最大似然法在折光率-组成数据拟合中的应用

Application of maximum likelihood method in data fitting of refractivity and composition

【目的】通过实例阐述最大似然法在折光率-组成数据拟合中的应用。【方法】配制不同碘甲烷浓度的碘甲烷-丙酮标准溶液和碘甲烷-醋酐标准溶液,采用2WA型阿贝折光仪测定标准溶液在25oC下的折光率,分别采用最小二乘法的多项式模型、修正多项式模型、加和模型以及最大似然法的加和模型对碘甲烷-丙酮体系和碘甲烷-醋酐体系的折光率-组成作数据拟合。【结果】对于碘甲烷-丙酮体系,用最小二乘法的多项式模型、加和模型以及修正多项式模型对折光率-组成数据拟合结果比较接近,误差比较小,采用最大似然法拟合误差最小,最大偏差为0.0003。对于碘甲烷-醋酐体系,直接采用最小二乘法多项式拟合最大偏差达0.0145,采用加和模型最大偏差降到了0.0046,仍不能满足高精度试验要求;采用最大似然法拟合最大偏差为-0.0012。【结论】对某些体系(如碘甲烷-醋酐体系)的折光率-组成数据,不能直接采用最小二乘法多项式模型拟合,否则折光率测量的微小误差可能造成组成计算很大的偏差。折光率-组成曲线最好的数据拟合方法是最大似然法,拟合误差非常小,能够满足高精度试验的要求。

[Objective] Expounding the application of maximum likelihood method in data fitting of refractivity and composition by examples. [Method] Preparing methyl iodide-acetone and methyl iodide-acetic anhydride standard solution with different concentration of methyl iodide,determining the refractivity of the standard solution by 2WA abbe＇s refractometer.Making data fitting of refractivity and composition for methyl iodide-acetone and methyl iodide-acetic anhydride system by polynomial model, modified polynomial model, summation modle of least square method and summation modle of maximum likelihood method. [Result] The result is close that make data fitting of refractivity and composition for methyl iodide-acetone system by polynomial model, modified polynomial model, summation modle of least square method and the error is minimum that make data fitting by maximum likelihood method with a maximum deviation of 0.0003.The maximum deviation reachs 0.0145 that make data fitting of refractivity and composition for methyl iodide-acetic anhydride system by polynomial model of least square method and it reduces to 0.0046 by summation motile of least square method,it still can not suffice experiment demands high accuracy. The error is minimum that make data fitting by maximum likelihood method with a maximum deviation of-0.0012. [Conclusion] Polynomial model of least square method is not applicable for data fitting of refractivity and composition in some solution system（e.g, methyl iodide-acetic anhydride system） or a minute error of determination of refractivity will take large calculation error of composition. The best method in data fitting of refractivity and composition is maximum likelihood method with a error small enough which can suffice experiment demands high accuracy.