Immunological Algorithm-based Neural Network Learning for Sales Forecasting

Abstract

This study proposes a variation immunological system (VIS) algorithm with radial basis function neural network (RBFN) learning for function approximation and the exercise of industrial computer (IC) sales forecasting. The proposed VIS algorithm was applied to the RBFN to execute the learning process for adjusting the network parameters involved. To compare the performance of relevant algorithms, three benchmark problems were used to justify the results of the experiment. With better accuracy in forecasting, the trained RBFN can be practically utilized in the IC sales forecasting exercise to make predictions and could enhance business profit.

INTRODUCTION

Traditionally, time series forecasting dominates financial market prediction, which tries to create linear prediction models to trace patterns in historic data (Zhang, Wang, and Zhao 2007). Although autoregressive (AR) integrated moving average (MA; ARIMA) models (Box and Jenkins 1976) are quite flexible in that they can represent several different types of time series (i.e., pure AR, pure MA, and combined AR and MA (ARMA) series), their major limitation is the preassumed linear form of the model. Therefore, it is very important for decision makers to focus on alternative models when movement and nonlinearity play a significant role in the forecasting (Sattari, Yurekli, and Pal 2012).

In the presence of complexity and nonlinearity of the input dataset, neural networks (NNs) are more likely to outperform other popular forecasting methods as the nature of the input dataset gets fuzzier (Azadeh et al. 2012). Further, the radial basis function (RBF) neural network (RBFN) is the most suitable for a complex structure because the adaptive learning capability can approximate the nonlinear system. The transfer function of the hidden layer is generally a nonlinear Gaussian function. It has been shown that the RBFN can estimate any continuous function mapping with a reasonable level of accuracy (Wu and Liu 2012). However, a drawback of the RBFN is that its learning strategy randomly selects from input datasets the RBF centers to be in the hidden layer, and such a mechanism is not adequate for building more ideal RBFNs (Liao 2010).

The bioinspired optimization methods of atomic structures have become very popular in the last few years (Mrozek, Kuś, and Burczyński 2015). Among the most popular nature-inspired approaches, when the task is optimization within complex domains of data or information, are those methods representing successful animal and microorganism team behavior (Marinakis, Marinaki, and Dounias 2011) such as artificial immune system (AIS) algorithm schemes that mimic the characteristics of biological immune systems (De Castro and Timmis 2002). Moreover, we found that no effort has been made to integrate the AIS algorithm with trained RBFN, for which there is room for improvement in terms of the fitting accuracy on function approximation and a practical sales forecasting exercise. Accordingly, this study proposes a variation immunological system (VIS) algorithm for training RBFN.

LITERATURE REVIEW

Most of the traditional methods for dealing with the forecasting problems include simple regression, multivariate regression, time series analysis, and NNs, etc. (Chu 2008). NNs have been broadly applied to many forecasting problems because of their capability in discovering hidden relationships in data (Azadeh et al. 2012). To improve forecasting nonlinear time series events, researchers have developed alternative modeling approaches (Ghiassi, Saidane, and Zimbra 2005). The emergence of various NN topologies and efficient learning algorithms have led to a wide range of successful applications in forecasting (Beliaev and Kozma 2007). This section will briefly introduce the background related to this study.

RBFN for Forecasting

An NN is essentially a nonlinear modeling approach that provides a fairly accurate universal approximation to any function. Therefore, an NN can be trained to predict the future values of a dependent variable (Rojas et al. 2008), the main reason being that NN models can capture more nonlinear patterns hidden in the data, thus leading to performance improvement (Yu et al. 2010). Moreover, RBFNs have a number of advantages compared with other types of NNs, such as better prediction accuracy, simpler network structure, and faster learning process (Tian, Li, and Chen 2010). Accordingly, we are confident to conclude that the RBFN has been recognized as a good approach for forecasting.

