2015 AASRI International Conference on Industrial Electronics and Applications (IEA 2015) Study on Optimization for Grey Forecasting Model.

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1 05 AASRI International Conference on Industrial Electronics and Applications (IEA 05 Stud on Optimiation for Gre Forecasting Model Ying Li Min-an Tang Tao Liu Min-an Tang School of Automation and Electrical Engineering Lanhou Jiaotong Universit Lanhou Gansu China School of Mechanical and Electronical Engineering Lanhou Universit of Technolog Lanhou Gansu China Abstract In order to improve the prediction accurac of GM( model function transformation f ( = cln+ d was applied to improve the smoothness of the original sequence. Considering affecting of the initial value and the bacground value selection to forecasting precision of the model this paper puts forward the idea of optimiation value from the three aspects: smoothness the bacground value and the initial data sequence at the same time and obtained the improved GM model. The model was applied to the bearing sleeve wear prediction and compared with the condition of single models. The simulation rusults show that the improved model has smaller error and higher accurac the new model prediction accurac is above 99.8% and the validit and practicabilit of the method is illustrated which enriches the optimiation theor of gra model and broadens the application scope of gre model. Kewords-gre theor; GM( model; data transformation; initial condition; bacground value; prediction accurac. I. INTRODUCTION As one of the important part of gre sstem theor gre model has been widel used in a number of fields. GM( model with its small sample sie required for modeling simple and convenient calculation and therefore has more advantages than other traditional forecasting methods. In order to improve the accurac of the model fitting and forecasting man scholars have done a lot of researches on increasing the smoothness of original data series improving method of parameter estimation of the model and optimiing the bacground and initial conditions. For the research of improving gre model prediction in order to improve the smoothness of the original sequence man function transformation were presented in the literatures [-9] which include logarithmic function transformation sine transform cosine transform power function transformation tangent function transformation tangent-eponential function transformation power function-eponential function transformation negative eponential function transformation and achieved satisfactor results in practical applications. It can be seen from the time response equation that factors which influence simulation results and prediction accurac of the model depends on the selection of the initial value and parameters a and b. Yet the parameter are also dependent on bacground. Therefore the initial value select and whether the structure of bacground is reasonable directl affect simulation value and prediction accurac of the gre model. The traditional GM( model regards the first data in historical as the initial condition which is equal to fitting curve must be after the first data point in the historical data but that does not necessaril fit the facts. Literatures [0-7] studied the method of improving the prediction accurac of the model in the aspect of improving bacground and initial conditions. However the contribution to unilateral improve model accurac is limited and in man was the optimiation model accurac can be greatl improved. Based on the previous researches this paper combined function transformation technolog which literature [9] proposed with the method of optimiing initial conditions and original bacground which the literature [4] proposed to improve GM( model thereb the model accurac is further improved. II. TRADITIONAL GREY FORECASTING MODEL ESTABLISHMENT Assume that there is a set of non-negative original data sequence : ( = ( ( ( (3... ( 0 = 3... n. Then ( is once accumulated generation sequence: ( = ( ( ( (3... ( ( ( ( i i= = = 3... n. ( is mean generation of consecutive neighbors sequence of ( series : ( = ( ( (3 (4... (3 ( ( ( ( = ( ( + ( = n. To establish the first order whiteniation differential equation ( ( d ( of series : + a = b and solving parameters dt a and b b the least square method. ˆ ( α = B B B T YN = [ a b] T. (4 a and b are the parameters to be identified. 05. The authors - Published b Atlantis Press 75

