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Generating Radial Basis Function (RBF) models with PySMO

The pysmo.radial_basis_function package has the capability to generate different types of RBF surrogates from data based on the basis function selected. RBFs models are usually of the form where

$$\begin{equation} y_{k}=\sum_{j=1}^{\Omega}w_{j}\psi\left(\Vert x_{k}-z_{j}\Vert\right)\qquad k=1,\ldots,m\quad\label{eq:RBF-expression} \end{equation}$$

where zj are basis function centers (in this case, the training data points), wj are the radial weights associated with each center zj, and ψ is a basis function transformation of the Euclidean distances.

PySMO offers a range of basis function transformations ψ, as shown in the table below.

List of available RBF basis transformations, d =  ∥ xk − zj∥
Transformation type PySMO option name ψ(d)
Linear 'linear' d
Cubic 'cubic' d3
Thin-plate spline 'spline' d2ln (d)
Gaussian 'gaussian' e(−d2σ2)
Multiquadric 'mq' $\sqrt{1+\left(\sigma d\right)^{2}}$
Inverse mMultiquadric 'imq' $1/{\sqrt{1+\left(\sigma d\right)^{2}}}$

Selection of parametric basis functions increase the flexibility of the radial basis function but adds an extra parameter (σ)to be estimated.

Basic Usage

To generate an RBF model with PySMO, the pysmo.radial_basis_function class is first initialized, and then the function training is called on the initialized object:

# Required imports
>>> from idaes.surrogate.pysmo import radial_basis_function
>>> import pandas as pd

# Load dataset from a csv file
>>> xy_data = pd.read_csv('data.csv', header=None, index_col=0)

# Initialize the RadialBasisFunctions class, extract the list of features and train the model
>>> rbf_init = radial_basis_function.RadialBasisFunctions(xy_data, *kwargs)
>>> features = rbf_init.get_feature_vector()
>>> rbf_fit = rbf_init.training()
  • xy_data is a two-dimensional python data structure containing the input and output training data. The output values MUST be in the last column.

Optional Arguments

  • basis_function - option to specify the type of basis function to be used in the RBF model. Default is 'gaussian'.

  • regularization - boolean which determines whether regularization of the RBF model is considered. Default is True.

    • When regularization is turned on, the resulting model is a regressing RBF model.
    • When regularization is turned off, the resulting model is an interpolating RBF model.

pysmo.radial_basis_function Output

The result of pysmo.radial_basis_function (rbf_fit in above example) is a python object containing information about the problem set-up, the optimal radial basis function weights wj and different error and quality-of-fit metrics such as the mean-squared-error (MSE) and the R2 coefficient-of-fit. A Pyomo expression can be generated from the object simply passing a list of variables into the function generate_expression:

# Create a python list from the headers of the dataset supplied for training
>>> list_vars = []
>>> for i in features.keys():
>>>     list_vars.append(features[i])

# Pass list to generate_expression function to obtain a Pyomo expression as output
>>> print(rbf_init.generate_expression(list_vars))

Similar to the pysmo.polynomial_regression module, the output of the generate_expression function can be passed into an IDAES or Pyomo module as a constraint, objective or expression.

Prediction with pysmo.radial_basis_function models

Once an RBF model has been trained, predictions for values at previously unsampled points x_unsampled can be evaluated by calling the predict_output() function on the unsampled points:

# Create a python list from the headers of the dataset supplied for training
>>> y_unsampled = rbf_init.predict_output(x_unsampled)

Further details about pysmo.radial_basis_function module may be found by consulting the examples or reading the paper [...]

idaes.surrogate.pysmo.radial_basis_function

Available Methods

idaes.surrogate.pysmo.radial_basis_function.FeatureScaling

idaes.surrogate.pysmo.radial_basis_function.RadialBasisFunctions

References:

[1] Forrester et al.'s book "Engineering Design via Surrogate Modelling: A Practical Guide", https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470770801

[2] Hongbing Fang & Mark F. Horstemeyer (2006): Global response approximation with radial basis functions, https://www.tandfonline.com/doi/full/10.1080/03052150500422294

[3] Rippa, S. (1999): An algorithm for selecting a good value for the parameter c in radial basis function interpolation, https://doi.org/10.1023/A:1018975909870

[4] Mongillo M.A. (2011) Choosing Basis Functions and Shape Parameters for Radial Basis Function Methods, https://doi.org/10.1137/11S010840