The RBFN was proposed by Duda and Hart (1973); it exhibits good approximation, learning ability, and it is easy to train (Maqsood et al. 2005). The design of RBFN can be viewed as a curve-fitting problem in a high-dimensional space. The mathematical justification for the network is based on the approximation or simulation in the higher-dimensional space through a nonlinear process that is more likely to be linearly separable (Chen and Yan 2008). Furthermore, once an RBFN is determined and well trained, it can easily calculate the results for any new input vectors (Yao et al. 2006). In addition, Chen, Wu, and Luk (1999) have employed the orthogonal least squares (OLS) algorithm for training and configuring the RBFN (i.e., RBFN-OLS algorithm). The hidden units are allocated one by one, based on an error reduction ratio (Omidvar 2004). Once the centers and widths of the RBFs are determined, each weight (), used in the approximation may be determined either by direct numerical least-squares methods such as singular-value decomposition or by iterative methods such as the least mean square (LMS) algorithm (Widrow and Hoff 1960).

In the field of prediction, RBFN has received a considerable degree of attention recently due to its universal approximation properties and its ability to estimate simple parameters (Zemouri, Racoceanu, and Zerhouni 2003). RBFNs have the ability to rapidly learn complex patterns and tend to present data and then adapt to changes quickly (Guerra and Coelho 2008). For example, Yu et al. (2008) proposes an RBFN-ensemble forecasting model to obtain accurate prediction results and improve prediction quality further. Also, an RBFN model was developed to forecast the total ecological footprint (TEF) from 2006 to 2015 (Li et al. 2010). All of these represent the capability of RBFN for making good predictions in a wide variety of applications.

AIS-Based Algorithm

Among optimization methods, evolutionary algorithms (EAs), which are generally known as general purpose optimization algorithms, are capable of finding the near-optimal solution to the numerical real-valued test problems for which exact and analytical methods do not produce the optimal solution within a reasonable computation time (Karaboga and Basturk 2008). In the field of immune optimization computing, more and more researches indicate that, compared with EAs, AIS algorithms can maintain better population diversity, and thus, they do not easily fall in local optima (Qi et al. 2015). Further, AIS is a new type of computational system inspired by theoretical immunology and observed immune functions, principles, and models. In recent years, AIS has received a significant amount of interest from researchers and industrial sponsors (Qi et al. 2015).

The immunological algorithm (IA), or AIS, inspired by the theory of immunology (Khilwani et al. 2008), is one of the recently developed evolutionary techniques (Kuo, Chiang, and Chen 2014). The clonal selection (CS) principle (De Castro and Von Zuben 2002) is utilized to design the IAs because of its self-organizing and learning capability when solving combinatorial optimization problems (Khilwani et al. 2008). After that, the modus operandi associated with the CS principle can be explained, in simple terms, as follows. Every molecule that can be recognized by the immune system (IS) is an antigen. When a person is exposed to an antigen, the B-cells produce a population of antibody molecules whose aim is to recognize and bind to the antigen. The higher the antibodies’ ability to recognize the antigen is, the higher is their affinity. In simple terms, the antigenic stimulus causes a proliferation and maturation (controlled mutation) of B-cells capable of producing high affinity antibodies, followed by a natural selection mechanism to preserve the most adapted individuals, which compose the group of memory cells (Silva et al. 2015).

Also, the class of IAs used for numerical and combinatorial optimization is based on the CS principle. Further, the CS algorithm (CSA) is based on two principles (Ozsen and Yucelbas 2015): (1) only cells that recognize the antigen are selected for proliferation; (2) selected and proliferating cells increase their sensitivity to the antigen through a maturation process (Ozsen and Yucelbas 2015). Such CSAs have been applied in various fields of computation in the literature, such as function approximation (Diao and Passino 2002), machine learning, EA and programming, generation, and emergent behavior (Dasgupta and Gonzalez 2002), etc. In addition, biological systems have provided the inspiration for a number of novel biologically inspired computational approaches such as NNs (Twycross and Aickelin 2010). For example, Diao and Passino (2002) proposed an immune-based algorithm for the RBFN structure and adjustment of parameters. Furthermore, Chen and Mahfouf (2006) proposed a novel population-adaptive-based immune algorithm (PAIA) using CS and immune network theories for solving multiobjective optimization problems.

