The Baltic Dry Index Computer Science Essay

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Baltic Exchange based in London created the Baltic Dry Index (BDI) or Dry Bulk Index in order to measure trading costs in shipping raw materials (e.g. grains, metals, coal, cement and fossil fuels) worldwide. In other words, the index reflects the demand for shipping raw materials against the available capacity. It is a daily index which includes 23 dry bulk routes globally by four different sizes of ships, Supramax, Panamax, Capesize and Handysize. Historically, in 1985 the Baltic Exchange publishes daily the Baltic Freight Index (BFI), a general indicator of dry bulk market, which was replaced by BDI in 1999, continuing the time series of BFI. The last change of calculation of BDI was in 1st July in 2009, where some capesize voyage routes were excluded and the final calculation formula was defined. It is a composite of equal weighted averages of the four indices, Baltic Capesize Index -BCI, Baltic Panamax Index -BPI, Baltic Supramax Index -BSI and Baltic HandySize Index -BHSI.

The BDI can be used as an overall economic index, since it denotes where final prices are heading for items that use the raw materials that are shipped in dry bulk. Many economists and investors consider BDI as a leading indicator for economic growth since it is related to the first stage of production. It is one of the purest leading indicators and it indicates the demand to move raw materials to production. Other economic indicators examine what has already occurred, whereas BDI gives a real time look at global international demand of raw materials, since it shows how much a company or country is willing to pay to import raw materials immediately. 

An increase of the indicator reflects to an increase of global demand for commodities, but a decline of its price shows recession of the demand for the raw materials. For this reason, the index is a predictor of the changes in economic activities of the global market. It can be used as a guide for the world trade because it represents the future economic production. Therefore, it is widely accepted the importance of the index. A forecast of the prices of the index is useful to predict the fluctuations of the global market and production.

Forecasting methods

The financial time series forecasting is a very complicated process, since time series have the appearance of natural randomness, which make the prediction of them extremely difficult. Economists are developing patterns using financial data in order to make forecasts.

There are many methods that are used for forecasting. In econometrics, forecasts can be made by using time-series models, which examine the past behavior of the prices, in order to forecast their behavior in future. There are two types of these models, the deterministic models such as linear extrapolation and stochastic models such as autoregressive models (AR), moving average models (MA) and others. In addition, applications of fuzzy logic, genetic algorithms and neural networks are used for forecasting. In the last few years, combinations of the neural networks and fuzzy systems have become popular.

Forecasting with ANFIS

The most popular neuro-fuzzy model developed by Jang (1993) called ANFIS (adaptive neuro-fuzzy inference system), which is like a fuzzy inference system, using adaptive neural networks in order to minimize the errors. This hybrid model has the attribute to combine the advantages of these two methods. This project proposes the ANFIS model as a tool for forecasting the Baltic Dry Index, because the variables are non-stationary and non-linear distributed. The method is used by many researches in financial sector, especially in stock market. The selected data that are used for this study are the daily prices of the BDI covering the period 06/30/2009 - 04/27/2012. Thus, 708 observations are used for the analysis. We choose to examine the specific period because before this the calculation of the index was different. MATLAB software is used for the export of results. A comparison of the results is made among the ANFIS model and two other time-series models, an autoregressive model (AR) and an autoregressive moving average model (ARMA). The criteria that are compared are: MSE (Mean Square Error), MAE (Mean Absolute Error), RMSE (Root Mean Square Error), and MAPE (Mean Absolute Percentage Error), in order to examine the efficiency of our model. The model that is giving the smallest errors is the better model for forecasting.

The main purpose of the research is to provide an efficiency forecast using a neuro-fuzzy model for Baltic Dry Index. Forecasts for these types of indices are useful due to the significance of the index. As it is mentioned above, BDI fluctuations affect the global economic activity. Therefore, the question which derives from this research is whether the adaptive neuro-fuzzy inference system has a better performance in forecasting than the two other forecasting models.

The project study is organized as following. In the next section, an extensive literature overview is presented. The presented studies highlight the use and the importance of the neuro-fuzzy techniques, focused on the ANFIS model use in financial sector. In the main part an analysis of the theoretical background of ANFIS model is presented. Specifically, it is analyzed the fuzzy set theory, the adaptive network structure, as well as the composition of ANFIS model. In the following section, the used data for the forecast is given which are the daily prices of BDI index. Afterwards, the results of the forecast are illustrated and analyzed. Moreover, a comprehensive analysis and a comparison of the results are performed. Finally, the conclusions are exported from this study relatively with the efficiency of the presented model.

