Advanced Model Based Process Engineering Tools

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SC in data mining and process engineering
SC in modeling and control
Advanced Model Based Process Engineering Tools
www.fmt.vein.hu/softcomp
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System Identification using Delaunay
Tessellation of Self-Organizing Maps
János Abonyi and Szeifert Ferenc

• Introduction of our laboratory
• Introduction to SOM
• Why Delaunay Tessellation
• Example pH process
• Conclusions

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Steps of Data Mining
Database
Data warehouse
Data Mining
Model
How soft-computing can help ???
www.fmt.vein.hu/softcomp
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Tasks of Data Mining

• Classification
• Change and Deviation Detection
• Dependency Modelling
• Clustering (prototypes, codebook, signatures,
  prob. density estimation )
• Summation (inc. Visualisation, Feature
  extraction)
• Regression and time-series analysis

Simultaneously by one tool
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Clustering

• Detect groups of data
• Hierarchical (dendograms) or not
• Prototypes (signatures) are based on a similarity
  measure (distance)
• (semi)-supervised or unsupervised
• Can be fuzzy !!!

x2
x1
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Feature Extraction

• (Nonlinear) mapping of the input space into a
  lower dimensional one
• Reduction of the number of inputs
• Useful for visualisation Non-parametric (Sammon
  projection) or Model-based (principal curves,
  NN, Gaussian mixtures, SOM)

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Concept of the SOM I.
Input space Input layer
Reduced feature space Map layer
x3
s1
s2
x1
x2
Cluster centers (code vectors)
Place of these code vectors in the reduced space
Clustering and ordering of the cluster centers
in a two dimensional grid
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Concept of the SOM II.
We can use it for visualization
y f ( u )
mc1
x1
mc2
u u1, u2, u3
x2
Known inputs
mc
mc3
x3
mc4
x4
We can use it for regression
y y1, y2
mc5
Unknown inputs
x5
mc
Best Matching Unit
We can use it for clustering
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SOM for Regression

• The widely applied approach divides the input
  space by Voronoi diagrams
• But this well known DISCRETE approach does not
  look natural

u2
mi
u1
u2
y1
u1
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Possible solutions and our main idea

• Fuzzy response of the SOM
• Not really designed for this purpose
• Identification of local linear models for each
  Voronoi cell
• It needs lots of data
• The surface is still not continuous
• Our Idea Application of Delaunay Tessellation
  for optimal approximation

y1
u2
u1
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Why Delaunay Triangulation ?
Avoid skinny triangles, i.e. maximize minimum
angle of triangulation
Empty circle
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Application to a pH process
Acid
Base
The goal is to estimate the acidity in the tank
reactor Use the resulted model to understand the
process Apply the model to control
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SOM of a pH process
We can use it for visualization
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Surface of the problem space
We can use it for regression
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Analysis of the operating regimes
Real titration curve
Piecewise linear estimation of the titration
curve
Stable operating regions
We can use it for clustering
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Application in process control
J. Abonyi, R. Babuska, F. Szeifert, Fuzzy
Modeling with Multidimensional Membership
Functions Constrained Identification and
Control Design, IEEE Transactions on Sys., Man
Cyber. Part B Oct, 2001
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Discussion

• SOM for clustering, visualization and regression
• The Voronoi diagram based approach
• gives discrete approximation of the regression
  surface
• Local models can also be identified by local
  least squares method
• The paper presented a Delaunay tessellation based
  approach
• Local models are defined based on these
  partitions.
• Can be considered as a fuzzy model
• Can be used in process control

www.fmt.vein.hu/softcomp