User Guide... ___________________________________________________________ DataFit 1.0 (Demo version) High-powered function fitting and data modelling ___________________________________________________________ A Soft Answer, P.O. Box 1743, Macquarie Centre NSW 2113, AUSTRALIA C O N T E N T S =================== 1. Introduction 1 2. Brief Program Description 1 3. System Requirements 1 4. Installation 2 5. Summary of Features 2 a) Types of function you can fit 2 b) Parameters 2 c) Maths and Arithmetic 3 d) Choice of four major function fitting methods 3 e) Publication-quality graphics output 3 f) Statistical analysis 3 g) Huge data capacity 4 h) Min/Maximisation and solving 4 i) User-friendly interface 4 j) Detailed reporting of results 4 6. Starting and using DataFit 5 7. Quick Reference to Commands 5 8. Files on the distribution disk 6 9. Program Navigation 6 10. Arithmetic & Maths 7 11. Alt-A : The ABOUT screen 8 12. Alt-D : DATA setup screen 8 13. More on F4 - Auto data entry 9 14. Alt-E : The EXECDOS setup screen 10 15. Alt-F : The FIT setup screen 11 16. Alt-G : The GRAPH setup screen 15 17. Alt-M : The MINMAXSOLV setup screen 18 18. Alt-O : The OUTPUT window 18 19. Alt-P : The PARAMS setup screen 19 20. Alt-R : The REPORT setup screen 21 21. F1 - Help 21 22. F2 - Save work file 22 23. F3 - Load a new work file 22 24. Alt-F3 - Pick file 22 25. F5 - Zoom the function screen width 22 26. F7 - Auto-scale axis values 23 27. F8 - Printing the graph 23 28. F9 - Select a new graph type 24 29. Alt-F9 - Select next graph type 24 30. Ctrl-F9 - Select previous graph type 24 31. F10 - Menu 25 DataFit 1.0 - A few case studies 26 CASE STUDY 1 : DEMO1.FIT - Simple straight line fit 26 CASE STUDY 2 : DEMO2.FIT - Combination of 2 curves 30 CASE STUDY 3 : DEMO3.FIT - Selectivity curve 31 CASE STUDY 4 : DEMO4.FIT - Manual adjustment of parameters 32 CASE STUDY 5 : DEMO5.FIT - "Dangerous" Linear Regressions 33 CASE STUDY 6 : FAMILY1.FIT - Fitting a family of curves 34 CASE STUDY 7 : GAUSS1.FIT - 2 Gaussians 35 CASE STUDY 8 : PARAMET1.FIT - Parametric equations 36 CASE STUDY 9 : POLAR1.FIT - Polar coordinates 37 ==================================== CHAPTER 1. Introduction ==================================== Thank you for taking the time to have a look at the DataFit demo disk. DataFit is a function fitting/data modelling program which has been specifically developed to be both powerful and excep- tionally easy to use. This document (MANUAL.DOC) is an abbrev- iated user manual which contains all the information you need to fully evaluate the program. This manual also contains a number of case studies which you are advised to work through. ==================================== CHAPTER 2. Brief Program Description ==================================== DataFit 1.0 is a generalised function fitting program: * Fit any math function through your data points quickly and easily. * Highly sophisticated techniques are used, including linear and non-linear methods, algebraic manipulation, and interac- tive graphical manipulation. * Comprehensive statistical and error analyses give confidence limits on fitted parameters. * Publication-quality graphs of the fitted results can be printed on most printers. ================================ CHAPTER 3. System Requirements ================================ IBM-PC or close compatible, DOS 3.1 or later, 540K available RAM. Video monitor: CGA, EGA, VGA, Hercules, and some others. Printers: Postscript, HP LaserJet, most dot matrix printers. - 1 - ========================= CHAPTER 4. Installation ========================= a) Insert the distribution disk into drive A: (or B:) b) Type INSTALL from the A:>_ (or B:>_) prompt. c) Press the F2 key to read the file README.DOC, and p to print it. d) Press the F3 key to read the user manual MANUAL.DOC, p to print it. e) Press the F1 key to install the program files onto your hard disk, or the F10 key to quit without installing. The INSTALL program gives you the opportunity to specify a disk/directory to install the files, and to read and/or print the README.DOC and MANUAL.DOC files. It is possible to run directly from the distribution floppy disk, though the disk response times may be slower and you may not be able to use the "swap to disk" options for printing graphs and exe- cuting DOS commands. ================================ CHAPTER 5. Summary of Features ================================ a) Types of function you can fit --------------------------------- * Linear or non-linear math functions with up to 6 independ- ent variables; * Parametric equations in multiple dimensions; * Families of related curves; * Discontinuous (piecewise-smooth) functions; * Rectangular or polar coordinates. b) Parameters --------------- DataFit will fit any function in up to 20 parameters, and up to 6 independent variables (IV's). You can place lower and/or upper limits on fitted parameter values. - 2 - c) Maths and Arithmetic ------------------------ Use the standard math functions + - * / ^ ln, log, exp, trig and hyperbolic trig functions, plus several others such as factorial, gamma, random, normal, abs, sgn, max, min. Also use Boolean functions > < <> = >= <= which return 1 or 0 and can be used to specify compound functions (discontinuous or piecewise-smooth) acting over different ranges of values. Flip between up to 9 functions in each application, or import your own standard functions from a text file. And more. d) Choice of four major function fitting methods ------------------------------------------------- 1. Least squares linear regression. In many cases this method will even linearise the function for you before fitting. 2. Levenberg/Marquadt non-linear regression. DataFit calcu- lates analytical derivatives with respect to each parame- ter, to achieve unprecedented convergence speeds. 3. Nelder & Mead non-linear simplex method is a good work- horse, which is slower than L/M but will often find a solution where L/M fails due to instability or poor start- ing estimates. 4. And finally, an interactive graphical fitting method. You change the parameter values manually, and watch in real time how they affect the fitted function. Very powerful technique for estimating starting values if you have no idea what they should be. e) Publication-quality graphics output --------------------------------------- Support for PostScript, HP LaserJet & DeskJet, and most 24 and 9 pin printers. Plot the fitted function (or family of functions), or plot the error residuals vs the dependent or independent variables. Graphs are user-configurable to a high degree: rectangular/polar coordinates, log/linear scales, grid lines, error bars, font and symbol control, legend box, programmable insertion of text and results into axis titles and/or legend box, and more. Plot several graphs on a page by chaining, for example to plot the fitted function with a graph of the residual errors. f) Statistical analysis ------------------------ Sophisticated statistical analysis gives you standard devia- tions and confidence intervals for each fitted parameter, as well as covariances and correlation coefficients between parameters. Plus, a Monte Carlo simulation module provides the most powerful possible method for estimating confidence - 3 - intervals on the fitted parameters. Standard deviations on the data must of course be supplied. These can be set as a constant for the whole data file, or varied individually for each data point. g) Huge data capacity ---------------------- Data is stored in virtual arrays which are allocated automat- ically to RAM, EMS, XMS or disk as required, so that immense data tables can be created - even millions of data points if necessary. And a powerful automatic data entry facility is available for entering data from math functions directly into the table. Read and write data from/to disk. Copy/Move/Sort blocks of data. Fit the function to a marked block or the whole data file. h) Min/Maximisation and solving -------------------------------- Find min/max values of any function, or solve functions for specific values. An integrated pop-up calculator is also included. Type AltC to pop up the calculator. i) User-friendly interface --------------------------- Intelligent use of Alt and function keys, and mouse support, lets you move quickly between DataFit's functions. Backtrack with Esc. Most program options are displayed on screen as they become available. Pop-up help screens give interactive context-sensitive help at (almost) all times. Execute Dos commands or a Dos shell. j) Detailed reporting of results --------------------------------- Results are written as a user-configurable report, which can be scrolled in a browse window or directed automatically to disk or printer. Or highlight sections from the browse window and send to disk or printer. - 4 - ======================================== CHAPTER 6. Starting and using DataFit ======================================== Start