LITERATURE REVIEW

Introduction

In recent years, new methodologies have been developed with main aim the forecasting of financial markets. According to the available literature, there are many studies, using different models for forecasting, such as time-series models, casual models, neural techniques, applications of fuzzy logic or combinations of them.

The last few years, the neuro-fuzzy techniques became more popular due to their efficient forecasts. Several researches introduce the application of artificial neural systems and fuzzy logic in financial analysis. Jang (1993) suggested for the first time the Adaptive Network based Fuzzy Inference System (ANFIS) architecture. In this work, comparisons on prediction between the results of ANFIS and neural network approaches are presented, as well as between the results of ANFIS and statistical and connectionist approaches. The author argues that ANFIS outperforms against the other methods. After this work, many other researchers were interested to engage with neural fuzzy networks such as James J. Buckley and Yoichi Hayashi (1994), M.M. Gupta and D.H. Rao (1994), Sun Zengqi and Deng Zhidong (1996), etc.

Neuro-fuzzy systems

The majority of the available literature concerning forecasts by using neuro-fuzzy systems, takes place in the recently years. Arit Thammano (1999) proposes Neuro-Fuzzy architecture as a new forecasting approach. The standard backpropagation model compared with the neuro-fuzzy model, indicating that the results of the proposed neuro-fuzzy model yields a more accurate output than the basic backpropagation algorithm.

Dusan Marcek (2004) uses Statistical, Classical and Fuzzy Neural Network Approach for Stock Price Forecasting. It is used data for VAHOSTAV stock prices from January 1997 to August 1997, totally used 163 observations. Comparing the results of an AR model, a Neural Network approach and fuzzy neural network, using the criterion RMSE, the author concludes that FNN forecasting model gave better results than the other models. A similar research has been done by the same author using the same dataset, but Neural Network approach is replaced by a fuzzy linear regression model. Again RMSE criterion is used and FNN forecasting model outperforms against the others.

The same year Gwo-Ching Liao and Ta-Peng Tsao proposed an application of fuzzy neural networks and artificial intelligence for load forecasting. The dataset, that was used, was collected from a study performed on the Taiwan Power System in order to predict the consumption of daily energy covered the period 21 April 2000 to 20 July 2000. A comparison is made between the artificial neural network (ANN), genetic algorithm artificial neural network (GA-ANN) and artificial integrated fuzzy neural network (AIFNN), using MAPE and maximum MAPE as criteria. According to the results of research the AIFNN approach gives a more accurate load curve forecast.

Another study of Ebrahim Abbasi and Amir Abouec (2008) uses Neuro-Fuzzy Inference System for forecast Stock Price. The daily closing prices were collected from Tehran Stock Exchange for the stock IRAN KHODRO in period from 1997 to 2004. Due to the non-linearity of the stock, the author concludes that the proposed model is more efficient in forecasting stock price behavior and decrease the estimation error.

Ching-Hsue Cheng, Liang-YingWei and You-ShyangChen (2009) proposed ANFIS in order to forecast problems in Taiwan stock index. For the purpose of this study, data were collected from 1993 to 2003 for the three stock indices, the Dow Jones, NASDAQ and TAIEX. The results of the proposed model compared with two other models, Chen's model and Yu's model, in terms of RMSE. According to the outputs of the three datasets, the proposed model performs better in forecasting comparing to the two other models in terms of RMSE.

Tugba Efendigil, Semih Onut and Cengiz Kahraman (2009) presented a comparative analysis between artificial neural networks and neuro-fuzzy models for demand forecasting. The data was collected from a Turkish company which is active in durable consumer goods industry. The results of the research show the effectiveness of the ANFIS model for the forecasts compared to ANN model. The comparison was made based on the lower values of the two criteria, MSE and MAPE.

