prediction models, such as the autoregressive moving-average model, the bilinear model and the nonlinear autoregressive model, are only suitable for the precise number series. In many applications, such as the electric power load, the prices of oil, stock, or gold, the daily air temperature, and the exchange rate, the observed data fluctuate at any time. Then using interval fuzzy numbers to represent these data is more reasonable than using precise data. In , Zadeh introduced fuzzy number firstly, and fuzzy mathematics has been applied to many fields . The common types of interval fuzzy numbers are binary interval fuzzy number, triangular fuzzy number (or trinary interval fuzzy number) and trapezoidal fuzzy number. In this article, we consider the forecasting of interval and triangular fuzzy number series.
Abstract: Forecasting accuracy is one of the most favorable critical issues in Autoregressive Integrated Moving Average (ARIMA) models. The study compares the application of two forecasting methods on the amount of Taiwan export, the Fuzzytimeseries method and ARIMA method. Model discussed for the ARIMA method and Fuzzytimeseries method include the Sturges rules. When the sample period is extend in our models, the ARIMA models shows smaller than predicted error and closer predicted path to the realistic trend than those of the Fuzzy models, resulted in more accurate forecast of the export amount the Autoregressive Integrated Moving Average models. In the economic viewpoints, the amount of Taiwan export is mainly attributable to external factors. However, this impact reduces with time and export amount in the timeseries analysis. The ARIMA models can be utilized to predicted export value accurately, when all of value or data is available.
From Table 7, we can see that the proposed method by modifying Fourier series with number of interval is seven which has a smaller MSE value of 8.57 and MAPE value of 0.0175% than SCI model , the C96 model , the H01 model , S.R Singh model . To be clearly visualized, Fig. 1 depicts the trends for
Since the order of the model significantly affects the performance, the order of the model for a timeseries is determined by using ACF and PACF function of the fuzzified series. In order to determine the length of interval, computationally efficient single-variable constrained optimization based method is employed. In addition, a one-pass fast learning GRNN model is used to represent the fuzzy logical relationships. Ten timeseries data sets are considered for evaluating the efficiency of the proposed model. In addition to the proposed model, three recent fuzzyforecasting models Aladag et al., Egrioglu et al.  and Yolcu et al.  are implemented. Each model is applied on every timeseries dataset for fifty different times and forecast accuracy is measured. The proposed model statistically outperformed all other models in four timeseries datasets considering both RMSE and SMAPE measure. The proposed model achieved statistically significantly better RMSE than Aladag in four timeseries datasets, Egrioglu  in nine timeseries datasets and Yolcu  in six timeseries datasets. The proposed model achieved statistically significantly better SMAPE than Aladag in six timeseries datasets, Egrioglu  in nine timeseries datasets and Yolcu  in six timeseries datasets. Moreover, the proposed model only provided statistically poor performance than each of the considered model in one dataset. Furthermore, to identify the best model considering all timeseries datasets, a Friedman and Nemenyi hypothesis test is being conducted on the obtained SMAPE measures. The test results revealed that the proposed model has the lowest rank SMAPE and acquired the best rank. From experimental results it can be concluded that the proposed model is statistically superior in terms of forecasting accuracy when compared with other alternatives considered in this study. Additionally, it is computationally very efficient. This can be understood from the fact that, in addition to the use of one-pass fast learning GRNN model for representing FLR, we have used the computationally efficient method in every step found in the literature so far.
Figures 3 show the numerical results of the RMSE for the simulation testing data set for the variance error and 4 parameters by the ARIMA and fuzzytime series.the numerical results under the three interest timeseries models mentioned above are highly consistent (the values almost fall on the same line but the chen model is equivalent to the deterministic model in this case). For the Chen model showed that best model to be used for forecast because it has lowest RMSE value compared to other model in each variance error term being selected.From figure 4 above it can be seen that the best method to forecast the timeseries data is the Chen fuzzytimeseriesmodel, because both the RMSE and MAPE errors values for this method has lower value compared to other methods and also contained Bias statistics with higher values than Bias statistics of other methods.
Abstract — In the past years, most of the fuzzyforecasting methods based on fuzzytimeseries used the static length of intervals, i.e., the same length of intervals. The drawback of the static length of intervals is that the historical data are roughly put into intervals, even if the variance of the historical data is not high. In this paper, an improved forecastingmodel is used to forecast the student enrolment at the University of Alabama. Firstly, a method of unequal-sized intervals partitioning based on K-means clustering algorithm is proposed. Secondly, used fuzzy logical relationship groups in determination of fuzzy relations stage to overcome the defect of traditional fuzzification method. Finally , to verify the effectiveness of the approach, we apply the proposed method to forecast enrolment of students of Alabama University. The experimental results show that the proposed method get higher forecasting accuracy rates than the existing methods with various orders and under different number of intervals.
In the last decade, fuzzytimeseries has been widely used for forecasting data of dynamic and non-linear data in nature. Many previous studies have been discussed for forecasting used fuzzytimeseries such as enrollment [1-3, 12, 15, 17, 22, 23, 26], the stock index [14, 16, 18, 19, 21, 24, 25, 27], temperature  and financial forecasting , etc. The accuracy of the proposed forecasting methods is one of the main issues discussed. The enrollment forecasting studies have been presented in many methods and models such as regression models, timeseries models and artificial intelligences. Most of them were done for real data or numerical data. It is very important to make reasonably accurate estimates of the future enrollment for a university because many decisions can be elaborated from them. However, obtaining accuracy is not easy task, as many variables have impacts on enrollment numbers. Many different methods and models have been written by researcher used fuzzytimeseries for enrollments forecasting . Song and Chissom  initiated a study on time- invariant and time-variant models for forecasting with fuzzytimeseries using enrollment data of Alabama University. In this paper, the study is concern to adopt the weighted on fuzzytimeseriesforecasting. Yu  proposed the used of weighted fuzzytimeseries models for Taiwan stock index (TAIEX) forecasting. It is assigned by the recurrent fuzzy logical relationships (FLRs) in fuzzy logical group (FLG). In establishing fuzzy relationship and forecasting are important step to consider the weighted factor. Cheng et al.  presented the trend-weighted fuzzytimeseriesmodel for TAIEX forecasting.