Evolutionary Functional Link Interval Type-2 Fuzzy Neural System for Exchange Rate Prediction

Transcription

1 Evolutionary Functional Link Interval Type-2 Fuzzy Neural System for Exchange Rate Prediction 3. Introduction Currency exchange rate is an important element in international finance. It is one of the chaotic, volatile financial time series. Hence, forecasting currency exchange rate poses difficulty for researchers. Over the years, various models have been applied to forecast it [7, 58, 59, 6, 6, 62, 63, 65, 66, 67, 68, 69, 7, ]. However, in this thesis, a hybrid Evolutionary Functional Link and Interval Type-2 Fuzzy Neural System (EFLIT2FNS) is used to predict the currency exchange rate data of five different currencies (Japanese Yen, Chinese Yuan, Indian Rupee, South Korean Won and Switzerland Franc vs. US Dollar) for six different time horizons, starting from one day to one year. The antecedent part of each fuzzy rule of FLIT2FNS model is an Interval Type-2 fuzzy set and the fuzzy rules are of the Takagi Sugano Kang (TSK) and the consequent part comprises a Functional Link Artificial Neural Network (FLANN). The parameters of both the antecedent and consequent part of the fuzzy rules are optimized by the gradient descent algorithm. Further, to overcome the limitations of the above said algorithm, two evolutionary algorithms i.e., enetic Algorithm (A) and Differential Evolution () are used to optimize all the parameters used in the model. In Chapter 2, the FIT2FNS model has been discussed in detail. So, in this Chapter, the evolutionary techniques like A and used to optimize the parameters of the models have been highlighted. Secondly, two other models () FLANN and (2) FLFNS have been used to measure the performance of EFLIT2FNS model. The figure of the model is given below (3.). 67

2 Fig 3. EFLIT2FNS Model 3.2 The Learning Algorithms used for weight optimization 3.2. Evolutionary Learning Algorithm Evolutionary learning Algorithm (EA) or evolutionary computing technique (discussed in chapter one) comes from the idea of applying the biological principle of natural evolution to artificial systems. EAs are successfully applied to numerous problems from different domains, including optimization, automatic programming, machine learning, economics, operations research, ecology, and population genetics. The details of enetic Algorithm (A) and Differential Evolution () are discussed below. 68

3 3.2.. enetic Algorithm (A) In A, the evolution starts from a population of completely random individuals and occur in generations. In each generation, the fitness of the whole population is evaluated; multiple individuals are stochastically selected from the current population (based on their fitness), and modified (mutated or recombined) to form a new population [6, 36, 37, 64, 7, 72, 73]. The new population is then used in the next iteration of the algorithm Selection Chromosomes are selected from the population to become parents to crossover. The problem is how to select these chromosomes. There are many methods to select the best chromosomes, such as, roulette wheel selection, Boltzman selection, tournament selection, rank selection, steady state selection and many others. Every method has some merits as well as some limitations. In this thesis, Roulette wheel selection is used to select the chromosomes. Lastly, elitism is used to copy the best chromosome (or a few best chromosomes) to new population. Elitism helps in increasing the performance of A, because it prevents losing the best found solution Crossover Crossover selects genes from parent chromosomes and creates a new offspring. The simplest way to do this is to choose randomly some crossover point and interchange the value before and after that point. Crossover rate taken for the proposed model is given in Table

4 Mutation Mutation takes place after crossover. Mutation changes randomly the new offspring. For binary encoding, we can switch a few randomly chosen bits from to or to. Mutation ensures genetic diversity within population. This entire process is continued until the convergence criterion is satisfied. This is pictorially represented in Fig Fig. 3.2 Flowchart of A The parameters used for all models with A as the optimization algorithm are given in Table 3.. 7

5 Table 3. Parameters used for all three models (A) Parameters Values Population size P 3 Maximum No. of generation 2 Dimension or No. of variables D (FLANN) 9 Dimension or No. of variables D(FLFNS) 39 Dimension or No. of variables D(EFLIT2FNS) 74 No. of consecutive generation for which no improvement is observed (FLIT2FNS Model) 25 Lower bound. Upper bound Crossover rate.8 Mutation rate Differential Evolution [39] is a population based stochastic search which can be efficiently used as a global optimizer in the continuous search domain. has been successfully applied in diverse fields such as large scale power dispatch problem [43], global numerical optimization [44], power loss minimization [42] and pattern reorganization. also has been extensively used in different types of clustering like image pixel clustering [4], text document clustering and dynamic clustering for any unknown datasets [4]. Like any other evolutionary algorithms, also starts with a population of NP D-dimensional parameter vectors. Two other parameters used in are scaling factor 7