George S. Atsalakis and Kimon P. Valavanis (2009) proposed neuro-fuzzy system for forecasting stock market short-term trends. The historical data were collected from data from the Athens Stock Exchange (ASE) and the NYSE for the stock of the National Bank of Greece (ETE) and the General Electric (GE) stock. The proposed methodology uses an ANFIS technique in order to predict the next day's trend of a stock. The outputs of the proposed model compared with thirteen other similar soft computing results such as regression models, neural network models, neuro-fuzzy models. The comparisons based on three error measures (MAE, RMSE, MSE). Concluding, the performance of the proposed model was more solid and superior of prediction accuracy of stock market trend.

Another research of the same authors was published in the same year, concerning surveying stock market forecasting techniques based on soft computing methods (part II). This study surveyed more than 100 related published articles where neural and neuro-fuzzy techniques were applied in order to forecast stock markets. The conclusion of the study is that the neural networks and neuro - fuzzy models outperform conventional models, so they are suitable for stock market forecasting.

Furthermore, the application of fuzzy-neural networks is used by Jandaghi G. et al. (2010) in order to forecast SAIPA's stock price (member of Iranian stock). According to the medium of the square of error (MSE), medium of the absolute deviation (MAD) and deterministic coefficient (R-square) criteria, the fuzzy neural models are superiors in comparison to ARIMA linear models for the predictions in stock prices.

Akbar Esfahanipour and Werya Aghamiri (2010) also proposed ANFIS model for forecasting the returns of Tehran Stock Exchange Indexes (TEPIX). 614 observations were collected for TSE index from 18 July 2003 to 31 December 2005 and 614 observations for the TEPIX from 20 April 2006 to 31 January 2009. The proposed ANFIS model performance is better than other models such as BPN (Back Propagation Neural networks), multiple regression analysis or TSK fuzzy rule mode, according to the MAPE value.

Ansari T., et al. (2010) made a research in order to identify uncertainties during recession period using statistical analysis, econometrical analysis and Adaptive Neural-Fuzzy networks. The prices of NASDAQ Stock Market over the years 2008-2009 are used for the tests and analysis. The main purpose of the study is to get an efficient Adaptive Neural-Fuzzy network in order to test the data properly and use it for forecasting the stock index.

Melek Acar Boyacioglu and Derya Avci (2010) exam the case of the Istanbul Stock Exchange, using an Adaptive Network-Based Fuzzy Inference System (ANFIS) for the prediction of stock market return. This study uses ANFIS as a tool for forecasts the monthly return of ISE National 100 Index. The results of the error terms are very satisfying as well as the performance of the model. The authors suggest the application of ANFIS for stock price forecasts and generally in financial market due to its capability of learning and predicting.

Nair B., et al. (2010) proposed a hybrid Neuro-fuzzy system for stock market prediction using prices of the Bombay Stock Exchange sensitive index (BSE-SENSEX). The experimental results indicate that the proposed system gives higher accuracy in comparison to two other systems.

George S. Atsalakis, Emmanouil M. Dimitrakakis and Constantinos D. Zopounidis (2011) published a research titled: "Elliott Wave Theory and neuro-fuzzy systems, in stock market prediction: The WASP system". In this work, the Wave Analysis Stock Prediction system (WASP) is presented based on the neuro-fuzzy architecture in order to forecast the trend of the stock market. The WASP is a system that selects 9 sub-ANFIS models, based on the hit-rates. The results of 400 trading days were remarkable because the most of sub-models achieved a hit-rate above 75% for a sample of 60 trading days. This fact indicates the effectiveness of neuro-fuzzy techniques in stock market forecasting.

Liang-Ying Wei, Tai-Liang Chen and Tien-Hwa Ho (2011) in order to forecast Taiwan stock market proposed a hybrid model based on adaptive-network-based fuzzy inference system. The experimental data were prices of the TAIEX index for six years. The root mean squared error (RMSE) was used as a performance indicator in order to compare the results of the proposed model with to two other fuzzy time-series models (Chen's and Yu's models). According to the experimental results the performance of the proposed model is superior in comparison to the two listing models.

Another research of the above authors (2011) was made in order to forecast stock price problems of the weighted stock index (TAIEX). This paper proposed a hybrid ANFIS model that is based on AR and volatility of the TAIEX. In addition, comparing again the proposed model with Chen's model and Yu's model, with regard to root mean squared error, show that the proposed model outperforms the other methods.