2 a is development coefficient and b is gre action quantit which are reflect the relationship between the growth rate and the data of original data sequence. ( ( ( ( (3 B = M M (3 YN =. (5 M ( Determining parameters a b and solving the differential equations then we can get model GM( : $ ( b a b ( + = ( + = 3... n. (6 a a The reduction value is: $ a b a $ ( $ ( ( + = ( e ( = ( + ( where = 3... n. (7 III. OPTIMIZED GM( MODEL A. Original Sequence Data Transformation The original data sequence b some function transforming to reduce its smooth ratio however inadequate date transformation can satisf smoothness requirement but not necessaril get higher accurac. The non-negative function transformation f ( = cln+ d c ma( =... n > e which meet the requirement of decreasing its smooth ratio adjusting compression ratio eeping character of concave and conve unchanged reduction error does not increase was put forwards in literature [9]. It is verified that the function transformation to deal with the original data sequence which can improve the smoothness of the initial sequence under the circumstance of remaining the basic rules of the original data sequence unchanged. B. Bacground Value and Initial Value Optimied For optimiing bacground value the literature [0] ( ( ( ( ( used = instead of ( ( ln ( ln ( ( ( ( ( = ( ( + ( to minimie the error caused b the bacground value and enhance precision of modeling sequences to a certain etent. While initial value is another important factor of effecting simulation and prediction accurac of GM( model. The literature [3] based on new information priorit principle and minimum information theor of gre sstem theor the initial condition of the original GM( model ( b instead ( with which is the nth component of series ( so that original sequence information can be full reflected and thus overcomes the disadvantages of GM( modeling has nothing to do with (. Since the bacground value and the initial value optimiation can improve the accurac of the model and the two are independent with each other. Therefore optimie the bacground value firstl and then replace the original date ( in model $ a b a ( ( + = ( e ( with and gets the new time response sequence $ ( ( ( a b a + = e. The literature [4] was studied the method of optimiing initial conditions and bacground value which get a new model was suitable for low growth inde sequence modeling as well as applies to high growth inde sequence modeling and the fitting precision was ver high. C. The Modeling Process of the Optimied GM( Model The new modeling process of GM( model is as follows: Suppose a non-negative original data sequence : ( = ( ( ( (3... = 3... n. Do once an accumulation of data sequence and gets the data ( sequence : ( = ( ( ( (3... = 3... n. Inspect smoothness and quasi-eponent of the original series and anales the feasibilit of modeling. Smooth ratio test formula: ρ = ( ( ρ (00.5. Quasieponential test formula: ( ( ( δ = δ ( (.5. Do function ( transportation for the original sequence f ( = cln+ d to get the new squence : ( = ( ( ( (3... = 3... n. Do once a cumulative to the series and get its generating sequence: = (... where ( ( ( (3 ( ( = ( i 3... n i= =. To the series ( do 76

3 mean generation of consecutive neighbors sequence = (... where ( ( (3 (4 ( ( ( ( ( = ( ( ln ( ln ( ( : = n. Establish the whiteniation differential d equation dt corresponded b using the first order accumulation ( generating sequence and using the method of least squares solution parameters a and b : ˆ T α = B B B T Y = [ a b] T where ( ( + a = b which GM( model ( ( ( ( (3 B = M M Y ( ( Mae an initial value of N N ( (3 =. M get the time response sequence solution of the differential equation: $ ( ( a b a + = e ( = 0... n. The reduction value is: $ ( ( ( a b a + = e. = 0... n. ˆ ˆ( d c Do inverse function transformation: = e and get the date series ŷ : $ $ $ $ $ ( = ( ( ( (3... = 3... n. Use the mean square error ratio C and S small error probabilit p for model testing: C = S n where S = ( n = n S = ( ε ε n = $ ( ( ε ε ε = ; p = P( ( < S. Mean square error ratio is as small as possible and small error probabilit is bigger the better. Model precision grade test table is as shown in table. TABLE I THE MODEL PRECISION TEST TABLE Model accurac level Mean square error ratio C (Good (Qualified 3 (Basic eligibilit 4 (Unqualified C <C <C 0.65 C <0.65 Small error probabilit p p > p < p <0.8 p <0.7 IV. GM ( MODEL IN PREDICTION OF BEARING SLEEVE WEAR In this section we present wear the bearing sleeve with time prediction as an eample. The bearing sleeve has been running in period measuring and recording the amount of wear ever half a month the data is as shown in table. μ m TABLE II BEARING SLEEVE WEAR VOLUME [ ] No Wear volume Firstl test the smooth and level of original data sequence. When > 3 smooth ratio are both within the range of (0 0.5 and smooth ratio are decreasing which satisf with quasi smooth condition. When > 3 the quasi eponent respectivel are both within the range of (.5. So an accumulated generating sequence of the original sequence satisfied with quasi eponent and GM( model can be established. A. Traditional GM ( Model Established Traditional GM ( model: $ ( 0.09 ( + = e where = B. Function Transformation GM ( Model To establish GM ( model b function transforming f ( = cln+ d get transformed model: $ ( e ( + = where c= ma( =.7 d = 60. = C. Optimied GM ( Model Establishment Using function transformation technique to transform the original data sequence and then combined with the optimiation method of bacground value and initial value obtain the optimied GM( model: $ ( ( 8 ( + = e where =