the demo program by typing datafit at the DOS prompt. Finding your way around the program is very easy once a few rules are understood: * Use the Alt-keys to activate the options shown on the top line of the screen. eg. AltF to Fit, AltX to eXit etc. * Function keys F1 to F10 (though not all of them) are shown on the bottom line of the screen. * If you have a mouse installed, try clicking on various parts of the screen. Click on arrows, keywords, data entry windows, etc. The mouse tries very hard to find something intelligent to do whenever you click it. Left button selects, right button is usually equivalent to pressing the Esc key (to cancel, or backtrack through previous options). * Press the F1 key for help at any stage. The help screens are hypertext - ie. you can move the highlight (if one is visi- ble) with the arrow keys, to select more help on various topics by pressing Enter on a highlighted phrase. Press Esc to return to the program. ========================================= CHAPTER 7. Quick Reference to Commands ========================================= Alt-D Use the Data table Alt-E Execute a DOS command or shell Alt-F Fit the function Alt-G Graph (display graph on screen) Alt-O Read the program output Alt-P Define data columns and parameters Alt-R Define options for the output report Alt-X Exit the program F1 Help F2 Save work file F3 Load a new work file Alt-F3 Load a previously-used file F4 Auto data entry into the data table F5 Zoom the function screen width F7 Auto-scale axis values F8 Print Graph F9 Select graph type Alt-F9 Select next graph type Ctrl-F9 Select previous graph type - 5 - ============================================ CHAPTER 8. Files on the distribution disk ============================================ readme .doc Read this file install .exe The installation program startup .txt Text file for startup information datafit .exe The DataFit program datafit .hlp Hypertext help file for DataFit blank .fit DataFit data file: Blank (new) demo1 .fit data file: Simple straight line fit demo2 .fit data file: Combination of two curves demo3 .fit data file: Selectivity curve demo4 .fit data file: Manual adjustment of parameters demo5 .fit data file: "Dangerous" linear regressions family1 .fit data file: Fitting a family of curves gauss1 .fit data file: Two Gaussians paramet1.fit data file: Parametric equations polar1 .fit data file: Polar coordinates general .fns Short list of standard functions prtgrf .exe For printing graphs when "swapping to disk" deffont .glf SansSerif graphics font file ================================ CHAPTER 9. Program Navigation ================================ Finding your way around the program is very easy once a few rules are understood: * Use the Alt-keys to activate the options shown on the top line of the screen. eg. AltF to Fit, AltX to eXit etc. * Function keys F1 to F10 (not all though) are shown on the bottom line of the screen. * If you have a mouse installed, try clicking on various parts of the screen. Click on arrows, keywords, data entry windows, etc. The mouse tries very hard to find something intelligent to do whenever you click it. Left button selects, right button is usually equivalent to pressing the Esc key (to cancel, or backtrack through previous options). * Press the F1 key for help at any stage. The help screens are hypertext - ie. you can move the highlight (if one is visi- ble) to select more help on various topics. Press Esc to return to the program. - 6 - ================================= CHAPTER 10. Arithmetic & Maths ================================= Arithmetic: + addition, 2+3 = 5 - subtraction, 2-3 = -1 * multiplication, 2*3 = 6 / division, 2/3 = 0.666.. ^ raising to a power, 2^3 = 8 Trigonometric: sin, cos, tan, asin, acos, atan. (radians) Hyperbolic Trig: sinh, cosh, tanh, asinh, acosh, atanh. Other1: ln(x) - Naperian logarithm log(x) - Base 10 logarithm exp(x) - exponential sqrt(x) - square root sqr(x) - x^2 cube(x) - x^3 pow4(x) - x^4 abs(x) - absolute value int(x) - integer part of x round(x) - x rounded to nearest whole number sgn(x) - returns 1 when x>=0, -1 when x<0 gamma(x) - gamma function gammaln(x) - ln(gamma(x)) fact(x) - factorial (x rounded to whole number) factln(x) - ln(fact(x)) random(x) - random number between 0 and x normal(x) - normally-distributed random number with a mean of 0, peak of 1, and standard deviation x. Other2: max - maximum of two numbers, (3 max 5) returns 5 min - minimum, (3 min 5) returns 3 div - integer division, (14 div 3) = 4 mod - remainder of integer division, (14 mod 3)=2 Logical: = > >= <> < <= These logical functions have two arguments, much like + - * / and ^, and the result is returned as 0 or 1 for FALSE or TRUE. Examples: (5+3)>2 returns the result 1 sin(x)>=2 returns the result 0 Boolean: and or xor not(x) These Boolean functions have two arguments ("not" has one), much like the logical functions. The arguments are considered - 7 - to be TRUE if they are greater than 0, otherwise FALSE. The result is returned as 0 or 1 for FALSE or TRUE. Examples: ((5+3)> 2 or sin(x)>=2) returns the result 1 not(a>b) is the same as a<= b (5 and 3) returns the result 1 Logical and Boolean functions can come in handy to describe a combination of two or more functions acting over different ranges. ========================================= CHAPTER 11. Alt-A : The ABOUT screen ========================================= Typing AltA displays a copyright message and general information about the developer of the program. ========================================== CHAPTER 12. Alt-D : DATA setup screen ========================================== The Data Table holds the data to which the function/s will be fitted. It can also be used to hold standard deviations of individual data points, some of the results of the function fitting, and it holds temporary data used for Monte Carlo simu- lations. The data table can have up to 24 columns, and any number of rows. The table will be located in RAM memory if enough is available, otherwise it will be allocated to EMS or XMS (if available) or to virtual memory on a hard disk. EMS and XMS are slower than RAM, and a disk-based data table is MUCH slower than either. So the moral is to make your data table just big enough for the job, to keep it in the fastest memory location. The default size of the data table is 1000 rows by 6 columns. This default size can be changed in two ways: 1) When starting DataFit: Use the /d option to set the maximum rows and columns for the data table. Minimum 100 rows and 2 columns. Examples: datafit /d500 (500 rows, 6 columns) datafit /d5000:10 (5000 rows, 10 cols) 2) While using DataFit: Type F7 to change the number of columns, or AltF7 to change the number of rows. You will have to write any data in the table to a disk file first, then delete the data, change the number of rows and/or columns, and finally read the data back in from the disk file. - 8 - A number of control and function keys can be used to help make your data entry more efficient. Either press the key or click on the phrase with a mouse: Ctrl left/right arrow - Change column CtrlD - Delete the line the cursor is on CtrlI - Insert a line at the cursor position CtrlT - Mark the current line as "top of block" CtrlB - Mark the current line as "end of block" CtrlC - Copy a marked block to the cursor position CtrlK - Unmark a marked block CtrlV - Move a marked block to the cursor position CtrlX - Delete a marked block CtrlS - Sort a marked block on the current column. F4 - Auto data entry. See next section for details. F5 - Read data from a disk file F6 - Write block to a disk file F7 - Change the number of columns AltF7 - Change the number of rows Note: If a block is marked when AltF is pressed then fitting will be done only on the data in the marked block. Unmark the block with CtrlK to fit the whole data file. =========================================== CHAPTER 13. More on F4 - Auto data entry =========================================== Type F4 while using the data table to pop up an "Auto Data Entry" setup screen. This is a very powerful utility which lets you enter data sequentially into the data table. Very useful for inserting data which follows a math function. Say for example you want to overwrite the data in columns 1, 2 and 4, for rows 21 to 90, and leave the other columns unaltered. And say you want to enter 101 to 170 in column 1, the square root of column 1 into column 2, and the square of the differences between them into column 4. Then you move the cursor to row 21 (any column) - 9 - and type F4. Then you would set up the table like this: FORMULAE: Enter f(i) for each column....i=1 to [70 ] (lines) --------------------------------------------------------------- ( ) Insert lines of data (else overwrite existing data) Enter functions in i. Only marked columns will be modified. (*) Column1 =[100+i ] (*) Column2 =[sqrt(c1) ] ( ) Column3 =[ ] (*) Column4 =[sqr(c2-c1) ] ( ) Column5 =[ ] ( ) Column6 =[ ] The first line defines an index, i, to run from 1 to 70. The "Insert lines..." line is switched off to indicate overwriting of existing data. Then the setup lines for columns 1, 2 and 4 are marked (*), and the others left unmarked. Finally the three formulae for columns 1, 2 and 4 are entered. Note that you cannot use any variables or parameters from the "Params" screen here. You can only use constants, the normal math functions, the index variable i, and variables c1 to c24 which will return the value of the data in columns 1 to 24 of the row being altered. When you have set up the data press F4 again to proceed with entering the data into the data table, or Esc to return to the data table without entering the data. ================================================= CHAPTER 14. Alt-E : The EXECDOS setup screen ================================================= You can execute any DOS command, or enter a DOS shell while using DataFit. Type AltE to open a DOS command window. Type in the DOS command, and press Enter to execute it. When the command has been completed you will be instructed to press any key to return to the program. If no command was entered (the DOS command was a blank line) then you will enter a DOS shell. You will see the normal DOS command line. Continue executing DOS commands as required, and type "exit" as the DOS command to return you to the DataFit program. The above method gives a quick way to execute small DOS applica- tions using whatever memory is available at the time. If there is not enough memory to execute the DOS command, then type AltE again instead of Enter. This will swap out DataFit to make more memory available before executing the DOS command or DOS shell. - 10 - ============================================= CHAPTER 15. Alt-F : The FIT setup screen ============================================= a) The top line of the "Fit" window shows the current function number (1 to 9) and the list of independent variables which have been defined and assigned data column numbers in the AltP "Param" window. f(x) in this example. b) Enter up to 9 separate functions, and use Alt-1 to Alt-9 to select one for fitting. Type F5 to zoom to wide screen for long functions. Use a semicolon to mark the end of the func- tion, if you want to add comments. The function must use all the independent variables (see paragraph a), and can be up to 255 characters in length. Example. 1. f(x) = [a+b*x ;straight line ] You can also define a system of parametric equations: the coordinate in each dimension varies with a parameter which must be called t. This is just slightly tricky, and requires a bit of setting up in the "Param" window. t must be defined as a parameter (constant), and its data column must be speci- fied in the "f(),Data...[ ]" line. There must also be one independent variable defined for each dimension of the fit. And finally, you indicate a parametric fit by using commas in the "Fit" function line. Example: 1. f(x,y) = [a*cos(t),b*sin(t) ;ellipse ] See the case study on "PARAMET1.FIT" for a detailed descrip- tion on fitting parametric equations. c) You can also create your own library of standard functions. The AltT command opens a text file, and lets you select one line of the file to be entered as the current function. A sample file GENERAL.FNS is included on the distribution disk. d) Once the function is entered, select a type of method to use for fitting: 1. Linear regression: Much more than a simple straight line fit. Any function which can be transformed so that it can be expressed in a form which is linear in its variable parameters can be fitted by linear regression. The trans- formation is even done for you provided you write the function in a standard form: f(x,y,z,...) = f2([..] + a*[..] + b*[..] + ...) - 11 - where [..] indicates a constant or any function of the independent variables (parameters being fitted not al- lowed), and f2 can be blank or a function such as exp, ln, sqr, or sqrt. eg. f(x) = ln(a+b*x+c*x^2). See DEMO1.FIT and DEMO5.FIT. Linear regression is very fast as it is a once-through (non-iterative) method. But you may not always be happy with the results, especially when a linearising transfor- mation is applied as in the DEMO5 example. The reason is that the function is fitted to a transformation of the data rather than the original data itself. When both are transformed back again after fitting, the fit may appear to be biased in some way. If this happens you should select the Levenberg/Marquadt method. 2. Levenberg/Marquadt: This is the fastest and most powerful procedure for fitting non-linear functions - ie any func- tion that you can type in on the function line. This method is difficult to generalise as it requires the analytical derivatives with respect to each fitted parame- ter. No problem for DataFit though, as sophisticated algebraic routines do the differentiation for you. This feature is believed to be unique among all generalised function fitting procedures. Levenberg/Marquadt can be a bit quirky though, especially on extremely non-linear functions or when poor starting estimates are supplied. If it fails, try Nelder & Mead first, and try Manual adjustment if you still have trou- ble. It is often a good idea to use Nelder & Mead until it appears to be approaching a solution, then interrupt the fit (press any key) and switch to Levenberg/Marquadt to finish off rapidly. 3. Nelder & Mead: The method proposed by Nelder & Mead is actually a function minimisation method rather than a curve fitting method. Here though the curve fitting is achieved by minimising a function defined as the sum of the squares of the differences between the fitted function and the data, to achieve the same result as a function fitting procedure. The method is known as a downhill simplex method (unrelat- ed to linear programming, which is also described as a simplex method, though for different reasons). It is crude and slow in comparison to the Levenberg/Marquadt method, but it exhibits exceptional stability and will often find a solution where L/M fails due to poor starting estimates, or a highly non-linear function. It is often efficient to use Nelder & Mead until it seems to be getting quite close to a solution, then switch to L/M to zero in quickly. Note that Nelder & Mead may give a slightly different solution to L/M, as it uses different - 12 - criteria to decide when a solution has been reached. You, the researcher, must decide which solution is "best". This is usually be st done by inspecting the graph of the fitted function plotted on the data points, in conjunction with the residuals graphs. The NORMALISE option can be selected as a variation of the standard Nelder & Mead procedure. In this case the sum of the squares of [errors as a fraction of the data value] is minimised, rather than the SSQ of [actual errors]. This biases the fit in favour of small data values. You might consider this option for example when you are plotting your data on log scales, and they cover several decades of values on the axes. It will give a more even-looking fit when plotted on log scales, where the other methods may seem to fit poorly at small values and well at larger values. 4. Manual (Graphical) adjustment: If you can't get any of the above procedures to work, it may be that your starting estimates are simply unreasonable. This option is a very powerful method for seeing the effects of individual parameters on the shape of the function you are fitting. More importantly though, it lets you use the best (though not necessarily the fastest) non-linear function fitting procedure of all - your own brain. When you initiate a fit with this option selected, you are presented with a graphic image showing the data, the function with its current parameter values, and at the top right of the screen a window showing the current parameter values. The left/right arrow keys decrease or increase the value of the factor F, shown in the window. Up/down arrows select a parameter, then use + - * / to add, subtract, multiply or divide the parameter by the factor F. You can also use A, S, X, D for + - * /, which is more convenient once you get used to it. Every time you change a parameter's value the function is re-drawn at the new value, so you can quickly see how the function is affected by the parameters. Press Esc when done. The normal report is compiled at the current parameter values, and if you feel your function now is a ball-park fit then you can select one of the other methods to complete the fit. e) Monte Carlo simulation is the most powerful available method for estimating standard deviations and confidence limits on your fitted parameters, PROVIDED you can specify accurate standard deviations on your experimental data. See "Params" for specifying these standard deviations. If you specified