AISs are based on the ISs of vertebrates and derive from various immunological theories, among those, the CS principle, negative selection, immune networks, and danger theory. Even though anomaly detection and classification are the most natural applications for AISs, they are also often applied to the problem of optimization (Jansen and Zarges 2011), because the CSA is capable of optimizing multimodal problems and keeping a varied set of local optimal solutions along a single run (De Castro and Von Zuben 2002). However, in the process of evolution, the CSA is easily trapped in the local optimum and lacks the global search capability (Tien and Li 2012). As such, this study proposes an immunological-based algorithm with RBFN learning for adjusting the network parameters involved and applied to function approximation.

METHODOLOGY

Compared to other metaheuristics, the advantages of AIS are as follows (Zhang, Su, et al. 2015):

1. AIS can incorporate many properties of natural ISs, including diversity, distributed computation, error tolerance, and dynamic learning in the evolutionary process (Aickelin, Dasgupta, and Gu 2014).

2. AIS performs hypermutation to effectively prevent premature convergence.

3. AIS is easy to implement and adjust with a small number of parameters (Zhang, Su, et al. 2015).

As such, this study focuses on training and adjusting the relevant parameters for RBFN. Then, the best solution of the parameter values set can be obtained through the proposed VIS algorithm and used in the RBFN to solve the problem for function approximation. The objective is to obtain the maximum of a fitness function with respect to the parameters of the RBFN (i.e., the hidden node center, width, and weight between the hidden and output layers). The inverse of the root mean squared error (1/RMSE) is used as a fitness function (DelaOssa, Gamez, and Puetra 2006). The fitness values for relevant measured algorithms in the experiment are computed by maximizing the fitness function (i.e., 1/RMSE) (Lee 2008) defined as Equation (1):

(1)

where is the actual output, is the predicted output of the learned RBFN model for the jth testing pattern, and is the number of the testing set (Lee 2008).

Moreover, the nonlinear function that the RBFN hidden layer adopted is the Gaussian function. Next, a typical hidden node in an RBFN is characterized by its center, which is a vector with dimension equal to the number of inputs to the node. The Gaussian basis functions are the most frequently used RBFs (Feng 2006), and the generalized Gaussian function (i.e., ) can be defined as:

(2)

where , represents the distance between the input vector and the jth center (Lee and Verleysen 2005), and is the shape parameter reflecting the decay rate of the density function of the jth RBF (Lu, Hu, and Bai 2015). The flowchart for the proposed VIS algorithm is shown in Figure 1.

FIGURE 1 The flowchart of the proposed VIS algorithm.

The Detailed Description of the Proposed VIS Algorithm

When the CSA is applied to optimization, each element of the search space (i.e., each possible solution) corresponds to the structure of an antibody, and the objective function for this antibody represents its affinity with a certain antigen (Silva et al. 2015). The learning algorithm with respect to training the RBFN with approaches based on the immunological algorithm will be described in the following section. In addition, the pseudocode for the VIS algorithm is illustrated in Figure 2. Furthermore, a detailed description of how the evolutionary procedure for the VIS algorithm was performed and summarized follows.

FIGURE 2 The pseudocode for the proposed VIS algorithm.

Step 1. Initialization:

1. Generate a population that has a random number of antibodies. Each antibody exists as its own center point.

2. With the averaged distance from itself to the center points of all the other antibodies, adjust the width of the center point in each antibody through Equation (3) as below:

   (3)
   where the default width is a constant that exists dependent on the resolution of the problem in this initialization stage. Additionally, the range of the width value is between one to two times the default width. Once the width value is set according Equation (3), it could then ensure that the antibodies are able to maintain a proper distance and avoid the autoimmune response in between.

3. Resolve the value of the weight parameter for each antibody through the LMS method proposed by Widrow and Hoff (1960).

