Fibers and Polymers 2013, Vol.14, No.7, 1179-1183
DOI 10.1007/s12221-013-1179-z
A Modified Grey Verhulst Model Method to Predict Ultraviolet Protection Performance of Aging B.mori Silk Fabric Jinfa Ming1,2, Zhihai Fan3, Zonggang Xie3, Yaoxing Jiang1,2*, and Baoqi Zuo1,2 1
National Engineering Laboratory for Modern Silk, Soochow University, Suzhou 215123, China 2 College of Textile and Clothing Engineering, Soochow University, Suzhou 215021, China 3 Department of Orthopedics, The Second Affiliated Hospital of Soochow University, Suzhou 215006, China (Received August 12, 2012; Revised January 4, 2013; Accepted January 18, 2013) Abstract: In this paper, an improved grey model by using Fourier series of error residuals to predict the tendency of ultraviolet protection performance of aging B.mori silk fabric is proposed. The ultraviolet protection factor of B.mori silk fabric under natural weathering was used as the test data set. In addition, the relative percentage error (RPE), the average relative percentage error (ARPE) and the root mean square error (RMSE) were used to compare the performance of the forecast models. The results showed that the improved EFGVM(1,1) model is more accurate and performs better than the traditional GM(1,1) model and grey Verhulst model. Keywords: Grey Verhulst model, B.mori silk, Aging, Ultraviolet protection performance
Introduction
Literature Review
Public awareness of the hazards of solar ultraviolet radiation (UVR) has increased in many countries. Consequently, more and more people have begun to take positive steps to protect themselves against ambient UVR during summer by using protective gear including hats, sunglasses and clothing [1]. For example, silk fabric is one of the most important summer costume fabrics. Ultraviolet (UV) radiation is one of the major causes of degradation of textile materials, which is due to excitations in some parts of the polymer molecule [2]. Thus, the good resistance to UV radiation is required for the textile materials, especially summer costumes. At the same time, a method is needed to predict ultraviolet protection performance in the using process. Grey system theory is an interdisciplinary scientific area that was first introduced by Deng [3]. Since then, the theory has become quite popular with its ability to deal with the systems that have partially unknown parameters. As superiority to conventional statistical models, grey models require only a limited amount of data to estimate the behavior of unknown systems [4]. In grey system theory, grey Verhulst model (GVM) has avoided the request of big sample in traditional one. Therefore, it has the specific superiority in small sample [5]. To make the grey predicting model more adaptive and precise, an improved grey model by using fourier series of error residuals to predict the tendency of ultraviolet protection performance of aging B.mori silk fabric is proposed. The ultraviolet protection factor (UPF) of B.mori silk under natural weathering was used as our case study to examine the model reliability and prediction accuracy.
GM(1,1) Model The grey model is a time series predicting model. It has three parts: accumulated generation, inverse accumulated generation and grey modeling. The grey model uses the operations of accumulated to construct differential equations. Generally speaking, it has the characteristic of requiring less data [6]. The grey GM(1,1) model is summarized as follows [7]: X
(0)
(0)
(0 )
(0)
(0)
= ( X ( 1 ), X ( 2 ), X ( 3 ), …, X ( n ) ) , n ≥ 4
(1)
where X (0) is a non-negative sequence and n is the sample size of the data. When this sequence is subjected to the accumulating generation operation (AGO), the following sequence X(1) is obtained. X
(1)
(1)
(1 )
(1)
(1)
= ( X ( 1 ), X ( 2 ), X ( 3 ), …, X ( n ) ) , n ≥ 4
(2)
where (1)
X (i) =
i
∑X
(0)
( j ) , j = 1, 2, 3, …, n
(3)
j=1
The first order differential equation of GM(1,1) model is as follows: (1)
( t -) dX (1 ) ----------------+ aX ( t ) = b dt
(4)
In the equation (4), a and b are the coefficients, and the parameters are obtained by using the least square method. The least square method is described below: a b
*Corresponding author:
[email protected] 1179
T
–1
T
= (B B) B Y
(5)
1180
Fibers and Polymers 2013, Vol.14, No.7 ( 1) 1 ( 1) –--- ( X ( 2) + X ( 1 ))…1 2 ( 1) 1 ( 1) –--- ( X ( 3) + X ( 2 ))…1 2
and B =
Jinfa Ming et al.