standard deviation data on your data points then the Linear Regression and Levenberg/Marquadt methods will also give estimates of standard deviations and confi- - 13 - dence limits on your fitted parameters. And if your errors are normally distributed then you can be reasonably confident in the error estimates on the fitted parameters. The Nelder & Mead method is unable to return any error estimates of this nature. But Monte Carlo takes the process of error estimation consid- erably further. And in the case of Nelder & Mead this is the only method for estimating errors in the fitted parameters. It works like this: First do the simulation as normal, using the method selected, and store the values of the fitted parameters. Then repeat the fit a number of times, but with each repetition the original data to be fitted are varied by a normally- distributed random variable with the standard deviation specified in the "Params" section. In other words, this is about the closest you can come to simulating a large number of fits on different samples of the data, all varying by some standard deviation. Each fit generates a new set of fitted parameters, and if enough fits are done then the fitted parameters can have their standard deviations and confidence limits accurately determined. Clearly the original data mustn't be destroyed, so a "Monte Carlo" data column must be allocated in the "Params" section. This must be a data column which is not used for any other purpose in the program. Also, you really should do at least 100 iterations (set alongside the Monte Carlo check box) for the results to be meaningful. This could take a long time of course, so do your fit first without Monte Carlo simulation, then when you are ready you can select Monte Carlo for a final run-through to get the parameter error estimates. f) Maximum iterations provides the option for stopping a simula- tion after some maximum number of iterations has been done. g) The convergence criterion tells the program how hard to try to find a solution. It is used only by the non-linear proce- dures, and it is quite important. If it is too large t hen the program may decide prematurely that it has converged. This is often ( but not always) immediately noticeable when plotting - the function quite obviously does not fit through the points as well as it could. If it is too small, the program may fail to converge. It may reach a solution, but be unable to satisfy the convergence criterion. This will sometimes be obvious by watching the iterations window and the parameter values. So what is "too large" and what is "too small"? Unfortunately that depends on both the problem and the type of fitting method chosen, and that is why it is provided as a value to be entered by the user. But, that having been said, it should also be noted that a value of 1e- 4 to 1e-6 will be satisfactory in most cases. h) The error limit on data is a safety valve designed to help protect against fatal evaluation errors when the program runs out of control (which it can do quite easily on some types of - 14 - functions, especially when very poor starting estimates are supplied). The fit will be terminated if the value of the function with current parameters exceeds the data by the factor supplied. 1e6 seems to be a reasonable value for most cases, but you can change the value, or deactivate this option by unchecking the box. =============================================== CHAPTER 16. Alt-G : The GRAPH setup screen =============================================== The "Graph" setup screen not only gives you tremendous control over the presentation of the graph of the fit, it also gives you instant access to several types of graph. The primary type is of course a plot of the fitted function vs the data points. If the fitted function has more than one independent variable (IV) then the curve and data are plotted vs the first IV defined in the "Params" setup screen. 3-D plots for 2 IV's are felt to be of limited value due to the perspective problem of estimating errors between data and function on a flat image. And for more than 3 IV's it becomes effectively impossible to represent any sort of function vs data plot. This is one major reason for using the residuals plots... It is informative (and even necessary in the case of multi- dimensional data) to plot the residuals vs either the independ- ent variables (IV's) or the function response values, f(). Type F9 and a window pops up showing the available graph types. The number of available graph types depends on the number of IV's, as there is one for the fitted function, one for residuals vs function, and one each for residuals vs each IV. Select one and press Enter to switch to that graph type. The current graph type is shown in the top border of the "Graph" setup screen. Once you have selected the type of graph, you need to fill in the data to describe the graph layout. Most of it is pretty well self-explanatory (or at least it is intended to be!) if you are familiar with scientific graphs. But the text entry lines (for axis titles and legend table) are certainly NOT entirely self-evident, as they can make use of a powerful facility for automatically entering items such as the function itself, and values of parameters used in the function (or even math functions of the parameter values). You can also add sub- and superscripts. The process works like this: 1. If the whole text string is a valid math expression, such as a, or sqrt(a/b) etc, then the value of that math expression is output (in the format specified in the "Report" setup screen). - 15 - 2. If the string is not recognised as a valid math expression, then it is written exactly as is. 3. What about a combination of text and math expressions then? You enclose the text b its in "quotes" and the math expres- sions outside the quotes, each entry separated by commas. For example, the text expression ["a = ",a] returns the string [a = 1.23] or whatever the value of a is (the square brackets are for clarity here, and are not actually entered in the setup screen). If there are any errors in the line then the line is printed literally. 4. You can also include the text description of your function automatically in these text entry lines. The current function entered into the "Fit" screen can have an optional comment attached (a semicolon ; indicates the end of the function and the start of comments). Insert ~F to include the whole function line including comments into the text. Insert ~f to include the function only, without comments. These MUST be within "quotes". Any errors will result in the line being printed literally. 5. Finally, you can introduce sub and superscripts into the text lines. Use ~+ to change to superscript, ~- for subscript, and ~# to return to normal. As before, these must be within "quotes" or the line will be written literally. Some of the other options are explained very briefly here: * Polar coordinates assumes your data is in (theta,r) format, and transforms them onto polar coordinates: [x=r*cos(theta),y=r*sin(theta)] * Overlay Graph lets you plot several graphs on the page by chaining another graph onto the current one. Several graphs can be chained together on the page in this way. See GAUSS1.FIT for an example. Type F9 to see the graph numbers. * Fast screen text does not affect the printed output; it draws text in a fixed but very fast font for faster screen plot- ting. * SansSerif/Helvetica font can be selected as an alternative to the default stroked font. This is better with high resolution printers, but may be less legible with lower resolution printers. Also, the default font is a fixed-pitch font and will line up columns of text, whereas the SansSerif/Helvetica font is a variable-pitch font. * The font size can be specified separately for the axis values, the axis titles, the symbol size, and (see later) the legend table. * You can choose to join the data points by straight line segments or one of five data symbols. 