Potsdam, Univ., Diss., 2000. / Computerdatei im Fernzugriff.

Potsdam, Univ., Diss., 2000. / Computerdatei im Fernzugriff.

Potsdam, Universiẗat, Diss., 2000.

In nonlinear regression statistical analysis based upon interpretation of the parameter estimates may be quite different from linear regression. An important point is that for finite samples the least squares estimator (LSE) is not unbiased. nor is it a minimum variance estimator: for nonlinear models, the LSE has these properties under some assumptions only asymptotically and many statistical conclusions are based upon this asymptotic theorie. But there are a lot of nonlinear models where the asymptotic properties are poorly approximated for finite samples. Assessing the nonlinearity can show us if statistical tests the justification of which rests on the assumption of linearity are valid. Better parameterizations and experimental design are good possibilities to reduce the non-neglible nonlinearitv of certain models. A case study shows that experimental design can reduce the nonlinearity considerably. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Diese Arbeit stellt die nichtlineare Parameterschätzung als alternative Auswertemethode von Kuchenfiltrationsexperimenten vor. Anhand eines größeren Datensatzes werden die Vorteile dieser Methode gegenüber der verbreiteten Auswertung mittels einer linearisierten Form der Kuchenfiltrationsgleichung für den Fall konstanten Drucks gezeigt. Zur Bewertung der Anpassungsgüte werden Residuenplots erläutert und verwendet. Die Unterschiede der Ergebnisse bewegen sich im Bereich von 5 bis 15% bei der Bestimmung des spezifischen Kuchenfiltrationswiderstands, welcher der wichtigste Parameter bei der Auslegung von Filtrationsapparaten ist. Weitere Möglichkeiten der Auswertung werden aufgezeigt, die durch die nichtlineare Parameterschätzung möglich werden, darunter die Auswertung von Experimenten bei variablem Druck, die Bestimmung des Kuchenwiderstands kompressibler Feststoffsysteme sowie eine Bewertung der anfänglichen Verblockungsvorgänge am Filtermedium.:1 Introduction 2 Cake Filtration Theory 2.1 Historical Development 2.2 Derivation of the Cake Filtration Equation 2.3 Fit Procedures for Cake Filtration Data 2.4 Additional Methods for Finding the Time Offset 3 Materials and Methods 3.1 Materials 3.2 Filter Medium 3.3 Laboratory Pressure Filters 3.4 Example Dataset 3.5 Preparation of Example Dataset 3.6 Residual Plots and Chi-Squares 3.7 Bootstrapped Statistics 4 Proposed Fit Procedure 4.1 Nonlinear Regression 4.2 Region of Best Fit 5 Results and Discussion 5.1 Constant-Pressure Filtration 5.2 Hermans & Bredée Models 5.3 Residual Plots of Fit Results 5.4 Nonconstant Filtration 5.5 Compressibility Effects 5.6 Optimal Parameter Definition 5.7 The Role of the t/V-V-Diagram 6 Conclusions 7 Outlook 7.1 Constant-Flux Filtration 7.2 Inline Resistance Measurements 7.3 Parameter Estimation in Chemical Engineering A Appendix A.1 The Concentration Parameter A.2 Obsolete Fit Methods A.3 Residual Statistics A.4 Bootstrapped Statistics Data A.5 Fit Example in Microsoft Excel A.6 Experimental Data and Metadata B References / This thesis presents nonlinear parameter estimation as an alternative method for the evaluation of cake filtration experiments. A dataset of 225 constant-pressure filtration experiments is used to highlight the advantages of this method compared to the widely used evaluation method which uses a linear transformation of the cake filtration equation. The goodness-of-fit is tested through the means of residual plots, which are introduced and discussed. The difference in results for the two methods for the specific cake resistance parameter, which is the most important parameter in the dimensioning of filtration apparatused, lies between 5 and 15%. Further possibilities of evaluation are presented, which become possible through the use of nonlinear parameter estimation, such as: evaluation of filtration experiments with nonconstant pressure, the determination of cake resistances for compressible systems, and the investigation of the processes present in the beginning stages of cake filtration.:1 Introduction 2 Cake Filtration Theory 2.1 Historical Development 2.2 Derivation of the Cake Filtration Equation 2.3 Fit Procedures for Cake Filtration Data 2.4 Additional Methods for Finding the Time Offset 3 Materials and Methods 3.1 Materials 3.2 Filter Medium 3.3 Laboratory Pressure Filters 3.4 Example Dataset 3.5 Preparation of Example Dataset 3.6 Residual Plots and Chi-Squares 3.7 Bootstrapped Statistics 4 Proposed Fit Procedure 4.1 Nonlinear Regression 4.2 Region of Best Fit 5 Results and Discussion 5.1 Constant-Pressure Filtration 5.2 Hermans & Bredée Models 5.3 Residual Plots of Fit Results 5.4 Nonconstant Filtration 5.5 Compressibility Effects 5.6 Optimal Parameter Definition 5.7 The Role of the t/V-V-Diagram 6 Conclusions 7 Outlook 7.1 Constant-Flux Filtration 7.2 Inline Resistance Measurements 7.3 Parameter Estimation in Chemical Engineering A Appendix A.1 The Concentration Parameter A.2 Obsolete Fit Methods A.3 Residual Statistics A.4 Bootstrapped Statistics Data A.5 Fit Example in Microsoft Excel A.6 Experimental Data and Metadata B References

