Verhulst Model of Interval Grey Number Based on Information Decomposing and Model Combination

Grey Verhulst models are often employed to simulate the development tendency with the characteristic of saturated process of S curve. However, the uncertainty of interval grey numbers will be increased since the boundaries of interval grey number are extended by the Axiom of nondecreasing grey degree in the existing Verhulst modeling method. In this paper, the interval grey number is divided into two real number parts, that is, “white” and “grey” parts.Then the “white” and “grey” parts are simulated and forecasted by building the grey Verhulst model and DGM (1, 1) model, respectively. To some degree, this method resolves the issue of amplifying the range of interval grey number. Finally, an example is used to compare the simulation performance between the new model and the traditional model, and the results show that the new model is superior to the other model.


Introduction
The grey theory, which was founded by a famous Chinese scholar Deng [1] based on the idea of "Grey Box" in 1982, is a novel method employed to study the uncertain problems with small sample and poor information [2].As one of the important constituents of grey theory, grey prediction models attract dramatic attentions and are always studied actively by many scholars.Accumulating generation of sequence is an important method in the process of building a grey prediction model, and one can potentially uncover a development tendency existing in the process of accumulating grey quantities so that the characteristics and laws of integration hidden in the chaotic original data can be sufficiently revealed [3].Based on the properties of sequence accumulative generation, the grey exponent laws of sequence can be founded and excavated, and the expressions of simulation and prediction can be deduced.
GM (1,1) model, first proposed by Professor Deng, is a classical grey prediction model, and currently it is of the most significant advantages in its influence, research findings, and extensive applications [4].After 30 years development, grey prediction models have been expanded from original GM (1, 1) model to a great many of novel prediction categories, such as GM (1, N), GM (0, N), GM (2, 1), and DGM (1, 1) [5].The prediction objects are extended to sequence predictions, interval predictions, grey disaster predictions, grey waveform predictions, system predictions, and so forth.
However, the modeling objects of conventional grey prediction model are sequences with the rules of exponent or approximate exponent.Hence the precision of grey prediction is inferior with the nonmonotone increasing (or decreasing) sequence [6].It is important to note that not all the actual systems in the world meet the rules of approximate exponent growth, such as the process of ground subsidence: slow subsidence in beginning, then suddenly quick subsidence, and finally from quick subsidence to slow subsidence evenly sometimes it stops [7].The variation tendency of the whole subsidence process presents the S curve.The common probabilistic methods have the characteristic of large samples modeling, and the samples need to satisfy typical probability distribution.Otherwise the effect of simulation or prediction is inferior.Therefore, the conventional GM (1, 1) model or its derived models and the probabilistic methods cannot effectively simulate in the process of saturation state.In this context, the grey Verhulst model is built to solve such problems.
Grey Verhulst model [8] is a special grey prediction model which is developed to deal with the simulation for small sample data sequence with the characteristic of approximate single peak.This model is capable of simulating the time sequence data with the characteristic of saturated S curved.At present, the grey Verhulst model has been widely used in the fields of commodity economy life analysis [9], landslide prediction [10][11][12], biological breeding prediction [13], traffic accidents [14], energy demand [15], and so forth.In order to improve the simulation and prediction performance of grey Verhulst model, various researches have been done, including the optimization of initial and background values [16,17] and the analysis of modeling mechanism [18,19].Those studies have boosted the development and improvement of grey Verhulst model.
However, grey Verhulst model is only able to be constructed based on real number sequence according to the existing research findings.Though interval grey number sequence is more common in grey system theory, the Verhulst model is not capable of building on such sequence.Along with the development of science and technology, increasing the complexity in society, this trend leads to a larger amount of information in grey degree used to describe such systems.Under this context, the conventional grey Verhulst and its expanded models based on real sequence cannot meet the actual needs of systems modeling.Therefore, a modeling method of grey Verhulst model of interval grey number sequence is developed based on kernel and grey degree by Professor Yang [20].Such method expands the modeling objects of grey Verhulst model from real number to interval grey number.It is of an important significance for enriching the model system of grey number prediction theory.
However, the simulation and prediction of the "kernel" of interval grey number are achieved by building the DGM (1, 1) model of kernel sequence, and the grey degree of simulated or forecasted interval grey number is determined by Axiom of nondecreasing grey degree [21].It is actually a simplified approximate measure, and the grey degree of the aim interval grey number is added; the value range of the simulated or forecasted interval grey number is enlarged to some extent; resulting in the increasing uncertainty of the future grey number.
In this paper, the interval grey number is divided into two real number parts, that is, "white" and "grey" parts.Then the "white" and "grey" parts are simulated and forecasted by building the Verhulst model and DGM (1, 1) model, respectively.To some degree, this method solves the issue of amplification in the value range of interval grey number caused by Axiom of nondecreasing grey degree and has a positive significance for developing and improving the model system of Verhulst model of interval grey number.

Information Decomposing of Interval Grey Number
Definition 1 (see [20]).Assume that an interval grey number ⊗() ∈ [  ,   ],   ≥   ,  = 1, 2, . . ., ⊗() can be decomposed equivalently to the following form: where Formula ( 1) is called the standard form of interval grey number ⊗(). the interval grey number expressed by corresponding standard form is called the standard interval grey number.In Formula (1),   and   , respectively, are the white and grey part of the interval grey number ⊗().

