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CHAPTER 2
REVIEW OF RELATED WORKS 2.1 INTRODUCTION
Time series plays a vital role in the branches of Stock prices,
Global surface temperature, Economy Population growth and
agricultural production. Time series is the sequence of observations
ordered in time. Mostly these observations are collected at equally
spaced and discrete time interval. The basic assumption of time series
analysis is some aspects of the past pattern which will continue to
remain in the future. In recent years, many researchers have used
fuzzy time series to handle forecasting problems.
In this thesis, Markov modeling is a major statistical tool used
to predict the fuzzy time series data. The advantage of using Markov
modeling gives better forecasting accuracy for predicting time series
data. Many works are related to fuzzy time series in the applications of
stock price prediction, agriculture commodities prediction, and
Economy and university enrollment data. The works related to Markov
modeling fuzzy time series can be discussed.
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2.2 MARKOV MODEL FUZZY TIME SERIES (MM-FTS)
Fuzzy Time Series (FTS) is introduced by Song and
Chissom [1993a]. Several fuzzy time series model is developed in the
literature during the last two decades. Zadeh [1965] discussed fuzzy
sets and it is characterized by a membership function which assigns
the value to each object grade membership ranging from zero to one.
Song and Chissom [1993b] have initiated the study of time
invariant and time variant forecasting models using fuzzy time series
for enrollment data of University of Alabama. Also compare the
predicted values of fuzzy time series with those of non linear
regression models.
Song et al., [1993a] have used time-invariant fuzzy time series
model, while Song and Chissom [1994] used time-variant model for
the same problem. In their example, the crisp data is fuzzified into
linguistic values to illustrate the fuzzy time series method using fuzzy
set theory and its modeling by applying fuzzy relations equations and
approximate reasoning.
Sullivan et al., [1994] reviewed the first-order time-variant fuzzy
time series model and the first-order time-invariant fuzzy time series
model presented by Song and Chissom [1993b], where their models
are compared with each other and with a time-variant Markov model
using linguistic labels with probability distributions. The results of
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this model enrollment data are compared with three traditional time
series models such as a first order auto regressive model and second
order auto regressive models. However, the Markov model has an
advantage that any repeated transitions in the data are taken into the
account of estimating the model parameters, whereas the fuzzy time
series model based on the usual max-min operations cannot perform
the same.
Song et al., [1994] proposed an approach for developing the
time-variant fuzzy time series models with an example which
presented the process of forecasting the enrollment for the University
of Alabama. To defuzzify the output of the model, a 3-layer back
propagation neural network was trained and used as the defuzzifier.
Among the three different defuzzification methods, neural network
method yielded the best result.
Song et al., [1995] have constructed fuzzy time-series model by
means of defining some new operations on fuzzy numbers.
Chen [1996] developed a simplified method for time series forecasting
using the arithmetic operations rather than using complicated
maximum – minimum composition operations. The proposed method
not only gives good forecast for the University enrollments, but also
gives robust forecast even when the historical data are not accurate.
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Ranaweera et al., [1996] proposed a fuzzy logic model for short
term load forecasting which was applied to historical weather and load
data. Since Conventional fuzzy systems cannot operate with random
phenomena, the Control processes in real-life plants were used.
They have deal the stochastic processes as Markov modeling
approach.
Song et al., [1997] have discussed fuzzy stochastic fuzzy time
series and three different models. Ping-Teng Chang [1997] has
developed a fuzzy technique for trends and seasonality forecasting
through fuzzy regression and fuzzy trend of time-series model. Fuzzy
forecasting and the analysis of the fuzziness of the forecasts are the
features of this method which differs from the traditional forecasting
techniques for seasonality.
Jeng- Ren Hwang et al., [1998] have proposed new method for
handling forecasting problems based on time variant fuzzy time series.
In the proposed method where the variation of enrollment of the
current year is related to the trend of the enrollments of past years.
Song [1999] has developed a seasonal forecasting model for fuzzy time
series with minor modifications of fuzzy logical relationship and the
membership functions. The model can be used directly for seasonality
forecast.
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Sullivan and Woodall [1999] have discussed three methods for
estimating Markov transition matrices when observed state
probabilities are not all either zeros or ones and a simulation-based
comparison of the performance of the estimators.
Fang-Mei Tseng et al., [1999] have proposed the method of fuzzy
seasonal time series for forecasting the production value of the
mechanical industry in Taiwan. This method provided the decision
makers with insight regarding the possible future situation. Chen et
al., [2000] have discussed fuzzy time series method and adapted to
temperature data in Taiwan.