Applying the inverse AGO, the formula is obtained as follows:
( 0)
X (2)
( 0)
( 0)
, Y=
… ( 1) 1 ( 1) –--- ( X ( n) + X ( n – 1 ))…1 2
X (3) …
(6)
(15)
According to the equation (4), the data of X ( t ) at time (7)
To obtain the predicted value of the primitive data at time (k+1), the inverse accumulating generation operation (IAGO) is used to establish the following grey model. (8)
(0)
(1)
(9) (1)
(0)
(1)
2
(1)
(10)
X ( k ) = – aZ ( k ) + b ( Z ( k ) )
2
b
T
–1
( 1)
2
( 1)
( 1)
2
( 1)
( 1)
–Z ( n ) ( Z ( n ) ) (0)
( 0)
(0)
(0)
(0)
can be defined as:
(0)
= ( e ( 2 ), e ( 3 ) , … , e ( n ) )
(16)
where (0)
(0)
(17)
The error residuals in equation (17) can be expressed in Fourier series as follows [4,8-9]: z (0) 2πi- k⎞ + b sin ⎛ ------2πi- k⎞ e (k) ≅ 1 --- a0 + ∑ ai cos ⎛ ------i ⎝ ⎠ ⎝ 2 T T ⎠ i=1
k = 2, 3, …, n
(18)
– 1-⎞ – 1 --------T = n – 1 and z = ⎛⎝ n 2 ⎠
(19)
It is obvious that T and z will be selected as an integer number. The above equation (18) can be rewritten as follows: (0)
e ≅ PC
(12)
( 1)
B = –Z ( 3 ) ( Z ( 3 ) ) …
e
(0)
(8)
(20)
P and C matrixes can be defined as follows: P=
T
= (B B) B Y –Z ( 2 ) ( Z ( 2 ) )
Then, the error sequence of X
(11)
Similar to the GM(1,1) model, a and b are also the coefficient and are obtained by using the least square method. a
b –ak (0) (0) a Xp ( k + 1 ) = X ( 1 ) – --- e ( 1 – e ) a
(0)
(1)
X ( k ) + aZ ( k ) = b ( Z ( k ) )
Residual Modification (0) First, the X sequence and the predicted values given by the above models:
e ( k ) = X ( k ) – Xp ( k ) , k = 2, 3, …, n
Grey Verhulst Model Verhulst model was introduced by Germany biologist Verhulst to describe some increasing process like “S” curve which has a saturation region [8]. Grey Verhulst model is also a time series predicting model. It can be established by using a first order differential equation. The grey Verhulst model can be defined as: d X (1) (1) 2 ----------+ dX = b ( X ) dx
a (k – 2 )
( 0)
k:
b –ak (0) (0) a Xp ( k + 1 ) = X ( 1 ) – --- e ( 1 – e ) a
a
X (n) (1)
b –ak b (1) (0) Xp ( k + 1 ) = X ( 1 ) – --- e + --a a
(0 )
aX (1)(a – bX (1))(1 – e )e (0 ) Xp (k) = ----------------------------------------------------------------------------------------------------------------------------------------(0) ( 0) a( k – 1) (0 ) (0) a( k – 2) (bX (1) + (a – bX (1))e )(bX (1) + (a – bX (1))e )
1 2π 2π 2π2 2πz 2πz --- cos⎛2------⎞ sin⎛2------⎞ cos⎛2---------⎞ sin⎛22π2 ---------⎞ … cos⎛2--------⎞ sin⎛2--------⎞ ⎝ T⎠ ⎝ T⎠ ⎝ T ⎠ ⎝ T ⎠ ⎝ T ⎠ ⎝ T ⎠ 2 1 2π 2π 2π2 2π2⎞ 2πz 2πz --- cos⎛3------⎞ sin⎛3------⎞ cos⎛3---------⎞ sin⎛3-------- … cos⎛3--------⎞ sin⎛3--------⎞ ⎝ T⎠ ⎝ T⎠ ⎝ T ⎠ ⎝ T ⎠ ⎝ T ⎠ ⎝ T ⎠ 2 …
,
( 0)
…
…
…
…
…
1 2πz⎞ sin⎛n2πz --- cos⎛n2π ------⎞ sin⎛n2π ------⎞ cos⎛n2π2 ---------⎞ sin⎛n2π2 ---------⎞ … cos⎛n---------------⎞ ⎝ T⎠ ⎝ T⎠ ⎝ T ⎠ ⎝ T ⎠ ⎝ T ⎠ ⎝ T ⎠ 2
2
Y = ( X ( 2 ), X ( 3 ), …, X ( n ))
…
…
T
(13)
(1)
The data of X ( t ) at time k:
(21)
One can use the least squares method to solve, and calculate the matrix C:
(0)
aX ( 1 ) (1) X ( k + 1 ) = -------------------------------------------------------------(0) (0) ak bX ( 1 ) + ( a – bX ( 1 ) )e
C = [ a0 a1 b1 a2 b2 … an bn ] T
(14)
T
–1
T (0)
C ≅ ( P P) P e
(22)
A Modified GVM Method to Predict UV Protection Performance
Fourier series correction can be obtained as follows: (0)
(0)
(0)
Xpf ( k ) = Xp – ep ( k ) , k = 2, 3, …, n
(23)
Experimental Data In this study, the used type of B.mori silk fabric was crepe de chine, which was purchased from Jiangsu province of China. The specification of B.mori silk fabric was as follows: Yarn count (Tex): warp, 2.53; weft, 2.06. Fabric density (counts per cm): 30.8×35.2. B.mori silk fabric was placed outdoor for natural weathering aging. The time of natural weathering was from March to May (2010). In the natural weathering process, the UPF of B.mori silk was characterized by UV-1000F fabric ultraviolet tester (Labsphere Company, USA). An average of twenty specimens as the mean±standard deviation was tested for each sample. The UPF of each specimen was calculated from its transmission data with the following equation: 400
