A PieceWise Linear ModelBased Algorithm for the Identification of Nonlinear Models in RealWorld Applications
Abstract
:1. Introduction
2. Methodology
 ${y}_{t}$ is the output of the model at time t;
 ${x}_{t}$ is the input vector of the model at time t, including both the autoregressive and the exogenous parts;
 ${G}_{1}$, …, ${G}_{K}$ are the regions into which the input space ${x}_{t}$ is divided and are used to select the parameters to compute the output of the model at time t;
 ${\theta}^{1}$, …, ${\theta}^{K}$ are the parameter vectors to be estimated during the identification phase, with each vector ${\theta}^{i}$ associated with region ${G}_{i}$.
 
 K is the number of regions;
 
 N is the number of considered time instants (tuples in the identification dataset);
 
 ${\overline{y}}_{t}$ is the measured data of the output model at time t;
 
 ${\overline{u}}_{t}^{l}$ is the value of the exogenous input l at time t;
 
 ${n}_{a}$ is the autoregressive order of the linear systems;
 
 ${n}_{l}$ is the exogenous order of the lth input of the linear systems;
 
 L is the number of the exogenous inputs of the different models;
 
 $L{B}_{{\theta}^{i}}$, $U{B}_{{\theta}^{i}}$, $L{B}_{{c}^{i}}$, $U{B}_{{c}^{i}}$ are the lower and upper bound for the decision variables;
 
 ${y}_{t}$ is the output at time t;
 
 ${x}_{t}$ is the input vector of the model, including both the previous output values (${y}_{t1},\dots ,{y}_{t{n}_{a}}$) and the measured input values (${\overline{u}}_{t}^{l},\dots ,{\overline{u}}_{t{n}_{l}}^{l},l=1,\cdots ,L$);
 
 ${\theta}^{i}$ are the parameters of the ith model to be estimated;
 
 ${c}^{i}$ are the centroids of the different clusters (to be optimized during the identification);
 
 $g({x}_{t},{c}_{i})$ is a check function that determines if ${x}_{t}$ belongs to region i, based on a distance measure.
Algorithm 1 Objective function 

3. Experimental Results
 clustermse: the model parameters are computed firstly applying the cluster analysis to the model input, then splitting the data on the basis of the different clusters and finally computing a model for each cluster minimizing the mean square error;
 clustermse35: is a variant of clustermse, where only the tuples with NO${}_{2}$ concentrations higher than 35 $\mathsf{\mu}$g/m${}^{3}$ are considered in the objective function. The idea is to focus only on the highest (most critical) concentration values;
 clusterOptmse: the model parameters are computed by the proposed methodology, thus, jointly optimizing the centroid positions and the model parameters and dynamically splitting the dataset on the basis of the distance of the input data from the optimized centroids;
 clusterOptmse35: is a variant of clusterOptmse considering only the tuples with NO${}_{2}$ concentrations higher than 35 $\mathsf{\mu}$g/m${}^{3}$ in the objective function.
 Normalized Mean Absolute Error$$NMAE=\frac{{\sum}_{t=1}^{N}\left(y\left(t\right)\overline{y}\left(t\right))\right}{{\sum}_{t=1}^{N}y\left(t\right)}$$
 Correlation Coefficient$$Corr=\frac{{\sum}_{t=1}^{N}(y\left(t\right){\mu}_{y})(\overline{y}\left(t\right){\mu}_{\overline{y}})}{\sqrt{{\sum}_{t=1}^{N}{(y\left(t\right){\mu}_{y})}^{2}}\xb7\sqrt{{\sum}_{t=1}^{N}{(\overline{y}\left(t\right){\mu}_{\overline{y}})}^{2}}}$$
 Hit Ratio:$$HR=\frac{TP}{TP+FN}$$
 False alarm fraction:$$FA=\frac{FP}{TP+FP}$$
 True Skill Score:$$TSS=HRFA$$
3.1. ARX Model Validation
3.2. Validation and Comparison of Solution Methods
4. Conclusions and Future Works
Author Contributions
Funding
Conflicts of Interest
Abbreviations
ARX  AutoRegressive model with eXogenous input 
FA  False Alarm 
HR  Hit Ratio 
NO${}_{2}$  Nitrogen Oxides 
PLM  Piecewise Linear Model 
TSS  True Skill Score (index) 
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Paper  Approach  Model Type  Minimized  MIMO  Real World 

Error  Applicability  Applications  
Dolanc and Strmcnik, 2005 [20]  Fixed Intervals + RLS with forgetting factor  Hammerstein  Forecasting     
Ipanaque and Manrique, 2011 [22]  2 step: interval definition + Recursive Least Square  Wiener  Forecasting    PH control 
Westra et al., 2011 [23]  Fixed intervals + model parameter estimation based on optimization algorithm  State Space + discrete state  Forecasting  MIMO   
Zhang et al., 2018 [28]  Online clustering + Least Square  Hammerstein  Forecasting  MIMO  Stirred track control 
Lassoued and Abderrahim, 2019 [29]  Static Clustering based on SVM reconstruction of regions + least square  PWARX  Forecasting  MISO   
Yang et al., 2019 [24]  Plane clustering (number of plane selected through optimization algorithm)  Non dynamical regression  Regression error    UCI data 
Hadid et al., 2020 [21]  Static Clustering + Least Square  PWARX  Forecasting  MISO  River flood 
Liu et al., 2022 [25]  Time partitioning + optimal identification  PWARX  Forecasting  MISO  Injection Molding 
Carnevale et al., this work  Full dynamical selection of cluster centroids + constrained optimization  PWARX  Simulation  MISO  Air quality simulation 
Observed Values  

≥35 $\mathsf{\mu}$g/m${}^{3}$  <35 $\mathsf{\mu}$g/m${}^{3}$  
Model  ≥35 $\mathsf{\mu}$g/m${}^{3}$  $TP$  $FP$ 
<35 $\mathsf{\mu}$g/m${}^{3}$  $FN$  $TN$ 
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Carnevale, C.; Sangiorgi, L.; Mansini, R.; Zanotti, R. A PieceWise Linear ModelBased Algorithm for the Identification of Nonlinear Models in RealWorld Applications. Electronics 2022, 11, 2770. https://doi.org/10.3390/electronics11172770
Carnevale C, Sangiorgi L, Mansini R, Zanotti R. A PieceWise Linear ModelBased Algorithm for the Identification of Nonlinear Models in RealWorld Applications. Electronics. 2022; 11(17):2770. https://doi.org/10.3390/electronics11172770
Chicago/Turabian StyleCarnevale, Claudio, Lucia Sangiorgi, Renata Mansini, and Roberto Zanotti. 2022. "A PieceWise Linear ModelBased Algorithm for the Identification of Nonlinear Models in RealWorld Applications" Electronics 11, no. 17: 2770. https://doi.org/10.3390/electronics11172770