Non-Linear Least Squares Analysis with Excel

Transcription

1 Non-Linear Least Squares Analysis with Excel 1. Installation An add-in package for Excel, which performs certain specific non-linear least squares analyses, is available for use in Chem 452. The package, named nlls11.xla, might be loaded automatically when you launch Excel. (The file extension.xla identifies the file as an add-in file to Excel: it is loaded into memory, and its functions available for use, but it is not otherwise obviously opened.) If the package has been opened, you will find an extra item on the Tools menu, NLLS Fit... If the package has not opened, the Tools menu will end with Data Analysis..., and you will need to find and open the file yourself. (It should be located in the directory zeus\\c:\chem452.) See Fig. 1 for a display of the Tools menu, after nlls09.xla has been opened. Figure 1. Tools menu of Excel, showing non-linear least squares fit add-in item. 2. Import Data File Several experimental procedures you perform this semester will result in the generation of a data file in ASCII format. Such a file can be imported into most all applications. When such a file is opened from within Excel, the Text Import Wizard is automatically run. This procedure has three steps: First, you can choose the number of lines of the file to skip. This is an easy way to skip over spurious lines in the data file. Step 2, shown in Fig. 2 below, lets you set the delimiters for the data file. Delimiters are used to separate items on each row of the data file: depending upon the application which created the data file, the delimiter could be the tab key (used by most spreadsheet applications, such as Excel), or the comma, or a space, etc. The data preview window shows how Excel will treat the data file, using the delimiters Chem 452 NLLS Notes Page-1

2 selected. The vertical bars in Fig. 2 indicate that this particular data file will be imported as two columns of data. Step 3 allows you do define the format for each column. Figure 2. Step 2 of Text Import Wizard. 3. Data Manipulation In many cases, you will need to perform some type of data manipulation before performing an analysis. Please consult the Excel manuals, experiment write-ups, and/or your instructor for help. The particular data being analyzed here requires all time values to be increased by a value measured independently in the lab. Referring to Fig. 1, note that column B was inserted between the time and mass columns imported by the Text Import Wizard. This column contains the manipulated time values necessary for data analysis. 4. Non-Linear Least Squares Analysis Select NLLS Fit... from the Tools menu (or type ctrl-l). This starts the analysis package, which presents two screens requesting user input. First, you identify the arrays containing the independent (x) and dependent (y) values, the array containing the weights (if desired), the type of fit desired, and the weighting method desired. This screen is shown in Fig. 3. The array range can be typed directly, or can be identified by mouse click-and-drag (or mouse click and shift-click) on the data worksheet. A weight array must be specified if instrumental weighting is chosen. The arrays need not be contiguous. The specific non-linear fit is selected from the left side scroll box: presently, you can chose amongst exponential, kinetic half-order, and kinetic second order. Mouse click on the desired fit. Three weighting methods are available: none, instrumental (i.e., user-supplied in a third array), or statistical (i.e., dependent upon the measured value). Mouse click on the desired weighting method. After making all selections, mouse click the OK button. If the arrays are of unequal size, or if Excel has trouble retrieving the data, an error message will be generated. Chem 452 NLLS Notes Page-2

3 Figure 3. Step 1 of Non-Linear Least Squares Analysis Routine. After defining the data arrays, type of fit and weight method, the program prompts you for suitable seed values for each coefficient. Fig. 4 shows the initial choices obtained by the user s analysis of the data. Note that the analysis can perform either a single or double exponential fit. If coefficient c4 is 0 (zero) (the default value), a single exponential fit is performed. Select whether each coefficient is allowed to vary, or held constant, during the fitting routine. There are many cases where you specifically need to hold a coefficient constant (for example, the baseline value)! Mouse click on the fit or hold button to select the desired action for each coefficient. This analysis package implements the Levenberg-Marquardt method for a dataset which depends nonlinearly on a set of parameters. The success of the procedure is highly dependent upon good initial values for the coefficients. The values calculated by the program may be grossly in error for your particular dataset. A poor fit is often the result of poor initial values; it does not necessarily indicate that the data does not conform to the fitting function chosen. A Singular Matrix error may result from poor initial values. Try the fit again with different values. Manual analysis of the predicted relationship of the data should enable you to make very good estimates of the coefficients. Keep a record of the initial values used, and observe how the fit changes as these values are modified. Max # is the maximum number of iterations the program will perform, trying to converge. When the difference in residuals for two successive sets of coefficients is less than Convergence, the iteration routine is judged complete. You should not have to change these from the default values. Figure 4. Step 2 of Non-Linear Least Squares Analysis Routine: Seed Values for Coefficients. Chem 452 NLLS Notes Page-3

4 After setting the initial values for the coefficients, mouse click the OK button. The program will then attempt to find the best set of coefficients. If the fit converges, a new sheet will be added to the current Excel workbook; see Fig. 5. The name will be a combination of the name of the sheet containing the data, and a number added by the program. (Note that the data being analyzed here is on a sheet named KCl, and observe the various names of the analysis sheets generated. The data is copied to the new sheet, along with the calculated (fitted) y values. A table containing the best estimate found for each coefficient, its standard error, upper and lower limit, and Q value is generated. The statistical parameters chi squared, r squared, and the variance are reported. Two charts are also placed on this sheet. One contains the original data (shown as symbols if fewer than 100 points) and fitted data (always shown as a line). The second chart contains the residuals. Double-click on the chart to access Excel s chart menus if changes in the display are desired. Note that if a coefficient is held constant, its standard error will be 0 (zero). Figure 5. Analysis Sheet Created by NLLS09. You will then be asked if another fit should be performed (using new seed values for the coefficients; perhaps the values obtained from the current fit) or if you are finished. If another fit is chosen, the program will begin to delete the current analysis sheet. You will be prompted to confirm this deletion: if Chem 452 NLLS Notes Page-4

5 you want to keep this sheet, click on Cancel ; otherwise, click on Yes. In this example, a careful examination of the data and fitted values, especially examination of the residuals, indicates that a single exponential decay is not a good representation of the KCl diffusion data. Using the values obtained for c2 and c4 in the single exponential decay (Fig. 5) and the relationship predicted by theory between c2 and c4, and c3 and c5, a double exponential fit was performed. The results, as shown in Fig. 6, indicate a much better fit. Note, among other criteria, that the variance is only 5.43 X 10-8, compared to 3.21 X 10-7 for the single exponential fit. (Note the name of this sheet - KCl31. Many fits were performed before the appropriate initial values were obtained.) y = c1 + c2*exp(-c3) + c4*exp(-c5*x) Figure 6. Result of Double-Exponential Fit of KCl Diffusion Data. As this nlls11.xla help document shows, you can copy/paste data, pictures, text, etc., between a variety of applications. Make use of these features to help prepare your lab reports! Chem 452 NLLS Notes Page-5