6 F and cross over rate CR. Four steps in are: Initialization, Mutation, Crossover and Selection Initialization generates a population of NP D-dimensional parameter vectors called individual that encodes the candidate solutions D { x,... x } X i, i,..., i, = (3.) where i=,., NP towards the global optimum. For each parameter, there will be a certain range within which value of the parameters should lie. The parameters of upper and lower bound are respectively X D D = { x,... x } and X { x x } max max, max =. min min,..., min The initial value of j th parameter in the i th individual at generation = is generated by x j i. j j j = x + rand,).( x x ) (3.2) min ( max min where rand(,) represents a uniformly distributed random number lying between and, j=,2, D Mutation For each target vector, one mutant vector V i, has to be produced by using the mutation operation. For each target vector X i. at the generation, its associated 2 D mutant vector is { v, v,... v } V i, i, i., i, =. The mutant vector can be generated using any mutation strategy. Five most frequently used mutation strategies are given below. i. /rand/ : 72

7 .( X i X i ) V i, X i + F r, r, r, = (3.3) 2 3 ii. /best/ : ( X i X i ) V i, X best, + F. r, r, = (3.4) 2 iii. /rand-to-best/ : ( X best, X i, ) + F. ( X i X i ) V i, X i, + F. r, r, = (3.5) 2 iv. /best/2 : ( X i X i ) + F. ( X i X i ) V i, X best, + F. r, r, r, r, = (3.6) v. /rand/2 : ( X i X i ) + F. ( X i X i ) V i, X i + F. r, r, r, r, r, = (3.7) The indices i i i i i r, r2, r3, r4, r5 are mutually exclusive integers randomly generated within the range to NP. These indices are randomly generated once for each mutant vector. F is the scaling factor. /rand/2 has been chosen as the mutation strategy for our experiment Crossover In this phase, a trial vector U i, is generated from each pair of target vector X i, and its corresponding mutant vector i, 2 D { u u,..., u,} U =. i,, i, i, V i, after going through the crossover operation. In this thesis, employs binomial crossover as defined below:- ( rand [,) CR) or( j j ) j u if = j i,, j rand ui, = (3.8) j xi,, otherwise 73

8 j=[, 2, 3,., D] and CR is the crossover rate i.e. a user defined constant within the range of (,). j rand, an integer, which is randomly chosen in the range of [, D]. The crossover operator copies the j th parameter of the mutant vector V i, to the corresponding element in the trial vector U i,, if [,) Otherwise, it is copied from the corresponding target vector X i,. ( rand CR) or( j = ). j j rand Selection In selection operation phase, checks the upper and lower bounds of each parameter of the newly created trial vector. If they exceed, again they have to be reinitialized within the pre-specified range. The objective function values of all trial vectors are evaluated and selection operation is carried out. The objective function value of each trial vector f(u i, ) is compared to that of its corresponding target vector f(x i, ) in the current population. If the vector has less or equal objective function value than the corresponding target vector, the trial vector will replace the target vector and will enter the population of next generation. Otherwise, the target vector will remain in the population for the next generation. It can be defined as:- { U if ( f ( U )) f ( X ) X i, = i,, i, i, + (3.9) The above steps are repeated until some specific termination criteria are satisfied. The steps of are pictorially represented in Fig. 3.3 and the parameters used for all models with as the optimization algorithm are given in Table

9 Fig. 3.3 Flowchart of Table 3.2 Parameters used for all three models () Parameters Values Population size P Maximum No. of generation 2 Dimension or No. of variables D (FLANN) 9 Dimension or No. of variables D (FLFNS) 39 Dimension or No. of variables D (EFLIT2FNS) 74 75

10 No. of consecutive generation for which no improvement is observed (FLIT2FNS Model) 36 Lower bound. Upper bound Scaling Factor.5 Scaling Factor 2.3 Crossover Probability Different Currency Exchange Datasets The foreign exchange data used in this research are daily and monthly data of five different currencies such as, Japanese Yen, Chinese Yuan, Indian Rupee, South Korean Won and Switzerland Franc vs. US Dollar. Each dataset is normalized by dividing each value by the maximum value of each dataset such that the normalized value of each element in each dataset is less than or equal to unity. The daily and monthly normalized values of all five datasets vs.us Dollar are given in Fig.3.4 and Fig. 3.5, respectively. The total number of samples present in each dataset and the number of samples used for training as well as testing for daily and monthly are given in Table 3.3 and Table 3.4, respectively. 76

