Overview of Nonlinear Designs
Construct designs to fit models that are nonlinear in their parameters using the Nonlinear Design platform. You can construct optimal designs or optimally augment existing data for nonlinear models. Nonlinear designs based on information that is descriptive of the underlying process can yield more accurate estimates of model parameters and prediction of process behavior than is possible with standard designs for polynomial models. For background on nonlinear models, see “Nonlinear Models”.
The efficiency of a design for a nonlinear model depends on the unknown values of the parameters that the design is intended to estimate. For this reason, JMP uses a Bayesian approach to construct designs that are efficient over a wide range of likely parameter values. You can specify a range of values for the unknown parameters and a distribution for the prior. The prior distribution choices include Uniform, Normal, Lognormal, and Exponential.
The Nonlinear Design platform uses a Bayesian approach, optimizing the design over a prior distribution of likely parameter values that you specify. The Bayesian D-optimality criterion is the expectation of the logarithm of the determinant of the information matrix with respect to a sample of parameter vectors that represents this prior probability distribution. The information matrix entries depend on the prediction variances at the design points. Little information is contributed by observations with low variance, where the response is almost certain. It follows that an optimal design places some design settings at high-variance points. For more information, see Gotwalt et al. (2009).
A principal of optimal design is that, over the feasible region of experimentation, the optimal design places points in locations with the highest variance of prediction. Though this may seem counter-intuitive, if an alternative design put points at other locations, the prediction variance at the design points of the optimal design would be even higher. For models that are linear in the parameters, the high-variance points tend to be at the vertices of the experimental region. But this is not necessarily true for models that are nonlinear in the parameters.
Note: Nonlinear designs are computed using a random starting design. For this reason, nonlinear designs that you obtain for identical specifications usually differ.
To use the Nonlinear Design platform, you must have an existing data table. That data table must contain the following:
• A column for the response.
• A column for each factor.
• A column that contains a formula showing the relationship between the factors and the response. This formula must include the unknown parameters.
Note: This is the same format as is required for a data table used in the Nonlinear platform for modeling.
Your table can come in one of two forms:
• It might be a template, containing only column information and no rows. See “Create a Nonlinear Design with No Prior Data”.
• It might contain rows with predictor information. In this case, the predictor values are included in the nonlinear design. See “Augment a Design Using Prior Data”.
Examples of Nonlinear Designs
This section contains the following examples:
Create a Nonlinear Design with No Prior Data
This example shows how to create a design when you have not yet collected data, but have a guess for the unknown parameters. In this example, you model the fractional yield (Observed Yield) of an intermediate product in a chemical reaction. The fractional yield is a function of reaction time and temperature. See Box and Draper (1987).
Create the Design
1. Select Help > Sample Data Library and open Design Experiment/Reaction Kinetics Start.jmp.
Notice the following:
‒ The data table contains no rows because no data have been collected.
‒ The columns for the predictors, Reaction Temperature and Reaction Time, have the Coding, Design Role, and Factor Changes properties. To see these properties, click 
in the Columns panel. They tell JMP how to treat these predictors when constructing a design. For information about how to save these column properties, see the “Column Properties” chapter.
in the Columns panel. They tell JMP how to treat these predictors when constructing a design. For information about how to save these column properties, see the “Column Properties” chapter.‒ The Observed Yield column will contain response data obtained by running the experiment.
‒ The Yield Model column contains the formula that relates the predictors to the response, Observed Yield. Click 
in the Columns panel to see the formula. The formula is nonlinear in the parameters t1 and t3.
in the Columns panel to see the formula. The formula is nonlinear in the parameters t1 and t3.2. Select DOE > Special Purpose > Nonlinear Design.
3. Select Observed Yield and click Y, Response.
4. Select Yield Model and click X, Predictor Formula.
5. Click OK.
In this example, the values 510 and 540 for Reaction Temperature and 0.1 and 0.3 for Reaction Time were specified using the Coding column property. Alternatively, you can specify a reasonable range of values directly in the Factors outline.
6. Change the values of the parameter t1 to 25 and 50, and t3 to 30 and 35.
These new values represent a reasonable range of parameter values for the experimental situation. The default values were constructed based on the initial parameter values that were specified in the definition of the prediction formula. For information about constructing formulas, see the Formula Editor chapter in the Using JMP book.
Notice that the prior distribution, shown under Distribution, for each of t1 and t3 is set to Normal by default.
