非线性模型对数回归的偏差校正及与加权回归的对比分析
Bias Correction in Logarithmic Regression and Comparison with Weighted Regression for Non-linear Models
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摘要: 本文结合大样本的立木生物量实测数据,对非线性模型对数回归的偏差校正问题进行了探讨,并与加权回归结果进行了对比分析。首先,分析了对数回归产生偏差的内在原因,并提出了一个新的校正因子,同时对另外3个偏差校正因子一并进行了检验,结果表明本文和Baskerville(1972)提出的校正因子,能保证与加权回归估计结果趋于一致;然后,对非线性加权回归中基于普通回归残差推导的权函数与通用权函数(W＝1/f(x)2)的拟合效果进行了对比分析,结果表明二者基本相当,而通用权函数更具有广泛的适应性。建议对带有异方差的非线性模型,最好直接采用加权回归进行估计;当按照通用权函数进行估计其总相对误差超出一定范围时,应该根据普通回归估计的残差推导效果最佳的权函数后再进行加权回归。Abstract: Non-linear models with heteroscedasticity are commonly used in forestry modeling, and logarithmic regression and weighted regression are usually employed to estimate the parameters. Using the single-tree biomass data of large samples, the bias correction in logarithmic regression and comparison with weighted regression for non-linear models are studied in this paper. The immanent cause producing bias in logarithmic regression is analyzed, and a new correction factor is presented with which three commonly used bias correction factors are examined together, and the results show that the correction factors presented here and by Baskerville (1972) should be recommended which could insure the corrected model to be asymptotically consistent with that fitted by weighted regression. Secondly, the fitting results of weighted regression for non-linear models, using the weight function based on residual errors of the model estimated by ordinary least squares (OLS) and the general weight function (W=1/f(x)2) presented by Zeng (1998) respectively, are compared with each other that show two weights works well and the general function is more applicable. It is suggested that the best way to fit non-linear models with heteroscedasticity would be using weighted regression, and when the total relative error of the estimates from the model fitted by the general weight function is more than a special limit such as ±3%, a better weight function based on residual errors of the model fitted by OLS should be used in weighted regression.
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Key words:
- non-linear model
- / biomass model
- / logarithmic regression
- / weighted regression
- / bias correction
- / heteroscedasticity
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