The Gray Statistical Method Of Economic Cycles

This paper put forward the fundamental idea of gray statistical method and made some exploration on theory and practical application of gray principal component analysts and gray factor analysis,1 THE FUNDAMENTAL IDEA OF GRAY STATISTICAL METHODThe nature included in less-data sequence is called available nature. Studying available nature by the generation and dig of sequence is the main task of gray system theory meanwhile studying the historical nature of sequence is the main task of statistics. Although the two ones have different studying sequence, they have the same goal which is to found the implicit law of the sequence with the explicit character of the sequence.Apparently what we select to deal with the gray sequence is the method of gray system. But the results of gray system such as absolute degree of gray incidence and matrix of absolute degree of incidence can be taken as the pleased solution of correlation coefficient and correlation matrix in the frame of statistics, as they have the statistical connote, so we can use the statistical method to handle the results until we abstain the pleased solution of what we want.2 THE FUNDAMENTAL THEORY OF GRAY PRINCINPAL COMPONENT ANALYSISThe gray principal component analysis refers to gray principal component analysisof degree of gray incidence. Its basic thought is that it is according the degree of similarity of two series' preface curve shape to judge how much they contact with each other.Let î–¬j be absolute degree of gray incidence between Xt and Xj then satisfies: Write its matrix of absolute degree of incidence is denoted by , that isapparently, V is positive definite and symmetrical. We take V as the pleased solution of correlation matrix for X.Let , we define u'Vv as gray covariance between x = u'X and y = v'X which is denoted by Gcov(x,.y) Because Gcov(x,y) is the pleased solution of covariance between x = u'X and y = v'X in the frame of statistics, the linear operational rule isstill available.In the same way, we define the gray variance of x and denote it by G var(x), Gvar(x) = G vai(u'X) = u'Vu . Find vector satisfying:Theorem 1 For every p -dimension unit vector if , then a'Va = i. is a orthogonal matrix, , that is uj is the jth column of U' . Theorem 2 Write Based on theoreml and theorem2, yt=ui'X is defined as the /th principal component of X. In addition, be the proportion of the /th principal component, be the cumulative proportion of the first principal components. The number of principal components we usually adopt is r which is the smallest one satisfying cr > 0.80 . We composed the coincident index of macroscopic economic prosperity of Jinan cityusing the gray principal component analysis and made the analysis on the circulation of prosperity, and then we got the time series of gray principal component of economic prosperity index as figure 1 shows.3 GRAY FACTOR ANALYSIS ?COMMON FACTOR MODELLet random vector be written as common factors, they are unobservable random variables. s1,s2,...sp are said to be specific factors. From ( i X ( ii ) , the common factors are independent with each other, st only act on yi, The aij of matrix is called loading of factor, A = (aij) is called the loading matrix of factors; because Cov. is thecorrelation coefficient of yi and fj.THE ESTIMATE METHOD ON LOADING MATRIX observations on Y , the determination of factor analysis is to describe the covariance structure of p correlated variables with a few (assume as q) common factors : R = D(y) = D(Af + s) = D(Af) + D(s)That is to estimate the number of common factors, loading matrix A and the covariance of special factors: The estimate method this paper used is the principal component method: Assume the are characteristic roots of which is the estimate of the sample correlation matrix, corresponding normalized characteristic vectors of.If the are small, can be approximated by...