Abstract

The grey forecasting model has been successfully applied in numerous fields since it was proposed. The nonhomogeneous discrete grey model (NDGM) was approximately constructed based on the nonhomogeneous index trend; it increased the applicability of discrete grey model. However, the NDGM required accurate data and better effect when the original data did not meet the conditions and fitting and prediction errors were larger. For this, the NDGM with the fractional order accumulating operator (abbreviated as ) has higher performance. In this paper, the matrix perturbation bound of the parameters was used to analyze the stability of and the can decrease the disturbance bound. Subsequently, the parameter estimation method of was studied and the Particle Swarm Optimization algorithm was employed to optimize the order number of and some steps were provided. In addition, the results of two practical examples demonstrated that the perturbation of was smaller than that of NDGM and provided remarkable predication performance compared with the traditional NDGM model and DGM model.

1. Introduction

Forecasting the future values of time series data plays a very important role in our research; thus, many forecasting methods have been developed for many years, such as the Rough sets theory proposed by Pawlak (see [1, 2]) and fuzzy mathematics proposed by Zadeh (see [3]). However, because of limited knowledge and information, only part of system structure could be fully known. To address this problem, professor Deng proposed grey forecasting models to catch the system development tendency [4, 5].

As a core model of grey prediction, GM (1, 1) model has been widely used in some fields such as transportation, agriculture, economy, and management [6–10]. Meanwhile, many scholars have improved the GM (1, 1) model a lot [11–19], thus enhancing its simulative accuracy and predictive accuracy. However, during the processes of both practical application and theoretical research of the GM (1, 1) model, GM (1, 1) model directly jumped from discrete form to continuous form, which resulted in failing in completely fitting homogeneous exponential sequence in simulation and prediction. Then the discrete grey model [20] was put forward to solve the transformation from discrete to continuous GM (1, 1) model. However, GM (1, 1) model and DGM model were constructed based on a hypothesis that the original data sequence was a homogeneous index sequence. However, the fact was not consistent, and the most original data sequence was the nonhomogeneous index sequence.

Hence, Xie and Liu [21] come up with nonhomogeneous discrete grey model (NDGM), and the model was constructed based on the approximate nonhomogeneous index trend. The results indicate there is no error between original value and simulative value based on pure nonhomogeneous index sequence. And NDGM model was the extension of DGM model while the latter was the special case of the former. NDGM model increased the applicability of discrete grey model. According to improve simulation and prediction accuracies, there were some the results [22, 23] of nonhomogeneous discrete grey model. However, these models had a higher requirement for data; when the data did not meet the requirements, the errors of both model-fitting and prediction were larger. The actual value cannot always meet the definition of monotonic increasing (decreasing) concave (convex) function. So the original data were accumulated to increase the exponent law according to the classical grey modeling mechanism. However, the accumulation of integer order was not suitable for some data, as modeling, and the effects of simulation and prediction were poor. Nevertheless, the current fraction grey model which used factional order accumulation [24–27] has the important significance for improving the performance of grey model.

Therefore, Wu et al. [28] defined the actual data with fractional order accumulation and defined the NDGM model with fractional order accumulation (). But they only presented the algorithm of model and some random fractional order values without considering the initial value of the model and presenting the nature of model. When a system was studied, the stability must be considered. They did not study the stability of the model; hence, the parameters of matrix perturbation bound were used to analyze the stability of model; then it was concluded that the solution of the model perturbation bound of solution was smaller than that of NDGM model. When , the solution of model perturbation bound of solution was smaller and could decrease the disturbance bound. And they did not conclude which order number was the best as well as how to obtain the best order number and other properties. In order to solve these issues, the parameter estimation method of was studied. Furthermore, the Particle Swarm Optimization algorithm was employed to optimize the order number of the model and obtained the better simulation and prediction results. Finally, the results from previous works demonstrated that the perturbation bound solution of that model was smaller than traditional NDGM model in Case 1, and the results the Case 2 practical demonstrated that provided better predication performance than the traditional NDGM model and DGM model.

This paper is organized as follows. In Section 2, we introduced the NDGM model with integer order accumulating operator and the model. In Section 3, the stability of the model was discussed. In Section 4, the Particle Swarm Optimization algorithm was employed to optimize the order number of model; some steps were provided. Finally, in Section 5, the paper was concluded.

2. NDGM Model with Fractional Order Accumulating Operator

Definition 1 (see [29]). Assume that the sequenceis an original data sequence, and the symbol represents a kind of mathematical operational method. When is applied once on the sequence , we have thatis the accumulated generation sequence of , where Then is called the first-order accumulating generation operator of , denoted by 1-AGO. If is applied times on , we obtain where Then is called the -order accumulating generation operator of , denoted by -AGO.
Accordingly, the inverse accumulating generation operator is the inverse operation of the accumulating generation process and plays a role in recovery from the acts of accumulating operators; the inverse accumulating operator is defined as follows:where In Definition 1, the order , and we call the two operators accumulating and inverse accumulating generation operators with integer order.
The sequence is the mean sequence of , where

Definition 2 (see [29]). The equation is called a model, where , The whitenization equation of model is solved to obtain

Definition 3 (see [20]). The sequences and are defined as (1) and (2). Then the equation is called discrete grey model (DGM). Similar with GM (1, 1) model, and are the parameters of DGM model. The equation is called the recursive function of DGM model.

Definition 4 (see [21]). The sequences and are defined as (1) and (2). Then the equation is called nonhomogenous discrete grey model (NDGM). is the simulative value of and is the iterative value of the NDGM model. , , , and are parameters of NDGM model. The equation is called the recursive function of DGM model.

Definition 5. Let the order accumulated generating operator of the original nonhomogeneous index sequence be . Let , ; then , , where order inverse accumulated generating operator of is

Definition 6 (see [28]). Assume thatis established.
The sequences and are defined as (2) and (18). Then (18) is called nonhomogenous discrete grey model with fractional order accumulation (abbreviated as model). is the simulative value of and is the iterative value of the NDGM model. , , , and are parameters of model.

That is least square method. So we can get the expressions of parameters in Proposition 7.

Proposition 7. Based on the least square method the first level parameters , , and satisfy the matrix equationwhere

Proposition 8. The recursive function of model is

Proof. By Definition 6, we have We use the least square method to calculate the value of parameter . By minimizing the error of and , . We can construct a nonrestraint optimized model; then where .

3. The Stability of the Solution of NDGM Model with Fractional Order Accumulation

3.1. The Stability of Model

In order to illustrate the stability of the model, the theorem related to the matrix perturbation analysis was introduced as follows.

Lemma 9 (see [29]). Let , , , and . Suppose and satisfy and . If and , where is the pseudo-inverse of matrix , then where , , and are the tolerance norm and , , and .

Theorem 10. For the model of original data based on nonhomogeneous index sequence, if the rth data is disturbed, that is, , , based on the least square method , let be a solution of the Lemma 9; then the perturbation bound is and where , and when , then where mathematical notation is the same as Lemma 9.

Proof. If is regarded as a disturbance of , then thus, we have Because is maximum of eigenvalue, then By Lemma 9 and based on the least square method , is the disturbed solution of equation , and we have When , then According to the proof of , we have According to this rule, if is regarded as a disturbance of , , both and changed; thus, If is regarded as a disturbance of , we have When , let be the perturbation bound of the NDGM model. By Theorem 10, we have