Esfahanipour A. and Mardani P. (2011) examine the case of Tehran stock exchange using an ANFIS model for prediction. The dataset covers the period from 25 March 2001 to 25 September 2010 for the prices of TEPIX. In order to approve the superiority of the proposed ANFIS model, a comparison has been done with ANN model and ANFIS models with no clustering and FCM clustering.

More recently, Patricia Melin, Jesus Soto, Oscar Castillo andJose Soria (2012) published a paper proposing a new approach for time series prediction using ensembles of ANFIS models. The time series that were considered are: the Mackey-Glass, Dow Jones and Mexican stock exchange. In conclusion the authors argued that the results obtained with the architecture were satisfying, since they achieved high percentages of accuracy with the time series. Therefore, the proposed models offer efficient results in the prediction of such time series.

In addition, the literature includes several studies for the use of neural-fuzzy techniques, not only in financial sector but also in other sectors such as chemical industry, hydropower, weather prediction, etc., where the researches have proved the efficiency of these techniques.

Dry bulk market

As far as it concerned the forecasting of the Baltic dry Index and generally the dry bulk market, many researches can be found in the available literature. In 1997 Veenstra and Franses developed a vector autoregressive model for a sample of ocean dry bulk freight rates, but due to the stochastic nature of the freight rate series, a large part of them cannot be forecasted. Ming-Tao Chou (2008) applied a fuzzy time-series model to forecast the Baltic Dry Index (BDI) index for the next month. The results (using RMSE as criteria) indicate that the model is efficient for the BDI prediction. Okan Duru (2010) developed an improved fuzzy time series approach for dry bulk shipping index forecasting. Also, Okan Duru (2012) proposed a multivariate model of fuzzy integrated logical forecasting method (M-FILF) for dry cargo freight market. Tsakonas, Nikolaidis and Dounias in order to exam the efficiency of the short-term predictions used two approaches, the fundamental analysis and a computational intelligence based on neuro-fuzzy technique for the Baltic Panamax Index, concluding that the combination of these approaches is a useful forecasting tool.

As it is presented by the studies mentioned earlier, the last few years, more and more researches use the neuro-fuzzy networks in order to get efficient forecasts for the stock market. Undoubtedly, based on all these previous researches, neuro-fuzzy techniques are a useful tool for forecasts.

THEORETICAL BACKGROUND

Fuzzy set theory

The concept of Fuzzy Logic (FL) was firstly introduced by Zadeh (1965) and presented as a way of processing data by allowing partial set membership. In contrast to usual logic which is based on "true and false" (Boolean logic, 1-0), fuzzy logic computes problems based on a degree of "truth," or "fuzzy sets of true and false" (Nikravesh, M., 2004). A fuzzy inference system (FIS) is a computing framework that combines the concepts of fuzzy logic, fuzzy decision rules, and fuzzy reasoning (Jang, 1993).

Fuzzy sets

According to Jang (1993), a classical set is a set with a crisp boundary. For instance, a classical set A can be represented as

A = {x I x > 6} (1)

where there is a clear, unambiguous boundary point 6 such that if x is greater than this number, then x belongs to the set A, otherwise z does not belong to this set. In contrast to a classical set, a fuzzy set, as the name implies, is a set without a crisp boundary. That is, the transition from "belonging to a set" to "not belonging to a set" is gradual, and this smooth transition is characterized by membership functions that give fuzzy sets flexibility in modeling commonly used linguistic expressions, such as "the water is hot" or "the temperature is high." As Zadeh pointed out in 1965 in his seminal paper entitled "Fuzzy Sets" such imprecisely defined sets or classes "play an important role in human thinking, particularly in the domains of pattern recognition, communication of information, and abstraction." Note that the fuzziness does not come from the randomness of the constituent members of the sets, but from the uncertain and imprecise nature of abstract thoughts and concepts.

Definition of Fuzzy Sets and Membership Functions

If X is a collection of objects denoted generically by z, then a fuzzy set A in X is defined as a set of ordered pairs:

If X is a collection of objects x, then a fuzzy et A in X is defined as a set of ordered pairs:

(2)

is called the membership function (or MF for short) for the fuzzy set A. The MF puts each element of X to a membership grade (membership value) between 0 and 1. Obviously the definition of a fuzzy set is a simple extension of the definition of a classical set in which the characteristic function is permitted to have continuous values between 0 and 1. If the value of the membership function is restricted to either 0 or 1, then A is reduced to a classical set and is the characteristic function of A.

