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Research Papers

Using Support Vector Machines to Formalize the Valid Input Domain of Predictive Models in Systems Design Problems

[+] Author and Article Information
Richard J. Malak

Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843rmalak@tamu.edu

Christiaan J. J. Paredis

G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332chris.paredis@me.gatech.edu

J. Mech. Des 132(10), 101001 (Sep 30, 2010) (14 pages) doi:10.1115/1.4002151 History: Received October 02, 2009; Revised July 01, 2010; Published September 30, 2010; Online September 30, 2010

Predictive modeling can be a valuable tool for systems designers, allowing them to capture and reuse knowledge from a set of observed data related to their system. An important challenge associated with predictive modeling is that of describing the domain over which model predictions are valid. This is necessary to avoid extrapolating beyond the original data, particularly when designers use predictive models in concert with optimizers or other computational routines that search a model’s input space automatically. The general problem of domain description is complicated by the characteristics of observational data sets, which can contain small numbers of samples, can have nonlinear associations among the variables, can be nonconvex, and can occur in disjoint clusters. Support vector machine (SVM) techniques, developed originally in the machine learning community, offer a solution to this problem. This paper is a description of a kernel-based SVM approach that yields a formal mathematical description of the valid input domain of a predictive model. The approach also provides for cluster analysis, which can lead to improved model accuracy through the decomposition of a data set into multiple subsets that designers can model independently. This paper includes a mathematical presentation of kernel-based SVM methods, an explanation of the procedure for applying the approach to predictive modeling problems, and illustrative examples for applying and using the approach in systems design.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

An illustration of the one-dimensional case

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Figure 2

An illustration of two independent variables with a valid domain that occupies less than the rectangular region defined by their upper and lower bounds

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Figure 3

Two convex hull domain descriptions. In (a), the convex hull envelops the data somewhat tightly. In (b), a large region without any data falls within the domain description.

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Figure 4

A synthetic data set with internal voids and distinct clusters

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Figure 5

SVDD results for different settings of the Gaussian kernel width parameter, q. The regularization constant is held at C=0.4. Support vectors are indicated by boxes.

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Figure 6

Cluster membership associations. Points A, B, and C are in the same cluster; point D is in a different cluster.

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Figure 7

SVC on the data set from Fig. 4 for two different settings of the kernel width parameter

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Figure 8

A visualization of the adjacency information corresponding to the graphs in Fig. 4. Two distinct clusters are evident in (a). In (b), six distinct clusters are evident, including two that contain only one point each.

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Figure 9

SVDD execution time on different synthetic data sets with varying data set sizes and kernel width parameter values

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Figure 10

Number of support vectors identified using SVDD on different synthetic data sets with varying data set sizes and kernel width parameter values

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Figure 11

Number of support vectors per data point in each synthetic data set

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Figure 12

Execution time of SVC for a synthetic data set with varying numbers of data points and kernel width parameter values

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Figure 13

Visualization of engine data domain. Small dots are not part of the valid domain.

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Figure 14

Hydraulic log splitter: (a) physical layout, (b) functional configuration, where white boxes correspond to components associated with predictive models, and (c) circuit diagram

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