Nonlinear Data-driven Process Modelling using Slow Feature
Analysis and Neural Networks
Jeremiah Corrigan and Jie Zhang
School of Engineering, Merz Court, Newcastle University, Newcastle upon Tyne NE1 7RU, U.K.
Keywords: Slow Feature Analysis, Neural Network, Soft Sensor, Dynamic Process Modelling, Data-driven Modelling.
Abstract: Slow feature analysis is a technique that extracts slowly varying latent variables from a dataset. These latent
variables, known as slow features, can capture underlying dynamics when applied to process data, leading to
improved generalisation when a data-driven model is built with these slow features. A method utilising slow
feature analysis with neural networks is proposed in this paper for improving generalisation in nonlinear
dynamic process modelling. Additionally, a method for selecting the number of dominant slow features using
changes in slowness is proposed. The proposed method is applied to creating a soft sensor for estimating
polymer melt index in an industrial polymerisation process to validate the method’s performance. The
proposed method is compared with principal component analysis-neural network and a neural network
without any latent variable method. The results from this industrial application demonstrate the effectiveness
of the proposed method for improving model generalisation capability and reducing dimensionality.
1 INTRODUCTION
The advanced process monitoring and control of
many industrial processes require robust and reliable
models and measurements. The use of hardware
sensors to provide measurements can be too costly or
the samples may not be frequent enough for
acceptable control or monitoring. Soft sensors (Tham
et al., 1991) (software sensors) provide the alternative
to hardware sensors through the use of a process
model. Kadlec et al. (2009) provided a
comprehensive review of the soft sensor design
process, detailing some of the commonly used
techniques and associated issues with soft sensor
applications. Mechanistic and data-driven modelling
approaches make up the two main areas for the
development of soft sensors. Figure 1 illustrates the
key advantages of data-driven modelling over
mechanistic modelling. Mechanistic models are
developed by utilising first principle mathematical
equations and fundamental scientific and engineering
concepts to describe the process. However, many
modern processes are highly complex and so the
development of mechanistic models can be very
expensive and time consuming. Data-driven
modelling makes use of process data to create models
through a variety of techniques. Data-driven models
make use of easy to measure process variables to
predict more difficult to measure variables, such as
polymer quality variables.
When using data-driven techniques, the
complexity of process can determine the necessary
technique that is required. In its most basic form, a
data-driven model can be produced using linear
regression, though this can often lead to poor
generalisation in many real world problems,
particularly those that display nonlinearities that
simply cannot be modelled adequately using linear
techniques.
Figure 1: Key differences between data-driven and
mechanistic process modelling.
Artificial neural networks are a nonlinear data-
driven technique that have been applied to nonlinear
Corrigan, J. and Zhang, J.
Nonlinear Data-driven Process Modelling using Slow Feature Analysis and Neural Networks.
DOI: 10.5220/0007958904390446
In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 439-446
ISBN: 978-989-758-380-3
Copyright
c
2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
439
process modelling and control (Bhat and McAvoy,
1990; Chen et al., 1990; Willis et al., 1991) and in
particular in the polymerisation industry (Gonzaga et
al., 2009; Zhang et al., 2006; Zhang et al., 1997).
Bishop (1995) delivered a detailed overview of neural
networks and their function. The most common form
of neural network is the single hidden layer feed
forward network. Although neural networks possess
universal approximation capability (Cybenko, 1989),
poor performance can still be observed in may
applications due to a variety of issues, as discussed by
Qin (1997). One of the main issues associated with
neural network modelling, and modelling in general,
is overfitting. This is where the model performs well
on the data it was trained on, but performs poorly on
unseen validation data. The ability to perform well on
unseen data is known as the generalisation capability.
Many techniques have been used to improve
generalisation in neural networks, such as
regularisation and early stopping (Bishop, 1995),
ensemble methods (Breiman, 1996; Yang et al., 2013;
Zhang, 1999), and combining neural networks with
latent variable techniques.