Fuzzy if - then rules

A fuzzy if-then rule (also known as fuzzy rule, fuzzy implication, or fuzzy conditional statement) assumes the form:

if x is A then y is B, (3)

where A and Î’ are linguistic values defined by fuzzy sets on universes of discourse X and Y, respectively. Often "x is A" is called the antecedent or premise, while "y is B" is called the consequence or conclusion. Examples of fuzzy if-then rules are widespread in our daily linguistic expressions, such as the following:

- If demand is high, then prices will be increased.

- If there are sharks in this coast, then there are not swimmers.

- If business loan demand is low, then the interest rate will be decreased.

- If the inflation is high, then prices will be increased.

Before we can use fuzzy if-then rules to model and analyze a system, first we have to formalize what is meant by the expression "if χ is A then y is Β", which is sometimes shortened as A →Β. In essence, the expression describes a relation between two variables χ and y, this suggests that a fuzzy if-then rule be defined as a binary fuzzy relation R on the product space X x Y. Generally speaking, there are two ways to interpret the fuzzy rule A → B. If we interpret A →Β as A coupled with B, then

(4)

where is a fuzzy AND (or more generally, T-norm) operator and A →Β is used again to represent the fuzzy relation R. On the other hand, if A -> Β is interpreted as A entails B, then it can be written as four different formulas:

•Material implication:

•Propositional calculus:

•Extended propositional calculus:

•Generalization of modus ponens:

Where R = A →Β and is a T-norm operator.

Though these four formulas are different in appearance, they all reduce to the familiar identity when A and B are propositions in the sense of two-valued logic. Fig. 3.1 illustrates these two interpretations of a fuzzy rule A →Β.

Figure 3.1: Two interpretations of fuzzy implication: (a) A coupled with B, (b) A entails B.

Fuzzy Reasoning

Based on Jang (1993), fuzzy reasoning (also known as approximate reasoning) is an inference procedure used to derive conclusions from a set of fuzzy if-then rules and one or more conditions. Before introducing fuzzy reasoning, we shall discuss the compositional rule of inference, which is the essential rationale behind fuzzy reasoning.

Let A, A', and Β be fuzzy sets of Χ, X and Y, respectively. Assume that the fuzzy implication A → Β is expressed as a fuzzy relation R on Χ x Y. Then the fuzzy set Β induced by "χ" is "A" and the fuzzy rule "if x is A then y is B" is defined by

(5)

or, equivalently,

(6)

Equation (6) is a general expression for fuzzy reasoning, while (5) is an instance of fuzzy reasoning where min and max are the operators for fuzzy AND and OR, respectively.

Now we can use the inference procedure of fuzzy reasoning to derive conclusions, provided that the fuzzy implication A → Β is defined as an appropriate binary fuzzy relation.

Fuzzy Models

The Fuzzy inference system is a popular computing framework based on the concepts of fuzzy set theory, fuzzy if-then rules, and fuzzy reasoning. It has been successfully applied in fields such as automatic control, data classification, decision analysis, expert systems, and computer vision. The basic structure of a fuzzy inference system consists of three conceptual components: a rule base, which contains a selection of fuzzy rules, a database or dictionary, which defines the membership functions used in the fuzzy rules, and a reasoning mechanism, which performs the inference procedure (usually the fuzzy reasoning) upon the rules and a given condition to derive a reasonable output or conclusion. The steps to create a fuzzy inference model are as follows:

(a) Fuzzification: the input variables are compared with the MFs on the premise part of the fuzzy rules to obtain the probability of each linguistic label.

(b) Combine (through logic operators) the probability on the premise part to get the weight (fire strength) of each rule.

(c) Application of firing strength to the premise MFs of each rule to generate the qualified consequent of each rule depending on its weight.

(d) Defuzzification: Aggregate the qualified consequents to produce a crisp output

Fig. 3.2: Block diagram for a fuzzy inference system

In what follows, we will first introduce three types of the most commonly used fuzzy inference systems.

