HS3000: Fit Method Comparison  Approximation on the Arm Model
Learn how to create approximations for the output responses of the arm example introduced in tutorial, and review the differences between different Fit methods.
In HS2000: DOE Method Comparison: Arm Model Study, you learned that instead of using the nine input variables, you could continue additional studies just as effectively with six shapes since the others did not have a great influence on the output responses. This will save computational effort.
 Length1: Lower Bound = 0.5, Initial Bound = 0.0, Upper Bound = 2.0
 Length2: Lower Bound = 0.0, Initial Bound = 0.0, Upper Bound = 2.0
 Length3: Lower Bound = 1.0, Initial Bound = 0.0, Upper Bound = 1.0
 Length4: Lower Bound = 1.0, Initial Bound = 0.0, Upper Bound = 1.0
 Length5: Lower Bound = 1.0, Initial Bound = 0.0, Upper Bound = 1.0
 Height: Lower Bound = 1.0, Initial Bound = 0.0, Upper Bound = 1.0
Run MELS DOE Study
In this step you will create a Modified Extensible Lattice Sequence (MELS) DOE. You will use this matrix to create the Fits for both output responses.
MELS is a space filling DOE designed to equally spread out points in a space by minimizing clumps and empty spaces. The minimal required number of points to create a second order polynomial with N variables is 1.1*(N + 1)*(N + 2)/2.
In order to create the approximations to be used as surrogate models, you must perform specific DOEs that will serve as the input matrix. You will need to run a DOE suitable to be used in response surface creation, such as MELS.

Add a DOE.

Modify input variables.
 Go to the step.
 In the work area, Active column, clear the radius_1, radius_2, and radius_3 checkboxes.

Define specifications.
 Go to the step.
 In the work area, set the Mode to Modified Extensible Lattice Sequence (MELS).
 In the Settings tab, verify that the Number of Runs is set to 31.
 Click Apply.

Evaluate tasks.
 Go to the step.
 Click Evaluate Tasks.
 Go to the step.

Click the Scatter tab to review a 2D scatter plot of the results from the MELS
DOE.
Optional: Run DOE with Less Runs
In this step you will create an optional second DOE with less number of runs to be used as a Validation matrix in the Fit approach.
In this tutorial, you will use the Hammersley method to create the Validation matrix.

Add a DOE.

Define specifications.

Evaluate tasks.
 Go to the step.
 Click Evaluate Tasks.
Run Fits
In this step you will use the 31 runs from the MELS DOE as an Input matrix and the 12 runs from the Hammersley DOE as a Validation matrix to create four Fits using Least Square Regression (LSR), Moving Least Square Method (MLSM), HyperKriging (HK), and Radial Basis Function (RBF).

Add a Fit.
 In the Explorer, rightclick and select Add from the context menu.
 In the Add dialog, select Fit Existing Data and Setup, and click OK.

Import matrix.
 Go to the step.
 Click Add Matrix twice.
 In the work area, define Fit Matrix 1 and Fit Matrix 2 by selecting the options indicated in Figure 4.
 Click Apply.

Define specifications.

Evaluate tasks.
 Go to the step.
 Click Evaluate Tasks.
 Go to the step.

Click the Scatter tab to compare the original Max_Stress
output response to the Fit Max_Stress.
The scatter shows the Fit accuracy. The closer together the points are along the diagonal, the better the fit. In the Max_Stress vs Max_Stress_LSR plot, you can see some dispersed points, which indicates the Fit has some inaccuracy. In comparison, the points in the Max_Stress vs Max_Stress_MLSM plot follow the diagonal more closely, which indicates it provides better Fit accuracy on Max_Stress.
You will not compare HyperKriging and Radial Basis Function using scatter plots, because the results will be misleading. HyperKriging and Radial Basis Function go through the exact points by default, therefore the scatter plot comparing the original output response vs. the Fit output response will produce a straight line. However, this does not necessarily mean that the Fit has good predictive capability. 
Click the Diagnostics tab to review the diagnostics of
the Fit study.
The RSquare value measures how much of the variability of the response data around its mean is captured. If the model perfectly predicts the known values, RSquare will have a maximum possible value of 1.0.The Rsquare value for an Input Matrix in HyperKriging and Radial Basis Function has no meaning because the runs will always go through the exact data points, which will result in a value of 1.0. Although the value is 1.0, this does not mean the Fit will be accurate. In HyperKriging and Radial Basis Function, the only meaningful diagnostic values are for Crossvalidation Matrix and Validation Matrix.

Click the Residuals tab to review the Error (and Percent
Error) between the original output response and the Fit output response for each
run of the Input and Testing matrices.
The Input Matrix Residual errors are slightly smaller with Least Square Regression, than they are with Moving Least Square Method, but the Testing Matrix Residual errors are much smaller with Moving Least Square Method.The Input Matrix Residuals are meaningless for HyperKriging and Radial Basis Function, as indicated in the Testing Matrix Residuals below.
Fit Comparison
Overview of the max percent of errors for Input and Testing matrices.
LSR (Interaction Regression Model)  MLSM  HK  RBF  

Max_Disp  1.18%  2.79%     
Max_Stress  6.72%  10.92%     
LSR (Interaction Regression Model)  MLSM  HK  RBF  

Max_Disp  7.26%  3.00%  9.57%  2.45% 
Max_Stress  34.74%  19.72%  35.06%  17.99% 
It can be seen that the percent of errors for Max_Disp are smaller than Max_Stress. These results indicate the Fit approach works well for Max_Disp, but is not very efficient for Max_Stress.