HS-3000: Fit Method Comparison - Approximation on the Arm Model

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.

1. Add a DOE.
   1. In the Explorer, right-click and select Add from the context menu.
      The Add dialog opens.
   2. For Definition from, select an approach.
   3. Select DOE and click OK.
2. Modify input variables.
   1. Go to the Setup > Definition > Define Input Variables step.
   2. In the work area, Active column, clear the radius_1, radius_2, and radius_3 checkboxes.

   
   
   Figure 1.

3. Define specifications.
   1. Go to the DOE 5 > Specifications step.
   2. In the work area, set the Mode to Modified Extensible Lattice Sequence (MELS).
   3. In the Settings tab, verify that the Number of Runs is set to 31.
   4. Click Apply.
4. Evaluate tasks.
   1. Go to the DOE 5 > Evaluate step.
   2. Click Evaluate Tasks.
5. Go to the DOE 5 > Post-Processing step.
6. Click the Scatter tab to review a 2D scatter plot of the results from the MELS DOE.

   
   
   Figure 2. Typical Sampling of the MELS DOE with 31 Runs (length_1 vs. length_2). This visualization is a projection of 31 points distributed in 6 dimensions onto a 2 dimensional plane.

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.

1. Add a DOE.
   1. In the Explorer, right-click and select Add from the context menu.
      The Add dialog opens.
   2. For Definition from, select DOE 5.
   3. Select DOE.
   4. Click OK.
2. Define specifications.
   1. Go to the DOE 6 > Specifications step.
   2. In the work area, set the Mode to Hammersley.
   3. In the Settings tab, change the Number of Runs to 12.

      
      
      Figure 3.

   4. Click Apply.
3. Evaluate tasks.
   1. Go to the DOE 6 > Evaluate step.
   2. Click Evaluate Tasks.

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).

1. Add a Fit.
   1. In the Explorer, right-click and select Add from the context menu.
   2. In the Add dialog, select Fit and click OK.
2. Import matrix.
   1. Go to the Fit > Specifications step.
   2. Click Add Matrix twice.
   3. In the work area, define Fit Matrix 1 and Fit Matrix 2 by selecting the options indicated in Figure 4.
   4. Click Apply.

   
   
   Figure 4.

3. Define specifications.
   1. In the work area, Fit Type column, select the appropriate Fit method.
      Important:

      For the Least Sqaure Regressions (LSR) Fit, in the Settings tab, set Regression Model to Interaction.

      An Interaction regression model enables linear and cross terms to be considered in the function f(x,y)=A+Bx+Cy+Dxy; where the first three terms are linear, and the last term is a cross term between the variables.

   2. Click Apply.
4. Evaluate tasks.
   1. Go to the Fit > Evaluate step.
   2. Click Evaluate Tasks.
5. Go to the Fit > Post-Processing step.
6. 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.

   
   
   Figure 5. Max_stress and Max_stress, LSR Comparison

   
   
   Figure 6. Max_stress and Max_stress, MLSM Comparison

7. Click the Diagnostics tab to review the diagnostics of the Fit study.
   The R-Square value measures how much of the variability of the response data around its mean is captured. If the model perfectly predicts the known values, R-Square will have a maximum possible value of 1.0.

   
   
   Figure 7. Diagnostics for Max_Stress, LSR

   
   
   Figure 8. Diagnostics for Max_Stress, MLSM

   The R-square 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 Cross-validation Matrix and Validation Matrix.

   
   
   Figure 9. Diagnostics for Max_Stress, HK

   
   
   Figure 10. Diagnostics for Max_Stress, RBF

8. 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.

   
   
   Figure 11. Input Matrix Residuals on Max_Stress, LSR

   
   
   Figure 12. Testing Matrix Residuals on Max_Stress, LSR

   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.

   
   
   Figure 13. Input Matrix Residuals on Max_Stress, MLSM

   
   
   Figure 14. Testing Matrix Residuals on Max_Stress, MLSM

   The Input Matrix Residuals are meaningless for HyperKriging and Radial Basis Function, as indicated in the Testing Matrix Residuals below.

   
   
   Figure 15. Testing Matrix Residuals on Max_Stress, HK

   
   
   Figure 16. Testing Matrix Residuals on Max_Stress, RBF