Abstract: Applying the problem of lower precision as well as lower adaptability in non-equidistant multivariable MGM(1,n)model, based on index characteristic of grey model GM(1,1), the characteristic of integral, the function with non-homogeneous exponential law was used to fit the accumulated sequence and the formula of background value was given, taking the mean relative error as objective function and taking the modified values of response function initial value as design variables, based on accumulated generating operation of reciprocal number a non-equidistant multivariable optimizing MGRM(1,n) model was put forward which was taken the first component as the initialization. The new non-equidistant multivariable optimizing MGRM(1,n) model can be used in non-equal interval and equal interval time series and has the characteristic of high precision as well as high adaptability. Example validates the practicability and reliability of the proposed model.
INTRODUCTION
Grey model is the important part in grey theory and have been applied in many fields. MGM(1,N) model is the form of the GM(1,1) model in the n dimensional variables which is not simply composed by GM(1,1) model and is not different from the one order differential equations in the GM(1,n) model. They are differential equations of n dimensional variable and the solutions are got by united solution, in which the parameter can reflect relations of the affection and constraint between every variable (Luo and Xiao, 2009). Zhai et al. (1997) took the first vector of sequence x(1) as the initial condition and establish optimized MGM(1,N) model. Based on the principles of the prior new information He and Luo (2009) took the nth vector as the initial conditions and established the multi-variables new information MGM(1,N) model. Luo and Li (2009) took the nth vector of x(1) as the initial conditions and establish the multi-variables new information MGM(1,N) model, in which the initial values and backgrounds values are optimized. But these model is the equal interval model. Wang (2007) adoped homogeneous index function to fit background values and establish the unequal interval MGM(1,N) model but the non-homogeneous index functions are more common and the above establishment mechanism of the model has some defects. Xiong et al. (2011) established the multi-variables unequal MGM(1,N) model and the background values are got by middle values which make the accuracy need to be improved further. Xiong et al. (2010) adoped the non-homogenous index functions to fit background values and establish the unequal multi-variables grey MGM(1,N) model and the accuracy is improved greatly. Wang et al. (2008a) and Dai and Li (2005) provided many different forms of background values and establish a few non-equal GM(1,1) model. For the grey model accumulated operation is the key and reciprocal is the supplement. For the non-negative discrete data x(0) the data after AGO process is monotonic increasing. It is reasonable that the curve is monotonic increasing which is used to fit x(1). But if x(0) is monotonic decreasing that the AGO operation determines x(1) is monotonic increasing. So the fitting
THE NON-EQUIDISTANT MULTIVARIABLE OPTIMIZATION MGRM(1,N) BASED ON ACCUMULATED GENERATING OPERATION OF RECIPROCAL NUMBER
Definition 1: given sequence:
if Δtj = tj-tj-1≠cons(, i = 1,2,..., n, j = 2,..., m, where n is the variables number and m is the sequence number of variables, then
then:
is called the reciprocal sequence of
Definition 2: Given sequence:
if:
then
Supposed the original data matrix of multi-variables is:
| (1) |
where,
is the objective values at tj for every variable X(0)(tj) (j = 1,2,... m); the sequence:
is the non-equal which means tj-tj-1 is not const.
In order to establish the model we accumulate the original data firstly and get a new matrix:
| (2) |
where,
| (3) |
Based on the reciprocal a ccumulated operation of multi-variables non-equal MGRM(1,n) is n variables one-order differential equations>:
| (4) |
Given:
and the Eq. 4 is:
| (5) |
Taking the first vector
| (6) |
Taking the first column data as the initial values and fix it by substituting X(0)(t1) to X(0)(t1)+β, where β has the same column with X(0) (t1), namely, β = [β1, β2,..., βn]T. The fitting data after restoring the original sequence is:
| (7) |
where,
is the unit matrix. In order to identify A and B, integrating Eq. 4 between [tj-1, tj] and get:
| (8) |
Given:
In the traditional equation the background values can be got by trapezoid area
as the background values between [tj-1, tj] for
where, ai, Gi, Ci is undetermined coefficient which satisfies:
So we can establish the grey model as:
After accumulate subtraction for
| (9) |
Where:
When, ai and Δtj is small, the first two polynomial after expanding
Then:
| (10) |
Put Eq. 10 into 9:
| (11) |
From the initial condition:
and get:
| (12) |
Putting Eq. 10 and 12 into background values formula:
then get:
| (13) |
Given ai = (ai1, ai2,..., ain, bi)T (i = 1, 2, ..., n),by the least square method and get the estimated âi for αi:
| (14) |
Where:
| (15) |
| (16) |
Get identified A and B:
| (17) |
Based on the reciprocal accumulated and optimized operation the formula of MGRM(1,n) model is:
| (18) |
the fitting values of the original data after restoring is:
| (19) |
by the definition 1 the model values for the original sequence
Define the absolute error of ith multi-variable:
The relative error (%) for ith variables is:
The average error (%) for ith variables is:
The average error (%) for all data is:
| (20) |
Taking the average error f as the objective function and β is the design variables, optimized function Fmincon in Matlab 7.5 is adopted to solve it.
EXAMPLES
Example 1: the data of PA66 mechanical properties versus absorbance water ratio (Xiong et al., 2011; Xiong et al., 2010) is as Table 1.
Based on the reciprocal accumulated operation the unequal optimized model is established and the parameters is:
| Table 1: | Affection of absorbance water to the mechanical properties of PA66 |
The fitting data of
The absolute error of X(0) (t3) is:
q = [0, -1.2673,-3.8503,-2.5419,-0.21171,4.2698,1.0743, -0.0043601,2.1584]
The relative error (%)of
e = [0, -1.5015,-4.4615,-3.0153,-0.26041,5.7007,1.4191, -0.0059565,3.2263]
The mean value of the relative error for
Example 2: the experimental data of TiN film as the loading is 600 N and the relative sliding speed is 0.314, 0.417, 0.628, 0.942 and 1.046 m sec-1 (Table 2).
If taken the sliding speed, friction coefficient and wear loss ratio as
The fitting data of
The absolute error of
q = [0, 0.00037413, -0.00062244, 0.00034973,-0.0066412]
| Table 2: | Experimental data of TiN film (Youxin and Xiaoyi, 2008) |
The relative error of
e = [0, 0.14501,-0.23488, 0.12811, -2.306]
The mean values of relative error is 0.56279% and the mean values of the relative error of the model is 0.76992%, indicating the high accuracy of the mode.
CONCLUSION
In the system of multi-variables non-equal sequence which the multi-variables are affected and constrained. Based on index characteristic of grey model, the characteristic of integral, reconstructing background value in non-equidistant multivariable optimization MGRM(1,n) was researched and the discrete function with non-homogeneous exponential law was used to fit the accumulated sequence and the equation of reconstructing background value was given. Taking the mean relative error as objective function and taking the modified values of response function initial value as design variables, based on accumulated generating operation of reciprocal number, a non-equidistant multivariable optimization MGRM(1,n) model was put forward which was taken the first component as the initialization. The optimum MGRM(1,n) model can be used in non-equal interval & equal interval time series and has the characteristic of high precision as well as high adaptability. Example validates the practicability and reliability of the proposed model.
ACKNOWLEDGMENT
This research is supported by the grant of the 12th Five-Year Plan for the construct program of the key discipline (Mechanical Design and Theory) in Hunan province (XJF2011(76)), the National Natural Science Foundation of China (No. 51075144).