Example of a Poisson Loss Function
Use the Nonlinear platform to fit a Poisson model to count data. The Poisson distribution has the following parameterization:
where μ can be a single parameter, or a linear model with many parameters. Many texts and papers show how the model can be transformed and fit with iteratively reweighted least squares (Nelder and Wedderburn 1972). However, in JMP it is more straightforward to fit the model directly.
1. Select Help > Sample Data Folder and open Ship Damage.jmp.
2. Select Analyze > Specialized Modeling > Nonlinear.
3. Assign model to the X, Predictor Formula role.
4. Assign Poisson to the Loss role.
5. Click OK.
6. Set the Current Value (initial value) for b0 to 1, and the other parameters to 0.
Figure 15.18 Enter New Parameters
7. Click Go.
8. Click the Confidence Limits button.
The Solution report appears. The results include the parameter estimates and confidence intervals, and other summary statistics.
Figure 15.19 Solution Table for the Poisson Loss Example
Note: The standard errors, confidence intervals, and hypothesis tests are correct only if least squares estimation is done, or if maximum likelihood estimation is used with a proper negative log-likelihood.