0=line segments, - 16 - 1=circle, 2=square, 3=triangle, 4=inverted triangle, 5=dia- mond. And they can be filled or hollow. * You can add error bars if standard deviation data were sup- plied in the "Params" setup screen, and specify the width of the top and bottom caps on the bars. * Specify lengths of x and y-axes, and their offset, or dis- tance in inches from the left or bottom of the page. * Specify min/max values on x and y-axes; the INTERVAL between major tick marks; the NUMBER of minor ticks per major tick interval; decimal places and notation; grid lines on major and/or minor tick marks; and log scales on the axes. * The 'Plotting Range', to the right of the min & max values line, lets you plot your function over just part of the x- axis range. Leave these values as 0 and 0 to plot over the full axis range. * You can include a table of contents with optional drop-shad- ow. This will be overlaid over the graph axes and grid lines, but will not clear any data points if they happen to lie on the legend table. You can specify the size and location of the legend table, font and font size, and several lines of text in the table. The lines of text can include the math expressions etc described earlier in this section. Example: "a = ",a * Finally, if you are plotting a function in more than one independent variable then you will see an additional line at the end of the "Graph" setup screen. This line lets you enter values for the additional independent variables: The graph of the fitted function is always plotted with respect to the first independent variable, as defined in the "Param" setup screen. When more than one IV is used, the additional IV's must be given specific values for the plot. Values for sever- al IV's on the same curve are separated by commas. Values for multiple curves on the same plot are separated by semicolons. Example: if the function has 3 IV's, such as f(x,y,z)=..., then you might define: [y=1, z=1; y=1, z=2; y=2, z=2; ] See the demo file FAMILY.FIT for an example of plotting several curves. - 17 - ==================================================== CHAPTER 17. Alt-M : The MINMAXSOLV setup screen ==================================================== The "MinMaxSolv" setup screen is used to find the minimum or maximum value of a function, or to solve a function for a spe- cific value. a) Type Alt1 to Alt9 to copy function 1 to 9 from the "Fit" setup screen to the current "MinMaxSolv" function. b) Select the type of solution you require: Minimise, Maximise, or Solve. If you select Solve then enter the function value you require on the "Solve for f(..)" line. c) The solution is found by non-linear regression using the Nelder & Mead routine (see "Fit"). This is certainly not the fastest or the most efficient method, but it is a very gener- al method which will solve most problems provided you supply a reasonable starting estimate. d) The "error limit" serves a similar purpose to that described for the "Fit" setup screen. It prevents a run-away solution. e) Make sure to enter a "Start value" as your starting guess for the solution. A poor starting estimate may mean no solution is found, or even the wrong solution (eg. for functions with multiple local minima or maxima). If your function has more than one independent variable then you need to enter a start- ing guess for each independent variable. ========================================== CHAPTER 18. Alt-O : The OUTPUT window ========================================== Program results are written to an internal buffer, so you can scroll through them later. Use the obvious commands such as Home, End, PgUp, PgDn, and arrow keys. Also use: b - Mark bottom screen line as bottom of block c - Clear the output buffer f - Find (search) a text string g - Find again (repeat last Find) h - Hide the marked block - 18 - p - Print marked block, or all if not marked r - Read any file from disk t - Mark top screen line as top of block w - Write marked block to disk, or all if not marked mouse- click on a line to mark block begin/end ================================================ CHAPTER 19. Alt-P : The PARAMS setup screen ================================================ The "Params" setup screen is used to: * define the independent variable/s which your function uses; * specify which columns in the data table will hold various data; * specify what standard deviation data are available, if any; * define any parameters to be used in the function; * specify which parameters are constant, and which will vary; * assign values to constants, & parameter starting estimates; Some instructions are shown to the left of the "Params" setup screen, indicating various required and optional data. You need to define at least one independent variable (IV), though your function can have up to 6 IV's. Example: f(x) = a+b*x has 1 IV, called x f(r) = c/r has 1 IV, called r f(x,y,z) = a+x+b/y^2+c*ln(z) has 3 IV's (x,y,z) Each IV must be assigned a column number in the data table, where the individual data vectors are located. The function value, f(x) etc, must also be assigned a data column number to hold the function value at each data vector. Enter this in section (b) at the line which says f(),Data...[--]. Below that line are three additional (and optional) data column allocations. You can leave them as 0 to ignore them, or fill in a valid column number to activate them: 1. 'Fitted' is a column which will hold the fitted function value corresponding to each data vector after the fit has been completed. 2. 'Residuals' is a column which will hold the residual error (ie the difference between the fitted function and the f() data) corresponding to each data vector after the fit has been completed. - 19 - 3. A 'MonteCarlo' data column is required only if you are doing Monte Carlo simulations for error analysis on fitted parameters. The Monte Carlo procedure requires an addi- tional data column for temporary storage of data during the simulations. To the right of these data column allocations is the standard deviation section. If you want a statistical analysis on the fitted parameters then you need to supply DataFit with infor- mation on how much error is associated with the individual data points. This is done by supplying standard deviations for each individual f() data value. Standard deviation data can be entered in one of three ways: 1. If you don't know the standard deviation, then check the 'Unknown' row by clicking on it with the mouse or pressing Enter while the cursor is there. 2. If the standard deviation is constant for all data points, then check the centre row and fill in the value of the standard deviation. 3. If the standard deviation varies for each data point, then check the third row, 'Column', and fill in a data table column number. Then go to the data table with AltD and fill in the standard deviations in each row of that col- umn. Note that if the standard deviation is a simple function of the function value, or one of the independent variables, then you can use the F4 command in the data table to automatically enter the standard deviation into the table. Finally, you need to enter the names of any constants and parameters which will be used in the function. Click or press Enter on the left of the line to mark (*) the parameters which will be varied during the fit. Any unmarked parameters will be treated as simple constants. Then assign values for each constant, and starting values for each variable parame- ter. (Linear Regression does not require starting estimates for the variable parameters). You can place lower and/or upper limits on parameters to be fitted. In the column where you enter the initial estimate of the parameter values, follow the value with one or two num- bers separated by commas. But please take note: It is strong- ly recommended that you don't put limits on parameters unless you really need to. Limits can have a destabilising effect on the fitting process, and may even result in failure to converge. Nelder & Mead will usually be the most stable method in these cases. Examples: [1.234,0,100 ] limits the value between 0 and 100, [1.234,0 ] limits the value to >0, [1.234,,100 ] limits the value to <100. - 20 - IMPORTANT: Take care with data column assignments, as check- ing is not done to any great extent to determine conflicts. If for example you allocated the 'Residuals' data column to the same as the 'x' data column then the 'x' data will be overwritten by the residuals. ================================================ CHAPTER 20. Alt-R : The REPORT setup screen ================================================ The "Report" setup screen lets you control the format of the output generated by the program. Specify page width in columns for text output, left margin width, and whether to include the list of data points in the output from function fitting. If data points are included in the output report then you can choose any or all of errors (between fitted function and data), percent errors, and standard deviations to be written alongside each data point. Output always goes to the internal buffer for subsequent scroll- ing, but it can also be sent directly to printer and/or a disk file. The internal buffer has a limited size, so very big prob- lems may need to send output to printer or disk. You can specify a format for representing numbers (natural, scientific or auto) in the output report, plus decimal places (1 to 9), field width for lining up columns, and whether to show trailing zeroes. Finally, the output report can include several lines of com- ments, either at the beginning or the end of the report. =========================================== CHAPTER 21. F1 - Help =========================================== Type F1 while using DataFit to enter these Help screens. When you are using the Help screens F1 takes you to the main