Chapter 5 Experimental Results
5.1 Prediction of Mackey-Glass Time Series
To verify the proposed RGLS-HCCA, Mackey-Glass time series is utilized to compare RGLS-HCCA with that of other methods. The initial parameters of the proposed RGLS-HCCA are determined by parameter exploration methods ([98] and [99]). As shown in [98], a small population size is good for the initial performance, a large population size is good for long-term performance and a low mutation rate is good for on-line performance, a high mutation rate is good for off-line performance. Moreover, in [99], parameters for genetic
algorithms can be adjusted by exploring the predefined range in increments of a small value.
For instance, the population size has the range from 10 to 100 in increments of 10. Thus, this study adjusts parameters of RGLS-HCCA according to the criteria mentioned in parameter exploration methods. The results of parameters used in this study are listed in Table 5.1 where
“none” in SLE indicates “not used” in the learning phase.
Moreover, since AT
A (with size of 50 × 50 under conditions of 10 fuzzy rules and four
input in a TNFN) in Eq. (3.5) is singular (rank of ATA is about 47) for the example of Mackey
Glass time series prediction, this dissertation incorporates RGLS to make (ATA+λI) is
non-singular. To consider the RGLS parameter (λ), this paper adopts the cross-validation method [100] to adjust it. The notion of the cross-validation method is to divide the training data set into training data and validation data and increase λ with small increments to balance the error of training data set and validation set. Thus, this paper uses cross-validation method to optimize the RGLS parameter (λ) and final adjusted λ of this example is listed in Table 5.1.Table 5.1: Initial parameters of RGLS-HCCA before training.
Value
Parameters PLE SLE
Psize 30 20
Nc 20 none
Selection_Times 40 none
NormalTimes 10 none
ExploreTimes 15 none
Crossover Rate 0.6 0.6
Mutation Rate 0.2 0.3
[Mmin, Mmax] [6, 15] [6, 15]
[mmin, mmax] [-5, 5] [-5, 5]
[σmin, σmax] [3, 20] [3, 20]
Minimum_Support TransactionNum/2 none
researches [102] followed Lapedes and Farber’s work to be a benchmark to examine algorithms. Thus, we utilize such Mackey-Glass time series to perform an analysis on our proposed algorithm and other evolutionary algorithms.
The Mackey-Glass time series is generated from the following delay differential equation:
For this time series prediction problem, Jang [103] extracted 1000 input-output data pairs {x, yd} from t=118 to t=1117, which consisted of four past values of x(t), that is
[
x
(t
−18),x
(t
−12),x
(t
−6),x
(t
);x
(t
+6)], (5.2) where τ=17 and x(0)=1.2 and x(t)=0 for t<0. The reason choosing four past values to predict time series is from Jang’s [103] work which wanted to allow comparison with other researches’ algorithms (Lapedes and Farber [101], Moody [104], Crower [102]). Thus, there are four input to RGLS-HCCA, corresponding to these values of x(t), and one output representing the value x(t+Δt), where Δt is a time prediction into the future. The first 500 pairs [from x(118) to x(617)] are the training data set, and the remaining 500 pairs [from x(618) tox(1117)] are the testing data set used for validating the proposed method. The values are
floating-point numbers assigned using the RGLS-HCCA initially. The fitness function in this case is defined in Eq. (3.26) and (3.27) to train the neural fuzzy network. The evolution learning processes 500 generations and it is repeated 50 times. For comparative analysis, the present study adopts the root mean square error (RMSE), which is defined as follows:1 ( ( 6) ( 6)) 1/2, predicted value by the model with four inputs and one output.
In this example, RGLS-HCCA is compared the performance with the HESP [23], ESP
[14], and SANE [13]. In these models, the learning parameters, which are determined according the parameter exploration method [98] and [99], are shown in Table 5.2. To perform training, the evolution learning processes for 500 generations. Figure 5.1(a)-(d) show the prediction results of the three models. The symbol “o” represents the desired output of the time series, and the symbol “*” represents the output of the four models. Figures 5.2(a)-(d) illustrate the error between the desired and four models’ outputs. As shown in Fig. 5.1-2, the performances of the RGLS-HCCA are better than those of others. Fig. 5.3 shows the learning curves of the four models. As shown this figure, the proposed RGLS-HCCA model converges faster than those of other three models.
(a) (b)
(a) (b)
(c) (d)
Figure 5-2: Prediction errors of the (a) proposed RGLS-HCCA, (b) HESP, (c) ESP, and (d) SANE.
Figure 5-3: Learning curves of the proposed RGLS-HCCA, HESP, ESP, and SANE.
In addition HESP, ESP, and SANE, to further show the effectiveness and efficiency of the proposed RGLS-HCCA model, we also apply MGCSE [15], and traditional genetic algorithm (TGA) [16] to the same problem. To compare with theses algorithms, according the
parameter exploration method [98] and [99], 14, 13, 12, 14, and 12 fuzzy rules are set for HESP, MGCSE, ESP, SANE and TGA, respectively. In addition, the population size has the range of 10 to 250 in increments of 10, the crossover rate has the range of 0.1 to 1 in increments of 0.1, and the mutation rate has the range of 0 to 0.4 in increments of 0.01. To this end, the parameters used for HESP, MGCSE, SANE and TGA are listed in Table 5.2. In addition, as same with RGLS-HCCA, the evolution learning of each method processes for 500 generations and is repeated 50 times. Table 5.3 lists the generalization capabilities of the proposed RGLS-HCCA, HESP, MGCSE, ESP, SANE, and TGA. Clearly, as shown in Table 5.3, RGLS-HCCA obtains a lower RMSE than other methods. In TGA, according to [13], cooperative coevolutionary algorithms can find solutions faster and solve harder problems than TGA. Thus, RGLS-HCCA and other methods (HESP, MGCSE, ESP, and SANE) exhibit lower RMSE than TGA. In SANE, symbiotic evolution is adopted. Since symbiotic evolution only used one population to evaluate every partial solution, the evaluation would cause partial solutions too similar. Instead, the proposed RGLS-HCCA provides several groups to evaluate each partial solution. Thus, the proposed model has more chance to obtain optimal solution.
The explanation can specify that the proposed method has better performance than SANE. To consider group-based evolutionary algorithms (HESP, MGCSE, and ESP), when faced with complex problems, the dimension of chromosomes is still high such that low convergence rate occurs. Thus, this dissertation incorporates RGLS to reduce the dimension of chromosomes and proposes HCCA to self adjust the parameters and structure of TNFN. Based on this fact,
Table 5.2: Initial parameters of four learning models.
Table 5.3: Performance comparison of various existing models.
RMSE Method
Best Mean Worst STD
RGLS-HCCA 0.0017 0.0023 0.0026 0.0005
HESP 0.0118 0.0149 0.0193 0.0017
MGCSE 0.0100 0.0158 0.0190 0.0019
ESP 0.0110 0.0172 0.0219 0.0026
SANE 0.0145 0.0219 0.0313 0.0039
TGA 0.0192 0.0271 0.0747 0.0079
Furthermore, this example also compares the running time of RGLS-HCCA with that of other methods. The running time defined in this case is used to measure the time when the fitness of the algorithm exceeds the predefined value (0.85). The results of four algorithms over 50 runs are reported in Table 5.4. As shown in this table, the proposed RGLS-HCCA is faster than HESP, MGCSE, ESP, SANE, and TGA.
Table 5.4: Comparison of the running time of various algorithms.
Method Best(seconds) Worst(seconds) Mean(seconds) RGLS-HCCA 6.07 43.02 23.28