7. Change the number of runs to 12 in the Design Generation panel.
Figure 23.2 Completed Outlines for Reaction Kinetics Experiment
8. Click Make Design.
9. Click Make Table.
Figure 23.3 Design Table
Your design should be similar to the one shown in Figure 23.3. The runs might be in a different order, and the values for Reaction Temperature and Reaction Time, and consequently those computed for Yield Model, can be slightly different. Notice that values appear in the Yield Model column because the column contains the formula for the model. Also notice that the table contains a Model script that you can use to fit a nonlinear model to your observations.
Now that you have created your design table, run your experiment, and record the responses in the Observed Yield column. From this point on, work with the data table Reaction Kinetics.jmp, found in the Design Experiment folder.
Explore the Design
Before analyzing your results, construct a plot to see the design settings.
1. Select Help > Sample Data Library and open Design Experiment/Reaction Kinetics.jmp.
1. Select Graph > Graph Builder.
2. Drag and drop Reaction Temperature into the Y zone.
3. Drag and drop Reaction Time into the X zone.
4. Click the second icon above the graph to deselect the Smoother.
Figure 23.4 Design Settings
Notice that the points are located in three areas. There are no points at low temperature and high time (the lower right corner of the graph). Unlike orthogonal designs, nonlinear designs do not necessarily place design points at the corners of the design region. In this example, design points at low temperature and high time would be inefficient.
To see the density of design points in the remaining three corners, use the Contour tool.
5. Click 
to turn on the Contour tool.
to turn on the Contour tool.6. Click Done.
Figure 23.5 Design Settings with Density Contours
Notice that there are comparatively few points at low time and high temperature. From the design table, you can see that there are only three such points. Because of the model and the parameter specifications, the optimal design places more design points at high time and high temperature.
Analyze the Results
Now that you visually explored your design, analyze your results.
1. Select Analyze > Specialized Modeling > Nonlinear.
2. Select Observed Yield and click Y, Response.
3. Select Yield Model and click X, Predictor Formula.
Notice that the model appears in the Options for fitting custom formulas panel.
4. Click OK.
5. Click Go in the Control Panel.
The iterative search for a solution proceeds until one of the Stop Limit values is reached. Then, the Solution and Correlation of Estimates reports appear.
6. Click the Nonlinear Fit red triangle menu and select Profilers > Profiler.
7. To maximize the yield, click the Prediction Profiler red triangle menu and select Optimization and Desirability > Maximize Desirability.
Figure 23.6 Time and Temperature Settings for Maximum Yield
The estimated maximum yield is approximately 63.5% at a reaction temperature of 540 degrees Kelvin and a reaction time of 0.1945 minutes.
Augment a Design Using Prior Data
This example shows how to create a nonlinear design when you have prior data. In this example, the data pertain to a chemical reaction. You want to model the rate of uptake (velocity) of available organic substrate as a function of the concentration of that substrate. See Meyers (1986). You have already run an experiment, but you want to leverage your results to obtain more precise estimates of the parameters.
Obtain Prior Parameter Estimates
Use your existing experimental data to obtain better estimates of the parameter values.
1. Select Help > Sample Data Library and open Nonlinear Examples/Chemical Kinetics.jmp.
2. Click the plus sign next to Model (x) in the Columns panel.
3. Click Table Columns in the top left of the formula editor window and select Parameters.
The parameter values in the formula element panel (VMax = 1 and k = 1) are your initial guesses. They are used to compute the Model (x) values in the data table. For your next experiment, you want to replace these with better estimates.
4. Click Cancel to close the formula editor window.
5. Select Analyze > Specialized Modeling > Nonlinear.
6. Select Velocity (y) and click Y, Response.
7. Select Model (x) and click X, Predictor Formula.
Notice that the formula given by Model (x) appears in the Options for fitting custom formulas panel.
Figure 23.7 Nonlinear Analysis Launch Window
8. Click OK.
9. In the Control Panel, click Go.
The iterative search for a solution proceeds until one of the Stop Limit values is reached. Then, the Solution and Correlation of Estimates reports appear. Also, an option appears in the Control Panel enabling you to add confidence limits to the Solution report.
10. In the Control Panel, click Confidence Limits.
Confidence intervals for the parameters VMax and k appear in the Solution report.
Figure 23.8 Nonlinear Fit Results
The Lower CL and Upper CL values for VMax and k define ranges of values for these parameters. Next, use these intervals to define a range for the prior values in your augmented nonlinear design.
Note: Do not close the Nonlinear Fit report because these results are needed in the next steps.
Augment the Design
Now, create a design to estimate the nonlinear parameters more precisely.