1) Mamdani Fuzzy Model: The Mamdani fuzzy model was proposed as the very first attempt to control a steam engine and boiler combination by a set of linguistic control rules obtained from experienced human operators.

Fig. 11 is an illustration of how a two-rule fuzzy inference system of the Mamdani type derives the overall output z when subjected to two crisp inputs x and y.

Fig. 3.3: The Madmani Fuzzy Inference System using min and max for fuzzy AND and OR operators, respectively.

If we adopt product and max as our choice for the fuzzy AND and OR operators, respectively, and use max-product composition instead of the original max-min composition, then the resulting fuzzy reasoning is shown in Fig. 12, where the inferred output of each rule is a fuzzy set scaled down by its firing strength via the algebraic product. Though this type of fuzzy reasoning was not employed in Mamdani's original paper, it has often been used in the literature. Other variations are possible if we have different choices of fuzzy AND (T-norm) and OR (Tconorm) operators.

Fig. 3.4: The Madmani Fuzzy Inference System using product and max for fuzzy AND and OR operators, respectively.

In Mamdani's application, two fuzzy inference systems were used as two controllers to generate the heat input to the boiler and throttle opening of the engine cylinder, respectively, in order to regulate the steam pressure in the boiler and the speed of the engine. Since the plant takes only crisp values as inputs, we have to use a defuzzifier to convert a fuzzy set to a crisp value. Defuzzification refers to the way a crisp value is extracted from a fuzzy set as a representative value. The most frequently used defuzzification strategy is the centroid of area, which is defined as:

(7)

Where μC'(z) is the aggregated output MF. This formula is reminiscent of the calculation of expected values in probability distributions. Other defuzzification strategies arise for specific applications, which include bisector of area, mean of maximum, largest of maximum, and smallest of maximum, and so on. Fig. 3.5 demonstrates these defuzzification strategies. Generally speaking, these defuzzification methods are computation intensive and there is no rigorous way to analyze them except through experiment-based studies Both Figs. 3.3 and 3.4 conform to the fuzzy reasoning defined previously via centroid defuzzification is equal to the weighted average of each rule's crisp output, where the weighting factor for a rule is equal to its firing strength multiplied by the area of the rule's output MF, and the crisp output of a rule is equal to the centroid defuzzified value of its output MF. This reduces the computation burden if we can obtain the area and the centroid of each output MF in advance.

Fig. 3.5: Various defuzzification schemes for obtaining a crisp output.

2) Sugeno Fuzzy Model: The Sugeno fuzzy model (also known as the TSK fuzzy model) was proposed by Takagi, Sugeno, and Kang in an effort to develop a systematic approach to generating fuzzy rules from a given input-output data set. A typical fuzzy rule in a Sugeno fuzzy model has the form

if x is A and y is B then z = f ( x ,y ) (8)

where A and B are fuzzy sets in the antecedent, while x = f(x,y) is a crisp function in the consequent. Usually f(x,y) is a polynomial in the input variables z and y, but it can be any function as long as it can appropriately describe the output of the system within the fuzzy region specified by the antecedent of the rule. When f(x,y) is a first-order polynomial, the resulting fuzzy inference system is called a first-order Sugeno fuzzy model. When f is a constant, we then have a zero-order Sugeno fuzzy model, which can be viewed either as a special case of the Mamdani fuzzy inference system, in which each rule's consequent is specified by a fuzzy singleton (or a predefuzzified consequent), or a special case of the Tsukamoto fuzzy model (to be introduce later), in which each rule's consequent is specified by an MF of a step function crossing at the constant. Moreover, a zero-order Sugeno fuzzy model is functionally equivalent to a radial basis function network under certain minor constraints. It should be pointed out that the output of a zero-order Sugeno model is a smooth function of its input variables as long as the neighboring MF's in the premise have enough overlap. In other words, the overlap of MF's in the consequent does not have a decisive effect on the smoothness of the interpolation; it is the overlap of the MF's in the premise that determines the smoothness of the resulting input-output behavior.

