高维观测数据的投影寻踪回归法及其应用

On Improving Projection Pursuit Regression Method

高维观测数据的非线性建模是工程实际中普遍存在的问题。本文首次给出了岭函数为多项式形式的投影寻踪回归(简称PPR)的新方法。应用PPR新方法,本文对全向攻击导弹数据进行处理,建立了有效攻击距离R_(min)和R_(max)的非线性回归模型,摸型精度达到了实际工程要求。同时,本文建立了飞机性能参数换算功率N_(op)的数学模型,其相对误差小于2.5%,使模型精度明显提高。该方法适用面广,所需的条件弱,建立的模型形式简单,精度高,易于在计算机上实现,对于工程实际应用有着普遍的实用价值。

In engineering applications there exist a very large number of problemswhose various nonlinear models for high-dimensional observation data needto be established. Although the Projection Pursuit Regression (PPR) methodwas presented by Friedman and Stuetzle[2], Hall[10, 11], no specific forms ofthe ridge functions for PPR have been proposed up to now; so it is difficultto employ the PPR method to solve practical problems. In this paper, theauthors propose polynomials as ridge functions. Thus nonlinear models can beestablished for practical problems. In this paper, nonlinear models for twoimportant practical problems are established and useful results are obtained. 1. The first important problem treated is the determination of airplaneconversion power N_(op), one of many parameters encountered in evaluation ofairplane performance. Up to now it has always been the practice in China tolook up a huge table to determine N_(op). Often straight-line interpolation needsto be employed and this leads to errors that have to be tolerated but arereally not quite satisfactory, especially when such values of N_(op) are used inturn to determine other parameters. The authors establish the nonlinear modelfor 200 tabular values of N_(op) and obtain an imitation curve that deviatesfrom tabular values by less than 2.5% (see fig.1 of full line). Test engineersare very impressed by the accuracy of the authors imitation curve. 2. The second important problem treated is the valid attack distance R_(max)for one kind of full orientation attack missile. The model of R_(max) establishedby the traditional regression methods is far away from meeting the precisionrequired in engineering applications. Using the improved PPR method, theauthors establish the model of R_(max) which satisfies the error bounds. Models established by the improved PPR algorithm are rather simple andprecise. The improved PPR algorithm can be easily performed on computerand be widely used in engineering applications. The improved PPR method hasfollowing features: (a) Nonlinear models for high dimensional observation data can be easilyestablished and the parameters can also be estimated. (b) One only needs to know that the model is a multiple nonlinearregression model before establishing it. (c) The precision of the model is quite good.The L_2-convergence of the improved PPR method has been proved by thefirst two authors elsewhere[2].