最小二乘估计的递推公式

RECURSION FORMULA OF LEAST SQUARE ESTIMATION

设数据是按时间顺序一批一批地取得的,再添进一批新的数据,同时就要剔出一批最早的数据。让数据的批数永远保持不变。根据农业上要用前5年的产量预报下一年产量的要求,现在考虑只用最近这5批数据来计算回归系数向量和残差平方和的递推公式。再设第t批数据向量和数据矩阵分别为数据批号:t=1,2,…1。第t批数据的样本数:n_t,Yk(x_t)=m,n_t>m。于是,我们知道,用这第t数据算出的回归系数向量和残差平方和分别为 B_t=(X_′X)^(-1)X_t′Y_t(?)C_tX_t′Y和Q_t=(Y_t-X_tB_t)′(Y_t-X_tB_t)又设从第i批数据开始依次与下一批数据合并,一直合并到第j批,合并数据向量和数据矩阵以分块形式表出,分别为 我们又知道,由此合并资料算得的回归系数向量和残差平方和为 B_(i…j)=(X′_(i…j)X_(i…j))^(-1)X′_(i…j)Y_(i…j)(?)C_(i…j)X′_(i…j)Y_(i…j) Q_(i…j)=(Y_(i…j)-X_(i…j)B_(i…j))′(Y_(i…j)-X_(i…j)B_(i…j)) 本文指出:在上述假设条件下,则只用最近5批数据计算回归系数向量和残差平方和的递推公式为 B_(23456)=C_(23456)(C^(-1)_(12345)B_(1 345)-C_1^(-1)B_1+C_6^(-1)B_6) Q_(23456)=Q_(12345)-Q_1+Q_6+B′_(12345)C^(-1)_(12345)B_(12345)-B_1′C_1^(-1)B_1+B_6′C_6^(-1)B_6-B′_(23456)C^(-1)_(23456)

Suppose the sampled data are obtained in time order in batches,we must reject the earliest batch of data when we get a fresh one sothat the number of batches can hold constant. According to the requirement in agriculture, the next year yield will be predicted from thepreceding five year yield, thus we consider the recursion formula forcalculating regressive coefficient vector and sum of square of residualusing only the data of last immediate five batchesIf the ith batch data vector and matrix are respeetively;Wheret=1, 2,…, l, the batch number;n_t, the number of samples of the ith batch data;Since rk(x_t)=m, n_t>m: then, by use of the ith batch data, we haevthe calculated regressive coefficient vector and sum of square of residual as follows respectively.B_t=(X_t′X_t)^(-1)X_t;f Y_t(?)C_tX_t′Y_t and Q_t=(Y_t-X_tB_t)′(Y_t-X_tB_t)Suppose we merge every batch data with next batch data successively beginning from the ith batch data down to the jth batch data,merged data vector and matrix shown in blocked groups may be writtenreSpectively: From this merged data, we get the regressive coefficient recto and sam of square of residual:B_(i…j)=(X′_(i…j X_(i…j)^(-1)X′_(i…j)Y_(i…j)(?)C_(i…j)X′_(i…j)Y_(i…j)Q_(i…j)=(Y_(i…j)-X_(i…j)B_(i…j))′(Y_(i…j)-X_(i…j) B_(i…j))It is pointed out in this paper that on the above mentioned condition, we can only use the latest five batches data to calculate recursion formula of the regressive coefficient vector and sum of square of residual as:B_(23456)=C_(23456)(C_(12345)^(-1) B_(12345)-C_1^(-1)B_1+C_6^(-1)B_6)Q_(23456)=Q_(12345)-Q_1+Q_6+B′_(12345)C_(12345)^(-1) B_(12345)-B_1′C_1^(-1)B_1+B_6~1C_6^(-1)B_6-B′_(23456)C_(23456)^(-1)B_(23456)