
The Default Solution



Each datafitting command in the Statistics package attempts to fit a model function to a given set of data by determining values of the model parameters that minimize the leastsquares error. The model function may be specified in algebraic, operator or Matrix forms, with algebraic form being the most commonly used form. (See Statistics/Regression/InputForms for more details about the input forms.)


When the model function is given in algebraic form, then an algebraic expression, representing the model function with the final parameter values, is returned by default. Otherwise, the model parameters are returned in a Vector.



The output Option



The output option can be used to control the form of the returned solution. If the output=solutionmodule option is provided, then a solution module, as described in the following section, is returned. The output option can also take as value a single name (or string) or a list of names (or strings), and the associated results are returned. The acceptable names are described in the Results section below.



The Solution Module



When the output=solutionmodule option is provided to one of the fitting commands, a module containing two exports, Settings and Results, is returned. Each export is a procedure that accepts a name (or string) or a list of names (or strings) and returns the associated values. The acceptable names are described in the Settings and Results sections below. For example, if the module is assigned to variable m, then the call m:Results(residualsumofsquares) returns the sum of squares of the final residuals.


If no argument is provided to the Settings or Results export, then all available information is returned as a list of equations nm=val, where nm is a name and val is the associated value.



Settings



The following names can be given to the Settings export of the solution module.


Linear Fitting



These settings are available for the linear fitting routines only.

•

confidencelevel  Confidence level.

•

svdtolerance  Tolerance controlling when a singularvalue decomposition takes place.



NonlinearFitting



These settings are available for the nonlinear fitting routines only.

•

initialpoint  Initial point used by the nonlinear optimization solver.

•

parameternames  The names of the parameters, if they are available.




Results



The following names can be given to the Results export of the solution module or used as a value of the output option.


General Fitting



These settings are available for both the linear and nonlinear fitting routines.

•

leastsquaresfunction  The model function containing the computed leastsquares parameters. This result is available only when the input is in algebraic form.

•

parametervalues  The values of the computed parameters. If the input is in algebraic form, then a list of equations of the form $\mathrm{nm}=\mathrm{val}$ is returned, where nm is the name of the parameter and val is its value. Otherwise, a Vector of parameter values is returned.

•

parametervector  A Vector containing the values of the computed parameters. If the model function is given in algebraic form, then the parameternames option must be provided so that the order of the values in the parameter Vector is specified.

•

residualsumofsquares  The sum of squares of the residuals.

•

residuals  The residual Vector.

•

degreesoffreedom  The degrees of freedom in the leastsquares problem.

•

residualmeansquare  The residual mean square (residual sum of squares divided by degrees of freedom).

•

residualstandarddeviation  The residual standarddeviation (square root of residual mean square).



Linear Fitting



These settings are available for the linear fitting routines only.

•

standarderrors  Standard errors for the parameters.

•

confidenceintervals  Confidence intervals for the parameters using the confidencelevel setting.

•

leverages  Leverages.

•

variancecovariancematrix  Variancecovariance Matrix.


The following are available only if certain conditions are met by the solution of the leastsquares problem.

•

internallystandardizedresiduals  Internally standardized residuals.

•

externallystandardizedresiduals  Externally standardized residuals.

•

CookDstatistic  Cook's D statistic (also known as Cook's distance).

•

AtkinsonTstatistic  Atkinson's T statistic.

•

rsquared  The coefficient of determination, often used to indicate how well data fits a model.

•

rsquaredadjusted  The adjusted coefficient of determination that accounts for the number of variables in the model.

•

tprobability  The pvalue for the hypothesis test for which the t value is the test statistic.

•

tvalue  The value of the t statistic for testing whether the corresponding regression coefficient is different than 0.




