The NonlinearFit command fits a model function that is nonlinear in the model parameters to data by minimizing the least-squares error.  If you are not sure if the model function is linear or not, it is recommended to use the Statistics[Fit] command, which will call this command or Statistics[LinearFit] depending on linearity of the model.

The NonlinearFit command minimizes the error in a local sense; see the Notes section below, in particular the advice regarding the initialvalues option.

This help page describes how to use the NonlinearFit command with algebraic-form and operator-form input.  An advanced form of the command is described in the Statistics/NonlinearFitMatrixForm help page. For more information about the input forms, see the Input Forms help page.

Consider the model $y=f\left(x_1\,x_2\,\mathrm{...}\,x_n;a_1\,a_2\,\mathrm{...}\,a_m\right)$, where y is the dependent variable and f is the model function of n independent variables $x_1,x_2,\mathrm{...},x_n$, and m model parameters $a_1,a_2,\mathrm{...},a_m$. Given k data points, where each data point is an (n+1)-tuple of numerical values for $\left(x_1\,x_2\,\mathrm{...}\,x_n\,y\right)$, the NonlinearFit command finds values of the model parameters such that the sum of the k residuals squared is minimized.  The ith residual is the value of $y-f\left(x_1\,x_2\,\mathrm{...}\,x_n;a_1\,a_1\,\mathrm{...}\,a_m\right)$ evaluated at the ith data point.

In the first two calling sequence, the first parameter falg is an algebraic expression in the independent variables $x_1,x_2,\mathrm{...},x_n$ and the model parameters $a_1,a_2,\mathrm{...},a_m$. In the last two calling sequences, the first parameter fop is a procedure having n input parameters representing the independent variables $x_1,x_2,\mathrm{...},x_n$ followed by m input parameters representing the model parameters $a_1,a_2,\mathrm{...},a_m$ and returning the single value $f\left(x_1\,x_2\,\mathrm{...}\,x_n;a_1\,a_2\,\mathrm{...}\,a_m\right)$.

The parameter X is a Matrix containing the values of the independent variables.  Row i in the Matrix contains the n values for the ith data point while column j contains all values of the single variable $x_j$.  If there is only one independent variable, X can be either a Vector or a k-by-1 Matrix.  The parameter Y is a Vector containing the k values of the dependent variable y. The parameter XY is a Matrix consisting of the n columns of X and, as last column, Y. For X, Y, and XY, one can also use lists or Arrays; for details, see the Input Forms help page.

The parameter v is a list of the independent variable names used in falg.  If there is only one independent variable, then v can be a single name.  The order of the names in the list must match exactly the order in which the independent variable values are placed in the columns of X.

By default, either the model function with the final parameter values or a Vector containing the parameter values is returned, depending on the input form.  Additional results or a solution module that allows you to query for various settings and results can be obtained with the output option.  For more information, see the Statistics/Regression/Solution help page.

Weights for the data points can be supplied through the weights option.

The options argument can contain one or more of the options shown below.  These options are described in more detail on the Statistics/Regression/Options help page.

initialvalues = set(equation), list(equation), list(realcons) or Vector(realcons) -- Provide initial values for the parameters.

output = name or string --  Specify the form of the solution.  The output option can take as a value the name solutionmodule, or one of the following names (or a list of these names): degreesoffreedom, leastsquaresfunction, parametervalues, parametervector, residuals, residualmeansquare, residualstandarddeviation, residualsumofsquares.  For more information, see the Statistics/Regression/Solution help page.

parameternames = list(name) -- Specify the order of parameter names in the model function.  This determines the order in which values are placed in Vector-valued results.

parameterranges = list(name=range), list(range) -- Specify the allowable range for each parameter.

weights = Vector -- Provide weights for the data points.

The underlying computation is done in floating-point; therefore, all data points must have type realcons and all returned solutions are floating-point, even if the problem is specified with exact values.  For more information about numeric computation in the Statistics package, see the Statistics/Computation help page.