Help menu. - 21 - =========================================== CHAPTER 22. F2 - Save work file =========================================== Type F2 while editing the text file to save it to disk. It is good practice to do this regularly when entering data, and especially just before starting to fit a function to the data. NOTE: The DataFit demo version will not save your data. This section is included only for completeness. =========================================== CHAPTER 23. F3 - Load a new work file =========================================== Type F3 to load a work file from disk, or to create a new file. If the name you enter contains wildcard characters (eg. *.hls) another window will be opened showing all files on disk which match that filespec. Select one with the arrow keys and Enter, or Esc to quit and return to the editor. =========================================== CHAPTER 24. Alt-F3 - Pick file =========================================== Type Alt-F3 to select from a list of files which have been accessed during this work session. Move the cursor to select one and type Enter to load it. To load a file not on the list, type F3 or select the -- load file -- option at the bottom of the list. ================================================= CHAPTER 25. F5 - Zoom the function screen width ================================================= F5 toggles the zoom setting on the edit window between full size and the "zoom-in" size when appropriate. - 22 - =========================================== CHAPTER 26. F7 - Auto-scale axis values =========================================== Type F7 while you are editing the Graph data entry screen (Alt- G) to set sensible values for the axis ranges, divisions along the axes, subdivisions, and number of decimal places for numeric axis labels. When you press the F7 key the program scans through the data table to find the minimum and maximum data values on the x and y axes, then sets the graph axis values to appropriate limits. ====================================== CHAPTER 27. F8 - Printing the graph ====================================== Type F8 at any stage to enter the "Print Graph" setup screen. This lets you specify printer type, output destination, hi/lo resolution, and a "swap to disk" option to overcome certain memory limitation problems. a) Printer: A list of printers is shown at the left of the setup screen. Select the one closest to yours by clicking on it with the mouse or moving the cursor to it with the arrow keys and pressing Enter. b) Output Port: Select the output port your printer is installed on. This will normally be the default option. You can also direct the graphics output to a disk file. c) Resolution: The default is hi-res. Select the low-res option for rapid printing of draft-quality graphs. The resolution option has no effect with PostScript printers. d) Swap to disk : Note the "swap out to disk" option at lower right. High resolution printers such as LaserJets require a lot of memory to compose the plot. EMS memory will be used if it is available, but if not then you may not have enough RAM available to complete the plot. In this case, check the "swap" option, and this is what will happen: When you print, DataFit first creates a temporary disk file which contains primitive graphic instructions for the graph to be plotted. Then it unloads all but a small kernel of itself to the hard disk. Then it loads in a smaller program which reads the temporary file and plots the graph using all the RAM that was freed by swapping out DataFit. When printing is completed it loads the DataFit program back in, to continue from where you were. The "swap" option is NOT required for PostScript printers, which use an entirely different method for generating the - 23 - graphs, and which should not run out of memory. e) Font error: If you get a font file error while printing, the program has failed to find a file called DEFFONT.GLF in the directory containing the DataFit program. Check that it is there, or de-select the "Use SansSerif/Helvetica font" option in the "Graph" setup screen. ============================================ CHAPTER 28. F9 - Select a new graph type ============================================ DataFit lets you create several graph types from the function fitting results. You can plot the fitted function with the experimental data, or the residual errors vs either the function response or any of the independent variables. Type F9 to open a window which lets you select from the avail- able types of graph. =============================================== CHAPTER 29. Alt-F9 - Select next graph type =============================================== DataFit lets you create several graph types from the function fitting results. You can plot the fitted function with the experimental data, or the residual errors vs either the function response or any of the independent variables. Type Alt-F9 when using either the Graph setup screen (Alt-G) or the Print Graph setup screen (F8) to change the current graph type to the next available type. ==================================================== CHAPTER 30. Ctrl-F9 - Select previous graph type ==================================================== DataFit lets you create several graph types from the function fitting results. You can plot the fitted function with the experimental data, or the residual errors vs either the function response or any of the independent variables. Type Ctrl-F9 when using either the Graph setup screen (Alt-G) or the Print Graph setup screen (F8) to change the current graph type to the previous available type. - 24 - =========================================== CHAPTER 31. F10 - Menu =========================================== The F10 key transfers control to the main menu options displayed on the top line of the screen. Select an option with the left and right arrow keys and press Enter to activate it (or click on an option with the mouse). - 25 - ****************************************************************** DataFit 1.0 - A few case studies ****************************************************************** Several demo data files have been provided on the distribution disk to help you get started, and to act as case studies to demonstrate some of the major features of DataFit. It is strong- ly recommended that you go through them all, as each has some major features to demonstrate. ==================================================== CASE STUDY 1 : DEMO1.FIT - Simple straight line fit ==================================================== a) Introduction: This shows how to fit a simple straight line through a set of data points. This is the simplest type of problem to solve with DataFit, though you will also be shown how to estimate standard deviations and confidence limits on the fitted parameters - features not commonly available with other straight line fitting programs. b) Loading: First type F3 to load in a data file. You will see a small window showing the current DOS filespec (eg. *.FIT) for the type of file to load from disk. Press Enter when *.FIT is displayed, and a second window is opened to display all the files ending in .FIT. Move the cursor to DEMO1.FIT, and press Enter to load it. c) Description: When loaded, the file contains a data list of 30 points, which was generated from a straight line with normally-distributed random errors. The errors have a stand- ard deviation of 2. This is equivalent to measuring data points which are expected to fall on a straight line, but where the measurement error has a standard deviation of 2. d) General : The main screen shows the function, f(x)=a+b*x, and that Linear Regression has been selected. The inactive "Param" window shows that just one independent variable has been defined (x) by having a data column defined for it, and that these x-data are in column 1 of the data table. It also shows that the function response data, f(x), are to be found in column 2 of the data table. Additionally, data column 3 has been allocated to hold the corresponding fitted values at each of the x-data values. Data column 4 will hold the residuals of the fit (the differ- ences between the data and the fitted function), and data column 5 has been allocated to be used as a temporary storage column for when we come to do the Monte Carlo simulations - 26 - later on. The standard deviation on the f(x) data values is known to be 2, and this is shown as such towards the centre-right of the "Params" window. Finally, parameters a and b have been checked (marked) as variables to be fitted. Linear regression does not require starting estimates of the parameter values as it is not an iterative procedure. So if you enter values for a and b they will just be ignored. e) Fitting: When the "Fit" window is active (indicated by a double border, and the word "Fit" highlighted on the top line) then just type AltF to start fitting. Or click on the word "Fit" or any phrase starting with "AltF...". The screen changes to display the progress of the fit. It will be fairly quick as this is a very simple problem to solve. When done, a report is generated in the style speci- fied in the "Report" setup screen. In this case, you will see the fitted parameter values, error estimates and confidence limits on the fitted parameters, and a table showing fitted vs actual function values at each value of the independent variable x. Scroll through the output window to examine the solution. You can also highlight a block to be printed or written to a disk file. f) Graph the Fit: Type AltG or click on "Graph" to open the graph setup screen. Sensible values have been supplied ready to use, though you can change them if you wish. Type AltG again to display the graph on screen. The graph page is shown scaled to the page dimensions defined in the "Graph" setup screen, and positioned on the page as it will be printed out. Note that the function itself is included in the graph title. This is accomplished by including ~f in the title in the Graph setup screen - described in more detail earlier in this document. Similarly, the parameter values have been automati- cally entered Type AltG to display the graph of the residu- als (or whichever type you selected), and Esc to return to the " Graph" setup screen. As you get used to the program you may find it easier to switch between graph types with the CtrlF9 and AltF9 keys. Switch back to the graph of the fitted function in prepara- tion for printing it. g) Print graph: Type F8 to enter the "Print Graph" setup screen. Select a printer closest to yours from the list shown at left. Select a printer output port if your printer is not - 27 - installed as the default printer. Set the "swap to disk" option if necessary (see the F8-Print Graph setup screen for more details). When all ready, type F8 again to start plotting. h) Monte Carlo: The original Linear Regression fit will already have returned standard deviations and confidence limits for the fitted parameters, as standard deviations for the data were specified (StdDev=2 for all data points in this case). There is a very important reason why the estimates may not be accurate - the error estimation relies on the fact that the data errors are indeed normally distributed (with StdDev=2 in this case). It is possible that the errors on your data may not be normally distributed, and may be skewed or biased in some way. This is where Monte Carlo simulation can provide useful additional information. Monte Carlo simulation was described earlier, but to recap very briefly it simulates fitting the function to a large number of sets of data points, each varying from your own original data by the standard deviation which you specified in the "Params" setup screen. Each fit returns a new set of fitted parameters, and the Monte Carlo routine returns some statistics about how those parameters varied. If the standard deviation/s of errors on your data (which you supplied in "Params") were accurate, then Monte Carlo simula- tion will return similar values for the parameters and their statistical deviations and correlations. If not, then the Monte Carlo results will vary from the initial estimates of these quantities, and you can ponder on the implications of that for your own specific function fitting problem. By the way, a second reason for using Monte Carlo simulation is in conjunction with the Nelder & Mead method. This method does not return error estimates on fitted parameters, even if standard deviations on the data points are available. So if you are fitting with Nelder & Mead then Monte Carlo simula- tion is the only way to get error estimates on fitted parame- ters. See the next section (called "No StdDev?") for details on how to do this. To do the Monte Carlo simulation first set the (*)Monte Carlo option, then set the number of iterations you want to do. It is strongly recommended that you do at least 100 iterations, more if you have the time and patience. Then type AltF as normal to begin fitting. The fit proceeds as before, but when it is done a small window opens to show the progress of the Monte Carlo iterations. When done the results are added to the end of the output report. No StdDev? If you don't have any information on the standard deviations of your data you can still estimate confidence - 28 - limits etc. for the fitted parameters, though with some limi- tations. Do it like this. First fit the function using any of the methods. At the end of the report, the standard deviation between the fitted function and your data is listed. This can be used as an approximation to the standard deviation on the data points IF, AND ONLY IF, THE FITTED MODEL IS A GOOD ONE. You must plot the fit against the data points and decide whether the fitted function looks like a good representation of the data points. If so, then enter the value of the stand- ard deviation between model and data into the "Params" setup screen, and do the fit again. If you are using Linear Regression or Levenberg/Marquadt then they will automatically report error estimates on fitted parame- ters once the data standard deviation has been entered in the "Params" screen. If you are using Nelder & Mead then you will have to use Monte Carlo simulation for error estimates on fitted parameters. Note by the way that no additional information will be acquired by using Monte Carlo simulation with Linear Regression or L/M when the data standard deviations are estimated as explained in this section. It may or may not be obvious, but if the standard deviation between the original fitted model and the data is used as the standard deviation for Monte Carlo simulations, then Monte Carlo will return the same result as the original fit (provided you do enough iterations to be statistically meaning- ful). - 29 - =================================================== CASE STUDY 2 : DEMO2.FIT - Combination of 2 curves =================================================== a) Introduction: This demo shows how to fit two separate func- tions through different ranges of your data points. In this case the function to be fitted is a straight line of negative slope, or a logarithm curve, depending on which has the greater value. The function is defined as f(x) = a+b*x max c*ln(x)-d. In words this means f(x) at any x-value is either [a+b*x] or [c*ln(x)-d)], depending on which has the greater value. We need to find values for a, b, c, and d which provide the best fit through the data points. b) Solving: First type AltG twice to see what the data points look like. Then Esc twice to return to the "Fit" setup screen, and type AltF to start fitting with Levenberg/Mar- quadt. Then look at the graphs of fitted function and residu- als. c) More: You can also fit separate function segments over specific ordinate ranges using the logical functions < > <> = <= and <=. These let you fit piecewise smooth, dis- continuous functions. For example, say you want to fit a+b*x for x<3, c*x^2 for 3<=x<6, and d+e*exp(x) for x>=6. Then... f(x)=(a+b*x)*(x<3) + (c*x^2)*((x>=3) and (x<6)) + (d+e*exp(x))*(x>=6) A bit cumbersome perhaps, but it works. You will almost certainly have to use Nelder & Mead for these "strange" functions, though it would be worth giving Levenberg/ Mar- quadt a try just in case (especially if you have good start- ing estimates of the fitted parameters). - 30 - ============================================= CASE STUDY 3 : DEMO3.FIT - Selectivity curve ============================================= a) Introduction: This is a simple demo illustrating the use of a hyperbolic trig function (atanh) in a selectivity curve. It also illustrates log axes on the graph. b) Solving: Nelder & Mead is used for solving this example. Levenberg/ Marquadt gives a "singular matrix" error when starting with a=1, b=1 and c=1, but Nelder & Mead is able to find the solution from these starting estimates. - 31 - =========================================================== CASE STUDY 4 : DEMO4.FIT - Manual adjustment of parameters =========================================================== a) Introduction: Demo 4 is an exponentially decaying sine wave, which is very difficult to fit: f(t) = sin(a+b*t)*exp(-c*t). In addition, the starting estimates of a=1, b=1, c=1 are very poor, and neither of the non-linear methods will find the solution from these values. b) Solving: This is a good chance to illustrate using the fourth fitting method - Manual (Graphical) adjustment. Type AltF to start fitting. The function is plotted with the (bad) current estimates of parameters, and the data points. A table at the top right shows a factor F and the current values of the parameters. The Up/Down arrows move a marker to one of the parameters, and pressing + - * / performs that operation on the marked parameter (+ adds F to the parameter, - sub- tracts F, * multiplies and / divides