4 V. MODEL PRECISION COMPARISON is as shown in table 3. A. Three Forecasting Models Precision Comparison Compare and analsis the above three methods results TABLE III. COMPARISON OF THREE MODELS SIMULATION VALUE AND RELATIVE ERROR No Measured Original model simulated Literature [9] model simulation New method simulated The average relative error % B. Results of Three Methods Forecasting Comparison Figure shows the comparison of three methods forecast performance. Wear volume Three models simulation value and relative errors comparison Actual value Traditional GM( value Literatual [9] simulation value New method simulation value value at the same time and get a new optimum gre model. Application eamples compared the new improved models results with improved single condition models proves that the optimum model is better than improved single condition models indicates that the new method this paper reported is practical and effective which enrich the optimiation theor of the gra model and broadens the application scope of the gre model. VII. ACKNOWLEDGEMENTS This wor was financiall supported b the National Natural Science Research Foundation ( Innovation Planning Program of Gansu Province Science and Technolog Research and Development Program (090GKCA GKCA03 Science and Technolog Research Projects of Lanhou cit ( value Figure. Predicted value and actual value comparison From table 3 and figure we can see the mean relative error obtained b the method proposed in this paper is smaller than the original model and literature [9] presented and the new model prediction accurac is above 99.8%. The new model mean square error ratio C = < 0.35 small error probabilit p = > 0.95 the accurac of the model belongs to grade one. Therefore we can conclude that the method proposed in this paper has higher simulation and prediction accurac and certain practical value. VI. CONCLUSIONS This paper we have presented a new method of optimiing the traditional GM( model b improving the smooth ratio of data series the bacground value and initial REFERENCES [] Chen T.J. A New Development of Gre Forecasting Model J. China. Sstems Engineering. 8 ( [] Cao C. Fang C.J. and Hu Z.L. Gre Forecasting Model and its Application Based on The Sine Function Transformation J. China. Journal of Mathematics. 33 ( [3] Zheng F. and Wei Y. A new method to improve the smoothness of gra modeling data J. China. Statistics and Decision. 9 ( [4] Li Q. The Further Generaliation for Gre Forecasting Model J. China. Sstem Engineering Theor & Practice. 3 ( [5] Li C.F. and Dai W.Z. An Approach of the Gre Modeling Based on Transformation J. China. Sstems Engineering. 3 ( [6] Guan Y.Q. a and Liu S.F. An Approach to Gre Modeling Based on cot Transformation J. China. Sstems Engineering. 6 ( [7] Chen J. and Xu C.X. An improvement of gre forecasting model J. China. Journal of Liaoning Normal Universit (Natural Science Edition. 8 ( [8] He B. and Meng Q. Stud on generaliation for gre forecasting model J. Sstem Engineering Theor & Practice. (

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