The NonlinearFit command relies on the Matrix-form version of the Optimization[LSSolve] command, which in turns uses various methods implemented in a built-in library provided by the Numerical Algorithms Group (NAG).  More information is available on the Optimization[LSSolveMatrixForm] help page.  Additional options listed on that page may be provided to the NonlinearFit command and are passed directly to the LSSolve command.

The Optimization[LSSolve] command computes only local solutions to nonlinear least-squares problems.  The parameter values returned by NonlinearFit minimize the sum of the residuals squared in a local sense.  When the results returned are unexpected, it is highly recommended that you provide initial values for the parameters using the initialvalues option.

To obtain more details as the least-squares problem is being solved, set infolevel[Statistics] to 2 or higher.  To obtain details about the progress of the Optimization solver, set ${\mathrm{infolevel}}_{\mathrm{Optimization}}$ to 1 or higher.

For fitting a data sample to a distribution, see Statistics[MaximumLikelihoodEstimate].

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{Vector}\left(\left[1\,2\,3\,4\,5\,6\right]\,\mathrm{datatype}=\mathrm{float}\right)\:$

$Y≔\mathrm{Vector}\left(\left[2.2\,3\,4.8\,10.2\,24.5\,75.0\right]\,\mathrm{datatype}=\mathrm{float}\right)\:$

$\mathrm{NonlinearFit}\left(a+bv+\exp \left(cv\right)\,X\,Y\,v\right)$

$6.49670407384335-4.54643947753519v+{\ⅇ}^{0.758384141232909v}$

$\mathrm{XY}≔\mathrm{Matrix}\left(\left[\left[2.2467\,5.2219\,6.5622\right]\,\left[2.0083\,6.0656\,6.3261\right]\,\left[5.8386\,1.1084\,11.942\right]\,\left[7.7071\,5.9855\,32.096\right]\,\left[4.6193\,4.6921\,15.297\right]\right]\right)$

$\mathrm{XY}≔\left[\begin{array}{ccc}2.2467& 5.2219& 6.5622\\ 2.0083& 6.0656& 6.3261\\ 5.8386& 1.1084& 11.942\\ 7.7071& 5.9855& 32.096\\ 4.6193& 4.6921& 15.297\end{array}\right]$

$\mathrm{NonlinearFit}\left(a{x}^b{y}^c\,\mathrm{XY}\,\left[x\,y\right]\right)$

$1.28883204859941x^{1.23745132612891}{y}^{0.383635147679040}$

$\mathrm{ExperimentalData}≔⟨⟨1\,1\,1\,2\,2\,2⟩\left|⟨1\,2\,3\,1\,2\,3⟩\right|⟨1\,2\,3\,4\,5\,6⟩|⟨0.531\,0.341\,0.163\,0.641\,0.713\,-0.040⟩⟩$

$\mathrm{ExperimentalData}≔\left[\begin{array}{cccc}1& 1& 1& 0.531\\ 1& 2& 2& 0.341\\ 1& 3& 3& 0.163\\ 2& 1& 4& 0.641\\ 2& 2& 5& 0.713\\ 2& 3& 6& \mathrm{-0.040}\end{array}\right]$

$\mathrm{NonlinearFit}\left(x^a+\frac{b{x}^2}{y}+cyz\,\mathrm{ExperimentalData}\,\left[x\,y\,z\right]\,\mathrm{initialvalues}=\left[a=2\,b=1\,c=0\right]\,\mathrm{output}=\left[\mathrm{leastsquaresfunction}\,\mathrm{residuals}\right]\right)$

$\left[x^{1.14701973996968}-\frac{0.298041864889394x^2}{y}-0.0982511893429762yz\,\left[\begin{array}{cccccc}0.0727069457676300& 0.116974310183398& \mathrm{-0.146607992383251}& \mathrm{-0.0116127470057686}& \mathrm{-0.0770361532848388}& 0.0886489085642805\end{array}\right]\right]$

