Nonlinear modelling of curvature by bi-linear metamodelling
Nonlinear modelling of curvature by bi-linear metamodelling
复制标题
DOI：
10.1016/j.chemolab.2011.04.010
复制
发表时间：
2012-08-01
影响因子：
3.9
通讯作者：
Martens, Harald
中科院分区：
文献类型：
--
作者：
Isaeva, Julia;Saebo, Solve;Martens, Harald
线曲率现象--一条光滑的线z = f（x）偏离直线--经常在科学数据中观察到。今天，将非线性数学模型拟合到曲线需要缓慢的迭代搜索过程，由于局部最优而容易出错。本文提出了一种新的通用方法-直接查找法，用于对这种曲率进行数学描述，具有较少的主观性和更简单的参数估计。新的模型测量方法是基于双线性元建模来模拟一整套能够描述曲率的潜在相关非线性模型。本文从不同的科学学科中收集了38个非线性数学模型。每个模型都可以根据其输入参数值集生成各种单调的S形或拱形输出形状。对于每一个非线性模型，其模型现象-其可能的输出曲线的剧目-是建立一次和所有的计算机模拟统计设计，以填补相关的模型参数空间在选定的分辨率。通过主成分分析对该模拟曲线集进行压缩。由此产生的一组38双线性元模型模拟的非线性模型的输入输出行为。然后，为了参数化新曲线，每条曲线的输入数据通过其元模型拟合到所有相关的非线性模型。具有足够好拟合的模型被列为合理模型，其未知参数值根据其最接近的已知模拟设计点进行估计。因此，缓慢的，迭代的非线性曲线拟合被替换为快速线性投影与简单的查找量化。该算法消除了传统的初值选取问题，避免了局部最优解。多元元建模允许处理广泛的非线性曲率描述。(C)2011 Elsevier B. V.保留所有权利。
The phenomenon of line curvature - that a smooth line z = f(x) deviates from being straight - is often observed in scientific data. Fitting nonlinear mathematical models to curves today requires slow iterative search processes prone to errors due to local optima. A new generic method, the direct look-up method, is presented for mathematical description of such curvature with less subjectivity and simpler parameter estimation. The new modelometric method is based on bi-linear metamodelling to emulate a whole set of potentially relevant nonlinear models capable of describing curvature. A comprehensive set of 38 nonlinear mathematical models was here collected from different scientific disciplines. Each model can generate a wide range of monotonous sigmoid or arched output shapes depending on their set of input parameter values. For each nonlinear model, its model phenome - its repertoire of possible output curves - was established once and for all by computer simulations statistically designed to fill the relevant model parameter space at a chosen resolution. This simulated curve set was compressed by Principal Component Analysis. The resulting set of 38 bi-linear metamodels emulates the input-output behaviour of the nonlinear models. Then, to parameterise new curves, the input data of each curve were fitted to all relevant nonlinear models via their metamodels. Models with good enough fit were listed as plausible, and their unknown parameter values were estimated from their closest known simulation design points. Thereby, the slow, iterative nonlinear curve fitting was replaced by a fast linear projection with a simple look-up quantification. The traditional problem of choosing initial values to avoid local optima was eliminated. The multivariate metamodelling allowed a wide set of nonlinear curvature descriptions to be handled. (C) 2011 Elsevier B.V. All rights reserved.