Specialized Models • Nonlinear Regression with Custom Models • Example of Fitting a Custom Model
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Example of Fitting a Custom Model
To fit a custom model, you must first create a model column with initial parameter estimates. This method does require a few more steps than fitting a built-in model, but it does allow any nonlinear model to be fit. Also, you can provide a custom loss function, and specify several other options for the fitting process.
This section provides an example of creating the formula column for a model, and fitting the model in the Nonlinear platform. The data is in the US Population.jmp data table. The response variable is the population (in millions) of the Unites States and the predictor is the year.
1.
Select Help > Sample Data Library and open Nonlinear Examples/US Population.jmp.
2.
Create a new column called Model.
3.
Right-click the Model column and select Column Properties > Formula.
The Formula Editor appears.
4.
Above the list of columns on the left, select Parameters.
5.
Select New Parameter.
6.
Use the default name of b0.
7.
Type 4 for Value. This is the initial estimate of the parameter.
8.
Click OK.
9.
Select New Parameter.
10.
Keep the default name and enter 0.02 for Value.
11.
Click OK.
12.
Enter the model formula using the Formula Editor functions, the column year, and the parameters. Completed Model Formula shows the completed model.
Completed Model Formula
13.
Click OK.
14.
Select Analyze > Modeling > Nonlinear.
15.
Assign Model to the X, Predictor Formula role.
16.
Assign pop to the Y, Response role.
17.
Click OK.
18.
Click Go on the Control Panel to fit the model.
A portion of the report is shown in Plot and Solution Report.
Plot and Solution Report
The final parameter estimates are shown in the Solution report, along with other fit statistics. The fitted model is shown on the plot.
Parameters for Models with a Grouping Variable
In the formula editor, when you add a parameter, note the check box for Expand Into Categories, selecting column. This option is used to add several parameters (one for each level of a categorical variable for example) at once. When you select this option a dialog appears that enables you to select a column. After selection, a new parameter appears in the Parameters list with the name D_column, where D is the name that you gave the parameter. When you use this parameter in the formula, a Match expression is inserted, containing a separate parameter for each level of the grouping variable.
Launch the Nonlinear Platform
To launch the Nonlinear platform, select Analyze > Modeling > Nonlinear. The launch window is shown in Nonlinear Platform Launch Window.
Nonlinear Platform Launch Window
The Nonlinear platform launch window has the following features:
Y, Response
Select the Y variable.
X, Predictor Formula
Select either the X variable or a column containing the model formula with parameters.
Group
Specify a grouping variable. The fitted model has separate parameters for each level of the grouping variable. This enables you to compare fitted models and estimated parameters across the levels of the grouping variable.
Weight
Specify a variable containing the weights of observations.
Freq
Specify a variable representing the frequency of an observation.
Loss
Specify a formula column giving a loss function.
By
Specify a variable to perform a separate analysis for every level of the variable.
Model Library
Launches the Model Library tool, which helps you choose initial values to create a formula column. See Create a Formula Using the Model Library.
Second Derivatives
Uses second derivatives as well as first derivatives in the iterative method to find a solution. With second derivatives, the method is called Newton-Raphson rather than Gauss-Newton. This method is useful only if the residuals are unusually large or if you specify a custom loss function and your model is not linear in its parameters. This option is used only when a formula column is provided in the X, Predictor Formula role.
Numeric Derivatives Only
Uses only numeric derivatives only. This option is useful when you have a model for which it is too messy to take analytic derivatives. It can also be valuable in obtaining convergence in tough cases. This option is used only when a formula column is provided in the X, Predictor Formula role.
Expand Intermediate Formulas
Tells JMP that if an ingredient column to the model is a column that itself has a formula, to substitute the inner formula, as long as it refers to other columns. To prevent an ingredient column from expanding, use the Other column property with a name of “Expand Formula” and a value of 0. This option is used only when a formula column is provided in the X, Predictor Formula role.