Class Objectives:

• Learn to use JMP's Nonlinear Fit platform and its most important options
  • To do nonlinear regression
  • To use loss functions other than squared residuals
  • To do maximum likelihood estimation
  • To fit models based on physical principles

Homework Assignments:

• Use JMP's Nonlinear Fit platform to fit the model:

to the following data:

Obtain likelihood ratio confidence intervals for these parameters.

Class Outline and Main Points:

• JMP's Nonlinear Fit platform: Review
  • Object function
  • Model function
  • Loss function
    • May depend on parameters that are not in the model function
    • May depend on variables that are not in the model function
    • Needs to have non-zero first and second order derivatives
      • Maximum likelihood estimation in the single sample case
    • Default loss function is the quadratic loss function
      • If in addition, the model function is linear we have ordinary least squares
  • Use of symbolic derivatives by JMP
• JMP's Nonlinear Fit platform: Optimization and numerical issues
  • Structure of the Hessian matrix for the platform's object function
  • When Second Derivative option is checked Newton-Rhason is used
  • When Second Derivative option is unchecked Gauss-Newton is used
  • For parameters in the loss function second derivatives are used even if the Second Derivative option is not checked.
• Techniques for getting good initial estimates for the parameters
  • Using graphical estimates
  • Use historical estimates
  • Try fitting the model using the square residual loss and use the estimates so obtained as initial values for fitting the model with a more complex loss function.
  • Throw censored data away in an initial model fit.
  • Linearize the model using a Taylor series expansion.
  • Don't use the second derivative option initially to fit the model even if you think you should. Use the estimates so obtained as initial estimates when using the Second Derivative option.
  • Remember to always hit Reset and Go as you refine estimates with the Nonlinear Fit Platform.

Example: Simple Regression Using JMP's Nonlinear Platform

To fix the ideas covered in this class we do a simple linear regression using both JMP's Fit Y by X and Nonlinear Fit platforms.

Consider the following data table:

We use the Fit Y by X platform:

We get the following estimates and confidence intervals for the regression parameters:

Now let's do the simple linear regression above using JMP's Nonlinear Fit platform.

The first step is to define a new column - named "Model" - using JMP's calculator:

We are ready to use JMP's Nonlinear Fit platform:

Notice that we don't have to specify a loss function since the default loss function is "squared residuals."

Notice that we do not need to check any options. The model is linear in the parameter so we don't need to check the Second Derivative option. (And even if the model had no been linear, the fact that we are using the "squared residual" loss would probably make checking this option unnecessary.)

Notice that we get the same answer we got using the Fit Y by X platform.

Study Questions

• Why is it important to optain good initial estimates of the parameter when using JMP's Nonlinear Fit platform?
• How do one obtain "good" initial extimates for the parameters?
• What it is frequently best not to check the second derivative option?
• Why is it that in reliability applications one frequently needs to derive the likelihood function of a model from first principles and then use a platform such as JMP Nonlinear Fit platform to maximize the likelihood and obtain confidence intervals?

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