$\mathrm{Input}≔\left[\mathrm{seq}\left(0.1..1\,0.1\right)\right]$

$\mathrm{Input}≔\left[0.1\,0.2\,0.3\,0.4\,0.5\,0.6\,0.7\,0.8\,0.9\,1.0\right]$

$\mathrm{Output}≔\left[1.932\,2.092\,2.090\,2.416\,2.544\,2.638\,2.894\,3.188\,3.533\,3.822\right]$

$\mathrm{Output}≔\left[1.932\,2.092\,2.090\,2.416\,2.544\,2.638\,2.894\,3.188\,3.533\,3.822\right]$

$\mathrm{ODE}≔\left[x\left(0\right)=-a\,\mathrm{diff}\left(x\left(t\right)\,t\right)=z{x\left(t\right)}^{-b}+1\right]$

$\mathrm{ODE}≔\left[x\left(0\right)=-a\,\frac{\ⅆ}{\ⅆt}x\left(t\right)=z{x\left(t\right)}^{-b}+1\right]$

$\mathrm{ODE\_Solution}≔\mathrm{dsolve}\left(\mathrm{ODE}\,\mathrm{numeric}\,\mathrm{parameters}=\left[a\,b\,z\right]\right)$

$\mathrm{ODE\_Solution}≔\mathbf{proc}\left(\mathrm{x\_rkf45}\right)...\mathbf{end\; proc}$

$\mathrm{ODE\_Solution}\left(\mathrm{parameters}=\left[a=-1\,b=-0.5\,z=1\right]\right)$

$\left[a=\mathrm{-1.}\,b=\mathrm{-0.5}\,z=1.\right]$

$\mathrm{ODE\_Solution}\left(1\right)$

$\left[t=1.\,x\left(t\right)=3.44630585135012\right]$

$\mathrm{ODE\_Solution}\left(\mathrm{parameters}=\left[a=1\,b=1\,z=1\right]\right)$

$\left[a=1.\,b=1.\,z=1.\right]$

$\mathrm{ODE\_Solution}\left(1\right)$

Error, (in ODE_Solution) cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up

f := proc(zValue, aValue, bValue) global ODE_Solution, a, b, z, x, t; ODE_Solution('parameters' = [a = aValue, b = bValue, z = zValue]); try return eval(x(t), ODE_Solution(1)); catch: return 100; end try; end proc;

$f≔\mathbf{proc}\left(\mathrm{zValue}\,\mathrm{aValue}\,\mathrm{bValue}\right)\mathbf{global}\mathrm{ODE\_Solution}\,a\,b\,z\,x\,t\;\mathrm{ODE\_Solution}\left(\'\mathrm{parameters}\'\=\left[a\=\mathrm{aValue}\,b\=\mathrm{bValue}\,z\=\mathrm{zValue}\right]\right)\;\mathbf{try}\mathbf{return}\mathrm{eval}\left(x\left(t\right)\,\mathrm{ODE\_Solution}\left(1\right)\right)\mathbf{catch}\:\mathbf{return}100\mathbf{end\; try}\mathbf{end\; proc}$

$f\left(1\,-1\,-0.5\right)$

$3.44630585135012$

$\mathrm{NonlinearFit}\left(f\,\mathrm{Input}\,\mathrm{Output}\,\mathrm{output}=\left[\mathrm{parametervector}\,\mathrm{residualstandarddeviation}\right]\right)$

$\left[\left[\begin{array}{c}1.\\ 1.00000002108371\end{array}\right]\,108.7701554\right]$

$\mathrm{NonlinearFit}\left(f\,\mathrm{Input}\,\mathrm{Output}\,\mathrm{output}=\left[\mathrm{parametervector}\,\mathrm{residualstandarddeviation}\right]\,\mathrm{initialvalues}=\left[-1\,-0.5\right]\right)$

$\left[\left[\begin{array}{c}\mathrm{-0.739350291476489}\\ \mathrm{-1.03096957779941}\end{array}\right]\,0.07109672265\right]$

The XY parameter was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.