Using latent variables (LVs) with regression
models has been applied before, most commonly with
methods such as principal component analysis (PCA)
(Jolliffe 2002) and partial least squares (PLS) (Geladi
and Kowalski 1986). Principal component regression
(PCR) has been applied to soft sensors (Ge et al.,
2011; Ge et al., 2014; Hartnett et al., 1998). PCA has
also been combined with neural networks (Dong and
Mcavoy, 1994). The reason for applying PCA to the
data first is to remove collinearity because a neural
network model trained on collinear data is only valid
when new data follows the same collinearity (Qin,
1997). Thus using PCA on the data before the neural
network training can improve generalisation.
Slow feature analysis (SFA) is a technique that
extracts slow varying trends from data in the form of
LVs known as slow features (SFs) (Wiskott and
Sejnowski, 2002). The slowest SFs capture the most
important trends, while the fastest mostly represent
noise. Applying SFA to process modelling means that
underlying dynamics of the process can be captured,
as well as offering a de-noising affect when looking
at the slower SFs. Additionally, by only selecting a
certain number of SFs to be considered for modelling,
the dimensionality can be reduced leading to a
decrease in model complexity. PCA also produces
dimensionality reduction in a similar way by retaining
only a certain number of principal components (PCs).
By using a reduced number of inputs that express the
key trends, combining SFA with data-driven
modelling can improve generalisation. This has been
demonstrated for soft sensing applications utilising
SFA with linear regression (Shang et al., 2015a;
Shang et al., 2015b).
However, since many process are nonlinear, using
linear regression with SFA will often lead to poor
model performance and so combining SFA with
neural networks is a promising approach for
improving generalisation capability.
A method combining SFA with neural networks
for nonlinear dynamic process modelling is proposed
in this paper. Dynamic SFA is first applied to the
process data and the number of retained (dominant)
SFs is selected via inspection of the changes in
slowness of the SFs. The dominant SFs are then used
as inputs for a single hidden layer feed forward neural
network.
The paper is organised as follows: Section 2
describes SFA, Section 3 defines the proposed
method and Section 4 presents the application of the
proposed method to a soft sensor for polymer melt
index in an industrial polymerisation process. Finally,
the conclusions of this work are presented.
2 OVERVIEW OF SLOW
FEATURE ANALYSIS
Slow feature analysis aims to transform a set of inputs
into outputs that are as slowly varying as possible.
This allows for the extraction of information that is
not overwhelmed by noise and it can reveal
underlying dynamics of the inputs.
Wiskott and Sejnowski (2002) defined the
optimisation problem, as described below, that
enables the extraction of the slow features from the
input signals. Given an input vector x(t), the objective
is to determine a function g(x) such that the output y(t)
varies as slowly as possible, without being constant,
so that relevant information can still be extracted, i.e.
y(t) = g(x(t)).
〈
〉
(1)
subject to three constraints that y has a zero mean,
unit variance and is decorrelated (identity covariance
matrix), i.e.
〈
〉
0
(2)
〈
〉
1
(3)
〈
,
〉
0
(4)
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where
〈
〉
represents temporal averaging
〈
〉
1
(5)
The first two constraints are included so that the
solution of a constant y is avoided. Constraint (4)
ensures that the extract slow features are decorrelated
and so are not duplicates of one another.
2.1 Linear SFA
For the linear case of SFA, the function g(x) is simply
a vector of weights W, such that the output y(t) is a
linear combination of all of the input variables:
(6)
The optimisation problem can therefore be
reduced to the following generalised eigenvalue
problem (Shang et al., 2016):
〈
〉
〈
〉
Ω
(7)
where
〈
〉
is the covariance matrix of the first order
derivative of X,
〈
〉
is the covariance matrix of X,
and Ω is a diagonal matrix of the generalised
eigenvalues, which are the optimal objectives of the
objective function.
Solving this problem first requires normalising
the input signal to zero mean and unit variance. The
normalised input, x(t), is then sphered (or whitened)
to remove underlying correlations, giving the sphered
matrix z(t). The next stage is performing singular
value decomposition on the matrix
〈
〉
:
〈
〉
Ω
(8)
From this, the slow features can be calculated:
(9)
In reality, process data typically has discrete
intervals and so the first order derivative can be
approximated by a first order difference
approximation.
