规范变换降维与误差修正结合的环境系统的一元线性回归预测

针对高维、非线性环境系统的传统预测模型存在结构复杂、收敛速度慢、求解精度低的局限,提出对环境系统预测量及其影响因子进行幂函数与对数函数相结合的规范变换.此规范变换能使变换后的各影响因子皆等效于一个线性化的规范因子,从而将多因子、非线性的预测建模简化为简单的一个"等效"规范因子的一元线性回归建模;并对预测样本的模型输出进行误差修正,以提高样本的预测精度.在规范变换的基础上,由有m个规范因子的每个建模样本生成一个规范因子的m个"等效"训练样本,n个建模样本共生成N=m×n个训练样本.应用最小二乘法,建立基于规范变换的一元线性回归预测模型.将基于规范变换的一元线性回归预测模型与相似样本误差修正法相结合,分别用于某市5个点位的SO_2浓度预测和南昌市城市降水酸度pH值预测及某河段COD_(Mn)预测,并与多种传统预测模型和方法及基于规范变换与误差修正的3种智能预测模型的预测结果进行了比较.结果表明:该预测模型用于3个实例预测的相对误差绝对值的平均值分别为1.14%、0.49%和1.45%;最大相对误差绝对值分别为2.22%、0.87%和1.85%,与基于规范变换与误差修正的3种智能预测模型的相应误差几乎没有差异,甚至还要小;均远小于多种传统预测模型和方法的相应误差,其预测精度甚至提高了一个数量级以上.基于规范变换与误差修正的一元线性回归预测模型简单、预测精度高、稳定性好,不存在"维数灾难",因而可广泛用于任意系统的预测建模.

A univariate linear regression prediction model for environmental systems based on normal transform dimension reduction and error correction

Aiming at the limitations of complex structure, slow convergence speed and low precision of traditional prediction models for high-dimensional and nonlinear environmental systems, a canonical transformation of prediction variables and their influencing factors for environmental systems was proposed. It combined power function with logarithmic function. The normalized transformation can make the transformed influence factors equivalent to a linearized specification factor, so that the multi-factor and nonlinear complex model can be simplified as a simple univariate linear regression model by a "equivalent" norm factor. Furthermore, the error correction method calculating the output results was proposed to improve the prediction accuracy of the samples. On the basis of the canonical transformation, each model sample, which has m norms factors, generates m "equivalent" training samples, which has only one-gauge factor. Therefore, n modeling samples were symbiotic to N=m×n training samples. Then, a univariate linear regression prediction model was established by the least-squares method based on canonical transformation. The univariate linear regression prediction model was combined with the error correction method of similar samples. concentration of SO₂ at 5 points in a city, the pH value of urban precipitation acidity in Nanchang and the COD_Mn prediction of a river section were all predicted with the model. Three intelligent prediction models were compared with the prediction results. Results show that the average of relative error of the prediction model of three examples are 1.14%, 0.49% and 1.45% respectively, and the maximum of relative error are 2.22%, 0.87% and 1.85% respectively. Compared with the corresponding errors of the three intelligent prediction models based on gauge transformation and error correction, there is almost no difference or even less. They are far less than the corresponding errors of many traditional prediction models and methods, and the accuracy of the prediction accuracy is improved by an order of magnitude. A univariate linear regression prediction model based on canonical transformation and error correction is simple, accurate and stable, and there is no "dimension disaster", so it can be widely used in prediction of arbitrary system.