the parameter by F). You can change the value of F with the Left/Right arrow keys, for Decrease/Increase. Each time you change the value of a parameter, the function is replotted. So you can see immediately how each parameter affects the function. With a bit of guesswork you should be able to home in to the approximate solution. At that stage you can exit with Esc, and switch to one of the non-linear fitting methods. For this example, the solution occurs near a=0.125, b=1, c=0.125. Change the parameter values to these values, and then Esc back to the "Fit" screen. Now select Levenberg/ Marquadt and fit with AltF to home in on the correct solu- tion. - 32 - ========================================================== CASE STUDY 5 : DEMO5.FIT - "Dangerous" Linear Regressions ========================================================== a) Introduction: This case study is an illustration of the dangers of using transformations to linearise non-linear functions. The function to be fitted is: f(x) = ln(a+b*x +c*x^2). It can be linearised into this form: exp(f(x)) = a+b*x+c*x^2 which is linear in the parameters to be fitted (a, b, c) once the function response has been transformed by the exponential function. It can therefore be fitted using Linear Regression, but this problem illustrates very well how this can lead to "less than best" results. b) Solving: First solve the problem with Linear Regression by typing AltF. The fitted parameters are a=230.72, b=-39.8 35, and c=5.5096. The output report indicates that the standard deviation between model and data is 0.5678. Plot the fit with AltG, and you will see that while the fit is good for higher values of x, it is poor near x=0. Now type AltF or Esc to return to the "Fit" setup screen, change the type of fit to Levenberg/Marquadt, and fit again with AltF. Notice that as far as L/M is concerned the results of the linear fit are so far from the solution that it can not converge. Nelder & Mead finds the same problem. So, type AltP to go to the "Params" setup screen, and reset the starting values to a=1, b=1 and c=2. Fit the function using Levenberg/Marquadt. The results are now returned as a=5.8459, b=1.3232, and c=3.9058. The standard deviation between model and data is now 0.1989. These results are considerably different to the linear regression results, and a plot of the fit with AltG shows that the fit is good for the entire range of x values. Going back to the linear regression case, the error estimates on the fitted parameters were based on the standard deviation of 0.2. The data fitted by non-linear regression was trans- formed by the exponential function though, and this really invalidates the standard deviation data. So the error esti- mates on the fitted parameters are incorrect (for Linear Regression only). Here is a good chance to use Monte Carlo simulation. Fit again by Linear Regression with Monte Carlo simulation selected, for say 100 iterations. Now check the error estimates on the fitted parameters. You will see at the end of the output report that while the parameter values differ substantially from the L/M results, they also have very large associated standard deviations (errors), which reconciles the two methods. Clearly Levenberg/Marquadt is the much better method to use in this case. - 33 - ======================================================== CASE STUDY 6 : FAMILY1.FIT - Fitting a family of curves ======================================================== a) Introduction: This is an interesting example of fitting a related family of curves through the data. In this case the dependent data f() is a function of 2 independent variables x and y. y takes on several discrete values only, and at each y-value there are several values of x, representing a func- tion in x for each y-value. Go to the data table with AltD to see the values of x, y and the function. You will see that y has the values 1, 2, 3, 4 and 5. Each y value has a different set of x-values. b) Solving: The solution is quite straightforward using Leven- berg/ Marquadt. c) Plotting: Type AltG to see the "Graph" setup screen. Notice right at the end of the screen is a line which lets you enter several values of the 2nd independent variable, so that several curves can be plotted. This line is always available when more than one independent variable has been defined in the "Params" setup screen. It is not restricted to parametric curves only. Type AltG again to plot the fitted family of curves. - 34 - ======================================== CASE STUDY 7 : GAUSS1.FIT - 2 Gaussians ======================================== a) Introduction: This case study is a nice example for using Manual Adjustment of parameters in fitting a function with a fairly large number of parameters. It also demonstrates how to plot both the fitted function and its residuals on the same page. The function is two Gaussians (normal distribu- tions): f(x) = a1*exp(-((x-b1)/c1)^2) + a2*exp (-((x-b2)/c2)^2) b) Solving: First try estimating the approximate parameter values by fitting with the Manual (graphical) Adjustment method. Hint: the function is made up of two normal distribu- tions, each of which has a peak height, peak position along x-axis, and width or spread. The peak height is determined by parameter a, b controls the horizontal position of the peak along the axis, and c controls the width. When you are getting close, type Esc to return to the "Fit" screen, select Nelder & Mead, and fit. Then plot the fit with AltG. - 35 - =================================================== CASE STUDY 8 : PARAMET1.FIT - Parametric equations =================================================== a) Introduction: Here is a simple example of fitting parametric equations to data. The method is not entirely trivial. Nor- mally, a two dimensional fit has one independent variable (say x) and a dependent variable or response f(x). Parametric equations in two dimensions require two independent variables (say x and y), each of which varies with a third parameter, usually t to indicate passage through time. The specific example given here is a simple ellipse centred on (x,y) = (0,0). The parametric equations are: x=a*sin(t), y=b*cos(t) b) Solving: So how do we set this up with DataFit? First go to the "Params" screen. Now define both x and y as independent variables and allocate their data columns (this is all al- ready done for you in PARAMET1.FIT). Make sure that a parame- ter called t is defined as a constant parameter. DataFit insists that the parametric equation parameter is called t. This needs to be assigned a data column, and an obvious spare one which is not otherwise required for fitting parametric equations is the "f()Data...[--]" data column. So assign this the data column which holds the time or t data for the (x,y) data pairs. Note that the actual values of t corresponding to each (x,y) value must be entered into the data table, or an error will result. Again, this is all done for you in PARA- MET1.FIT. Now go to the "Fit" screen with AltF. See how the parametric equations are entered into the equation line: f(x,y) = [a*sin(t), b*cos(t) ] The x and y are assumed, and the individual equations are separated by commas. In fact the commas are the trigger to tell DataFit that parametric equations are being fitted. The number of equations must match exactly the number of inde- pendent variable defined in "Params", and must be between 2 and 6. Now, all you need do is type AltF to do the fit. Then plot it out with AltG. - 36 - ============================================== CASE STUDY 9 : POLAR1.FIT - Polar coordinates ============================================== a) Introduction: Here is an example of a normal simple fit, with the result plotted in polar coordinates rather that the more usual rectangular coordinates. The function to be fitted is just a straight line, but when plotted in polar coordinates it becomes a spiral: f(theta) = a+b*theta b) Solving: Solve with Linear Regression, and plot with AltG. Notice the "Graph" setup screen has the 'Polar Coords' option set. This means that when plotting the function, the (x,y) data are assumed to be in (theta,r) format, and are trans- formed before plotting onto the rectangular coords of the graph: x=r*cos(theta), y=r*sin(theta) c) Plotting Range: Note that when plotting with polar coordi- nate transformations the theta values (in this example) run from 0 to 12.5 rad. By default the function plotting range normally reflects the x-axis range: -8 to 8 in this case, so the plotting range must be specified explicitly. To the right of the "minimum value" and "maximum value" settings in the "Graph" setup screen is a third column headed "Plotting Range". Enter 0 to 12.5 to plot from theta = 0 radians to 12.5 radians. ================================================================== End of Document ================================================================== - 37 -