Nguyen et al.,[2002] has developed the mathematical modeling
of domains of linguistic variables, which gives a unified algebraic
approach to the natural structure of domains of linguistic variables.
Tapio Frantti et al., [2001] have obtained an anticipatory Fuzzy Logic
Advisory Tool (FLAT) for predicting the demand for signal transmission
device. The design principle for a general predictor is also explained.
The algorithm is implemented as a practical computer program and it
is applied to real manufacturing data.
Fang-Mei Tsenga et al., [2001] have developed Fuzzy ARIMA
(FARIMA) model and applied it for forecasting the exchange rate of NT
dollars to US dollars. With the help of this model it is possible for
decision makers to forecast the best and worst possible situations
based on fewer observations than the ARIMA model.
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Fang-Mei Tsenga et al., [2002] have proposed fuzzy seasonal ARIMA
(FSARIMA) forecasting model. It is used to forecast two seasonal time
series data of the total production value of the Taiwan machinery
industry and the soft drink time series data.
Song [2003] has proposed the sample autocorrelation functions
of fuzzy time series and used in model selection. The main idea is to
select a number of different data sets from each fuzzy set and
calculate the sample auto correlation function for each data set. Three
different auto correlation functions are proposed and examples are
also presented. Javier Contreras et al., [2003] have proposed a method
to predict next-day electricity prices based on ARIMA methodology.
Yimin Xiong et al., [2004] have proposed a model-based method for
clustering univariate and simple multivariate ARMA time series.
Hsuan-Shih Lee et al., [2004] have presented an improved
method to forecast university enrollments based on the fuzzy time
series. The method proposed not only defines the supports of the fuzzy
numbers that represent the linguistic values of the linguistic variable
more appropriately, but also makes the RMSE smaller.
Melike Sah et al., [2005] have proposed a method to attain better
forecasting accuracy using time-invariant fuzzy time series. It should
be emphasized that it uses only historical data in the numerical form
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without any addition pieces of knowledge for forecasting. In addition
the method is tested on different number of fuzzy sets for the purpose
of examining the forecasting accuracy.
Ruey-Chyn Tsaur et al., [2005] have proposed fuzzy relation
matrix affecting the forecasting performance and proposed an
arithmetic procedure for deriving fuzzy relation matrix method using
Fuzzy relation analysis in fuzzy time series. Fuzzy relation is a crucial
connector in presenting fuzzy time series model. Also the concept of
entropy is applied to measure the degrees of fuzziness when a time
invariant matrix is derived.
Hui-Kuang Yu [2005] has proposed weighted models to tackle
two issues in fuzzy time series forecasting, viz., recurrence and
weighting. It is argued that recurrent fuzzy relationships, which are
simply ignored in previous studies, should be considered in
forecasting. It also recommended that different weights shall be
assigned to various fuzzy relationships. The weighted model is
compared with local regression models in which weight function plays
an important role in forecasting.
Ping-Feng Pai et al., [2005] have proposed autoregressive
integrated moving average (ARIMA) model which has become one of
the most widely used linear models in time series forecasting.
Nai-Yi-Wang et al., [2009] have presented a new method to predict the
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temperature and the Taiwan Futures Exchange (TAIFEX), based on
automatic clustering techniques and two-factor high-order fuzzy time
series. Tasha Inniss [2006] has applied this technique to weather and
aviation data to determine probabilistic distributions of arrival
capacity scenarios, which can be used for seasonal forecasting and
planning.
Alonsoa et al., [2006] have proposed time series method based
on the probability density of the forecasts using nonparametric kernel
estimator. Hao-Tien Liu [2007] has proposed improved time-variant
fuzzy time series method. The proposed method takes into
consideration of Window base, length of interval, degrees of
membership values, and existence of outliers. The improved method
provides decision makers with more precise forecasted values.
Sheng-Tun Li [2007] has proposed a deterministic forecasting
model to manage the crucial issues. In addition, an important
parameter, the maximum length of subsequence in a fuzzy time series
resulting in a certain state, is deterministically quantified.
Hao-Tien Liu [2009] has proposed a method to design an improved
fuzzy time series forecasting method in which the forecasted value
would be a trapezoidal fuzzy number instead of a single point value.
Also, the decision analyst can gather information about the possible
forecasted ranges under different degree of confidence.
