Tellinghuisen Joel
Department of Chemistry, Vanderbilt University, Nashville, TN 37235, USA.
Analyst. 2005 Mar;130(3):370-8. doi: 10.1039/b411054d. Epub 2005 Jan 28.
In univariate calibration by both the classical method and the standard additions method, calibration data are fitted to a response function y=f(x), from which the amount of an unknown x(0) is estimated by solving an equation of form y(0)=f(x(0)). Most such calibrations are limited to linear response functions f, for which the uncertainty in x(0) can be estimated from well-known expressions. The present work describes and illustrates one-step algorithms, in which x(0) is treated as an adjustable parameter in a nonlinear fit of the calibration data, with its standard error thus obtained numerically as a direct outcome of the fit. The computations are easily implemented with a number of data analysis programs and are illustrated here for a representative one, KaleidaGraph. The methods handle heteroscedastic data as easily as homoscedastic and nonlinear functions as easily as linear, permitting the analyst to experiment with different response functions in the quest for an optimum calibration. The estimates of sigma (x0) are obtained from the variance-covariance matrix V for the fit, so for weighted fitting with commercial programs, it is important to know which V--a priori or a posteriori--is being used.
在采用经典方法和标准加入法进行单变量校准时，校准数据被拟合到响应函数y = f(x)，通过求解y(0)=f(x(0))形式的方程来估计未知量x(0)。大多数此类校准仅限于线性响应函数f，对于线性响应函数，x(0)的不确定度可根据已知表达式进行估计。本工作描述并说明了一步算法，其中x(0)在校准数据的非线性拟合中被视为可调参数，其标准误差通过拟合直接以数值方式获得。这些计算可通过许多数据分析程序轻松实现，此处以一个具有代表性的程序KaleidaGraph为例进行说明。这些方法处理异方差数据与同方差数据一样容易，处理非线性函数与线性函数一样容易，这使得分析人员可以尝试使用不同的响应函数来寻求最佳校准。σ(x0)的估计值从拟合的方差 - 协方差矩阵V中获得，因此对于使用商业程序进行加权拟合，了解使用的是先验V还是后验V非常重要。
