A number of commands are available for fitting a model function that is linear in the model parameters to given data.  For example, the model function $b{t}^2+at$ is linear in the parameters a and b, though it is nonlinear in the independent variable t.

The LinearFit command is available for multiple general linear regression.  For certain classes of model functions involving only one independent variable, the PolynomialFit, LogarithmicFit, PowerFit, and ExponentialFit commands are available. The PowerFit and ExponentialFit commands use a transformed model function that is linear in the parameters.

The NonlinearFit command is available for nonlinear fitting.  An example model function is $ax+{\ⅇ}^{by}$ where a and b are the parameters, and x and y are the independent variables.

This command relies on local nonlinear optimization solvers available in the Optimization package.  The LSSolve and NLPSolve commands in that package can also be used directly for least-squares and general nonlinear minimization.

The general Fit command allows you to provide either a linear or nonlinear model function.  It then determines the appropriate regression solver to use.

The OneWayANOVA command generates the standard ANOVA table for one-way classification, given two or more groups of observations.

Various options can be provided to the regression commands. For example, the weights option allows you to specify weights for the data points and the output option allows you to control the format of the results.  The options available for each command are described briefly in the command's help page and in greater detail in the Statistics/Regression/Options help page.

The format of the solutions returned by the regression commands is described in the Statistics/Regression/Solution help page.

Most of the regression commands use methods implemented in a built-in library provided by the Numerical Algorithms Group (NAG).  The underlying computation is done in floating-point.  Either hardware or software (arbitrary precision) floating-point computation can be specified.

The model function and data sets may be provided in different ways.  Full details are available in the Statistics/Regression/InputForms help page. The regression routines work primarily with Vectors and Matrices.  In most cases, lists (both flat and nested) and Arrays are also accepted and automatically converted to Vectors or Matrices.  Consequently, all output, including error messages, uses these data types.

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

$X≔\mathrm{Vector}\left(\left[1.2\,2.1\,3.1\,4.0\,5.7\,6.6\,7.2\,7.9\,9.1\,10.3\right]\right)\:$

$Y≔\mathrm{Vector}\left(\left[4.6\,7.7\,11.5\,15.4\,22.2\,33.1\,48.1\,70.6\,109.0\,168.4\right]\right)\:$

$\mathrm{Fit}\left(ax+b\exp \left(x\right)\,X\,Y\,x\right)$

$6.02861839712210x+0.00380375570529786{\ⅇ}^x$

$\mathrm{Fit}\left(ax+b\exp \left(x\right)\,X\,Y\,x\,\mathrm{summarize}=\mathrm{embed}\right)$

$6.02861839712210x+0.00380375570529786{\ⅇ}^x$

$\mathrm{PolynomialFit}\left(3\,X\,Y\,x\right)$

$-3.37372868459017+9.90059487215674x-2.79612412098216x^2+0.336249676048196x^3$

$\mathrm{PolynomialFit}\left(3\,X\,Y\,x\,\mathrm{output}=\left[\mathrm{residualsumofsquares}\,\mathrm{standarderrors}\right]\right)$

$\left[47.8471318673565\,\left[\right]\right]$

$\mathrm{NonlinearFit}\left(ax+\exp \left(bx\right)\,X\,Y\,x\right)$

$2.12883148575966x+{\ⅇ}^{0.486510105685615x}$

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

$\mathrm{ExperimentalData}≔\left[\right]$

$\mathrm{infolevel}\left[\mathrm{Statistics}\right]≔2\:$

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

In NonlinearFit (algebraic form)

$\left[x^{1.14701973996968}-\frac{0.298041864889394x^2}{y}-0.0982511893429762yz\,\left[\right]\right]$

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

In Fit

In LinearFit (container form)

final value of residual sum of squares: .0537598869493245

Summary:
----------------
Model: .82307292*x-.16791011*x^2/y-.75802268e-1*y*z
----------------
Coefficients:
    Estimate  Std. Error  t-value  P(>|t|)
a    0.8231    0.1898      4.3374   0.0226
b   -0.1679    0.0940     -1.7862   0.1720
c   -0.0758    0.0182     -4.1541   0.0254
----------------
R-squared: 0.9600, Adjusted R-squared: 0.9201

$\left[0.823072918385878x-\frac{0.167910114211606x^2}{y}-0.0758022678386438yz\,\left[\right]\right]$

$\mathrm{infolevel}\left[\mathrm{Statistics}\right]≔0\:$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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;

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

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

$3.44630585135012$

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

Draper, Norman R., and Smith, Harry. Applied Regression Analysis. 3rd ed. New York: Wiley, 1998.