1. With the Chemical Kinetics.jmp data table active, select DOE > Special Purpose > Nonlinear Design.
2. Select Velocity (y) and click Y, Response.
3. Select Model (x) and click X, Predictor Formula.
4. Click OK.
Figure 23.9 Nonlinear Design Outlines for Factors and Parameters
In the Chemical Kinetics.jmp data, the values for Concentration range from 0.417 to 6.25. Therefore, these values initially appear as the low and high values in the Factors outline. You want to change these values to encompass a broader interval.
5. Click 0.417 and type 0.1.
6. Press Tab over to 6.25 and type 7.
Leave the prior Distribution for each parameter set to Normal.
The range of Values for the parameters reflects the uncertainty of your knowledge about them. You should specify a range that you think covers 95% of possible parameter values. The confidence limits from the Nonlinear Fit report shown in Figure 23.8 provide such a range. Replace the Values for the parameters in the Parameters outline with the confidence limits, rounding to three decimal places.
7. In the DOE Nonlinear Design window, enter these values into the Parameters for VMax and k:
‒ VMax: 0.568 and 3.158
‒ k: 6.858 and 45.830
Figure 23.10 Updated Values for Factor and Parameters
8. Enter 40 for the Number of Runs in the Design Generation panel.
9. Click Make Design.
The Design outline opens, showing the Concentration and Velocity (y) values for the original 13 runs and new Concentration settings for the additional 27 runs.
10. Click Make Table.
This creates a new JMP design table that contains the settings and results for the original 13-run design and settings for 27 new runs. Instead of creating a new data table, you can add the new runs to your existing data table by clicking Augment Table instead of Make Table.
The new runs reflect the broader interval of Concentration values and the range of values for VMax and k obtained from the original experiment, which are used to define the prior distribution. Both should lead to more precise estimates of k and Vmax.
Create a Design for a Binomial Response
In some applications, the only measurement type available is a pass/fail (binomial) measurement. In this example, two factors are of interest, X1 and X2, which you will vary between -1 and 1. You will construct a nonlinear design for the binomial response and then view it in the context of your proposed nonlinear model.
Logistic Model
Model the probability of success for your binomial response (Y) using a logistic model:
This model is nonlinear in the unknown parameters β0, β1, and β2. Your goal is to estimate these parameters using an experimental design.
Prior Knowledge
To construct a design using Nonlinear Design, you need to specify your prior knowledge (or uncertainty) about each parameter value using a distribution. You can specify a best guess for each parameter value, but you have a lot of uncertainty relative to these values. Your best guess about the values of the parameters and 95% ranges for them are as follows:
• β0 is 0, but might range from -2 to 2
• β1 is 5, but might range from 0 to 10
• β2 is 5, but might range from 0 to 10
For parameter values in these ranges, the logistic function is nonlinear. So you expect that a design constructed using Nonlinear Design will differ from an orthogonal design. In particular, you expect that the nonlinear design will place factor settings at points where the predicted response has high variance.
Data Table for Launch Window
To construct a nonlinear design, you must first have a data table containing columns for the predictors and a column containing a formula that represents the nonlinear model that you are fitting. The Binomial Optimal Start.jmp data table, found in the Design Experiment folder, contains the following:
• Columns X1 and X2 for the two predictors. The Coding property defined for each of these columns causes the initial factor settings to be -1 and 1.
• A column for the response, Y.
• A column called Logistic Model that contains a formula relating the predictors to the response. To view the formula, click on the plus sign to the right of Logistic Model in the Columns panel. See Figure 23.11.
• Your initial guesses for the parameters b0, b1, and b2. When you defined these parameters, you were asked to specify a value. You set this value to your initial guess. These values are shown in the formula element panel at the top left of the formula editor window. See Figure 23.11.
• A column called Variance that contains the formula for the variance of the predicted value based on the assumed logistic model. When you construct your design, this column indicates which design points have comparatively high variances.
Figure 23.11 Formula Relating Predictors to Binomial Probability
Create the Design
1. Select Help > Sample Data Library and open Design Experiment/Binomial Optimal Start.jmp.
2. Select DOE > Special Purpose > Nonlinear Design.
3. Select Y and click Y, Response.
4. Select Logistic Model and click X, Predictor Formula.
5. Click OK.
Figure 23.12 Nonlinear Design Window
The Factors outline shows the two factors with the appropriate factors settings specified as values. The Parameters outline shows the three parameters with each prior distribution set to a normal distribution. JMP computes default Values based on your initial guesses for the parameter values. You are comfortable assuming a Normal prior, but your uncertainty about the parameters requires that you specify a wider range of values.