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Guoqi Qian et al., [2007] have proposed a feasible computing
method based on the Gibbs sampler. By this method, model selection
is performed through a random sample generation algorithm and also
a model of fixed dimension in which the parameter estimation is done
through the maximum likelihood method.
Hye-young Jung et al., [2008] have proposed rearranged interval
method to reflect the fluctuation of historical data and to improve the
forecasting accuracy of fuzzy time series. This is discussed and the
forecasting accuracy is evaluated. Empirical analysis shows that this
method in forecasting accuracy is superior to existing methods and it
fully reflects the fluctuation of historical data.
Ashraf Abd-Elaal et al.,[2010] have proposed a fuzzy time series
technique based on fuzzy c- mean clustering. This method is adapted
to university enrollment data and it is improved the forecasting
accuracy compared with existing models.
Srivastava [2011] has presented a new method which gives
short term forecast of agricultural production based on fuzzy time
series. The study used fuzzy set theory and applied in fuzzy time
series model and artificial neural network to forecast the production of
food grains. Pushparani Suri et al., [2011] have presented a simple,
systematic and iterative methodology for forecasting gold price. This
method is realized on fuzzy clustering and weighted least square.
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Hemant Kumar [2011] has proposed relation matrix method and
it is used to reduce its calculation. This method is most
remarkable one which gives the most plausible range of forecasting
and it is in the form of interval rather than a single value.
Seyed Morteza Hosseini et al., [2011] have demonstrated the results of
test data from the currency market shows that this combined
approach have to be more careful from existing fuzzy time series
model. The practical results obtained show that this model have a
good application.
2.3 HIDDEN MARKOV MODEL FUZZY TIME SERIES (HMM – FTS)
Rabiner et al., [1986] have proposed the theory and
implementation of Markov modeling technique and applied it to
speech recognition problems. Rabiner [1989] has developed theory of
hidden Markov models from the simplest concepts to the most
sophisticated models and focused on physical explanations of the
basic mathematics. Rabiner also illustrated the applications of the
theory of HMM’s to simple problems in speech recognition and pointed
out how the techniques could be applied to more advanced speech
recognition problems.
Juang et al., [1991] have reviewed the statistical method of
HMM's. It reveals that the strengths of the method lie in the consistent
statistical framework that is flexible and versatile, particularly for
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speech applications, and the ease of implementation that makes the
method practically attractive. It is pointed out some areas of the
general HMM method that deserves more attention with the hope that
increased understanding will lead to performance improvements for
many applications.
Chen et al., [2000] have proposed two factor time variant
algorithms based on fuzzy time series model and applied it to forecast
the daily temperature. Both the algorithms have the advantage of
obtaining good forecasting results.
Rafiul Hassan et al., [2006] have proposed a single Hidden
Markov Model based on clustering method, which identifies a suitable
number of clusters in a given dataset without using prior knowledge
about the number of clusters. Initially, the dataset is partitioned into
windows of fixed size based on the HMM log likelihood values. This
model provides a framework for identifying the most appropriate
number of clusters (windows of varying sizes). After determining the
number of clusters, the data values are then labeled and allocated to
clusters.
Ching-Hsue Cheng et al., [2007] have discussed the fuzzy time
series method based on rough set theory and this method is applied to
stock price index forecasting problem. Sheng - Tun Li [2008] has
studied fuzzy time series which has increasingly
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attracted much attention due to its salient capabilities of tackling
uncertainty and vagueness inherent in the data collected.
Ching-Hsue Cheng et al., [2008] have introduced a new multiple-
attribute fuzzy time series method based on fuzzy clustering
technique. The methods of fuzzy clustering are integrated in the
processes of fuzzy time series to partition datasets objectively and
enable processing of multiple attributes.
Sheng-Tun Li et al., [2009] have proposed a new forecasting
model based on Hidden Markov Model for fuzzy time series to realize
the probabilistic state transition and also conducted experiments of
forecasting a real-world temperature application to validate the better
accuracy of the proposed model achieved over traditional fuzzy time
series models.
Rafiul Hassan [2009] has presented a novel combination of
hidden Markov model and fuzzy models for forecasting stock market
data. The model is tested by preparing forecasts for the financial time
series data of six stock prices.
Ming Dong et al., [2009] have proposed the states of hidden
semi Markov models are used to represent the PM2.5 concentration
levels. The model parameters are estimated through modified forward
backward training algorithm. It can be used to predict PM2.5
concentration levels. Jens Runi Poulson [2009] has been developed
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new fuzzy time series method which combines aggregation and
particle swarm optimization techniques. By combining these
techniques, forecast rule can be individually tuned to match the data.
