Grey Prediction Modeling Based On Fractional Order Operators

Grey prediction model is a very important part of grey system theory. With GM(1,1) as its core, grey prediction model has been widely used in social, economic, managerial areas etc. Based on GM(1,1), researchers have done a lot of optimizing work so as to improve the prediction accuracy and expand the application scope of the grey prediction model. In accordance with taking one order accumulating generation sequence as the modeling sequence and then taking one order reducing generation operator to reduce the predicted value of the original sequence in grey prediction model, this dissertation mainly studies on grey prediction model based on fractional order accumulating generation operator and fractional order reducing generation operator. The major achievements include the following points.(1) By utilizing expansion of Gamma function from integer factorial, I have deduced the analytical expression of integer order accumulating generation operator based on one order accumulating generation operator, then expanded integer order accumulating generation operator into fractional order accumulating generation operator, given the analytical expression of fractional order accumulating generation operator. Actually, one order accumulating generation operator and integer order accumulating generation operator are both special cases of fractional order accumulating generation operator.(2) Likewise, I have expanded one order reducing generation operator into integer order reducing generation operator and fractional order reducing generation operator, given the analytical expression of fractional order reducing generation operator. Actually, one order reducing generation operator and integer order reducing generation operator are both special cases of fractional order reducing generation operator.(3) Through theoretical proof and numerical simulation, I have proved that fractional order accumulating generation operator and fractional order reducing generation operator satisfy commutative law, exponential law and reciprocal nature. And the reciprocal fractional order accumulating generation operator and reducing generation operator provide theoretical base for building grey prediction model of fractional order operators.(4) Based on reciprocal fractional order accumulating generation operator and fractional order reducing generation operator, the modeling data of GM(1,1) and DGM(1,1), which are the most commonly used grey prediction model, have been expanded from one order accumulating generation operator sequences to fractional order accumulating generation operator sequences. And then I have built fractional order operator GM(1,1) and fractional order operator DGM(1,1). Through particle swarm optimization algorithm, I have put forward the order optimization algorithm under the condition that fractional order operator GM(1,1) and fractional order operator DGM(1,1) are at the level of minimum average relative error and minimum mean square error. A lot of examples in the literature indicate that both fractional order operator GM(1,1) and fractional order operator discrete GM(1,1) can get optimal fitting effects via order optimization.(5) My research generates homogenous exponential sequences, non homogenous exponential sequences, oscillation exponential sequences as testing data sequences to study on the application scope of fractional order operator of grey prediction model. I have tested the relation between simulation accuracy and sequence length, development coefficient as well as order of fractional order operator GM(1,1) and fractional order operator DGM(1,1) by means of mass numerical simulation.