2
,  = 2, 3, . . ., . ( Then is called the grey Verhulst model of the white part sequence of interval grey number. Theorem 4 (see [4]).If one has sequences ,  (1) , and  (1)   as described in Definitions 2 and 3 and if â = [, ]  is a sequence parameters and According to formula (7) and Theorem 4, the time response expression of Verhulst model of white part sequence of interval grey number can be derived as follows:
Theorem 6. Assume that sequences ,  (1) , and  (1)   are described in Definitions 2 and 5; if we have the sequence parameters β = ( 1 ,  2 )  of the DGM (1, 1) model and According to literature [5] and Theorem 6, the time response expression of the DGM (1, 1) model of the grey part sequence of interval grey number is derived as follows: According to literature [5], the final restore expression of the DGM (1, 1) model of grey part sequence is as follows: )   1 . (15)

Verhulst Model
Based on Combination Models.The simulation and prediction are achieved by method of combining the white part sequence Verhulst model and grey part sequence-DGM (1, 1) model.On this basis, the upper and lower limit of interval grey number can then be simulated and forecasted.
In the practical situation, if the white part sequence of interval grey number is S curved, the grey Verhulst and DGM (1, 1) model can be built by taking  (1) and  (1) as the raw data sequence.On the contrary, if the saturated S curve can only be achieved by accumulating generation process on the white part sequence of interval grey number, then the grey combing models can only be built after the accumulating generation gets conducted.Hence, two difference models will be built according to the two circumstances mentioned above, as follows.

White Part Sequence of Interval Grey Number: Saturation
S Curve.Based on the above analysis, if the white part sequence of interval grey number has the S curve, then grey Verhulst and DGM (1, 1) model can be built through taking  (1) and  (1) as the raw data sequence directly.

White Part Sequence of Interval Grey Number: Nonsaturation S Curve.
If the white part sequence of interval grey number is not saturation S curve, firstly, the operation of accumulating generation needs to be applied to white part sequence  and grey part sequence .Following by building the grey Verhulst and DGM (1, 1) model based on  (1) and  (1) , respectively.Finally, the simulation of white and grey part sequence is achieved by inverse accumulating generation.For â+1 = â(1) +1 − â(1)  , according to formula (10), â+1 can be derived as follows: 1 According to Definition 1, ĉ+1 = b+1 − â+1 ; then b+1 = â+1 + ĉ+1 Combining formula ( 19) and ( 20), we can get a formula group as follows: Formulas ( 18) and ( 21) are called the Verhulst model of interval grey number based on information decomposing and models combination.

Comparison and Analysis of Models
Exploiting groundwater is one of the main methods for providing people with living and industrial water in north China.However, overextraction of groundwater will lead to ground subsidence, which might cause disastrous consequences.In order to formulate the amount of groundwater mining scientifically and build the prediction and early warning system of ground subsidence, it is crucial to have the ability to forecast the future amount of ground subsidence according to historical monitoring data, so that the prediction data can be provided to support the research on the solutions of ground subsidence issues.
The process of ground subsidence is as follows: slow subsidence in beginning, then suddenly quick subsidence, and finally from quick subsidence to slow subsidence evenly sometimes it stops, and the variation tendency of the whole subsidence process presents the S curve.Hence, we plan on applying grey Verhulst model to forecast the amount of ground subsidence.Generally, the monitoring of ground subsidence amount for certain area is carried out in multiple continuous periods of time, and in each period the amount of ground subsidence will be monitored repetitively.Due to the monitoring errors and human factors, the collected monitoring data within the same area and period are not identical, neither are we able to identify the reliability of the data.At this time, in order to keep the data intact, an interval data will be defined according to the minimum and maximum values of the collected monitoring data.Assume that monitoring values of certain area are as shown in Table 1.
In this section, the Verhulst model of interval grey number based on information decomposing and models combination will be built according to data in Table 1.Then we will compare and analyze its simulation errors with other model in literatures [20].It is obvious that the data in Table 1 constitute an interval grey number sequence as follows: The scatter point broken line figure of real part sequence  and grey part sequence  can be shown in Figure 1, as follows.
In Figure 1, it can be shown that the real part sequence is saturation S curve and grey part sequence has the characteristic of approximate exponent laws.Hence, the grey Verhulst model of interval grey number sequence  ⊗ can be built by applying formula (18).
Then   According to formula (10), the grey Verhulst model of the real part sequence  of interval grey number sequence  ⊗ is as follows: According to formula (14),  [20], Model 2 for Short.In literature [20], the information scope of interval grey number of prediction and simulation is determined by the axiom of nondecreasing grey degree, and the kernel of interval grey number is forecasted by grey Verhulst model.On the basis, the grey Verhulst model of interval grey number is derived.

Definition of Information.
According to formula (1), it can be known that   =   −   and   which is the grey part of interval grey number is just the information scope that according to the axiom of nondecreasing grey degree, the maximum value in grey part sequence can be used to forecast , ⊗(4), ⊗(5), ⊗( 6)) of interval grey number sequence  ⊗ , according to the computing method of kernel in literature [22], we can get (2 ∘ ) Δ  () = |  ()|/  is the simulation relative error of the upper limit   ; (3 ∘ ) Δ  = (1/( − 1)) ∑  =2 Δ  () is the average simulation relative error of the upper limit sequence.

Figure 1 :
Figure 1: Scatter point broken line figure of real part sequence  and grey part sequence .

Table 1 :
Monitoring data of certain area within six continual periods.

Table 2 :
Simulation values and errors of the novel model.