Sheng-Tun Li et al., [2010] have proposed a novel forecasting
model based on the Hidden Markov model by enhancing Sullivan and
Woodall’s work to allow handling of two-factor forecasting problems.
It is built upon an HMM in which the fuzzy relationships are
formulated as state transitions. So that it can handle two factor
forecasting problems. Moreover, in order to make a nature of
conjecture and randomness of forecasting more realistic, the Monte
Carlo method is used to estimate the outcome.
Edmundo de Souza e Silva et al., [2010] have investigated the
performance of a hidden Markov model in predicting future crude oil
price movements. Additionally, they developed forecasting
methodologies consists of three steps. First, they employed wavelet
analysis to remove high frequency price movements, and then the
hidden Markov model is used to forecast the probability distribution of
the price return accumulated over the next F days.
Suresh et al., [2011] have analyzed Global surface temperature
with CO2 data using hidden Markov model in fuzzy time series. This
method is used to forecast successive year’s of global surface
temperature data.
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2.4 HIGHER ORDER MARKOV MODEL FUZZY TIME SERIES
(HOMM – FTS)
Adrian Raftery et al., [1994] have introduced the mixture
transition distribution model for higher order Markov chains. Also
proposed a computational algorithm for maximum likelihood
estimation which is based on a way of reducing the large number of
constrains.
Chen [2002] has presented present a new method for handling
forecasting problem using a high-order fuzzy time series model, where
the historical enrollments of the University of Alabama are used to
illustrate the forecasting process.
Vasek Chavatal [1983] has discussed linear programming
problem formulated from optimization problem and solved by simplex
method. Adrian Raftery [1985] has introduced higher order Markov
chain model which combines realism and parsimony.
Ching et al., [2002] have proposed a multivariate Markov chain
model for modeling multiple categorical data sequences and also
developed efficient estimation methods for the model parameters.
Ching et al., [2003] have proposed a new higher-order Markov model
whose number of states is linear and also developed a new parameter
estimation method based on linear programming.
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Tie Liu [2010] has adopted Markov chains model to analyze and
predict the time series. Some series can be expressed by a first-order
discrete-time Markov chain and others must be expressed by a higher-
order Markov chain model. Ching et al.,[2008] has proposed an nth-
order multivariate Markov chain model for modeling multiple
categorical data sequences such that the total number of parameters
are of O(ns2m2).
Yutong Li et al., [2008] have presented a stochastic simulation
approach to synthetic series of weather data to evaluate the
performance of open cycle solar desiccant air condition system of
Hong Kong. The results reveal that the open cycle desiccant system
can meet most of the latent load through out the cooling season if the
components are proper sized and energy savings.
Zhilong Wang et al., [2009] have presented a higher-order
multivariate Markov chain model combined with particle swarm
optimization algorithm, capable of searching for the optimal parameter
values η for level characteristics value to obtain a high accuracy model
for forecasting of multidimensional time series. Particle swarm
optimization algorithm is employed to optimize the coefficient of level
characteristics value.
Ersoy Oz [2011] has discussed the application of monthly
changes of the US Dollar selling rates and the monthly changes of the
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Euro selling rates. The two changes are taken into consideration as
two categorical data sequences and it is revealed with multivariate
Markov chain model to what degree these sequences affect each other
in Turkey.
Chi-Chen Wang et al., [2011a] have compared the applications
of the forecasting methods Auto Regressive Integrated Moving Average
(ARIMA) time series model and fuzzy time series model by heuristic
models on the amount of export values in Taiwan.
Chi-Chen Wang [2011b] has attempted to use information of
export values in Taiwan as an example to test whether the fuzzy time
series is indeed practical in its forecast of macro economic variables.
By comparing fuzzy time series with ARIMA time series method, better
understanding of the appropriateness of forecasting model could be
obtained.
Hongxing Yang et al., [2011] have introduced a new method to
generate the annual weather data by using the first order multivariate
Markov chain model. The weather variables are described in a
stochastic way and multiple categorical sequences are generated by
similar source.
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2.5 HIGHER ORDER COMPUTATIONAL FUZZY TIME SERIES
Huarng [2001a] has proposed an effective length of intervals to
improve the forecasting accuracy and introduced two methods such as
average based and distribution based methods. Distribution based
length is the largest length which is smaller than at least half of the
first differences of data. Average based length is set to one half of the
average of the first differences of data. Both can be used as effective
lengths to improve forecasting in fuzzy time series.
