GNUPlot Manual

22 FIT gnuplot 4.0 31
current parameter values, (2) continue the fit, (3) execute a gnuplot command as specified by the
environment variable FIT SCRIPT. The default for FIT SCRIPT is replot, so if you had previously
plotted both the data and the fitting function in one graph, you can display the current state of the fit.
Once fit has finished, the update command may be used to store final values in a file for subsequent
use as a parameter file. See update (p. 158) for details.
22.1 Adjustable parameters
There are two ways that via can specify the parameters to be adjusted, either directly on the command
line or indirectly, by referencing a parameter file. The two use different means to set initial values.
Adjustable parameters can be specified by a comma-separated list of variable names after the via
keyword. Any variable that is not already defined is created with an initial value of 1.0. However, the fit
is more likely to converge rapidly if the variables have been previously declared with more appropriate
starting values.
In a parameter file, each parameter to be varied and a corresponding initial value are specified, one per
line, in the form
varname = value
Comments, marked by ’#’, and blank lines are permissible. The special form
varname = value # FIXED
means that the variable is treated as a ’fixed parameter’, initialized by the parameter file, but not
adjusted by fit. For clarity, it may be useful to designate variables as fixed parameters so that their
values are reported by fit. The keyword # FIXED has to appear in exactly this form.
22.2 Short introduction
fit is used to find a set of parameters that ’best’ fits your data to your user-defined function. The fit is
judged on the basis of the sum of the squared differences or ’residuals’ (SSR) between the input data
points and the function values, evaluated at the same places. This quantity is often called ’chisquare’
(i.e., the Greek letter chi, to the power of 2). The algorithm attempts to minimize SSR, or more precisely,
WSSR, as the residuals are ’weighted’ by the input data errors (or 1.0) before being squared; see fit
error estimates (p. 32) for details.
That’s why it is called ’least-squares fitting’. Let’s look at an example to see what is meant by ’non-linear’,
but first we had better go over some terms. Here it is convenient to use z as the dependent variable
for user-defined functions of either one independent variable, z=f(x), or two independent variables,
z=f(x,y). A parameter is a user-defined variable that fit will adjust, i.e., an unknown quantity in the
function declaration. Linearity/non-linearity refers to the relationship of the dependent variable, z, to
the parameters which fit is adjusting, not of z to the independent variables, x and/or y. (To be technical,
the second {and higher} derivatives of the fitting function with respect to the parameters are zero for a
linear least-squares problem).
For linear least-squares (LLS), the user-defined function will be a sum of simple functions, not involving
any parameters, each multiplied by one parameter. NLLS handles more complicated functions in which
parameters can be used in a large number of ways. An example that illustrates the difference between
linear and nonlinear least-squares is the Fourier series. One member may be written as
z=a*sin(c*x) + b*cos(c*x).
If a and b are the unknown parameters and c is constant, then estimating values of the parameters is a
linear least-squares problem. However, if c is an unknown parameter, the problem is nonlinear.
In the linear case, parameter values can be determined by comparatively simple linear algebra, in one
direct step. However LLS is a special case which is also solved along with more general NLLS problems by
the iterative procedure that gnuplot uses. fit attempts to find the minimum by doing a search. Each step
(iteration) calculates WSSR with a new set of parameter values. The Marquardt-Levenberg algorithm
selects the parameter values for the next iteration. The process continues until a preset criterion is