In practical laboratory classes students traditionally receive data by reading from a measurement device (ruler, clock, voltmeter, etc.) or digitally as files in exchange formats such as CSV (comma separated value). In many cases these data have to be processed later using non-linear regression, here referred to as curve fitting. Therefore, analog data first have to be digitalised and imported to a data analysis and visualisation program, which is often commercial and requires installation. In this paper we present an alternative concept fusing open-source community tools into a single page web application facilitating data acquisition, visualisation, analysis via non-linear regression and further post processing usable for error calculations. We demonstrate the e-learning potential of this web application accessible at curvefit.tu-chemnitz.de in the context of acquired data as typically obtained in physical laboratory classes from undergraduate studies. A prototype workflow for the topic 'specific electric resistance determination' is presented along with a technical description of the basic web technology used behind. Restrictions, such as limited portability or cumbersome ways to share results electronically between student and supervisor as occurring in traditional software applications are overcome by enabling export via URL. The discussion is complemented by thorough comparison of curve fitting web applications with focus on their capability to be adaptable to user-specific models (equations) as faced by (undergraduate) students in the context of their education in laboratory classes in natural and life sciences, such as physics, biology and chemistry.

In practical laboratory classes students traditionally receive data by reading from a measurement device (ruler, clock, voltmeter, etc.) or digitally as files in exchange formats such as CSV (comma separated value). In many cases these data have to be processed later using non-linear regression, here referred to as curve fitting. Therefore, analog data first have to be digitalised and imported to a data analysis and visualisation program, which is often commercial and requires installation. In this paper we present an alternative concept fusing open-source community tools into a single page web application facilitating data acquisition, visualisation, analysis via non-linear regression and further post processing usable for error calculations. We demonstrate the e-learning potential of this web application accessible at curvefit.tu-chemnitz.de in the context of acquired data as typically obtained in physical laboratory classes from undergraduate studies. A prototype workflow for the topic 'specific electric resistance determination' is presented along with a technical description of the basic web technology used behind. Restrictions, such as limited portability or cumbersome ways to share results electronically between student and supervisor as occurring in traditional software applications are overcome by enabling export via URL. The discussion is complemented by thorough comparison of curve fitting web applications with focus on their capability to be adaptable to user-specific models (equations) as faced by (undergraduate) students in the context of their education in laboratory classes in natural and life sciences, such as physics, biology and chemistry.

The major aims of this paper are to present how we can improve the students understanding and involvement in mathematics by using a programming/graphic calculator. I will use differentials as examples such as differentiation ,integrals and differential equations, creating lines of slopes for differential equation of the type y’= f(x,y) . Find the solution of some differential equations by using regression and create the graph connected to the differential equation. As we have different approaches to solving a problem, it is a hope the students interest in mathematics will improve. The tools used will be programming, graphic commands as plot, f-line, etc. One goal is also to show how we can create small programs solving problems in mathematics. For many students this will be a stepping stone for further work with programming. The programs used can be copied using the program FA 124 that can be downloaded from Casios homepages. On request I can send you the programs.

The major aims of this paper are to present how we can improve the students understanding and involvement in mathematics by using a programming/graphic calculator. I will use differentials as examples such as differentiation ,integrals and differential equations, creating lines of slopes for differential equation of the type y’= f(x,y) . Find the solution of some differential equations by using regression and create the graph connected to the differential equation. As we have different approaches to solving a problem, it is a hope the students interest in mathematics will improve. The tools used will be programming, graphic commands as plot, f-line, etc. One goal is also to show how we can create small programs solving problems in mathematics. For many students this will be a stepping stone for further work with programming. The programs used can be copied using the program FA 124 that can be downloaded from Casios homepages. On request I can send you the programs.