A maximum of 25 fitting parameters are allowed in the expression. Variable fitting parameters are created with the SCALAR\VARY command, and can be converted to fixed value scalar variables with the SCALAR command. By default, variable fitting parameters are left at their last calculated values when fitting stops due to an unsuccessful fit or a control-c abort. Use the \RESET qualifier to cause the fitting parameters to be reset to their original values after an unsuccessful fit or a control-c abort. If the \VARNAMES qualifier is used, an array text variable named FIT$VAR will be made which will contain the names of the fitting parameter variables. The array length of FIT$VAR will be equal to the number of fit parameters.
Assume that each data point has an error that is independently random and distributed as a normal distribution. The weighted chi-square function, as a function of the fit parameters, is minimized using a Gauss-Newton method. Serious difficulties can arise when the expression is sufficiently nonlinear and chi-square is large at the minimum. If not entered, the weights, defined as the reciprocals of the variances, default to 1.0 for each data point.
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Assume that each data point has an error that is independently random and distributed as a Poisson distribution. The log likelihood function, as a function of the fit parameters, is minimized using a Gauss-Newton method. Since logarithms are involved, a good first approximation is required before starting the Poisson fit. Try a normal fit first, and use these derived parameter values to start off the Poisson fit. Weights do not have meaning, and so are not used, in a Poisson fit.
Indications of the accuracy of the fit are displayed in the output under the names E1 and E2. E1[i] is the root mean square statistical error of estimate, and the root mean square total error of estimate, or standard error, is displayed under E2. The w[k] for k=1,...,N are the weights which the user can specify for each data point. The accuracy of the parameters in a linear fit is p[i]+/-E2[i] for i=1,...,M. In theory, for the standard error E2 to be correct, the weights w[k] must be proportional to the variance of the probability distribution of y[k].
If the \CHISQ qualifier is used, a scalar named FIT$CHISQ will be made with value equal to chi-square.
If the \CL qualifier is used, a scalar named FIT$CL will be made with value equal to the confidence level, which is the probability that a random repeat of the given experiment would observe a worse chi-square, assuming the correctness of the model.
If the \E1 qualifier is used, then the root mean square statistical errors for each fit parameter are output into an automatically created vector named FIT$E1. The length of FIT$E1 will be equal to the number of fit parameters. If the \E2 qualifier is used, then the root mean square total error of estimate for each parameter are output into an automatically created vector named FIT$E2. The length of FIT$E2 will be equal to the number of fit parameters.
A matrix with the name FIT$CORR will be created if the \CORRMAT qualifier is used. This matrix will contain the correlation matrix for the fit. A matrix with the name FIT$COVM will be created if the \COVMAT qualifier is used. This matrix will contain the covariance matrix for the fit. The size of these matrices will be N by N, where N is the number of parameters in the fit. The E1's are the square roots of the diagonal elements of the covariance matrix.
Syntax: FIT\ITMAX n y=expression
FIT\WEIGHTS\ITMAX w n y=expression
FIT\WEIGHTS\ITMAX\TOLERANCE w n eps y=expression
FIT\ITMAX\TOLERANCE n eps y=expression
The \ITMAX qualifier allows the user to specify the maximum number of
iteration steps for the fit. When this maximum number is reached, the
fit will stop, and the variable parameters will be updated to their last
values. The fit will also stop if the fit is successful before this maximum
iteration number is reached. If the \WEIGHT qualifier is also used, the
weight array comes before the iteration number in the parameter list.
Syntax: FIT\TOLERANCE eps y=expression
FIT\WEIGHTS\TOLERANCE w eps y=expression
FIT\ITMAX\TOLERANCE n eps y=expression
FIT\WEIGHTS\ITMAX\TOLERANCE w n eps y=expression
The \TOLERANCE qualifier allows the user to specify the fitting tolerance
(default value 0.00001). This value is used in calculating the central
difference formula for the partial derivatives, that is, the partial
derivative of f(x,p) wrt p is approximated by
f(x,p*(1+eps))-f(x,p*(1-eps))/(2*eps*p)
This value is also used to determine when the fit is successful.
The parameter order is: weights, itmax, tolerance.
If the \FREE qualifier is used, then the number of degrees of freedom for the fit is output into an automatically created scalar named FIT$FREE. The number of degrees of freedom is either the number of data values minus the number of parameters, or, if the \-ZEROS qualifier is also used, the number of non-zero weights minus the number of parameters.
If the \NOMESSAGES qualifier is used, then there will be no informational messages output to the monitor screen.
Syntax: FIT\UPDATE yout The FIT\UPDATE command evaluates the previously fitted expression, that is, the expression to the right of the = in the last FIT command, for the current parameter values and stores this result in the vector yout. This is exactly equivalent to entering: yout=previously_fitted_expression and is provided to obviate the necessity of re-entering a complicated expression. Note that the vector yout usually differs from the name of the vector being fitted to avoid destroying the original data.
To fit a straight line to some data stored in vectors X and Y, use: SET PCHAR -1 ! use unconnected box plotting symbols GRAPH X Y ! plot the original data SCALAR\VARY A B ! create 2 variable fitting parameters FIT Y=A*X+B ! fit to the line FIT\UPDATE YF ! create vector YF SET PCHAR 0 ! turn off plotting symbol GRAPH\NOAXES X YF ! overlay the fitted line