2.2 Dynamic SFA
In many processes, the relationship between the
inputs and outputs may involve significant time
delay. Dynamic modelling includes inputs from
previous sampling times so that prediction
performance can be improved.
Dynamic SFA is simply the inclusion of time
lagged process inputs into the input signals for SFA.
For a given time lag d, the input matrix is as follows:
Xt
x
t
⋯x
td
⋮⋱ ⋮
x
tN1
⋯x
tNd1
(10)
where
is a vector of the process inputs at time t
and N is the number of samples.
2.3 Selection of Dominant Slow
Features
Given a set of m derived slow features, it is necessary
to select the number of dominant slow features, M,
that best capture the dynamics of the process since
many of the faster features represent mostly noise.
Including these faster features in model building
could decrease generalisation performance because
the model is fitting the noise as opposed to the
underlying trends. Additionally, decreasing the
number of slow features reduces the model
complexity, as with other latent variable methods,
such as PCA.
There are very few standard procedures for the
selection of the dominant slow features but there are
some ways this can be accomplished.
2.3.1 Cross Validation Slow Feature
Selection
A common method for selecting LVs in general is
through cross validation, which has been applied to
slow feature regression previously (Shang et al.,
2015b). This can work well enough when this is the
only parameter to be determined, however, in the case
of neural networks, cross validation is often already
used for the selection of the number of hidden
neurons. Therefore, also using cross validation for
slow features selection can lead to poor
generalisation, especially for complex processes
where it can be difficult to obtain the optimal values
for such hyper parameters.
2.3.2 Slowness Criterion based on
Reconstruction
Shang et al. (2015c) derived a slowness criterion
based on a de-noised reconstruction, suggesting to
discard slow features that are faster than all of the
input variables. This method gives good model
performance when the number of inputs is not too
large, however, when using dynamic SFA for a
system with a large number of process variables (such
as the case study in this work), the number of M slow
features calculated is too great and produces poor
generalisation.
Nonlinear Data-driven Process Modelling using Slow Feature Analysis and Neural Networks
441
2.3.3 Slowness Gradient Slow Feature
Selection
This work proposes using the changes in slowness of
the slow features to determine M. This is done by
observing the first relatively significant change in the
gradient of the sorted eigenvalues, which relate to the
Ω matrix from equation 7, and then selecting all of
the slow features up until this point. The values of the
eigenvalues directly relate to the slowness of the
features. Slow feature selection in this way ensures
that the selected features carry the most significant
slow varying trends without too much noise.
Figure
2
provides an example of the trend in eigenvalues. In
this case, the significant change in slowness occurs
after 11 slow features and so this is the number that
would be selected for M.
Figure 2: Eigenvalue trend for slow features used for
dominant slow features selection.
3 SLOW FEATURE ANALYSIS
WITH NEURAL NETWORK
METHODOLOGY
A simplified block diagram of the proposed method
is illustrated in Figure 3.
The neural networks used are single hidden layer
feed forward networks (SLFNN) that were trained
using the Levenberg-Marquardt algorithm with
regularisation and early stopping. The proposed
method utilising SFA and neural networks (SFA-NN)
is described in the following steps:
Step 1: Add d time lagged inputs to the input data
matrix for dynamic modelling, as detailed in Section
2.2.
Step 2: Partition data into a training and testing (TT)
set and an unseen validation set.
Step 3: Normalise and sphere TT data and apply SFA
as described in Section 2.1 to derive the P matrix and
obtain m slow features via
.
Step 4: Select the M dominant slow features by the
slowness gradient based method as described in
Section 2.3.3.
Step 5: Randomly partition this dynamic slow feature
TT data into training and testing data sets.
Step 6: The training data is used to train a neural
network model for each number of hidden neurons in
a given range, e.g. from 1 to 30.
Step 7: Optimal number of hidden neurons is selected
by cross validation using the testing data set.
Step 8: The unseen validation data set is normalised
and sphered to give
. Slow features for this data
set are calculated based on the previously derived P
matrix:
. These validation slow
features are applied to the trained neural network
model to assess the model’s performance on unseen
data.
Figure 3: Simplified block diagram of the SFA-NN method.