Step 2. Affinity Evaluation:

In immunology, affinity is the fitness measurement of an antibody (Peng and Lu 2015). Usually the antibody with high affinity is more likely to be retained (Qi et al. 2015). As such, Equation (4) is used to calculate the fitness value of each antibody in the population. Next, once each antibody is calculated through Equation (4), the global best (Gbest) solution can be gradually achieved.

(4)

Step 3. Clonal:

The main idea of CS theory lies in the phenomenon that the antibody can selectively react to the antigen. When an antigen is detected, the antibodies that can best recognize such antigen will proliferate by cloning. The newly cloned antibodies undergo hypermutations in order to increase their receptor population (Peng and Lu 2015). Thus, through Equation (5), we can generate a clone number to inherit the fitness value of each antibody in the parent population and have it as the initial local best (Lbest) solution.

(5)

where is the clone ratio and is the number of center points.

Step 4. Intercell Suppression:

During the clonal expansion and mutation process of antibodies, the average antibody affinity increases for the antigen that makes the immune response more effective. This phenomenon is called affinity maturation (Qiu and Lau 2014). Afterward, the antibody will be suppressed when the affinity between two antibodies is less than the suppression threshold. Meanwhile, the antibody with the highest fitness value will be kept to recognize the exhibition of the data character. This process is similar to the mechanism of the memory cells in a biological immune system. As long as the antibody recognizes the specific data character, there will be more antibodies generated around where the syndrome is more severe and they will be used to destroy the antigen.

Step 5. Recruitment:

CSA is a population-based search and optimization design imitating the learning and affinity maturation processes deployed by B-cells for solving a particular problem (Zhang, Yen, and He 2014). After that, to increase the diversity in the new generated population, a certain percentage of antibodies will be randomly generated through Equation (6) and then will be added into the initial population.

(6)

where is the recruitment number, max(training set/3, 100) is the number of the RBFN hidden node centers, is the recruit ratio, and is the decaying factor. As for the , it can enable the stability when implementing the VIS algorithm and accelerate the convergence speed. In which case the decaying factor is , where is the generation number to execute in the algorithm, and is the current generation.

Step 6. Update the Gbest Solution:

The adaptability of the immune response to an antigen can be conceptually formulated as a global optimization search with the antibodies being candidate solutions in the decision space and the antigen being the optima sought for (Zhang, Yen, and He 2014). As such, within the randomly generated population in the proposed VIS algorithm, the relevant evolutionary procedures would recruit more antibodies to where the data character has the largest RMSE in order that the Gbest solution can be gradually resolved. This process is similar to the secondary immune response in an immune system: when the antibody meets the similar antigen that it has encountered before, it is able to faster generate more antibodies with the assistance of the memory cells to rapidly destroy the antigen and lower the probability of pathogenesis.

Step 7. Termination:

The VIS algorithm will not stop returning to Step 2 until a specific number of generations have been achieved.

This study focuses on training and adjusting the relevant parameters for the RBFN. The objective is to obtain the maximum of a fitness function (i.e., RMSE⁻¹) with respect to the parameters of the network. As shown, it is promising to conclude that the proposed VIS algorithm can significantly increase the diversity of population-based solutions during the process of evolution and, thus, increase the possibility of solving the global optimal solution. Then, the best solution of the parameter values set can be obtained and used in the VIS algorithm with the network to further solve the exercise for sales forecasting.

EXPERIMENT RESULTS

This section focuses on training and adjusting the relevant parameters with the RBFN for the function approximation problem. The objective is to obtain the maximum of a fitness function with respect to the parameters of the network and to solve the proper values of the parameters from the setting domain in the experiment. The proposed VIS algorithm will gradually be able to train and, thus, obtain a solution to the parameter values set. Finally, the solution is validated by using the validation set, which has not been utilized throughout the entire training and testing procedures thus far.

