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International Journal of Modern Nonlinear Theory and Application, 2012, 1, 51-54 http://dx.doi.org/10.4236/ijmnta.2012.12007 Published Online June 2012 (http://www.SciRP.org/journal/ijmnta) 51 The Non-Equidistant Grey GRM (1, 1) Model and Its Application Ruibiao Zou1, Zouxin Mou2, Wei Yi2 1College of Sciences, Hunan Agriculture University, Changsha, China 2Sany Heavy Industry Co. Ltd., Changde, China Email: rbzou@163.com, yinanchang0@126.com Received April 5, 2012; revised May 4, 2012; accepted May 15, 2012 ABSTRACT Applying the modeling method of Grey system and accumulated generating operation of reciprocal number for the problem of lower precision as well as lower adaptability in non-equidistant GM (1, 1) model, the calculation formulas were deduced and a non-equidistant GRM (1, 1) model generated by accumulated generating operation of reciprocal number was put forward .The grey GRM (1, 1) model can be used in non-equal interval & equal in terv al time series and has the characteristic of high precision as well as high adaptability. Example validates the practicability and reliability of the proposed model. Keywords: Background Value; Grey Model GRM (1, 1) Generated by Accumulated Generating Operation of Reciprocal Number; Non-Equal Interval; Accumulation Generation Operatio n; Grey System 1. Introduction The grey model should be deemed as an important ele- ment of Grey System Theory. Since the Grey System Theory was firstly put forward by Professor Deng Julong, the grey model has been widely used in many fields [1]. Because of its research characteristics of “small sample” & “poor information” and its advantages of simplicity & practicability, the model GM (1, 1) has occupied an im- portant position during the test data processing & testing and the online monitoring [1-6]. The grey system model is mostly based on the equidistance sequence; however, the original data obtained in the practical work is mostly based on the non-equidistant sequence; therefore, the establishment of non-equidistant sequence model has the certain practical and theoretical significances. For the non-negative discrete po int range , its one-order ac- cumulated generating sequence should be mono- tonic; if the sequence is fitted by a curve , the curve should be also monotonic, so that t he grey model GM (1, 1) can be forecasted. If is decreased monotonically, due to the monotonic of , its value of model should be monotonic; if is used as the predictive value of accumulated decreasing generating reducing original sequence, there would be the unreasonable error of calculation. For the original sequence with the mono- tonic decreasing trend , the definition of reverse accumulated generating operation was put forward in the literature [7], so as to establish the grey model GOM (1, 0 X 1 X 1 X 0 XX X 0 X 1 1 1 X 1) based on the reverse accumulated generating operation; the definition of reciprocal generating operation was put forward in the literature [8], so as to establish the grey model GRM (1, 1) based on the reciprocal generating operation; the grey model GRM (1, 1) was improved in the literature [9], so as to establish the improved mod- el—grey model CGRM (1, 1) based on the reciprocal generating operation. The generated sequence with the monotonic decreasing trend can be use for the models established based on the reverse accumulated generating operation and the reciprocal generating operation; the value of model of can be gained after is fitted by the monotonic curve; when the value of mod- el is reduced as the predictive v alue of , there would not be any unreasonable error of calculation pro- duced by means of the modeling method as the same as the traditional accumulated and decreasing generating methods, so as to improve the modeling precision. In the literature [2], the spacing between sequ ences was used as the multiplier to establish the non- equidistance model GM (1, 1); this method could give tacit consent to the linear relationship between the data difference and the time difference; however, it was difficult to ensure whether the construction of model could be consistent with the actual condition. In the literature [3], the coeffi- cient of standard deviation of sequence was decreased by means of the functional transformation method, so that the original sequence could be converted into the new data sequence, so as to estimate the model parameters, 1 X 0 X 1 X 1 X 1 X 0 X C opyright © 2012 SciRes. IJMNTA R. B. ZOU ET AL. 52 and then to establish the model GM (1, 1), resulting in the more complex calculation. In order to improve