Modeling mechanism and extension of GM (1,1)

Firstly, the research progress of grey model GM （1,1） is summarized, which is divided into three development stages： assimilation, alienation and melting stages. Then, the matrix analysis theory is used to study the modeling mechanism of GM （1,1）, which decomposes the modeling data matrix into raw data transformation matrix, accumulated generating operation matrix and background value selection matrix. The changes of these three matrices are the essential reasons affecting the modeling and the accuracy of GM （1,1）. Finally, the paper proposes a generalization grey model GGM （1,1）, which is a extended form of GM （1,1） and also a unified form of model GM （1,1）, model GM （1,1,α）, stage grey model, hopping grey model, generalized accumulated model, strengthening operator model, weakening operator model and unequal interval model. And the theory and practical significance of the extended model is analyzed.

Modeling mechanism of a novel fractional grey mode based on matrix analysis

To fully display the modeling mechanism of the novelfractional order grey model （FGM （q,1））, this paper decomposesthe data matrix of the model into the mean generation matrix, theaccumulative generation matrix and the raw data matrix, whichare consistent with the fractional order accumulative grey model（FAGM （1,1））. Following this, this paper decomposes the accumulativedata difference matrix into the accumulative generationmatrix, the q-order reductive accumulative matrix and the rawdata matrix, and then combines the least square method, findingthat the differential order affects the model parameters only byaffecting the formation of differential sequences. This paper thensummarizes matrix decomposition of some special sequences,such as the sequence generated by the strengthening and weakeningoperators, the jumping sequence, and the non-equidistancesequence. Finally, this paper expresses the influences of the rawdata transformation, the accumulation sequence transformation,and the differential matrix transformation on the model parametersas matrices, and takes the non-equidistance sequence as an exampleto show the modeling mechanism.

Stability of GM(1,1) power model on vector transformation

The morbidity problem of the GM（1,1） power model in parameter identification is discussed by using multiple and rotation transformation of vectors. Firstly we consider the morbidity problem of the special matrix and prove that the condition number of the coefficient matrix is determined by the ratio of lengths and the included angle of the column vector, which could be adjusted by multiple and rotation transformation to turn the matrix to a well-conditioned one. Then partition the corresponding matrix of the GM（1,1） power model in accordance with the column vector and regulate the matrix to a well-conditioned one by multiple and rotation transformation of vectors, which completely solve the instability problem of the GM（1,1） power model. Numerical results show that vector transformation is a new method in studying the stability problem of the GM（1,1） power model.

Novel Early-Warning Model for Customer Churn of Credit Card Based on GSAIBAS-Cat Boost

As the banking industry gradually steps into the digital era of Bank 4.0,business competition is becoming increasingly fierce,and banks are also facing the problem of massive customer churn.To better maintain their customer resources,it is crucial for banks to accurately predict customers with a tendency to churn.Aiming at the typical binary classification problem like customer churn,this paper establishes an early-warning model for credit card customer churn.That is a dual search algorithm named GSAIBAS by incorporating Golden Sine Algorithm(GSA)and an Improved Beetle Antennae Search(IBAS)is proposed to optimize the parameters of the CatBoost algorithm,which forms the GSAIBAS-CatBoost model.Especially,considering that the BAS algorithm has simple parameters and is easy to fall into local optimum,the Sigmoid nonlinear convergence factor and the lane flight equation are introduced to adjust the fixed step size of beetle.Then this improved BAS algorithm with variable step size is fused with the GSA to form a GSAIBAS algorithm which can achieve dual optimization.Moreover,an empirical analysis is made according to the data set of credit card customers from Analyttica official platform.The empirical results show that the values of Area Under Curve(AUC)and recall of the proposedmodel in this paper reach 96.15%and 95.56%,respectively,which are significantly better than the other 9 common machine learning models.Compared with several existing optimization algorithms,GSAIBAS algorithm has higher precision in the parameter optimization for CatBoost.Combined with two other customer churn data sets on Kaggle data platform,it is further verified that the model proposed in this paper is also valid and feasible.