6. Enter the following under Values for the three parameters:
‒ b0: -2 and 2
‒ b1: 0 and 10
‒ b2: 0 and 10
Figure 23.13 Nonlinear Design Window with Parameter Values
The default number of runs, which you accept, is 14.
7. Click Make Design.
8. Click Augment Table.
This adds the 14 runs to Binomial Optimal Start.jmp. Your design table will be different because the optimization algorithm has a random component.
Figure 23.14 Augmentation of Binomial Optimal Start.jmp
Now that you have constructed your design, proceed to examine where the design points are located relative to the proposed logistic model. The Variance column gives the prediction variance at each design point, based on the logistic model.
View the Design
1. With Binomial Optimal Start.jmp active, select Graph > Graph Builder.
2. Select X1 and drag it to the X zone.
3. Select X2 and drag it to the Y zone.
4. De-select the Smoother, which is the second icon above the graph.
5. If you need to, drag each axis so that the -1.0 and 1.0 axis labels appear.
6. Click Done.
Because your design differs from the one in Figure 23.14, your plot will differ from the one in Figure 23.15.
Figure 23.15 Design Settings
Notice that there are no points at X1 = -1. The only point on a corner of the design region corresponds to X1 = 1 (more precisely, 0.996) and X2 = -1. There are several points in the central part of the design region.
To better see these points in relation to the model, construct a surface plot.
7. Select Graph > Surface Plot.
8. Select X1, X2, and Logistic Model and click Columns.
9. Click OK.
10. In the Dependent Variables outline, locate Logistic Model under Formula. In the Point Response Column Style list, click on none and select Logistic Model.
This adds points to the Surface Plot.
11. Right-click in the plot and select Settings.
12. Drag the Marker Size indicator to the right.
13. Click Done.
14. Rotate the plot to view the design points.
Because your design differs from the one in Figure 23.14, your plot will differ from the one in Figure 23.16.
Figure 23.16 Prediction Model with Design Points
Notice that many of the design points are in areas where the prediction model has a steep slope. These are high-variance points. Experimental results at these design points provide information to fit the model in areas where it is unstable. By way of contrast, an orthogonal design would place all design points at the corners of the design region. This would provide no information about nonlinear behavior within the design region.
Nonlinear Design Launch Window
To use the Nonlinear Design platform, you must have an existing data table that contains the following:
• A column for the response.
• A column for each factor.
• A column that contains a formula showing the relationship between the factors and the response. This formula must include the unknown parameters.
For information about formulas, see the Formula Editor chapter in Using JMP.
The table can contain values for the predictors and response. If it does, the design that you construct augments the design that is implicit in the table. There can be no row containing missing predictor values.
With your starting data table active, select DOE > Special Purpose > Nonlinear Design.
Figure 23.17 Nonlinear Launch Window
Y, Response
The numeric column for response values.
X, Predictor Formula
The numeric column that contains the formula for the nonlinear model. This formula must contain parameters.
Weight
(Optional) A numeric column that assigns weights to the observations.
Nonlinear Design Window
The Nonlinear Design window updates as you work through the design steps. The outlines, separated by buttons that update the outlines, follow the flow in Figure 23.18.
Figure 23.18 Nonlinear Design Flow
The initial design window shows the Factors, Parameters, and Design Generation outlines.
Figure 23.19 Initial Design Window
The next sections cover the following:
Factors
The column names used in the model formula are automatically inserted in the Name column of the Factors outline. Each factor’s role is set to Continuous.
For each factor, the Values are initially set to -1 and 1. Or, if you have defined a value using the Coding column property, those values are used instead. You can change these values in the factors outline.
Parameters
The parameter names used in the model formula are automatically inserted in the Name column of the Parameters outline.
For each parameter, the Values are initially set to a symmetric interval around the initial value specified in the parameter definition. This interval is obtained by taking the initial value’s distance to 0, and constructing an interval of this width around the initial value. These values are used in defining prior distributions for the model parameters.
Note: Adjust the Values for the parameters in conjunction with your choice of distribution to reflect your uncertainty about the model parameters.
Four families of prior distributions are listed under Distribution. The Values that you specify for the parameters determine which member of the family of prior distributions that is used. Denote the low value by low and the high value by high. Then the distributions are determined as follows:
• Uniform: The distribution is uniform on the interval (low, high).
• Normal, Lognormal, Exponential: The distribution is the one where low is the 0.025 quantile and where high is the 0.975 quantile.
Design Generation
JMP provides a suggested number of runs, determined as follows:
• If you are not augmenting a data table, the number of runs is four times the number of parameters plus two.
• If you are augmenting a data table, the number of runs is the number of runs in the data table plus two times the number of parameters.