Fig. 3.6: The Sugeno Fuzzy model

Fig. 3.6 shows the fuzzy reasoning procedure for a firstorder Sugeno fuzzy model. Since each rule has a crips output, the overall output is obtained via weighted average and thus the time-consuming procedure of defuzzification is avoided. In practice, sometimes the weighted average operator is replaced with the weighted sum operator (that is, Z = WlZ1 + W2Z2 in Fig. 3.6) in order to further reduce computation load, especially in training a fuzzy inference system. However, this simplification could lead to the loss of MF linguistic meanings unless the sum of firing strengths (that is, Σiwi) is close to unity.

3) Tsukamoto Fuzzy Model: In the Tsukamoto fuzzy models, the consequent of each fuzzy if-then rule is represented by a fuzzy set with a monotonical MF, as shown in Fig. 3.7. As a result, the inferred output of each rule is defined as a crisp value induced by the rule's firing strength. The overall output is taken as the weighted average of each rule's output. Fig. 3.7 illustrates the whole reasoning procedure for a two-input two-rule system. Since each rule infers a crisp output, the Tsukamoto fuzzy model aggregate each rule's output by the method of weighted average and thus also avoids the time-consuming process of defuzzification.

Fig. 3.7: The Tsukamoto Fuzzy model

Adaptive neural networks

An artificial neural network (ANN) or neural network (NN), is a computational method used to model data, derived from the field of artificial intelligence. Neural networks try to imitate the architecture of the human brain. Diagram 9 that follows shows a simple neural network with one neuron and n inputs. Every input is multiplied by a different parameter. All inputs are added in the neuron, giving the final result. There is no limit in the number of neurons used, although a very large number can make the network extremely demanding in computational power Neural networks are very efficient in modeling non-linear problems. Neural networks are trained from data with the help of adaptive algorithms such as the back propagation algorithm (Rumelhart, Hinton, & Williams, 1986; Werbose, 1974). The main advantages of the neural networks are the ability to learn by example, in other words to create knowledge from past data. Additionally, such networks are found to be extremely useful in pattern recognition. On the other hand, neural networks have also been criticized mainly due to the high computational power that is required, which limits the number of input variables that can be used. The main disadvantage of a neural network is the lack of information regarding the impact of every input on the output. Neural networks are commonly called a ''black box'' as there is no information given other than the output.

Fig. 3.8: Simple neural network

Architecture

As the name implies, an adaptive network (Fig. 3.9) is a network structure whose overall input-output behavior is determined by the values of a collection of modifiable parameters. More specifically, the configuration of an adaptive network is composed of a set of nodes connected through directed links, where each node is a process unit that performs a static node function on its incoming signals to generate a single node output and each link specifies the direction of signal flow from one node to another. Usually a node function is a parameterized function with modifiable parameters; by changing these parameters, we are actually changing the node function as well as the overall behavior of the adaptive network.

Fig. 3.9: A feed-forward adaptive network in layered representation.

In the most general case, an adaptive network is heterogeneous and each node may have a different node function. Also remember that each link in an adaptive network is merely used to specify the propagation direction of a node's output; generally there are no weights or parameters associated with links. Fig. 3.9 shows a typical adaptive network with two inputs and two outputs. The parameters of an adaptive network are distributed into the network's nodes, so each node has a local parameter set. The union of these local parameter sets is the network's overall parameter set. Adaptive networks are generally classified into two categories on the basis of the type of connections they have: feed-forward and recurrent types. The adaptive network shown in Fig. 3.9 is a feed-forward network, since the output of each node propagates from the input side (left) to the output side (right) unanimously. If there is a feedback link that forms a circular path in a network, then the network is a recurrent network; Fig. 3.10 is an example. (From the viewpoint of graph theory, a feed-forward network is represented by an acyclic directed graph which contains no directed cycles, while a recurrent network always contains at least one directed cycle.)

Fig. 3.10: A recurrent adaptive network.

In the layered representation of the feed-forward adaptive network in Fig. 3.9, there are no links between nodes in the same layer and outputs of nodes in a specific layer are always connected to nodes in succeeding layers. This representation is usually preferred because of its modularity, in that nodes in the same layer have the same functionality or generate the same level of abstraction about input vectors.

Another representation of feed-forward networks is the topological ordering representation, which labels the nodes in an ordered sequence 1, 2, 3, . . . , such that there are no links from node i to node j whenever i ≥ j. Fig. 3.11 is the topological ordering representation of the network in Fig. 3.9. This representation is less modular than the layer representation, but it facilitates the formulation of the leaming rule, as will be seen in the next section. (Note that the topological ordering representation is in fact a special case of the layered representation, with one node per layer.)

