Research On Matrix Sequence Oriented Grey Modeling

Grey System theory was proposed in the early1980s, which is a theory ofuncertainty researching for limited information and widely applicated in various fieldsincluding image processing, economy, environmental, control field, engineering, etc.Grey modeling is one of the improtant contents of Grey System theory, and it is theprocess of constructing an approximate differential equation model based on sequence.The theory system of Grey modeling includes Difference Information principle on thesequence, Synchronous Mapping, grey Cause and white Effect of derivative, greydifference equation, grey model, solution and parameter recognition of grey model, etc.Therefore, Grey model is modeling for small sample and limited information tosimulate and predict the development of things. Grey model has been applied to solvemany practical problems since it was proposed. Meanwhile, there are some improvedGrey Models according to the requirement of various applications. With thedevelopment of application fields and Information Technique, the traditional Greymodel based on small sample and point sequence can not satisfy the simulation andprediction requirement in practice. Thereby, the research of Grey modeling based onmatrix sequence is carried out in this dissertation.Combining with Grey modeling theory system, Grey Models based on matrixsequence are developed in this dissertation. The possible space and time relationships inmatrix sequence are further analyzed. Then, corresponding to these relationships,relative Partial Accumulating Generation Operations, Grey partial derivatives and Greypartial difference equations are defined, and relative Partial Grey Models areconstructed. The theory research and experimental results demonstrate that GreyModels based on matrix sequence not only inherit and extend the existing greymodeling theory system, but also make up the regret which traditional grey model cannot used well to model big sample data sequence, three-dimensional time and spacesequence, etc. The main innovation and content of this dissertation is concluded asfollows.Background of this dissertation is introduced in Chapter One. The researchbackground includes the research situation of grey modeling, application of traditionalGrey Model, basic concept and modeling principle, the significance of this research.Traditional grey model can not satify the application requirements of data modeling at the present stage. These requirements are to simulate and predict these data sequence:multi-stream time series, big sample data sequence, especially the time or spacesequence with planes as their elements. Moreover, the modeling and analysis of thesignal sequences, for example three-dimensional time or space sequence and imageframe sequence in videos, more and more becomes the key to the research work ofvarious fields.Matrix sequence is introduced into grey modeling in Chapter Two, and MatrixAccumulating Generation Operation (MAGO) is defined, and Matrix Sequence GreyModel (MGM) and Diagonal Transformation Matrix Sequence Grey Model (DTMGM)are constructed. MAGO is an important foundation of Grey Modeling based on matrixsequence. According to the method of traditional Grey modeling, MGM based onMAGO is an expansion on space of the sequence, and it makes more difficult toparameter setting and recognition. In addition, diagonal transformation is implementedon the matrix sequence to reduce the complexity of modeling computing. DTMGM isconstructed based on diagonal matrix sequence.Three kinds of possible grey relationships between time and space are analyzed inChapter Three, Four and Five. Then, corresponding Partial Accumulating GenerationOperation (PAGO) and grey partial differential equation are defined, and correspondingPartial Grey Model are constructed, relatively. The extending direction of the sequenceis time axis, and every matrix plane in the sequence has two space axes in the horizontaland vertical direction. The following three grey relationships between time and spaceare analyzed:(1) white correlations on the space axes, grey correlation on the time axis;(2) grey correlation in the diagonal direction of space axes, grey correlation on the timeaxis;(3) grey correlations on the space axes, grey correlation on the time axis.Generation Operation and grey partial differential equation is the theory and realizationfoundation of these models. So, the Generation Operation corresponding to three kindsof time and space grey relationships are Partial Accumulating Generation Operation(PAGO), Diagonal Transformation PAGO (DTPAGO), secondary-diagonal MeanTransformation PAGO (MTPAGO). Three models are Partial Grey Model (PGM),Diagonal Transformation PGM (DTPGM) and secondary-diagonal MeanTransformation PGM (MTPGM). Then, the three chapters give the solutions andparameter recognition methods of these models, respectively.These Grey Models and Partial Grey Models based on matrix sequence arecompared in Chapter Six. The relationship of these models is analyzed. The experiment results of these models and the time complexity in their modeling process are given.MGM and DTMGM are approximate differential equation models, PGM, DTPGM andMTPGM are approximate partial differential equation models. There are different timeand space grey relationships among last three partial grey models, so, they are differentbut interrelated. The experiment results can conclude that DTMGM and DTPGM aresuitable to simulate and predict periodic data sets, and MTPGM, MGM, PGM is apt tonon-predict periodic data sets. The time complexity in their modeling process showsthat the running time of these grey models based on matix sequence is acceptable.The application fields of the grey models based on matrix sequence are introducedin Chapter Seven. There is an example that they are used to predict the CT image framesfor3D Reconstruction in image processing. Finally, conclusion of this work and thefurther research concern are given at the end of this dissertation.