11 Exchange Rate Japan China India South Korea Swizerland Days Fig. 3.4 Normalized Daily Exchange Data Exchange Rate Japan China India South Korea Swizerland Months Fig. 3.5 Normalized Monthly Exchange Data 77

12 The chaotic behavior and the abrupt spikes are clearly visible in the graph. Due to this, the periods from the whole dataset of each currency have been randomly selected to train the models. It is seen that the prediction accuracy in the learning phase varies accordingly. So, in one day ahead prediction, after normalizing the data, samples from the beginning have been excluded from each dataset. Due to smaller sample size, it is not possible for month ahead prediction. The samples for validation are chosen just after the last samples used in training set. Two technical indicators along with the data are considered for all the models. The technical indicators those have been used and the formula are given in Table 3.5. Table 3.3 Total No. of Samples (Daily) Number of Samples Data range Datasets Total Training Testing From To Yuan to US$ Yen to US$ Rupee to US$ Won to US$ Francs to US$ Table 3.4 Total No. of Samples (Monthly) Number of Samples Data range Datasets Total Training Testing From To Yuan to US$ Yen to US$

13 Rupee to US$ Won to US$ Francs to US$ Table 3.5 Technical Indicators with Formula Indicators Formulae.Moving Average N N x i i=, N=no. of Days, x i =today s price; 2.Exponential Average Moving (P x A)+(Previous EMA x(-a)); 2 A= ; P=Current Price; A=Smoothing factor; N + N=Time period; 3.4 Experimental results and Model Outputs In this paper, all the models are implemented by MATLAB 7.7. (R28b) software package. Five different currencies, Yuan, Yen, Rupee, Won and Franc against US Dollar are taken as the experimental data Chinese Yuan vs. US Dollar Figs show the comparison between actual and predicted values during testing of EFLIT2FNS model integrated with all three learning algorithms-, A and for one day, one week and six months time horizons respectively. Table 3.6 depicts all the MAPE obtained during testing from all three models with, A and separately with all six time horizons chosen for this study for the dataset-yuan vs. US Dollar. 79

14 Chinese Yuan to US Dollar.8 Price Days Fig. 3.6 (One day) ahead EFLIT2FNS (Testing) Chinese Yuan to US Dollar Actual A.8 Price.6.4 Actual.2 A Days Fig. 3.7 (One Week) ahead EFLIT2FNS (Testing) 8

15 Chinese Yuan to US Dollar.8 Price Months Fig. 3.8 (Six Months) ahead EFLIT2FNS (Testing) Actual A Time Models Table 3.6 Chinese Yuan vs. US Dollar Horizon FLANN FLFNS FLIT2FNS MAPE (%) MAPE (%) MAPE (%) A A A One Day One Week One Month Two Months Six Months Twelve Months

16 3.4.2 Japanese Yen vs. US Dollar Similarly, Figs show the comparison between the actual and predicted values for the dataset Yen vs. US Dollar and the detail results are presented in Table Japnees Yen to US Dollar Price.6.4 Actual.2 A Days Fig. 3.9 (One Day) ahead FLIT2FNS (Testing) Japnees Yen to US Dollar.8 Price.6.4 Actual.2 A Days Fig. 3. (One Week) ahead FLIT2FNS (Testing) 82

17 Price Japnese Yen to US Dollar Actual A Months Fig. 3. (Six Months) ahead FLIT2FNS (Testing) Table 3.7 Japanese Yen vs. US Dollar Time Horizon Models FLANN FLFNS FLIT2FNS MAPE (%) MAPE (%) MAPE (%) A A A One Day One Week One Month Two Months Six Months Twelve Months

18 3.4.3 Indian Rupee vs. US Dollar Fig. 3.2 and Fig. 3.3 present the same comparison graph for EFLIT2FNS model for one day and six months ahead omitting all other graphs obtained for the dataset Rupees vs. US Dollar. Table 3.8 shows the MAPE during testing for the same dataset Indian Rupee to US Dollar Actual A Price Days Fig. 3.2 (One Day) ahead EFLIT2FNS (Testing) Indian Rupee to US Dollar Actual A Price Months Fig. 3.3 (Six Months) ahead EFLIT2FNS (Testing) 84

