基于对称百分误差的线性回归与印染工业应用

Linear regression based on symmetric percentage error and its application in printing and dyeing industry

印染工业现存的问题,是如何用染料在样品试验中的浓度预测真实面料中的浓度。理想情况下,染出相同颜色的染料浓度,无论在样品中还是真实面料中应该是等同的,或者至少呈现线性关系。虽然普通线性回归可以基本解决该问题,但其为最小化绝对误差的平方和,而工业上要求用百分误差,此时不能用代数方法求解线性模型。本文选用对称百分误差,并为对应的损失函数构造了梯度下降算法,其解优于用普通线性回归算法,从而提高预测性能。

There is a existing problem in the printing and dyeing industry:predicting the concentration of dyes in the real fabric via that in test samples. In the ideal case,the concentration of dyes that produces the same color should be equivalent in both the sample and the real fabric,or at least have linear relation. Ordinary linear regression can basically solve this problem. But ordinary linear regression minimizes the sum of squares of the absolute error,whereas the percentage error is demanded in industry. The problem cannot be solved with the linear model algebraically. In this paper we select symmetric percentage error,and construct a gradient descent algorithm for the corresponding loss function. Its solution is better than the ordinary linear regression algorithm,and sequentially improve the prediction performance.