∫290 Eλ × Sλ × dλ
UPF = --------------------------------------------400 × S × τ × d λ E λ λ λ ∫
(24)
290
Where Eλ was the relative erythemal spectral effectiveness,
Fibers and Polymers 2013, Vol.14, No.7
Sλ was the solar spectral irradiance in W/m2, τλ was the spectral transmission of the specimen, dλ was the bandwidth in nm, λ was the wavelength in nm. In the natural weathering process, the UPF of B.mori silk fabric shows in Figure 1. With natural aging time going on, the UPF values of B.mori silk fabric increased. This phenomenon was attributed to aromatic amino acid (phenylalanine, tyrosine, and tryptophan etc.) in B.mori silk molecular. These aromatic amino acids contained a large number of benzene ring structure. Due to benzene ring structure having good ultraviolet absorption, the UPF values of B.mori silk fabric increased in natural aging process.
Results and Discussion Table 1 shows the comparison between experimental data and the predicted values of GM(1,1) and GVM(1,1) model. At natural weathering 90 days, the relative errors of GM(1,1) and GVM(1,1) model are 1.1704 % and −0.7975 %, respectively. It is thus clear that the GM(1,1) and GVM(1,1) models have a certain predictive ability. However, when the natural weathering is 75 days, the predicted value of GVM(1,1) model and experimental data differ greatly, its relative error reaches up to −9.4489 %. At the same time, Table 2 depicts the relative errors of GM(1,1) and GVM(1,1) model modified by fourier series of error residuals, namely EFGM(1,1) and EFGVM(1,1) model. From the relative errors of predicted values, EFGM(1,1) and EFGVM(1,1) model exhibit better predictive ability, and its predictive accuracy has improved. To demonstrate the accuracy of the proposed forecasting models, three accuracy evaluation standards are used to examine the accuracy of the models in this study. First, RPE and ARPE represent the relative percentage error and the average relative percentage error, respectively. The formulas are defined as: (0)
(0)
X ( i ) – Xp ( i ) - × 100% RPE = -----------------------------------(0) X (i) (0)
Figure 1. The ultraviolet protection factor of B.mori silk fabric under natural weathering process.
1181
(25)
(0)
1 - n X ( i ) – Xp ( i ) ARPE = -------------------------------------------(0) n – 1 i∑ X (i) =2
(26)
Table 1. The predicted values of GM(1,1) and GVM(1,1) model Time (days) 05 10 15 20 30 45 60* 75* 90*
Measured value 4.15±0.25 5.82±0.09 6.19±0.18 6.38±0.18 6.41±0.42 6.88±0.06 7.47±0.02 7.53±0.30 8.10±0.05
GM value 5.7892 6.0636 6.3509 6.6519 6.9671 7.2972 7.6431 8.0052
Relative error (%) 0.5292 2.0420 0.4561 -3.7737 -1.2659 2.3133 -1.5019 1.1704
GVM value 1.9990 2.8467 3.9318 5.2069 6.5248 7.6324 8.2415 8.1646
Relative error (%) 65.6529 54.0113 38.3730 18.7691 5.1628 -2.1740 -9.4489 -0.7975
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Jinfa Ming et al.
Table 2. The predicted values of the GM(1,1) and GVM(1,1) model modified by using Fourier series of error residuals Time (days) 05 10 15 20 30 45 60* 75* 90*
Measured value 4.15±0.25 5.82±0.09 6.19±0.18 6.38±0.18 6.41±0.42 6.88±0.06 7.47±0.02 7.53±0.30 8.10±0.05
EFGM 5.7218 5.9738 6.2852 6.9304 7.0177 7.1608 7.7196 7.9469
Relative error (%) 1.6873 3.4927 1.4859 -8.1186 -2.0015 4.1392 -2.5179 1.8901
Table 3. The accuracy of the models Evaluation standards RPE ARPE RMSE
GM
GVM
EFGM
15.3952 % 208.0636 % 29.5395 % 0.0192 0.2601 0.0369 0.1227 1.9403 0.2381
EFGVM 29.2612 % 0.0366 0.2311
EFGVM 5.6209 6.3893 6.1808 6.6092 6.3687 7.5957 7.3308 8.0300
Relative error (%) 3.4210 -3.2197 3.1223 -3.1076 7.4317 -1.6827 2.6454 0.8642
improved on the basis of GM(1,1) model. Thus, it can be seen that this paper presents the EFGVM(1,1) model can preferable predicting the tendency of ultraviolet protection performance of aging B.mori silk fabric under natural weathering conditions.
