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作为路径积分的似然与成本AI 解读

SUMMARY We consider the optimal estimation, prediction and control etc. of time series. The essence of a path integral characterization is considered to be that optimal estimates and decisions etc. are obtained by the free extremization of some form. In Sections 2–5 it is shown for some stock situations that this property is ensured by the direct maximization of likelihood (rather than by appeal to the formally equivalent but operationally contrasting least square…

非平稳过程预测量的递推关系AI 解读

Summary It is shown that if a process {xt} is generated recursively by a relation of type ∑j=0pajtxt−j=∑k=0qbktϵt−k+λt, where ajt, bkt and λt, are time-dependent sequences and {ϵt} is a process of unautocorrelated variables, then in general the linear least-square predictor of xt+m based upon {xs; s≤t}, denoted vt, obeys a recursion of the type ∑k=0qβktξt−k=∑j=0τcjtxt−j+vt, where r = max(p – 1, q – m). A method is given for obtaining the coefficients βkt, cjt and vt, and…

多元自回归拟合与谱密度矩阵的近似规范分解AI 解读

The recursive method proposed by Durbin (1960) for the fitting of autoregreseive schemes of successively increasing order is generalized to the fitting of multivariate autoregressions, and of schemes with rational spectral density function. It is also shown that an auto regression fitted from the Yule-Walker relations, even if of insufficient order, has the necessary stability properties. This property holds in the multivariate case, too, and is important in connexion with a…

曲线与周期图平滑AI 解读

Summary The difficulty in constructing smoothing formulae is to express quantitatively the type of smoothness one expects of the curve one is estimating. An argument is given in Sections 1 and 3 for formulating this “smoothness hypothesis” in terms of the properties of a population of curves of which the curve being estimated is a member. In equation (20) we obtain a solution for the matrix of optimum weighting coefficients in terms of certain “population moments” of the…

多元平稳时间序列分析AI 解读

Summary After some preparatory work, the least square estimation equations are derived for a purely nondeterministic stationary multiple process (Th. 6). The asymptotic covariances of the parameter estimates are calculated for a normal process (Th. 9) and a test of fit derived (Th. 10). The testing of a sunspot model provides an illustration of the methods developed.