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Implementation of parameter magnitude-based information criterion in identification of a real system

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Model structure selection is one among crucial steps in system identification and in order to carry out this, an information criterion is needed. It plays an important role in determining an optimum model structure with the aim of selecting an adequate model to represent a real system. A good information criterion should not only evaluate predictive accuracy but also the parsimony of the model. In the past, there had not been, or scarcely have been, any information criterion that evaluates the parsimony of model structures (bias contribution) based on the magnitude of parameter or coefficient. However, recently, some efforts had been made that took into account such strategy and proved the criterion to be effective for simulated datasets. This paper presents the comparison between two information criteria that are based on parameter magnitude information in selecting a good model to represent a real system based on gas furnace. Genetic algorithm (GA) was used to optimise the implementation. The selected models were then tested using correlation tests for model validation. It is shown that parameter magnitude based information criterion 2 (PMIC2) is able to select a more parsimonious model than PMIC but with similar validation results. © Science & Technology Research Institute for Defence (STRIDE), 2018.
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VOL. 11 NUM. 1 YEAR 2018 ISSN
1985-6571
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Implementation of Parameter Magnitude-Based Information Criterion in Identification of a Real
System
Md Fahmi Abd Samad & Abdul Rahman Mohd Nasir
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Ministry of Defence
Malaysia
SCIENCE & TECHNOLOGY RESEARCH INSTITUTE
FOR DEFENCE (STRIDE)
99
IMPLEMENTATION OF PARAMETER MAGNITUDE-BASED INFORMATION
CRITERION IN IDENTIFICATION OF A REAL SYSTEM
Md Fahmi Abd Samad* & Abdul Rahman Mohd Nasir
Faculty of Mechanical Engineering, Universiti Teknikal Malaysia Melaka (UTeM), Malaysia
*Email: mdfahmi@utem.edu.my
ABSTRACT
Model structure selection is one among crucial steps in system identification and in order to carry out
this, an information criterion is needed. It plays an important role in determining an optimum model
structure with the aim of selecting an adequate model to represent a real system. A good information
criterion should not only evaluate predictive accuracy but also the parsimony of the model. In the
past, there had not been, or scarcely have been, any information criterion that evaluates the
parsimony of model structures (bias contribution) based on the magnitude of parameter or coefficient.
However, recently, some efforts had been made that took into account such strategy and proved the
criterion to be effective for simulated datasets. This paper presents the comparison between two
information criteria that are based on parameter magnitude information in selecting a good model to
represent a real system based on gas furnace. Genetic algorithm (GA) was used to optimise the
implementation. The selected models were then tested using correlation tests for model validation. It
is shown that parameter magnitude based information criterion 2 (PMIC2) is able to select a more
parsimonious model than PMIC but with similar validation results.
Keywords: Parameter magnitude; information criterion; system identification; linear regressive
model; genetic algorithm (GA).
1. INTRODUCTION
In many fields, nonlinear dynamic modelling is applied in approximating a wide range of systems.
Basically, system identification is used for estimating a model to represent their systems. System
identification can be considered a regression problem, where the relationship between input and
output variables of a dynamical system has to be estimated. This task is typically accomplished by
minimising a certain information criterion, which measures how well the estimated relationship
approximates the one that truly links the available input-output data pairs (Ljung, 1999). Its basic idea
is to compare the time dependent responses of the actual system and identified model based on a
performance function, hereby referred to as information criterion, giving a measure of how well the
model response fits the system response (Alfi & Fateh, 2010).
An identification procedure typically consists of estimating the parameters of different models, and
next, selecting the optimal model complexity within that set. Increasing the model complexity will
decrease the systematic errors, but, at the same time, the model variability increases (Riddef et al.,
2004). Model accuracy and parsimony, known as variance and bias: , are
important considerations in selecting a model structure (Ljung, 1999). Hence, selecting a model with
the smallest variance is not a good idea because when the number of parameters increase, the variance
will continue to decrease but will present a complex model. At a certain complexity, the additional
parameters no longer reduce the systematic errors but are used to follow the actual noise realisation on
the data. Often, in order to deal with the bias-variance trade-off, the information criterion is
augmented with a penalty term intended to guide the search for the “optimal” relationship penalising
undesired regressors, where regressors refer to possible terms and variables. Regularised estimation
has been widely applied in the context of system identification (Prando et al., 2015). Model validation
100
is the final step of system identification. It is used to check the goodness of the estimated model
(Zhang et al., 2005).
In Samad & Nasir (2017a, b, c, 2018a), parameter magnitude-based information criterion 2 (PMIC2)
was tested on simulated datasets generated from autoregressive with exogenous input (ARX) and
nonlinear autoregressive with exogenous input (NARX) models. Comparisons were made between
PMIC2, Akaike information criterion (AIC) (Akaike, 1974), Bayesian information criterion (BIC)
(Schwartz, 1978) and corrected Akaike information criterion (AICc) (Anderson et al., 1994). In the
simulation results, PMIC2 proved that it performed better than AIC, BIC and AICc for all the
simulated models (ARX and NARX). In this paper, the effectiveness of PMIC2 will be studied by
testing on real system data, in this case, gas furnace data. The models were generated as NARX
models. Since there are many possible models to test, genetic algorithm (GA) was used to optimise
the search. The models selected by PMIC2 were tested by correlation tests for model validation. The
results of PMIC2 were then compared to another parameter-magnitude based information criterion
(Samad et al., 2013), hereby denoted PMIC, in order to evaluate the performance between both
information criteria.
The next sections are as follows: Section 2 introduces information criteria; Section 3 explains about
GA, which is used as part of the search for model structure; Section 4 explains the model validation
tests; Section 5 provides simulation setup; Section 6 presents result and discussions; and lastly,
Section 7 concludes the findings of this paper.
2. INFORMATION CRITERIA
According to Samad et al. (2013), parameter magnitude-based information criterion (PMIC) evaluates
the bias contribution by the sum of squared residuals while the variance contribution is calculated by a
penalty function (PF). This is written as follows:

 (1)
where PF =ln, is the number of terms satisfying 1 while  represents the
absolute value of the parameter for term and  is a fixed value termed penalty function
parameter. The penalty function penalises terms with the absolute values of the estimated parameter
less than the penalty. The penalty parameter value will be set equal to or slightly lower than the
parameter value in cases where the smallest tolerable absolute parameter value is known or can be
roughly estimated.
By referring to Samad & Nasir (2018b), the effectiveness of PMIC was compared to AIC, AICc and
BIC, where, overall, PMIC was found to perform better in model structure selection, where it can
select the true model in the form of ARX and NARX models for given datasets.
The PMIC2 was developed from the approach of using PMIC. In order to overcome the hassle of
determining the suitable penalty function parameter of PMIC, it modifies the bias term or known as
penalty function, and begins as follows:
2 (2)
where RSS is the residual sum of square or defined as the maximised value of the likelihood function
for the estimated model. It can be written as follows:

 
 (3)
101
The basic theory of the penalty function for PMIC and PMIC2 revolves around the consideration that
the magnitude of parameter could have a big role in choosing whether a term is significant enough to
be included in a model. This assists one's judgment in choosing or discarding a term / variable. In
PMIC2:
Considering,   ⋯

One should give: ⋯
where θ is the value of parameter in a model and  is the value of penalty applied for having the
variable / term associated with the parameter. Applying a big penalty for having a variable / term that
has small parameter magnitude may cause the model to be unfavourable in comparison to other
models. In other words, the variable / term that was penalised is said to be insignificant to the model’s
accuracy in comparison to a variable / term that has big parameter magnitude. One way to realise this
is using the following equation:

