JMP's "Profiler" function allows you to visualize a predictive model and interactively examine the change in predicted value of the response (target variable) when factors (explanatory variables) are changed. It also has an "Optimization" function that finds the factor values that maximize or minimize the response.
However, if you examine the factor values unlimitedly, you may end up with settings that deviate significantly from the actual data and are impossible in reality, which is a risk of predictions called " extrapolation ." Therefore, when optimizing, you need to be careful because it may be possible to obtain predicted values using factor settings that are impossible in reality.
In the JMP Profiler, you can turn on Suppress Extrapolation to avoid unrealistic factor settings . In the following example, we compare the results with and without the Suppress Extrapolation feature.
Example of suppressing extrapolation
This time, we will take up an example of predicting the dissolution rate (Y) in a pharmaceutical manufacturing process. We will create a response surface model using process variables (X1, X2, X3) and perform optimization.
Data source: Pharmaceutical Quality by Design Using JMP®: Solving Product Development and Manufacturing Problems
We will fit a response surface model using JMP's "Fit Model" and optimize using the "Maximize Desirability" function in the "Prediction Profiler." We will compare the results when not suppressing extrapolation (OFF) and when suppressing it (ON).
Suppress Extrapolation Off
The default setting allows extrapolation, which finds factor settings that maximize the response (Y, dissolution rate), but may result in factor settings that are far removed from the original data.
Suppress Extrapolation On
Click the red triangle button to the left of the Prediction Profiler report and select Suppress Extrapolation > On before running the optimization. You can see that the factor settings obtained in this case are different from those obtained when extrapolation is turned off.
Let's compare the optimum values with extrapolation suppression turned off and on in a graph.
For three factors (X1, X2, X3), the optimal values when suppressing extrapolation are turned off and on are compared using a three-dimensional scatter plot and a scatter plot matrix. The red points are the optimal values when suppressing extrapolation are turned off, and the blue points are the optimal values when suppressing extrapolation are turned on.
When suppression of extrapolation is turned off, the optimum is located far away from the data, but when it is turned on, the optimum is located within the set of original data, confirming that extrapolation is being suppressed.
Do you see the results of suppressing extrapolation?
What does "suppression of extrapolation" mean?
So what does this suppression of extrapolation do?
In a nutshell:
"The maximum value of leverage in the data is used as a threshold, and factors with leverage below that threshold are considered."
Leverage in data is a measure of how much influence a point has on the model you have created. Points with high leverage have a strong influence on the response surface model in this example, and are also a way to find outliers and abnormal values.
In JMP, you can save the leverage values in the data table by selecting Save Columns > Hat from the red triangle in the Fit Model report.
The distribution of stored leverage values shows a maximum value of 0.587, which is the threshold for suppressing extrapolation.
To suppress extrapolation, you can display the area in the red frame in the figure below in the [Extrapolation Details] option. In the following figures, we will display "data points" on the profile for reference.
"Threshold" (=0.587) is the maximum leverage value found earlier. "Index" is the leverage value for the current factor settings (X1=16.51, X2=253.3, X3=386.67). The value at this time is 0.478, which does not exceed the threshold.
Let's change the factor settings and specify the data point (X1=40, X2=239, X3=370) where the leverage is at its maximum (=0.587). In this case, the index value and the threshold value match. This is the value at the very limit of the threshold.
As the curve displayed in the profiler is broken, factors with leverage exceeding the threshold cannot be set. Therefore, optimization is performed within the range that does not exceed the threshold.
When the Fit Model method is set to Standard Least Squares (which is the case for response surface models), the default extrapolation threshold is
K × maximum leverage (K = 1)
Since K=1, the "maximum leverage" is the threshold, as mentioned above. This K can be changed using the [Set threshold criteria] option.
You can adjust the strength of the extrapolation suppression by changing the value of K. For example, increasing K to 2 will expand the range of extrapolation, while decreasing K to 0.5 will further restrict extrapolation.
Note : In JMP Pro,In addition to the least squares method, there are also features for suppressing extrapolation in the profiles of neural networks, support vector machines, generalized regression, and the Profiler in the Graph menu. In these cases, regularized T^2 is used as the threshold criterion.
by Naohiro Masukawa (JMP Japan)
Naohiro Masukawa - JMP User Community