Benchmark Continuous Test Functions

Several benchmark continuous test functions have so many local minima that they are challenging enough for performance evaluation (Tsai, Chou, and Liu 2006). This study applies three continuous test functions that are frequently used in the literature to be the comparative benchmark of relevant measured algorithms. The experiment involves the following three benchmark problems: Rosenbrock, Griewank, and B2 continuous test functions (Shelokar et al. 2007), which are defined in Table 1.

Next, there are several parameter values within the RBFN that must be set up in advance to perform training for function approximation. The VIS algorithm is better than the trial-and-error method reported in the literature in that it determines the appropriate parameter values from the verified domain to train the RBFN. Relevant measured algorithms start with the selection of the parameters settings for three continuous test functions shown in Table 2.

TABLE 1 Three Continuous Test Functions Used in This Experiment

Continuous Test Function (Shelokar et al. 2007)	Equation
Rosenbrock
Griewank
B2

TABLE 2 Parameters Setup for Three Continuous Function Testing Experiments

		Continuous Test Function
Parameter	Description	Rosenbrock	B2	Griewank
DS	Search domain	[−5, 5]	[−100, 100]	[−300, 600]

Parameters Setup for the VIS Algorithm

In this study, all parameter settings for the proposed VIS algorithm are obtained according to the related literatures (De Castro and Timmis 2002). Subsequently, according to the Taguchi’s orthogonal arrays (Taguchi, Chowdhury, and Wu 2005), the setting of parameter values for the VIS algorithm are obtained. The VIS algorithm starts with the selection of the parameter settings shown in Table 3 to ensure consistency in the experiment. The maximum number of generations is set at 1000 as the termination condition in the experiment.

TABLE 3 The Setting of Parameters in the Proposed VIS Algorithm

Parameter	Description	Value
	The maximum number of generations	1000
	The number of the RBFN hidden node centers	min(training set/3, 100)
	Population size	50
	Clone ratio	0.1
	Recruit ratio	0.06
	Initial recruitment count	20
	RBF initial width	max(input domain width)/20
	Suppression threshold	||RBF initial width||

Performance Evaluation and Comparison

The advantage of NNs over other models is their ability to model a multivariable problem, given the complex relationships between the variables, and they can extract the implicit nonlinear relationships among these variables by means of learning with training data (Yao et al. 2006). After that, the relevant algorithms will carry out learning on several RBFN parameters’ solutions that are generated by the population during the operation of the evolutionary procedure in the experiment.

Accordingly, once an RBFN is determined and well trained, it can easily calculate the results for any new input vectors (Yao et al. 2006). Thus, Looney (1996) recommends 65% of the parent database to be used for training, 25% for testing, and 10% for validation. This technique helps ensure that the training, testing, and validation datasets are statistically representative of the same population in order that a fair comparison of the models developed can be made (Bowden et al. 2006). Consequently, we use the VIS algorithm to solve the Gbest solution of the RBFN parameter values set, and VIS algorithm randomly generates an unrepeated 65% of the training set from 1000 generated data and inputs the set to RBFN for learning. Then, with the same method, VIS algorithm randomly generates an unrepeated 25% testing set to verify the individual parameter’s solution in the population and calculates the fitness value. So far, the RBFN has used 90% of the dataset in the learning stage. After 1000 generations in the evolutionary process have progressed, the best RBFN parameter’s solution will have been obtained. Finally, the algorithm randomly generates an unrepeated 10% validation set to prove how each individual parameter’s solution approximates the three benchmark problems and records the RMSE values to confirm the learning situation of the RBFN.

Once the data processing steps presented here have been completed, relevant algorithms are ready to run. Finally, the VIS algorithm randomly generates an unrepeated 10% validation set to prove how the individual parameters’ solutions approximate three benchmark problems and records the RMSE values to justify the learning situation of RBFN. The learning and validation stages mentioned herein were implemented 50 times before the average RMSE values were calculated. The values of the average RMSE and standard deviation (SD) for the three algorithms are shown in Table 4.

