A
data-driven approach for parameter estimation of stochastic models by using
machine learning techniques

C.
Huang¹,F. Cattani², F. Galvanin^1,*

¹Department
of Chemical Engineering, University College London

Gower
Street, London WC1E 6BT (UK)

 ²Process
Studies Group, Global Sourcing and Production, Syngenta, Jealott’s Hill
International Research Centre,

Bracknell,
Berkshire, RG42 6EY (UK)

*
f.galvanin@ucl.ac.uk

Stochastic modelling [1]
is becoming increasingly popular and is widely used to study many systems whose
states are driven by dynamical processes. For stochastic models, mathematical
formulation of the dynamical processes with randomness usually involves some
model parameters that are assumed to describe the stochastic events in the
system. A well implemented stochastic model with adequate model identification
can present a good performance on the prediction of system evolution. To
improve the performance of stochastic models and to increase the quality of
model predictions, it is very necessary to characterise those parameters
postulated in the models by some sample data observed in relevant and realistic
processes.

Parameter estimation [2]
is an important procedure where sample measurements are employed to estimate
the parameter values and the corresponding variability in order to improve the
predictive capability of a model for a system of interest. The estimation
approach, however, relies on the feature of the model that is used to formulate
the system and to describe the processes observed in the system. From the
mathematical modelling point of view, models, in which some parameters are to
be estimated, can be classified into three classes in accordance with the model
features, namely knowledge-driven (white-box), data-driven (black-box) and
hybrid (grey-box) models [3].

Grey-box models are frequently
applied for systems whose physical mechanisms are partially understood, especially
for stochastic systems. Stochastic models, formulated with partial a priori
information and assumed model parameters, are different from white-box models
of which the parameters are estimated with sample data using different
parameter estimation techniques (e.g. least-squares, maximum-likelihood or
Bayesian estimation) [2]. To identify such models, data-driven approach is more
preferable.

We have set up a stochastic
on-lattice model to investigate the interaction behaviour between particles and
to study the steady state after a period of time in a particle coating system.
In this model, it is postulated that particles collide periodically with their
neighbours and in every collision event particles exchange some of their
coating material. The amount of coating material exchanged each time is assumed
to follow a linear interaction mechanism where two transfer parameters, k_P
and k_N, are defined to formulate the process. From the
experimental data observed in particle coating process, different amount of
material fed into the coater leads to different coating material distribution
at steady state, and these distributions can be obtained from stochastic
simulations based on linear exchange stochastic model by different values of k_P
and k_N, see Figure 1.

The objective of the work is to
estimate the transfer parameters, k_P and k_N,
by finding their feasible regions in the model for different amount of material
used in the coating process by using data-driven approach and machine learning
techniques. As one of the powerful and versatile tools in machine learning
techniques, support vector machine (SVM) [4] is of high capability of
performing nonlinear classification based on a given dataset.

Figure 2 presents the procedure of
finding feasible regions of transfer parameters Θ = (k_P,
k_N). Firstly we make initial random selection of different
parameters by which we execute several computational simulations. The
simulation outputs Ŷ is then compared to the corresponding
experimental observations y, and similarity scores between Ŷ
and y can be calculated as the input instance X_Θ
for SVM. SVM then classifies the Ŷ into different groups according
to their similarity scores and yields a parameter feasible region. Based on the
parameter feasible region, we analyse the classified points near the boundary
and select new parameter Θ accordingly so as to update the existing
feasible region until the estimated parameters Θ reach a desired
level of precision.

The advantage of using SVM is that
in each iteration the feasible region can be updated without using a lot of
simulation data and it therefore saves much computational expense for parameter
estimation. In addition, the refined feasible region of parameter is of great
benefit for further work involved. The result shows that the parameters in the
stochastic model can be effectively identified by refined feasible regions with
high precision and the model is capable of reasonably approximating the
particle coating process with corresponding operating condition.

References