导数外推和预测理论

THEORY OF DERIVATIVE EXTRAPOLATION AND PREDICTION

工程中存在大量外推和预测问题 ,而根据最小二乘方法求得的曲线和公式只能用于试验段之内 ,用于外推和预测往往会产生较大的误差。为了解决这一问题 ,文中提出一种导数最小二乘方法 ,该方法不但对函数本身 ,而且还能同时对它的各阶导数进行拟合 ,很好地描述曲线及其变化趋势 ,为曲线外推和预测奠定基础 ;并通过对观测值进行映射变换 ,使其呈现出较强的规律性 ,建立两种能描述这种规律的外推和预测模型 ,给出相应的误差估计 ,提出加权累加和累减方法 ,从而形成导数外推和预测理论。工程应用表明 ,本文方法与其他方法相比 。

The least square method has been widely used in research and engineering since it was established by Gauss in 19s century. It has played an important role in many area such as curve fitting, parameter estimating, empirical formula confirming. But the curves and formulas obtained by the least square method can only be used in the test section, extrapolation exceeded the test section will cause a great error. In order to solve this problem, a new derivative least square method is presented in this paper. The method can fit both the test data and its derivatives very well. Therefore it can rightly describe the development trend of the curve. Meanwhile, mathematical mapping is applied to enhance the inherent laws contained in the test data, and two models are also presented to represent the laws. Thus the derivative extrapolation and prediction theory is established herein. It shows that the present method has a higher precision than the other methods in extrapolation and prediction.