19 Table 3.8 Indian Rupee vs. US Dollar Time Horizon Models FLANN FLFNS FLIT2FNS MAPE (%) MAPE (%) MAPE (%) A A A One Day One Week One Month Two Months Six Months Twelve Months South Korean Won vs. US Dollar Figs. 3.4 and 3.5 show one day and two months ahead prediction for EFLIT2FNS model with, A and and MAPE during testing obtained from all three models for the dataset Won vs. US Dollar is given in Table

20 Price South Korean Won to US Dollar Actual A Days Fig.3.4 (One Day) ahead EFLIT2FNS (Testing) South Korean Won to US Dollar.8 Price Months Fig. 3.5 (Two Months) ahead EFLIT2FNS (Testing) 86 Actual A

21 Table 3.9 South Korean Won vs. US Dollar Time Horizon Models FLANN FLFNS FLIT2FNS MAPE (%) MAPE (%) MAPE (%) A A A One Day One Week One Month Two Months Six Months Twelve Months Swiss Franc vs. US Dollar Figs. 3.6 and 3.7 show the graphs obtained during testing from EFLIT2FNS model with all three learning algorithms for one day and one month in advance respectively. Table 3. provides the MAPE during testing for Franc vs. US Dollar. 87

22 Swiss Franc to US Dollar Actual A Price Days Fig. 3.6 (One Day) ahead EFLIT2FNS (Testing).8 Swiss Franc to US Dollar Actual A Price Months Fig. 3.7 (One Month) ahead EFLIT2FNS (Testing) 88

23 Table 3. Swiss Franc vs. US Dollar Time Horizon Models FLANN FLFNS FLIT2FNS MAPE (%) MAPE (%) MAPE (%) A A A One Day One Week One Month Two Months Six Months Twelve Months Results Analysis From the graphs and the tables given above, five observations can be made: () EFLIT2FNS model is producing better results as compared to the other two models, FLANN, FLFNS and FLIT2FLS irrespective of any time horizon chosen out of the six, as discussed in the study. This can be seen from Fig. 3.8 where MAPE during testing of three models M (FLANN), M2 (FLFNS) and M3 (FLIT2FNS) for day and month ahead prediction of Yuan vs. Dollar currency exchange rate dataset is presented; (2) The convergence speed of A and are also faster than the as is seen from Fig. 3.9; (3) The convergence speeds of all five currency exchange rate datasets are more or less similar. In Fig. 3.2 convergence speed of all five datasets in day ahead prediction for EFLIT2FNS with model is given; (4) Prediction accuracy is inversely proportional to time horizon: increase in time horizon shows 89

24 decrease in prediction accuracy. This can also be proved from another error measure i.e. RMSE, considered for this study. The average RMSE during testing obtained from EFLIT2FNS model integrated with for all datasets is.723 in case of one day ahead prediction where as it is.2443 for one month ahead prediction; Fig. 3.2 shows the convergence speed of all time horizons discussed in the study for Won vs. Dollar dataset; (5) As compared to A, maintains slightly better prediction accuracy and takes approximately 4% less time to optimize the parameters of all the models when all other required equipments like hardware and software packages remain constant. In other words, it can be said that is simpler to implement than A A %Error.5.5 M M2 M3 M M2 M3 One Day Ahead One Month Ahead Fig. 3.8 MAPE of M (FLANN), M2 (FLFNS) and M3 (FLIT2FNS) for One day and One month ahead prediction for Yuan vs. Dollar exchange rate data 9

25 2.5 A MSE No. of Experiments Fig. 3.9 Convergence Speed of A, and in One day ahead prediction of Yen vs. Dollar (FLIT2FNS) Yean vs. Dollar Yen vs. Dollar Rupee vs. Dollar Won vs. Dollar Franc vs. Dollar MSE No. of Experiments Fig. 3.2 Convergence Speed of all Datasets for Day ahead prediction (EFLIT2FNS with ) 9

26 Day week Month 6 Months 2 Months.2 MSE Summary No. of Experiments Fig. 3.2 Convergence Speed of Won vs. Dollar for EFLIT2FNS with (all time horizons) In this chapter, an EFLIT2FNS model is used to predict five currency exchange datasets. The application of evolutionary techniques like A and provide a robust prediction. Also, it is observed that is simpler than A to implement for the prediction of currency exchange time series data. MAPE and RMSE obtained from EFLIT2FNS model indicate that it can be used as a feasible alternative for exchange rate prediction for all stake holders in currency exchange market. 92