(4)
Hence, the equation of PMIC2 is:
2

(5)
where and are the k-step-ahead predicted output and actual output value at time
respectively, is the number of data, is the magnitude of parameter in the model, and j is the
number of parameters.
3. GENETIC ALGORITHM (GA)
Among all methods in evolutionary computation, GA is probably the most widely known. Its
application is recorded in various fields, including image processing, pattern recognition, operational
research, biology and computer sciences (see e.g., Haupt and Haupt, 2004, Kumar et al., 2010). There
are three main characteristics of GA (Holland, 1992, Eshelman, 2000):
i. The algorithm uses binary bit string representation.
ii. The selection method used is fitness-proportional selection.
iii. It uses crossover as its main genetic search operator.
Other genetic operators are mutation, which is used as a ‘background operator’ to prevent loss of
important gene information or allele, and inversion, which is particularly significant for permutation-
based coding. The outline of the algorithm was initially known as reproductive plan using genetic
operators (Holland, 1973).
The number of strings is known as the population size, typically denoted popsize. Each chromosome
consists of genes separated by different positions also defined as locus. The genes carry information
of the chromosome. In the simulation, the chromosomes represent different model structures and are
evaluated based on a specified information criterion. Based on the evaluation, the selection of highly
fit individuals and genetic manipulation stage, that includes the crossover and mutation then takes
place. The selection and genetic manipulation of these chromosomes are usually performed to a fixed
number of generations, denoted maxgen. Other related terminologies can be referred in Haupt &
Haupt (2004) and Bäck & Fogel (2000). The flow chart of a simple GA is given in Figure 1.
102
Figure 1: Flow chart of a simple GA.
4. MODEL VALIDATION TESTS
Once the final model is identified, the final step of system identification, which is model validation, is
carried out. A model can only be accepted as valid once it is proven that the selected terms and / or
variables do not contribute bias to its accuracy. The correlation tests were used to ensure that no other
significant terms and / or variables were omitted from the model. Nonlinear models require more tests
than linear models since nonlinear models contain polynomials of variables. The tests are as follows
(Billings & Voon, 1986):
i. Autocorrelation of residuals:

 (6)
where  is the Kronecker delta such that 01and 0, for 0.
ii. Cross correlation of residual with input:

0,∀ (7)
iii. 
0,0 (8)
iv. 


 0,∀ (9)
v. 


 0,∀ (10)
no
yes Output
solution
Initialiseandevaluate
p
o
p
ulation
Reproduceselectedparents
Crossoverandmutate
Evaluatenewsolutions
Updatepopulation
Termination
criterion
reached?
103
where  is the input, is the lag order and ∙ is the expectation operator that can be calculated
based on the formula
̅


 (11)
and

with set to zero. The residual, , is calculated by the following:
 (12)
with as the actual output and as the predicted output. The overbar denotes the time average
so that is given by:

 (13)
In all the tests above, the accepted bandwidth for a model’s fit to a system is approximately 1.96/
when allowed 95% confidence interval with as the number of data points.
5. SIMULATION SETUP
In this section, PMIC2 was tested with real data, in this case gas furnace data, through simulation
using MATLAB software. This input-output data was described as an actual process plant data
available in Jenkins & Watts (1968) and Box et al. (1994), consisting of a discrete stochastic input
series of gas feed rate in ft3/min and output series of carbon dioxide concentration in outlet gas. There
were 296 pairs of input-output data sampled at an interval of 9 s. In Jenkins & Watts (1968) and Box
et al. (1994), the process was found to be adequately represented by a second-order input and output
lags, but not tested using correlation tests. In Jamaluddin et al. (2007), PMIC was used for this data
set to choose a mathematical model to represent the gas furnace system and the model was also tested
using correlation tests.
To allow NARX modelling, the following specification, which was recommended in Ahmad et al.
(2004), was used in this study: output lag, = 2; input lag, = 2; and nonlinearity, = 2. With this
specification, the number of regressors amounts to 15 and the search space has 32,767 solution points.
The least squares method was used as parameter estimation method.
The specification of the GA is fixed where the population size, with popsize set to 500; the maximum
generation is 100; the mutation probability, pm = 0.01; and the crossover probability, pc = 0.6. The
algorithm implements roulette-wheel selection, single-point crossover and binary bit mutation (Bäck
et al., 2000).
For model validation test, the model selected by PMIC2 was tested using correlation tests. All five
tests listed in Section 4 were made.
6. RESULTS AND DISCUSSION
From the simulation test using GA, PMIC2 selected the model, represented in binary string as [111
010 000 000 000] as the best chromosome. Each bit in the string represents a specific variable or term.
By knowing the sequence of variable/term in the simulation program, the selected regressor may be
easily be traced back when bit 1 is found. Other than the calculated direct current level (which is
equivalent to a constant), the variables selected are 1, 2 and 2. From
Jamaluddin et al. (2007) and Samad (2017), PMIC selected the model [111 010 010 000 000] as the
104
best chromosome. The selected regressors and its parameter values selected by both PMIC2 and
PMIC are provided in Table 1. Comparisons could be made directly, where it shows that PMIC2
selected a more parsimonious model than PMIC. PMIC2 selected a model with 4 out of 15 regressors,
while PMIC selected more regressors, which is 5 out of 15 regressors. As can be seen from the table,
a nonlinear term, 11, is also selected by PMIC. With this additional term, considering
that a control system is to be developed for the gas furnace system, additional control plan needs to be
developed, which in turn, will require higher cost and thus undesirable. The model selected by PMIC2
promises simpler control plan as there are less variables to be controlled.
Table 1: Variables, terms and parameter values of selected model by PMIC2 and PMIC for gas furnace
data.
Variable / Terms Parameter Value
using PMIC2 Parameter Value
using PMIC
Constant or d. c. level 6.4103 7.1795
1 1.5073 1.4449
2 -0.6274 -0.5793
2 -0.3790 -0.6600
11 - 0.0047
Figure 2 shows the result of correlation tests on the model selected by PMIC2. It seems to be
acceptable where only a few points were out of the interval. Compared to the correlation test for a
model selected by PMIC in Figure 3 (Samad, 2017), it looks quite similar for all tests. According to
Jamaluddin et al. (2004), these validation deficiencies may be inherent from wrong selection of lag
orders or nonlinearity of the model. It is also plausible that based on the similarities of the situation to
the heat exchanger problem in Billings & Fadzil (1985), the deficiencies may be caused by lack of
noise terms.
Figure 2: Correlation tests of selected model by PMIC2 using GA for gas furnace data.
105
Figure 3: Correlation tests of selected model by PMIC using GA for gas furnace data
7. CONCLUSION
From the results, PMIC2 selected a more parsimonious model for gas furnace data than PMIC. This
was also achieved without having to try all possible models, but by utilising GA. The results of
correlation tests for the model seem good. It must be emphasised that PMIC2 overcomes the hassle of
selecting the suitable value of penalty function parameter before use in PMIC. Together with other
studies done using PMIC2 on simulated models, PMIC2 can be considered as a reliable information
criterion for model structure selection of discrete-time model. Future work shall be concentrated on
PMIC2 use for other real data, datasets with specifications of bigger search space and different forms
of discrete-time modelling such as linear, and nonlinear autoregressive moving average with
exogenous variable.
ACKNOWLEDGEMENT
The authors would like to acknowledge the support from Universiti Teknikal Malaysia Melaka and
Ministry of Higher Education Malaysia for research grant FRGS/1/2015/TK03/FKM/02/F000271.
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... Once a model has been identified, the parameter has been estimated using the least squares method. The OF used in evaluating the optimality of model has been Parameter Magnitude-based Information Criterion 2 (PMIC2) written as follows [28], [29]: ...