TABLE 4 Result Comparison Among Three Algorithms Used in This Experiment

Test Function	Rosenbrock Function		Griewank		B2
Algorithm	Training Set	Validation Set	Training Set	Validation Set	Training Set	Validation Set
RBFN-OLS(Chen, Wu, and Luk 1999)	11880 ± 1343.4	12731 ± 2555.7	26.394 ± 2.359	27.85 ± 3.777	5791.5 ± 403.7	5848.8 ± 673.66
AIS-Based(Diao and Passino 2002)	1176.2 ± 136.78	1970.4 ± 354.59	6.917 ± 11.099	50.944 ± 9.981	2199 ± 123.76	2778.7 ± 438.27
VIS	1039.1 ± 104.32	1785.8 ± 293.75	5.848 ± 10.153	25.337 ± 7.482	1277.6 ± 103.98	1422.5 ± 327.16

PRACTICAL EXERCISE FOR SALES FORECASTING

This research tries studies the sales data of the industrial computer (IC) product provided by one manufacturer of the IC industry in Taiwan. It then adopts the proposed VIS algorithm for sales data forecasting verification analysis and compares its accuracy against other algorithms in the literature. The analysis has assumed that the influence of external experimental factors does not exist, and the sales trend of the IC product is not interrupted by any special events. The sales data of the IC product from 2008 to 2009 are used in this exercise.

The experiment of this study was performed on a PC with Intel Xeon^TM CPU, running at 3.40GHz symmetric multiprocessing, with 2GB of RAM. Simulations were programmed on the Java 2 Platform, Standard Edition (J2SE) 1.5. In addition, software EViews^TM 6.0 and SPSS^TM 16.0 were also used in the analysis of Box–Jenkins models to calculate the numerical results.

Building the Box–Jenkins Models

This research carries out sales forecasting for the IC product based on the modeling strategy of an ARIMA model followed by Box–Jenkins methodology through the identification, estimation, diagnostic checking, and forecasting stages (Shukur and Lee 2015); the implementation of each stage is elaborated as follows.

Identification

This research precedes the data identification of ARIMA models through augmented Dickey–Fuller (ADF) testing (Dickey and Fuller 1981). The results reveal that the sales data for the IC product are stationary and, thus, differencing is not necessary. Thus, it can adopt ARIMA (p, 0, q) models to precede estimation and forecasting of sales data.

Estimation

The data eliminate the coefficient of each parameter item that is insignificantly different from zero, and then sequentially sift the candidate ARMA models out. Next, the Akaike information criteria (AIC) (Akaike 1974) are employed to sift the optimal model out (Engle and Yoo 1987). Thus, it infers that the AIC value (i.e., 14.8533) of ARMA (2, 1) mode is the smallest (R-square = 0.1347, adjusted R-square = 0.0981) among all candidate ARMA models, revealing that the ARMA (2, 1) mode is the optimal model and, thus, the most appropriate for the sales forecasting exercise.

Diagnostic Checking

This study adopts the Ljung–Box statistic (Kmenta 1986) to measure the residual values of ARMA models for white noise. The results of model diagnosis reveal that the values of the Ljung–Box statistic are greater than 0.05 in the result of the ARMA (2, 1) mode; the result is serial noncorrelation and it was suitably fitted.

Forecasting

This study adopted the ARMA (2, 1) mode to project forecasting on historical daily sales data for the IC product.

Parameters Setup for the Sales Forecasting Exercise

In this study, several parameter settings with respect to the VIS algorithm are obtained according to De Castro and Timmis (2002). Next, according to the Taguchi’s orthogonal arrays (Taguchi, Chowdhury, and Wu 2005), the setting of parameter values for the proposed VIS algorithm are obtained. The VIS algorithm starts with the selection of the parameters setting shown in Table 5 to ensure consistent basis in the experiment.

