Global Fitting Problem description Global fitting (related to Multi-response onlinear Regression) is a process for fitting one or more fit models to one or more data sets simultaneously. It is a generalization of ordinary nonlinear leastsquares regression, but we will show later how to tacle this general problem in SigmaPlot using special constructs in the Transform Language and the onlinear Regression Wizard. The Global Fit Wizard in SigmaPlot is a more convenient interface for solving a special case of the global fit problem and will be discussed below. To setup a global fit problem, we assume you are given a number of data sets, each having values for one or more independent variables and corresponding values for one dependent variable. The values of the dependent variable are the observed values in your study which are typically subject to measurement errors. Suppose a fit model is chosen for each data set so that at least one regression parameter is shared between some or all of the fit models. If there are no shared parameters, then global fitting reduces to the problem of performing ordinary nonlinear regression separately on each data set using the corresponding (local) fit model. The optimization problem that is used in global fitting is to determine the parameter values that minimize a sum of squares, as in ordinary least-squares regression. In this case, however, we construct the sum of squares of the differences between the observed values and the corresponding values of the fit model for each data set and then add these sums of squares over all data sets. An equivalent way to view this sum of squares is to thin of concatenating all of the data sets together and using a global fit model whose value at a data point is the value of the corresponding fit model for the data set containing that data point. We then consider one sum of squares over all data points for all data sets. With this global fit model (a mathematical description is given below), the problem can be interpreted in the context of an ordinary nonlinear regression problem and analyzed to compute the best-fit parameters, the predicted values and residuals, the diagnostics, and the statistics as for any regression problem. Some local results, computed for each data set, can also be obtained. For example, a local residual sum of squares and a local goodness of fit measure, lie the coefficient of determination R, to determine the quality of the fit per data set. Global Fit Wizard In SigmaPlot, a restricted but useful version of the global fit problem can be solved using the Global Fit Wizard. In this version, Only one fit model is selected and it is used for all data sets. The fit model must have only one independent variable. Any shared (global) parameters specified by the user are shared across all data sets.
The wizard allows you to select a fit model and the format for your data, to pic the data, to select shared parameters, and to compute the results. Most of the results in the report are global, meaning that they are computed from the fit problem described in the previous paragraph which is based on fitting all data sets simultaneously. There are also local results that are computed for each data set. More information on the computations is given in the sections below. Example of setting up a global fit problem As an example of the type of global fit problem solved by the Global Fit Wizard, suppose the problem is based upon the four-parameter, single independent variable equation family: bx f y0 ae cx If three data sets are being modeled, then there are at most 1 parameters in the corresponding global curve fit problem four for each data set. If parameters are shared across all data sets, then there could be as few as four parameters in the global curve fit problem. Suppose data sets 1,, and 3 share parameter c and the other parameters y0, a, b are allowed to vary independently among the data sets. In this case, there are ten independent parameters in the global fit model which taes the form: bx 1 01 1 if, is in data set1 bx 0 if, isin data set b3 x 03 3 if, isin data set 3 y a e cx x y F y a e cx x y y a e cx x y. This is how the global fit model is formulated in SigmaPlot. The goal is to estimate the true parameter values (the values of the parameters for which the expected value of each observation y equals the corresponding value of the model F ). These (consistent) parameter estimates are obtained by using least-squares optimization, following the same algorithm we use with ordinary nonlinear regression problems. Liewise, the global fit results in the report are obtained with procedures that follow the same algorithms used in ordinary nonlinear regression. The global fit report in SigmaPlot does not refer to the indexed values of the different parameters as shown above when displaying the output for each data set. Instead, the report uses the original parameter names to avoid confusion and to mae the setup of the global fit problem more transparent. When running this example in the Global Fit Wizard, you would begin by selecting the equation name - in this case Exponential Decay, Exponential Linear Combination. You would then select any shared parameters in this case, c. Finally, you would select the format for your data and then pic the data. Choosing the data format XY Pairs, particular data sets were chosen for this problem with data set 1 placed in columns 1 and, data set in columns 3 and 4, and data set 3 in columns 5 and 6. After running the global regression, a report was generated and the following graph of the curve fits per data set was obtained.
