Univariate Statistics (One Variable) : the Basics. Linear Regression Assisted by ANOVA Using Excel.

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1 Where are we in the grand scheme? Univariate Statistics (One Variable) : the Basics Examples of Calculations Made Using Excel. Analysis of Variance (ANOVA). ANOVA Using Excel. Introduction to Linear Regression. Linear Regression Using Excel. Linear Regression Assisted by ANOVA Using Excel. Nonlinear Regression. Nonlinear Regression Using Excel s Solver Tool and LabVIEW. Introduction to Chemometrics. Chemistry 5725 Nonlinear Regression 1 Linear regression tools: linest function in Excel regression tool in the data analysis toolbox in Excel Solver (brute force optimization) in Excel LabVIEW subroutines The use of a trend line is strongly discouraged, because it does not give statistics of the fit. Therefore, it is useless in typical multivariate calibrations. Nonlinear regression requires the use of an optimization or more sophisticated approach (e.g. Levenberg-Marquart algorithm). Solver in Excel LabVIEW subroutines Chemistry 5725 Nonlinear Regression 2 1

2 Regression vs Optimization Chemistry 5725 Nonlinear Regression 3 The Microsoft Office Excel Solver tool uses several algorithms to find optimal solutions. The GRG Nonlinear Solving Method for nonlinear optimization uses the Generalized Reduced Gradient (GRG2) code, which was developed by Leon Lasdon, University of Texas at Austin, and Alan Waren, Cleveland State University, and enhanced by Frontline Systems, Inc. The Simplex LP Solving Method for optimization in linear programming uses the Simplex and dual Simplex method with bounds on the variables, and problems with integer constraints use the branch and bound method, as implemented by John Watson and Daniel Fylstra, Frontline Systems, Inc. The Evolutionary Solving Method for non-smooth optimization uses a variety of genetic algorithm and local search methods, implemented by several individuals at Frontline Systems, Inc. Chemistry 5725 Nonlinear Regression 4 2

3 The Solver is an Optimization Tool which refines guesses at the best solution to a problem, evaluates the result, and makes a refined guess until a limit of precision for the fitted parameters has been reached. The process is called optimization. It searches a multidimensional response surface to arrive at an optimum solution. In general ther are a wide variety of searching approaches and algoritms. The minimization of the SS of the residuals is what is optimized. A model response must be known or proposed ahead of time to provide the surface which is to be explored. Chemistry 5725 Nonlinear Regression 5 A 2 D response surface grid. Optimization would maximize Y1. There are many algorithms and approaches to get there. Usually we optimize the fit to a model and evaluate it by minimizing the SS residuals. Chemistry 5725 Nonlinear Regression 6 3

4 Chemistry 5725 Nonlinear Regression 7 Simplex Optimization on a Surface: here defined by two variables: Mole % H 2 O and Temperature to Maximize reaction rate. X X= optimum reaction conditions (maximum rate) Chemistry 5725 Nonlinear Regression 8 4

5 (x 0 ) Chemistry 5725 Nonlinear Regression 9 Chemistry 5725 Nonlinear Regression 10 5

6 Chemistry 5725 Nonlinear Regression 11 A 2 D simplex is a guided search of the response surface of the model response. Guidance is provided by evaluation of gradients along this surface. Step 1 Step 2 Optimum midpoint in each case or trial Step 3 Chemistry 5725 Nonlinear Regression 12 6

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11 Chemistry 5725 Nonlinear Regression 21 Various Optimization Approaches: Chemistry 5725 Nonlinear Regression 22 11

12 Optimization in LabVIEW Chemistry 5725 Nonlinear Regression 23 Fitting of Gaussians Using Solver: For Multiple Peaks a Good Guess is Required for Convergence Chemistry 5725 Nonlinear Regression 24 12

13 Compare Nonlinear Least Squares from LabVIEW Chemistry 5725 Nonlinear Regression 25 Chemistry 5725 Nonlinear Regression 26 13

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