TABLE 5 The Setting of Parameters in the Proposed VIS Algorithm

Parameter	Description	Value
	The maximum number of generations	1000
	The number of the RBFN hidden node centers	min(training set/3, 100)
	Population size	40
	Clone ratio	0.25
	Recruit ratio	0.075
	Initial recruitment count	20
	RBF initial width	max(input domain width)/20
	Suppression threshold	||RBF initial width||

The Performance Evaluation for the Sales Forecasting Exercise

The approximation performance of the RBFN prediction is examined with the 10% validation set. This study elaborates how data is input to RBFN for forecasting through relevant algorithms in comparison with the ARMA (2, 1) mode. Consequently, the following predicted values were generated in turn by the moving window method: the first 90% of the observations were used for model estimation and the remaining 10% (Zou et al. 2007) were used for validation and one-step-ahead forecasting. Further, there are 227 observations of which 204 (90%) and 23 (10%) observations are for learning and forecasting, respectively. In addition, the RMSE, mean absolute error (MAE), and mean absolute percentage error (MAPE) are applied to evaluate the forecasting accuracy (Zhang, Zhang, and Zhang 2015) in this study. Afterward, three criteria were used to compare the sale forecasting performance of the VIS algorithm against other algorithms and were presented in Table 6. Among relevant algorithms, the results of the VIS algorithm have the smallest values. Next, the forecasting verification and the paired sample test (t-test) results among relevant algorithms are shown in Table 7.

TABLE 6 The Error Comparison of Relevant Algorithms

Error
Algorithm	RMSE	MAE	MAPE (%)
RBFN-OLS (Chen, Wu, and Luk 1999)	610.771	528.731	9.07%
AIS-Based (Diao and Passino 2002)	861.561	657.674	11.81%
ARMA (2, 1) mode	519.607	401.858	6.73%
VIS	472.579	363.289	5.46%

In Table 7, it is shown that the ARMA (2, 1) mode and VIS algorithm without statistical significance (p-value larger than 0.05, i.e., there is no significant deviation between the predicted values and actual values) are better in forecasting accuracy than RBFN-OLS (Chen, Cowan, and Grant 1991) and AIS-based (Diao and Passino 2002) algorithms.

TABLE 7 The Statistical Results for Paired Sample Test (t-test) Among Relevant Algorithms in This Exercise

	Paired Differences
Algorithm	Mean	Standard Deviation	t	Significant (2-Tailed)
RBFN-OLS (Chen, Wu, and Luk 1999)	−611.240	638.021	−3.030	0.014*
AIS-Based (Diao and Passino 2002)	−433.308	522.681	−2.622	0.028*
ARMA (2, 1) mode	−206.875	565.985	−1.156	0.278
VIS	−194.837	558.402	−1.036	0.681

*5% significance level.

CONCLUSIONS

This research adopted certain immunological-based algorithms and was applied on RBFN to provide the settings of network parameters such as node center, width, and weight. Consequently, further verification analysis and comparison among the relevant mentioned algorithms were executed as well. The evolutionary learning mechanism of the proposed VIS algorithm can be used to train and determine the optimal network parameters within the solution space of the individually generated population in RBFN. Additionally, this study attempts to verify the forecasting results on the sales data of the IC product provided by an international IC manufacturer in Taiwan. With better accuracy in forecasting, the proposed VIS algorithm for the trained RBFN can be practically utilized in the IC sales forecasting exercise to make predictions and could enhance business profit.

In the future, other EAs, such as ant colony optimization and artificial bee colony algorithms can be applied to further improve the accuracy for general forecasting problem. Additionally, the product characteristics in the IC industry have high variety, but each product has limited quantity with high customization. These characteristics increase the degree of variety in general product sales forecasting. Next, perhaps the sales data in the short term would be more stable and could be more beneficial toward improving prediction accuracy. Therefore, the accuracy of forecasting in product sales data within shorter periods could be further compared in the future. In addition, it is common to have significant fluctuation and change in general sales forecasting, and it could be the result of exogenous variables or unexpected variances such as sales force, promotional campaign, and exposure in international exhibitions. These exogenous variables were not considered in this study and thus could be considered in future work.

ORCID

Zhen-Yao Chen

http://orcid.org/0000-0002-2349-5486