Mathematical description of the Global Fit Problem The following discussion assumes an unweighted regression problem so that the observations for all data sets have been sampled from distributions with a common, but unnown, variance. Suppose we have M data sets, where each data set consists of pairs of values for the independent and dependent variables. Concatenating the data sets together, suppose the data values of our independent variable x are listed in sequence as nn 1 ote that each x. Let y n n1 be the corresponding sequence of observations. x n and x could be vectors if the problem has more than one independent variable. We can access each data set by partitioning the indexing set 1,,, into M segments defined by 1 D n : n n n for1 M, where n0 0 and n M For each, where1 M., it is assumed we have provided a function, f x A to model the data x x n where n is in D n D. For each model, A denotes its set of regression parameters. Shared parameters occur whenever at least two of the sets A intersect. Suppose a1, a,, a p is the set of all parameters contained in all of the sets A. The global fit model F is then defined as: f1 xn, A1 if n D1 f xn, A if n D F( xn, a1, a,, ap) fm xn, AM if n DM The global fit problem is to find values squares: * * * a1, a,, ap of the parameters that minimize the sum of M 1,, p n ( n, 1,, p) j j, SS a a y F x a a y f x A n1 1 jd As mentioned above, the global fit results, lie those that appear in the SigmaPlot report, are obtained from this formulation in the same way as would be obtained for any nonlinear regression problem. The SigmaPlot report also contains two local results that are computed for each data set. One of these is the residual sum of squares and the other is a goodness of fit measure nown as the coefficient of determination R. Using the notation above, we can give simple formulae for these quantities. First, the global residual sum of square is given by: SS SS a, a,, a res * * * 1 p
The local residual sum of squares for the data set is given by:, where A * are the values of the parameters in * SS y f x A res, j j, jd A corresponding to the best-fit values ina * * * 1, a,, a p. To compute the global value of R, we need the total sum of squares for all data sets. This is given by: total n1 SS y y, where y is the arithmetic mean of the observationsynn 1. Then we have R n SS SS res 1. To compute the local value of R for the data set, we need the total sum of squares for this data set. This is given by: total SS y y total, j jd, where y is the arithmetic mean of the observations in data set. The local value of R is then computed as: R SS res, 1. SStotal,
Using the onlinear Regression Wizard to perform more general global fit problems The onlinear Regression Wizard can be used to solutions to more general global fit problems than the Global Fit Wizard. Below is an example of an equation file, created in the Regression Wizard, for solving a global fit problem consisting of three data sets with multiple fit models. The data format is X Many Y so that each data set has the same values for the independent variable. It is assumed the data has been placed in the first four columns of the worsheet. This file could be used as a template for solving global fit problems where the data has the X Many Y format. [Variables] x={col(1),col(1),col(1)} y={col(),col(3),col(4)} ''Total of 3 data sets, using the X Many Y format [Parameters] ''Initial parameter values y0 =.000416667 L1 = 1.67391 L = 1.5 L3 = 1.5 a1 =.007 a =.0065 a3 =.00135 [Equation] =size(col(1)) X1 = x[data(1,)] X = x[data(+1, *)] X3 = x[data(*+1,3*)] ''Shared parameters are y0 and L1 f1=y0 + a1*exp(-l1*x1) f=y0 + a*(exp(-l*x) - exp(-l1*x)) f3=y0 + a3*(exp(-l3*x3) - exp(-l1*x3)) f= {f1, f, f3} fit f to y [Constraints] [Options] tolerance=1e-10 stepsize=1 iterations=100
otice the Equation section of this file. Although the data sets have been concatenated in the Variables section, they are then separated bac into the individual data sets for expressing the three fit models (actually the fit models for the second and third data sets are the same). otice how the fit models show which parameters are shared (global) or unshared (local). The crucial step is how the global fit model f is formed from the local models f1, f, and f3 - through the operation of concatenation. When running this example with the onlinear Regression Wizard, use From Code on the data format panel. The report will provide the global fit results, similar to those in the Global Fit Wizard report. Local results, lie those in the Global Fit Wizard report (the residual sum of squares and R per data set) can be obtained easily by writing a simple User-Defined Transform that operates on the predicted values that are output to worsheet after the example runs. You can also create a graph of the global fit results using the Plot Equation dialog (for the curve fit plots) and the Graph Wizard (for the raw data scatter plots). For particular data sets chosen for this problem, the graph of the global curve fits is shown below.