Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

Variance reduction by simultaneous multi-exponential analysis of data sets from different experiments

Abstract: The analysis of experimental data from the photocycle of bacteriorhodopsin (bR) as sums of exponentials has accumulated a large amount of information on its kinetics which is still controversial. One reason for ambiguous results can be found in the inherent instabilities connected with the fitting of noisy data by sums of exponentials. Nevertheless, there are strategies to optimize the experiments and the data analysis by a proper combination of well known techniques. This paper describes an applicable approach based on the correct weighting of the data, a separation of the linear and the non-linear parameters in the process of the least squares approximation, and a statistical analysis applying the correlation matrix, the determinant of Fisher's information matrix, and the variance of the parameters as a measure of the reliability of the results. In addition, the confidence regions for the linear approximation of the non-linear model are compared with confidence regions for the true non-linear model. Evaluation techniques and rules for an optimum experimental design are mainly exemplified by the analysis of numerically generated model data with increasing complexity. The estimation of the number of exponentials significant for the interpretation of a given set of data is demonstrated by using records from eight absorption and photocurrent experiments on the photocycle of bacteriorhodopsin.

Linear System Identification in a Nonlinear Setting - Nonparametric analysis of the nonlinear distortions and their impact on the best linear approximation

Abstract: This article addresses the following problems: 1) First, a nonlinearity analysis is made looking for the presence of nonlinearities in an early phase of the identification process. The level and the nature of the nonlinearities should be retrieved without a significant increase in the amount of measured data. 2) Next it is studied if it is safe to use a linear system identification approach, even if the presence of nonlinear distortions is detected. The properties of the linear system identification approach under these conditions are studied, and the reliability of the uncertainty bounds is checked. 3) Eventually, tools are provided to check how much can be gained if a nonlinear model were identified instead of a linear model. Addressing these three questions forms the outline of this article. The possibilities and pitfalls of using a linear identification framework in the presence of nonlinear distortions will be discussed and illustrated on lab-scale and industrial examples. In this article, the focus is on nonparametric and parametric black box identification methods, however the results might also be useful for physical modeling methods. Knowing the actual nonlinear distortion level can help to choose the required level of detail that is needed in the physical model. This will strongly influence the modeling effort. Also, in this case, significant time can be saved if it is known from experiments that the system behaves almost linearly. The converse is also true. If the experiments show that some (sub-)systems are highly nonlinear, it helps to focus the physical modeling effort on these critical elements.

Statistical linear approximation for environment compensation

Abstract: The statistical linear approximation (SLA) method is proposed as a novel way to approximate a nonlinear function by a linearized model. In the proposed method, an optimization criterion for approximation is defined in terms of statistical expectation. The SLA is applied to environment compensation where the speech contamination rule appears as a highly nonlinear function of the relevant variables.