Fig. 3.11: A feed-forward adaptive network in topological ordering representation.

Two of the most popular network paradigms adopted in the neural network literature are the back-propagation neural network (BPNN) and the radial basis function network (RBFN).

A backpropagation neural network (BPNN) is an adaptive network whose nodes (called "neurons") perform the same function on incoming signals; this node function is usually a composite function of the weighted sum and a nonlinear function called the "activation function" or "transfer function." Usually the activation functions are of either a sigmoidal or a hypertangent type which approximates the step function (or hard limiter) and yet provides differentiability with respect to input signals.

The Radial Basis Function Networks (RBFN's): The locally tuned and overlapping receptive field is a well-known structure that has been studied in the regions of the cerebral cortex, the visual cortex, and so forth. Drawing on the knowledge of biological receptive fields, Moody and Darken proposed a network structure that employs local receptive fields to perform function mappings. Similar schemes have been proposed by Powell, Broomhead and Lowe, and many others in the areas of interpolation and approximation theory; these schemes are collectively called radial basis function approximations.

ANFIS

An adaptive neuro-fuzzy inference system (ANFIS) is a fuzzy inference system formulated as a feed-forward neural network. Hence, the advantages of a fuzzy system can be combined with a learning algorithm. ANFIS has the ability to derive the expert's empirical knowledge, in form of rules, through the data using a learning algorithm. The hybrid learning algorithm proposed by Jang combine the error back propagation and

least square error method. The goal of ANFIS is to find a model that will correctly associate the inputs (initial values) with the target (predicted values).

ANFIS architecture

For simplicity, we assume the fuzzy inference system under consideration has two inputs x and y and one output z. For a first-order Sugeno fuzzy model, a typical rule set with two fuzzy if-then rules can be expressed as

Rule 1: If x is A1 and y is B1, then f1 = p1x + q1y + r1

Rule 2: If x is A2 and y is B2, then f2 = p2x + q2y + r2

Fig. 3.12 illustrates the reasoning mechanism for this Sugeno model. The corresponding equivalent ANFIS architecture is as shown in Fig. 3.13, where nodes of the same layer have similar functions, as described below. (Here we denote the output node i in layer 1 as Ol,i.)

Fig. 3.12: A two-input first-order Sugeno fuzzy model with two rules.

Fig. 3.13: A two-input first-order Sugeno fuzzy model with two rules; equivalent ANFIS architecture.

Layer 1: Every node i in this layer is an adaptive node with a node output defined by

(9)

where x (or y) is the input to the node and A, (or Bi-2) is a fuzzy set associated with this node. In other words, outputs of this layer are the membership values of the premise part. Here the membership functions for Ai and Bi can be any appropriate parameterized membership functions introduced in Section II. For example, A, can be characterized by the generalized bell function:

(10)

where {ai, bi, ci} is the parameter set. Parameters in this layer are referred to as premise parameters.

Layer 2: Every node in this layer is a fixed node labeled Π, which multiplies the incoming signals and outputs the product. For instance,

(11)

Each node output represents the firing strength of a rule. (In fact, any other T-norm operators that perform fuzzy AND can be used as the node function in this layer.)

Layer 3: Every node in this layer is a fixed node labeled N. The ith node calculates the ratio of the ith rule's firing strength to the sum of all rules' firing strengths:

(12)

For convenience, outputs of this layer will be called normalized firing strengths.

Layer 4: Every node i in this layer is an adaptive node with a node function

(13)

Where is the output of layer 3 and {pi, qi, ri} is the parameter set. Parameters in this layer will be referred to as consequent parameters.

Layer 5: The single node in this layer is a fixed node labeled Σi which computes the overall output as the summation of all incoming signals:

(14)

Thus we have constructed an adaptive network that has exactly the same function as a Sugeno fuzzy model. The structure of this adaptive network is not unique; we can easily combine layers 3 and 4 to obtain an equivalent network with only four layers. Similarly, we can perform weight normalization at the last layer; Fig. 3.14 illustrates an ANFIS of this type.

Fig. 3.14: Another ANFIS architecture for the two-input two-rule Sugeno fuzzy model.

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