the fitting and prediction accuracy of mode GM (1, 1), the structure method of multi-background values was put forward in the literature [4-6], so as to establish a variety of non-equidistance models GM (1, 1). The improvement method of background value of non-equidis- tance model GM (1, 1) was put forward in the literature [4,5]; by means of this method, the one-order accumulated gener- ating sequence could be fitted via the homogeneous ex- ponential function, so as to obtain the higher accuracy; however, it could be seen from the form of albino differ- ential equation solution of model GM (1, 1) that, the ex- ponential form of one-order accumulated gen- erated sequence should not be deemed as the non-ho- mogene- ous exponential form, but the homogeneous exponential form only after accumulated decreasing and reducing; if the one-order accumulated generating sequence was fit- ted via the homogeneous exponential func- tion, there would be the certain error of calculation . In the literature [6], the one-order accumulated generating sequence was fitted via the non-homogeneous exponential function to deduce the optimal back ground va lue calculatio n fo r mula, so as to establish th e non-equidis- tance model GM (1, 1). In this paper, the first component was used as the initial condition of differential equation of grey model; there- fore, on the basis of the literature [8, 9], the non-equidistance model GM (1, 1) could be established based on the reciprocal accumulating generating method, which was characterized by the following: High accuracy; good theoretical and practical values. The pra- cticality and reliability of the established model could be shown in the fatigue test data processing examples. 2. Non-Equidistance Model GRM (1, 1) Based on the Reciprocal Accumulated Generating Operation Definition 1. Supposing that the sequence is set as 00 000000 12 ,,, m x tx tx t X, if 1iii , , then should be referred to as the non-equidistance sequence. If ttt const 2, ,im 00 X 0 00 1 k k xt x t , 1, 2,,km then 00 00 12 ,,, m x tx tx tX 00 should be re- ferred to as the reciprocal sequence of . X Definition 2. Supposing that the sequence is set as 11 11 12 ,,, m x txtxt X if 10 11 x txt 0 11k , 11 kk 1k x tx ttxt 1 , , 1, ,1km then should be referred to as the first order recip- rocal accumulated generating operation of non-equidis- tance sequence . X 00 X Definition 3. Supposing that the original data sequence is set as 00 000000 12 ,,, m x tx tx t X (where, 00 j x t 1, 2,,jm shall represent the observed value of variable quantity at the moment j t, m shall represent the number of data), then the sequence 00 0 12 ,,, m x tx tx t should be referred to as the non-equidistant sequence, i.e. the spacing 1 j j tt should not be the con st a nt. In order to establish the model, the one-order recipro- cal accumulated generating operation shall be carried out firstly for the original data, to generate a new sequence. 11 11 12 ,,, m x txtxt X (1) where, 1 j x t 1, 2,,jm can meet the Defini- tion 2, i.e. 01 11 11 01 2, , 1 k jjj j k x txtttk xt xt k m (2) For based on the one-order reciprocal accumu- lated generating operation, the non-equidistance model GRM (1, 1) should be established as per the first order 1 X differential equation set 11 d d xa t b (in which 1 should be the background value), and its albino differen- tial equation should be: 11 d d xax b t (3) Its differential form should be: 01 kk x taztb (4) In the formula 111 1 0.5 kk ztxt xt k , , 0 00 1 k k xt x t , 1, 2,,km 1 k zt should be referred to as the background value of non-equidistance grey model GRM (1, 1) based on the reciprocal accumulated generating operation, i.e. the mean value of accumulated gen erat i ng se quence. If is the parameter of non-equidistance model GRM (1, 1), then the least square of non-equidis- tance grey model GRM (1, 1) based on the reciprocal ac- ˆ, abT a Copyright © 2012 SciRes. IJMNTA R. B. ZOU ET AL. 53 cumulated generating operation should be estimated as: 1 ˆ TT aBBBY (5) where, 11 21 11 32 11 1 11 2 11 2 11 2mm xt xt xt xt xt xt 2 03 0 m B 0 , x t x t x t Y , then the time response of differential Equation (4) of non- equidistance grey model should be: 1 10 1 ˆˆ ˆˆˆk at t k bb xtxt e aa (6) 1, 2,,km The fitted value of the recipro cal of original data gained after reduction should be 01 001 , ˆ ˆ1 ˆ,2,3,, k at k k xt k b xt xt e akm t 1 (7) The value of model of original sequence based on the Definition 1 should be 00 ˆk x t 1, 2,,km The absolute error of fitted data should be 00 00 ˆ kk qtx tx t k . (8) The relative error of fitted data (%) should be 0 00 00 00 ˆ() () 100 i kk k k xt xt et xt . (9) The average value of relative errors of fitted data col- umn should be 1 m i k ek . (10) After the analog values, predictive values, errors and other data can be gained, the non-equidistance model GRM (1, 1) shall be inspected as well [1 ]. 