Huarng [2001b] has demonstrated how the heuristic model
outperforms of Chen’s and other models using the enrollment
forecasting at the University of Alabama. This study assumes that
there is heuristic knowledge showing the increase or decrease in
enrollment for the next year. With this heuristic, the heuristic model
forecasts the enrollment better than the other models. In the heuristic
models, the heuristic knowledge is used to guide the search for
suitable fuzzy sets for index forecasting.
Chen [2002] has presented a forecasting method based on high-
order fuzzy time series. From the proposed model he developed an
algorithm to forecast the enrollments of the University of Alabama,
where the historical enrollment data at the University of Alabama are
used to illustrate the forecasting process.
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Chen et al., [2004] has proposed fuzzy time series belonging to
first order and time–variant methods. It is possible to get higher
forecasting accuracy rate for forecasting enrollments than the existing
methods.
Sheng –Tun -Li et al., [2004] have presented a new approach to
handling the issue by applying the natural partitioning technique,
which can recursively partitioned the universe of discourse level by
level in a natural way. Experimental results found that the enrollment
data of the University of Alabama results can forecast the data
effectively and efficiently and outperforms the existing models.
Chung-Ming Own et al., [2005] have proposed an enhanced
fuzzy time series model, called heuristic high-order fuzzy time series
model, to deal forecasting problems. The proposed method eliminates
ambiguities at forecasting and requires a vast memory for data
storage. The empirical analysis reveals that the proposed method yield
more accurate forecasts. Moreover, the forecasting model can be
restricted in the acceptable-order fuzzy time series to reduce the
memory needed for the data storage.
Huarng et al., [2006] have suggested that ratios, instead of
equal lengths of intervals, including distribution based length and
average based length can more properly represent the intervals among
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observations. The method also suggests that ratio-based lengths of
interval can be applied to improve fuzzy time series forecasting.
Ching-Hsue Cheng et al., [2006] have proposed two approaches
to improve the persuasiveness in determining the universe of
discourse, length of intervals and membership functions of fuzzy time
series. The first approach is using Minimize entropy principle
approach (MEPA) to partition the universe of discourse and build
membership functions, and the second is using Trapezoid fuzzification
approach (TFA).
Li-Wei Lee et al., [2006] have proposed a new method for
forecasting temperature and TAIEX, based on two factor high order
fuzzy time series. The two factor high-order fuzzy logical relationship
is used to increase the forecasting accuracy rate of prediction.
Shiva Raj Singh [2007] proposed a simple time variant method
for fuzzy time series forecasting. This method overcomes the deficiency
of ambiguity in trends of data and also does not need the heuristic
function. It provides simple computational algorithms for complexity
in linear order. It minimizes the time of generating relation equations
by using min-max composition operations and the time consumed by
the various defuzzification processes to be applied for getting crisp
forecast. The proposed algorithm is implemented for forecasting
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enrollments of university of Alabama and the results are compared
with the existing methods to show its superiority.
Tahseen Ahmed Jilani et al., [2008d] have presented new fuzzy
metrics for high order multivariate fuzzy time series forecasting for car
road accident casualties in Belgium. Sun Xihao et al. [2008] have
proposed Average – based fuzzy time series model, which can be used
to adjust the lengths of intervals determined during the early stages of
forecasting and that method is applied to daily stock index in
shanghai.
Tahseen Ahmed Jilani et al., [2008b] have proposed a new
method for time series forecasting, having simple computational
algorithm of complexity of linear order. The method first predicts the
trend of the future value and then use the quantile based fuzzy
forecasting approach.
Tahseen Ahmed Jilani et al., [2008a] have presented two new
multivariate fuzzy time series forecasting methods. These methods
assumed m-factors with one main factor of interest. Stochastic fuzzy
dependence of order k is assumed to define general methods of
multivariate fuzzy time series forecasting and control. This method
provided a general work for forecasting, that can be increased by
increasing the stochastic fuzzy dependence and the simplicity of
computation used triangular membership function.
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Tahseen Ahmed Jilani et al., [2008c] have proposed other
method based on frequency density partitioning applied for historical
enrollments of Alabama. The proposed method uses heuristic
approach to define frequency-density –based partitions of the universe
of discourse and also uses a trend predictor to calculate the forecasted
value. The trend predictor is used to adjust the weights of the
proposed fuzzy metric for forecasting. The method is found to be
robust and can handle the problem of inaccuracy in the data set.