Conclusion (0)
(0)
Where, X ( i ) is the experimental data and Xp ( i ) is the predicted value. Second, the root mean square error (RMSE) is used, which can be calculated using the following formula: n
∑ (X
j=1
(0)
(0)
( i ) – Xp ( i ) )
2
RMSE = ------------------------------------------------n
(27)
According to the above formula, it analyses the precision of comparisons among GM(1,1) model, GVM(1,1) model, and its modified EFGM(1,1), EFGVM(1,1) model. From Table 3, the RPE and ARPE of GM(1,1) model achieve 15.3952 % and 0.0192, respectively. At the same time, the RPE and ARPE of GVM(1,1) model reach up to 208.0636 % and 0.2601, respectively. It is thus clear that the prediction accuracy of GM(1,1) model is superior to GVM(1,1) model. However, after modified by using fourier series of error residuals, the ARPE of EFGM(1,1) is 0.0369 above 0.0192, which is the ARPE of GM(1,1) model. the prediction accuracy of EFGM(1,1) model has not been improved after modified by using fourier series of error residuals. At the same time, the ARPE of EFGVM(1,1) model is only 0.0366 less than 0.2601, which is the ARPE of GVM(1,1) model. This indicates the prediction precision of EFGVM(1,1) model has significantly improved after modified by using fourier series of error residuals. At the same time, the RMSE of EFGVM(1,1) model is only 0.2311 less than 1.9403, which is the RMSE of GVM(1,1) model. This RMSE results also confirmed the precision accuracy of EFGVM(1,1) model has improved. In addition, The ARPE and RMSE values of GM(1,1) are smaller than the EFGVM(1,1) model, but the relative error of GM(1,1) model is larger than the EFGVM(1,1) model. Moreover, the EFGVM(1,1) model is
This paper compares the performances of the various grey models prediction. It is shown that the performance of the grey predictors can be further improved by taking into account the fourier series of error residuals. The ultraviolet protection factors of B.mori silk fabric under natural weathering conditions are used as our case to examine the model reliability and prediction accuracy. The relative errors results show that the modified EFGVM(1,1) model is able to make higher accurate predictions, the ARPE and RMSE values of modified EFGVM(1,1) model are 0.0366 and 0.2311, respectively. Therefore, among these models, the EFGVM(1,1) model is applicable to predict the ultraviolet protection factors of B.mori silk fabric under natural weathering condition. But the optimization of the modified EFGVM(1,1) model is still not researched enough, which will be the key point of our further research.
Acknowledgements We are gratefully acknowledged the support of the First Phase of Jiangsu Universities’ Distinctive Discipline Development Program for Textile Science and Engineering of Soochow University, Doctoral Candidate Academic Award Project of Soochow University (No.5832003611), Jiangsu Province Ordinary Universities and Colleges Graduate Scientific and Innovation Plan (CXLX12_0815) and National Engineering Laboratory for Modern Silk.
References 1. P. Gies, C. Roy, A. McLennan, M. Pailthorpe, and R. Hilfiker, etc. Photochem. Photobiol., 77, 58 (2003). 2. D. Saravanan, Autex Res. J., 7, 53 (2007).
A Modified GVM Method to Predict UV Protection Performance
3. J. L. Deng, Syst. Control Lett., 1, 288 (1982). 4. E. Kayacan, B. Ulutas, and O. Kaynak, Expert Syst. Appl., 37, 1784 (2010). 5. Z. X. Wang, Y. G. Dang, and Y. M. Wang, Proceedings of 2007 IEEE International Conference on Grey Systems and Intelligent Services, 571 (2007). 6. M. Z. Mao and E. C. Chirwa, Technol. Forecast. Soc.
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Change, 73, 588 (2006). 7. J. F. Ming, Y. X. Jiang, and B. Q. Zuo, Fiber. Polym., 13, 653 (2012). 8. Z. J. Guo, X. Q. Song, and J. Ye, Journal of the Eastern Asia Society for Transportation Studies, 6, 881 (2005). 9. Y. H. Lin, P. C. Lee, and T. P. Chang, Expert Syst. Appl., 36, 9658 (2009).