3. Application Examples P. G. Forrest studied how the temperature would impact on the fatigue strength under the action of symmetrical cyclic load on many long-life materials. The experimen- tal data on the fatigue strength of titanium alloy with the change of temperature should be shown at Table 1, which should be deemed as the non-equidistance se- Table 1. Change relation of Ti alloy fatigue strength (σ–1) along with temperature (T). T 100 130 170 σ–1 560 557.54 536.10 T 210 240 270 σ–1 516.10 505.60 486.1 T 310 340 380 σ–1 467.4 453.8 436.4 quence. In this paper, the data indexed in the literature [2, 3] were applied, while the model was established by means of the method indexed in this paper, so as to ob- tain 0.00098951a 1 0 1.7852 k xte , , . 0.0017647b .00098951 1001.783 t ˆ4 The fitted value of original data is: 1 ˆ = [560, 557.7302, 538.7297, 517.8231, 500.2108, 485.5801, 469.0376, 453.0846, 437.6491]. The absolute erro r of fitted data is： k qt = [0, –0.19022, –2.6297, –1.7231, 5.3892, 0.51988, –1.6376, 0.71544, –1.24913]. The relative error of fitted data (%) is: k et = [0, –0.034117, –0.49053, –0.33387, 1.0659, 0.10695, –0.35037, 0.15766, –0.28623]. The average value of relative errors of fitted data col- umn is 0.31396%. In the literature [2], the original data was pre-proc- essed with 50 50tT , 01400 50 X, so that the maximum relative error of modeling was up to 4.86%, and the average relative error of modeling was up to 3.19%. In the literature [3], the model was established by means of the functional transformation method, so that the average relative error of modeling was up to 0.6587%. In the literature [5], the one-order accumulated generating sequence was fitted via the homogeneous ex- ponential function, so that the average relative error of modeling was up to 0.9765%. As a result, it could be seen that there should be the adaptive and scientific me- thods indexed in this paper. 4. Conclusion In this paper, applying the Grey System Theory with the first component of original data was used as the initial condition of grey differential equation, the non-equidis- tance model GRM (1, 1) based on the reciprocal accu- mulated generating operation was put forward; the MAT- LAB program was prepared. The model is suitable for both the modeling of equidistance sequence and the Copyright © 2012 SciRes. IJMNTA R. B. ZOU ET AL. Copyright © 2012 SciRes. IJMNTA 54 modeling of non-equidistance sequence with the charac- teristics of high precision, strong adaptability, etc. The correctness and validity of this model could be shown in the test data processing examples; due to its important practical and theoretical significances, this model should be widely used. REFERENCES [1] Y. X. Luo, L. T. Zhang and M. Li, “Grey Systems and Applications in Mechanical Engineering,” National Uni- versity of Defense Technology Press, Changsha, 2001. [2] Y. X. Luo and J. R. Zhou, “Non-Equidistance GM (1, 1) Model and Its Application in Fatigue Experimental Data Processing and On-Line Control,” Journal of Mechanical Strength, Vol. 18, No. 3, 1996, pp. 60-63. [3] Y. X. Luo, X. Wu and M. Li, “Function-Transfer Method of Parameters Estimation of Grey GM (1, 1) Model and Its Application,” Journal of Mechanical Strength, Vol. 24, No. 3, 2002, pp. 450-452. [4] W. Z. Dai and J. F. Li, “Modeling Research on Non- Equidistance GM (1, 1) Model,” Systems Engineering- Theory & Practice, Vol. 25, No. 9, 2005, pp. 89-93. [5] F. X. Wang, “Improvement on Unequal Interval Gray Forecast Model,” Fuzzy Information and Engineering, Vol. 6, No. 1, 2006, pp. 118-123. [6] Y. M. Wang, Y. G. Dang, Z. X. Wang, “The Optimiza- tion of Background Value in Non-Equidistant GM (1, 1) Model,” Chinese Journal of Management Science, Vol. 16, No. 4, 2008, pp. 159-162. [7] Z. M. Song and J. L. Deng, “The Accumulated Generat- ing Operation in Opposite Direction and Its Use in Grey Model GOM (1, 1),” Systems Engineering, Vol. 19, No. 2, 2001, pp. 66-69. [8] B. H. Yang and Z. Q. Zhang, “The Grey Model Has Been Accumulated Generating Operation in Rcciprocal Num- ber and Its Application,” Mathematics in Practice and Theory, Vol. 33, No. 10, 2003, pp. 21-25. [9] H. Zhou and X. G. Wang, “A Improvement of the Grey Model GRM (1, 1) Generated by Accumulattion Gener- ating Operation of Reciprocal Number,” Transactions of Shenyang Ligong University, Vol. 27, No. 4, 2008, pp. 84-86. |