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1. Experimental Modeling (Identification)


Vision: Automatic Modeling


Local Model Network


Model output is calculated by summing up the contributions of all M local models (LMs):




  • Separate input spaces for validity functions (rule premises) z and local models (rule consequents) x.
  • Local estimation of local model parameters much more robust → Regularization.
  • Local models can be of arbitrary type:
    - linear: number of parameters scale linearly with dim(x)
    - quadratic: optimization,
    - higher order polynomials: subset selection or ridge regression required,
    - from first principles: prior knowledge effectively utilized.
  • Exact shape of validity function not relevant.
  • Efficient greedy learning strategies are available.


  • Undesired interpolation side effect.
  • Undesired normalization side effects.
  • Suboptimal.

Separate Input Spaces for Validity Functions z and Local Models x

  • Additional flexibility.
  • Better interpretation.
  • z can be seen as scheduling variables (often low-dimensional).
  • x often is high-dimensional, particularly for dynamic systems.


Interpretation as Takagi-Sugeno Fuzzy System


Partion of Unity


Two Ways to Achieve This

1. Normalization based on Gaussians:  f12_b02
  • Defuzzification
  • Undesirable side effects

2. Hierarchy based on sigmoids
  • Binary tree
  • Knots weighted with Psi and 1-Psi

Algorithm 1: Local Linear Model Tree (LOLIMOT)

  • Partitioning: Axes-orthogonal.
  • Structure: Flat (parallel).
  • Splitting functions: Normalized Gaussian functions.
  • Splitting method: Heuristically, without nonlinear optimization.

LOLIMOT Construction Algorithm

  • Incremental (growing) algorithm: Adds one LM in each iteration.
  • Split of the locally worst LM.
  • Test of all splitting dimensions and selection of the best alternative.
  • Local least squares estimation of the LM parameters.
  • Use normalized Gaussian validity functions.

Demonstration Example (Hyperbola)



Algorithm 2: Hierarchical Local Model Tree (HILOMOT)


  • Partitioning: Axes-oblique.
  • Structure: Hierarchical.
  • Splitting functions: Sigmoid functions.
  • Splitting method: Nonlinear optimization of sigmoid parameters.

HILOMOT Optimization

  • Nested approach.
  • Separable nonlinear least squares.
  • Outer loop: Gradient-based nonlinear optimization of split.
  • Inner loop: One-shot least squares (LS) optimization of local model parameters.


  • Regularization of LS optimization (ridge regression).
  • Subset selection instead of LS can determine local structure.
  • Constraint nonlinear optimization of the splits guarantees for all local models f18_b02.
  • Analytical gradient calculation speeds up optimization by factor dim(u).
  • Derivative of inverse matrix (LS estimation) necessary.
  • Automatic sophisticated smoothness adjustment of sigmoid splitting functions.

Separable Nonlinear Least Squares


HILOMOT: Extension and Modification of LOLIMOT

  • Incremental (growing) algorithm: Adds one LM in each iteration → like LOLIMOT.
  • Split of the locally worst LM → like LOLIMOT.
  • Optimize splitting position and angle by nonlinear optimization → new.
  • Automatic smoothness adjustment algorithm → new.
  • Local least squares estimation of the LM parameters → like LOLIMOT.
  • Use sigmoid validity functions → different to LOLIMOT.
  • Build up a hierarchical model structure → different to LOLIMOT.

Demonstration Example (Hyperbola)


Properties of LOLIMOT

  • Axes-orthogonal splits.
  • Flat structure can be computed in parallel.
  • Strongly suboptimal in high dimensions.
  • Faster training.
  • Easier to understand and interpret.
  • Normalization side effects.

Properties of HILOMOT

  • Axes-oblique splits.
  • Hierarchical structure fully decouples local models.
  • Well suited for high dimensions.
  • Analytical gradient calculation for split optimization.
  • Superior model accuracy.
  • No normalization numerator (defuzzification). → No reactivation effects. Hierarchy automatically guarantee a ‘partition of unity’.
  • Sigmoids are easier to approximate than Gaussians for microcontroller implementation.

Summary: Advantages of Local Model Networks

  • Incremental → all models from simple to complex are constructed.
  • Local estimation:
    - All local model are decoupled.
    - Extremely fast.
    - Regularization effect → almost no overfitting.
  • Local model are linear in their parameters:
    - Global optimum is found.
    - Robust & mature algorithms for least squares estimation are utilized.
    - Structure selection technique can be applied.
    - Adaptive models → recursive algorithms can be applied.
  • Local model can be of any linearly parameterized type: constant, linear, quadratic, ...
  • Different input spaces for validity functions (rule premises) and local models (rule consequents) → new approaches to dimensionality reduction.
  • Complex problem is divided into smaller sub-problems (local models) → divide & conquer.
  • Many concepts from the linear world can be transferred to the nonlinear world → particularly important for dynamic models.



  • MATLAB toolbox.
  • Object-oriented programming.
  • Local linear and quadratic models.
  • LOLIMOT (axes-orthogonal partitions).
  • HILOMOT (axes-oblique partitions).
  • Model complexity chosen according to corrected Akaike information criterion.
  • Analytical gradient calculation for high training speed
  • Reproducible results
  • All fiddle parameters can keep their default values. No expertise required.
  • Trivial usage:  f23_b02

Next Chapter: 2. Design of Experiments (DoE)     Back to Overview