Note: If you are augmenting a design, the number of runs that JMP suggests or that you specify includes those runs corresponding to observations in your data table. Adjust the number of runs appropriately.
Design
When you click Make Design, JMP constructs the design and adds a Design outline to the Nonlinear Design window. In the Design outline, you can review the factor level settings.
Make Table or Augment Table
The Make Table button creates a new design table. If your original table included existing runs, the new table also includes the existing runs. The Augment Table button adds the new runs to your existing table.
Nonlinear Design Options
The red triangle menu in the Nonlinear Design platform contains these options:
Save Responses
Saves the information in the Responses panel to a new data table. You can then quickly load the responses and their associated information into most DOE windows. This option is helpful if you anticipate reusing the responses.
Load Responses
Not available.
Save Factors
Saves the information in the Factors panel to a new data table. Each factor’s column contains its levels. Other information is stored as column properties. You can then quickly load the factors and their associated information into most DOE windows.
Load Factors
Not available.
Save Constraints
Not available.
Load Constraints
Not available.
Simulate Responses
Adds response values to the design table. Select this option before you click Make Table. Then, in the resulting design table, the response columns contain simulated values.
Note: To set a preference to always simulate responses, select File > Preferences > Platforms > DOE and select Simulate Responses.
Number of Starts
Sets the number of times that a nonlinear design is created using the quadrature method. Among the designs created, the platform selects the design that maximizes the optimality criterion.
Advanced Options > Number of Monte Carlo Samples
Sets the number of octahedra per sphere used in computing the optimality criterion. The default value is one octahedron. See “Radial-Spherical Integration of the Optimality Criterion”.
Advanced Options > N Monte Carlo Spheres
Sets the number of nonzero radius values used in computing the optimality criterion. The default is two. See “Radial-Spherical Integration of the Optimality Criterion”.
Note: If N Monte Carlo Spheres (the number of radii) is set to zero, then only the center point is used in the calculations. This gives a local design that is optimal for the initial values of the parameters. For some situations, this is adequate.
Statistical Details
This section contains the following information:
Nonlinear Models
Denote the vector of n responses by Y = (Y1, Y2, ..., Yn)‘. A nonlinear model is defined by the following properties:
• The Yi are independent and identically distributed with an exponential family distribution.
• The expected value of each Yi given a vector of predictor values xi is a nonlinear function of parameters, θ. Denote this function as follows:
• Each Yi is expressed as follows:
• The vector of errors, ε = (ε1, ε2, ..., εn)‘ has mean 0 and covariance matrix σ2I, where I is the n x n identity matrix.
Denote the matrix of first partial derivatives of the function f with respect to the parameters θ by X. Under general conditions, the least squares estimator of θ is asymptotically unbiased, with asymptotic covariance matrix given as follows:
For the proof of this result, see Wu (1981) and Jennrich (1969).
Radial-Spherical Integration of the Optimality Criterion
The optimality criterion is the expectation of the logarithm of the determinant of the information matrix with respect to the prior distribution. Consequently, finding an optional nonlinear design requires minimizing the integral of the log of the determinant of the Fisher information matrix with respect to the prior distribution of the parameters. This integral must be calculated numerically. The approach used in the Nonlinear Design platform is based on Gotwalt et al. (2009).
For normal distribution priors, the integral is reparameterized into a radial direction and a number of angular directions equal to the number of parameters minus one. The radial part of the integral is computed using Radau-Gauss-Laguerre quadrature with an evaluation at radius = 0. This is done by constructing a certain number of hyperoctahedra and randomly rotating each of them.
If the prior distribution is not normal, then the integral is reparameterized so that the new parameters have a normal distribution. Then the radial-spherical integration method is applied.
Note: If the prior distribution for the parameters does not lend itself to a solution and the process fails, a message is added to the window that the Fisher information matrix is singular in a region of the parameter space. When this occurs, consider changing the prior distribution or the ranges of the parameters.
Finding the Optimal Design
The method used to find an optimal design is similar to the coordinate exchange algorithm described in Meyer and Nachtsheim (1995). For details about how the nonlinear optimal design is obtained, see Gotwalt et al. (2009). The general approach proceeds as follows:
• Random designs are tested until a nonsingular starting design is found.
• Iterations are conducted, where each iteration consists of a pass through all the runs.
• For each run, factors are optimized one at a time.
• The objective function is the Bayesian D-optimality criterion. This is the expectation of the logarithm of the determinant of the information matrix with respect to the prior distribution.
• Iterations terminate once the change in the objective function is small.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.