4 CASE STUDY: AN INDUSTRIAL
POLYMERISATION PROCESS
4.1 Process Description
The process used for application of the proposed
method is a propylene polymerisation process based
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in China. The data from this process has been used in
previous work on data-driven modelling using
bootstrap aggregated neural networks (Zhang et al.,
2006). Two continuously stirred tank reactors
(CSTR) and two fluidised bed reactors (FBR) in
series make up the main section of this process, as
illustrated in
Figure
4. The feed to the first CSTR
consists of a catalyst, hydrogen and propylene. The
melt index (MI) of polypropylene in the reactor is a
key variable of interest in assessing product quality,
however, it is difficult to measure and so creating a
model to estimate MI from easy to measure process
variables could lead to improvements in process
monitoring and product quality by providing MI
measurements at shorter intervals.
Figure 4: Simplified diagram of a polymerisation process
(Zhang et al., 2006).
4.2 Modelling of MI
The data provided covered 31 days and consisted of
MI measurements in reactors 1 and 4 that were made
every 2 hours, and measurements of 30 process
variables that were made every half an hour. All of
the process variables are shown in Figure 5. The melt
index in reactor 1 is shown in Figure 6. For
confidentiality reasons, the units of the variables are
omitted.
Table 1: Performance of SFA-NN for training, testing and
unseen validation data.
d
Validation
MSE / r
2
Testing
r
2
Training
r
2
M
0 188.8 / 0.9496 0.9373 0.9483 11
1 224.3 / 0.9401 0.9310 0.9461 11
2 243.7 / 0.9349 0.9404 0.9474 10
3
1271.5 /
0.6605
0.9433 0.9480 10
4 622.4 / 0.8338 0.9482 0.9623 17
5
2326.7 /
0.3788
0.9425 0.9540 16
Table 2: Performance of PCA-NN for training, testing and
unseen validation data.
d Validation
MSE / r
2
Training
r
2
Testing
r
2
#PCs
0 305.1 / 0.9186 0.9474 0.9229 23
1 636.9 / 0.8300 0.9783 0.9422 37
2 402.6 / 0.8925 0.9749 0.9459 50
3 785.5 / 0.7903 0.9813 0.9476 61
4 1059.5 /
0.7171
0.9943 0.9460 70
5 876.5 / 0.7660 0.9897 0.9363 81
Table 3: Performance of NN for training, testing and unseen
validation data.
d Validation MSE
/ r
2
Testing
r
2
Training
r
2
0 287.0 / 0.9234 0.9423 0.9593
1 708.9 / 0.8107 0.9385 0.9756
2 1088.5 / 0.7094 0.9443 0.9856
3
627.0 / 0.8326
0.9469 0.9921
4
471.2 / 0.8742
0.9400 0.9956
5
384.0 / 0.8975
0.9385 0.9815
Figure 5: Time series plots of all thirty of the process
variables.
Nonlinear Data-driven Process Modelling using Slow Feature Analysis and Neural Networks
443
Figure 6: Melt index data for reactor D201.
Often in complex chemical processes such as this,
there are time delays between some of the inputs and
outputs and so it is necessary to consider time lagged
inputs, hence a dynamic model is considered. The
form of the input matrix is as follow:
…
,…,
…
(11)
where X is the input matrix and
is the n
th
process
input.
The number of inputs is already large and
including time lagged inputs would only increase this
significantly. This is where the merit of latent variable
methods, which lead to dimensionality reduction,
comes in. To better understand and assess the effect
of adding in different time lagged inputs, models were
created for time lags d from 0 to 5 (representing 0,
0.5, 1, 1.5, 2, and 2.5 hours respectively).
Figure 7: Simplified block diagram of the PCA-NN
method.
The SFA-NN method, as described in Section 3,
was applied and compared to single neural network
(NN) and principal component analysis with neural
networks (PCA-NN). The number of dominant slow
features was selected by the gradient based method
that was described in Section 2.3.3. For PCA-NN, the
number of retained principal components (PC) was
selected as the PCs that captured up to 99% of the
variance. Retained PCs up to 90% variance was also
tested although using 99% produced much better
generalisation.