... The theoretical idea of AIC is based on Kullback-Leibler information and maximum-likelihood estimation theory, while BIC is developed from Bayesian arguments and it is related to Bayes factors. Another information criterion is known as parameter magnitude-based information criterion (PMIC), and later developed into PMIC2 [3]. Figure 2 provides a guide on how an information criterion may be developed. ...
... The theoretical idea of AIC is based on Kullback-Leibler information and maximum-likelihood estimation theory, while BIC is developed from Bayesian arguments and it is related to Bayes factors. Another information criterion is known as parameter magnitude-based information criterion (PMIC), and later developed into PMIC2 [3]. Figure 2 provides a guide on how an information criterion may be developed. ...
Article
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Information criterion is an important factor for model structure selection in system identification. It is used to determine the optimality of a particular model structure with the aim of selecting an adequate model. A good information criterion not only evaluate predictive accuracy but also the parsimony of model. There are many information criterion those are widely used such as Akaike Information Criterion (AIC), corrected Akaike Information Criterion (AICc) and Bayesian Information Criterion (BIC). Another information criterion suggesting use of logarithmic penalty, named as Parameter Magnitude-based Information Criterion (PMIC) was also introduced. The study presents a study on comparison between AIC, AICc, BIC and PMIC in selecting the correct model structure for simulated models. This shall be tested using computational software on a number of simulated systems in the form of discrete-time models of various lag orders and number of term/variables. As a conclusion, PMIC performed in optimum model structure selection better than AIC, AICc and BIC.
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Information criterion is an important factor for model structure selection in system identification. It is used to determine the optimality of a particular model structure with the aim of selecting an adequate model. There had not been, or scarcely have been, any loss function that evaluates parsimony of model structures (bias contribution) based on the magnitude of parameter or coefficient. The magnitude of parameter could have a big role in choosing whether a term is significant enough to be included in a model and justifies ones' judgement in choosing or discarding a term/variable. This study intends to develop a new information criterion such that the bias contribution is related not only to the number of parameters, but mainly to the magnitude of the parameters. The parameter-magnitude based information criterion (PMIC2) is demonstrated in identification of linear discrete time model. The demonstration is tested using computational software on a number of simulated systems in the form of discrete-time linear regressive models of various lag orders and number of term/variables. It is shown that PMIC2 is able to select the correct the model based on all of the tested datasets.
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Model structure selection is a problem in system identification which addresses selecting an adequate model i.e. a model that has a good balance between parsimony and accuracy in approximating a dynamic system. Parameter magnitude-based information criterion 2 (PMIC2), as a novel information criterion, is used alongside Akaike information criterion (AIC). Genetic algorithm (GA) as a popular search method, is used for selecting a model structure. The advantage of using GA is in reduction of computational burden. This paper investigates the identification of dynamic system in the form of NARX (Non-linear AutoRegressive with eXogenous input) model based on PMIC2 and AIC using GA. This shall be tested using computational software on a number of simulated systems. As a conclusion, PMIC2 is able to select optimum model structure better than AIC.
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Information criterion is an important factor for model structure selection in system identification. It is used to determine the optimality of a particular model structure with the aim of selecting an adequate model. A good information criterion not only evaluate predictive accuracy but also the parsimony of model. There are many information criterions those are widely used such as Akaike information criterion (AIC), corrected Akaike information criterion (AICc) and Bayesian information criterion (BIC). This paper introduces a new parameter-magnitude based information criterion (PMIC2) for identification of linear and non-linear discrete time model. It presents a study on comparison between AIC, AICc, BIC and PMIC2 in selecting the correct model structure for simulated models. This shall be tested using computational software on a number of simulated systems in the form of discrete-time models of various lag orders and number of terms/variables. It is shown that PMIC2 performed in optimum model structure selection better than AIC, AICc and BIC.
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Information criterion is an important factor for model structure selection in system identification. It is used to determine the optimality of a particular model structure with the aim of selecting an adequate model. This paper introduces a new parameter-magnitude based information criterion (PMIC2) for identification of linear discrete time model. It is shown that PMIC2 is able to select a suitable model in linear discrete-time model.
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