Shiva Raj Singh [2008] has proposed a method which minimizes
the complicated computations of fuzzy relational matrices and a
suitable defuzzification process and also provided the forecasted
values of better accuracy. The developed method is a generalized
method of forecast thereby proving the forecast of better accuracy
than the existing models.
Satyendra Nath-Mandal [2008] has proposed the method to
fuzzify the original data based on Gaussian function, triangular
function, s-function, Trapezoidal and L –function. After fuzzifying, all
fuzzified data were defuzzified to get normal form. The error analysis
indicates that the membership function was appropriate for
fuzzification of data and it was used to predict the short length at
maturity.
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Hui -Li Hsu et al., [2008] have presented evaluation techniques
for interval forecasting which can provide a more objective decision
space in interval forecasting to policy makers.
Adesh Kumar Pandey et al., [2008] have proposed a comparative study
of neural network and fuzzy time series forecasting techniques. It is
successfully implemented for forecasting wheat production at pant
Nagar farm.
Chen et al., [2008] has proposed a comprehensive fuzzy time-
series, which factors linear relationships between recent periods of
stock prices and fuzzy logical relationships (nonlinear relationships)
mined from time-series into forecasting processes.
Dug Hun Hong [2005] has considered the expanded results to the non
homogeneous fuzzy time series and the general fuzzy time series by
using the weakest t-norm based algebraic fuzzy operations and solved
the open problem.
Taylor [2008] has used minute-by-minute British electricity
demand observations to evaluate methods for prediction between 10
and 30 minutes ahead.
Muhammad Hisyam Lee et al., [2009] have proposed the
adoption for the weighted and the difference between actual data
toward midpoint interval based on fuzzy time series. The weights are
determined according to chronological number of fuzzy logical
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relations in fuzzy logic group and modification is also done in reversal
of weight elements of transpose matrix for forecasting rule.
Arkov et al., [2009] provided the method with high
computational speed because it utilizes only operations of move and
comparison. However, this approach allows the simulation of a limited
number of systems with states depending on state quantization.
Specific application is system state prediction.
Nai-Yi-Wang et al., [2009] have presented a new method to
predict the temperature and the Taiwan Futures Exchange (TAIFEX),
based on automatic clustering techniques and two-factor high-order
fuzzy time series. First, they had applied an automatic clustering
algorithm to cluster the historical data into intervals of different
lengths. Then, they applied the same based on two-factor high-order
fuzzy time series.
Kuo et al., [2009] have used the particle swarm optimization to
find the proper content of the main factors. A new hybrid forecasting
model which combined particle swarm optimization with fuzzy time
series is proposed to improve the forecasted accuracy. The
experimental results of forecasting enrollments of students of the
University of Alabama found that the new model is better than any
existing models, and it can get better quality solutions based on the
first-order and the high-order fuzzy time series, respectively.
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Chen et al., [2009] have presented a new method to forecast
enrollments based on automatic clustering techniques and fuzzy
logical relationships. First, he presented an automatic clustering
algorithm for clustering the historical enrollments of the University of
Alabama into intervals of different lengths. Then, each obtained
interval is divided into p sub-intervals.
Shiva Raj Singh [2009] has proposed a computational method of
forecasting based on higher order fuzzy time series. The developed
methods avoid the computations of complicated fuzzy logical relational
matrices and search for suitable defuzzification method.
Suresh et al., [2009] have presented a modified method of
forecasting in fuzzy time series using transition probability
membership function. In addition, the modified method is identified
outlier in time series data using cook’s distance method.
Okan Duru et al., [2010] have investigated the predictive
performance of fuzzy time series analysis method for dry bulk freight
method. The advantages of this method indicated the lack of several
diagnostic tests like as normality and stationarity.
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Narendra Kumar [2010] has developed time variant fuzzy time
series models and its implementation testing for forecasting of wheat
crop production. Also the results are compared with other known
existing methods.
Ismail et al., [2011] have discussed a modified weight for fuzzy
time series and it shows a significant reduction of mean square error
and average forecasting error when compared with known existing
method.
Lazim Abdullah et al., [2010] have presented a combination of
fourth-order fuzzy time series with the multi-period adaptation model.
It is adopted to forecast the KLCI stock index. The results found that
the forecasting model that combines with the multi-period adaptation
model produce lower RMSE compared existing approaches. It is
proves that the multi-period adaptation models can effectively improve
the forecasting performance.