Figure
7 shows a basic block diagram
of the main steps involved in the PCA-NN method.
The data was partitioned into training, testing and
unseen validation data sets as described in Section 3.
The first 55% of the data was used for training and
testing, with the remaining 45% used as unseen
validation data. The single hidden layer feedforward
neural networks were trained using the Levenberg-
Marquardt algorithm with early stopping and
regularisation. Regularisation was necessary for this
problem because the complexity of the system meant
that producing good generalisation performance on
the validation data was difficult without it. A
sigmoidal activation function was used for the hidden
layer neurons and the output layer used a linear
function. The optimal number of hidden neurons was
selected by cross validation on the testing data’s mean
squared error (MSE).
Tables 1-3 show the MSE on the validation data
along with the R
2
for all three data sets, for each time
lag for SFA-NN, PCA-NN and NN. The MSE values
in bold show the best model for that time lag across
the three techniques. It can be seen that SFA-NN
produces the best generalisation for the first three
time lags, with these models being the top three for
any model across all of the time lags and techniques.
The performance of SFA-NN decreases significantly
for time lags greater than 1. This is likely because the
number of inputs becomes extremely large (e.g. 120
model inputs for d = 3) and SFA struggles to capture
the relevant slow varying information with such a
number of inputs. PCA-NN and NN offer quite
similar performance, however, PCA-NN suffers a
similar drop in performance as SFA-NN for d > 1,
while NN seems to handle the increasing number of
inputs much better than the other two methods since
it displays consistent performance across the different
time lags. Another apparent advantage of SFA-NN is
that it reduces the dimensionality greater than PCA-
NN for all time lags, thus reducing model complexity.
The fact that d = 0 is the best model for SFA-NN
shows that it does not require the additional
information from added time lagged inputs since it
captures the key trends via the slow feature
extraction.
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Figure 8: Predictions of MI for full data set using SFA-NN
with d = 0.
Figure 9: Predictions of MI for full data set using PCA-NN
with d = 0.
Figure 10: Predictions of MI for full data set using NN with
d = 0.
Focusing on the models with d = 0, since these are
the best models across the three techniques, SFA-NN
provided a 34.2% and 38.1% improvement in
validation MSE when compared to NN and PCA-NN
respectively. Figures 8 to 10 show the predictions of
MI for the full data set using SFA-NN, PCA-NN and
NN respectively, all using d = 0. These figures
confirm that SFA-NN fits the data the best,
particularly on the unseen validation data, represented
by the last 45% of the samples. PCA-NN has some
predictions that produce a negative MI, highlighting
inadequacies in this model.
5 CONCLUSIONS
In this paper, combining slow feature analysis with
neural networks for nonlinear process modelling has
been presented. Slow feature analysis is used on
process data to extract underlying trends in the form
of slow features. By retaining a lower number of slow
features, model complexity can be reduced through
having fewer inputs. Selection of the dominant slow
features was performed by observing the slowness for
each slow features, represented by the eigenvalues
derived from the SFA. The slow features up until a
relatively large change in slowness were selected as
the dominant features. These dominant features were
used as the inputs for building a neural network
model. Many industrial processes are complex and
nonlinear, and so using neural networks as opposed
to linear techniques is often necessary. The proposed
SFA-NN method was applied to an industrial
polymerisation process for predicting polymer melt
index, which is a difficult to measure quality variable
with a relatively low sampling rate.
SFA-NN was compared to PCA-NN and NN.
Additionally, different time lags for the dynamic
inputs were assessed to see the effect on
generalisation capability for each technique. The
prediction error on unseen validation data was the
lowest for SFA-NN for the first three time lags. These
were also the best performing models across all of the
18 models that were created for the different time lags
and techniques. The d = 0 models had the best
generalisation performance for each technique and
when comparing these models, SFA-NN showed a
34.2% and 38.1% improvement in generalisation over
NN and PCA-NN respectively. SFA-NN also used a
lower number of latent variables than PCA-NN,
reducing the model complexity. Application of the
proposed SFA-NN method for the nonlinear
modelling of an industrial polymerisation process
shows its effectiveness in improving generalisation
capability and reducing dimensionality.
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