key: cord-0937937-jqutuvap
authors: dos Santos Gomes, Daiana Caroline; de Oliveira Serra, Ginalber Luiz
title: Interval type-2 fuzzy computational model for real time Kalman filtering and forecasting of the dynamic spreading behavior of novel Coronavirus 2019
date: 2022-04-08
journal: ISA Trans
DOI: 10.1016/j.isatra.2022.03.031
sha: 5f76f53d795a16e2b898dbfab9d6c4ba5dbf619b
doc_id: 937937
cord_uid: jqutuvap

This paper presents a computational model based on interval type-2 fuzzy systems for analysis and forecasting of COVID-19 dynamic spreading behavior. The proposed methodology is related to interval type-2 fuzzy Kalman filters design from experimental data of daily deaths reports. Initially, a recursive spectral decomposition is performed on the experimental dataset to extract relevant unobservable components for parametric estimation of the interval type-2 fuzzy Kalman filter. The antecedent propositions of fuzzy rules are obtained by formulating a type-2 fuzzy clustering algorithm. The state space submodels and the interval Kalman gains in consequent propositions of fuzzy rules are recursively updated by a proposed interval type-2 fuzzy Observer/Kalman Filter Identification (OKID) algorithm, taking into account the unobservable components obtained by recursive spectral decomposition of epidemiological experimental data of COVID-19. For validation purposes, through a comparative analysis with relevant references of literature, the proposed methodology is evaluated from the adaptive tracking and forecasting of COVID-19 dynamic spreading behavior, in Brazil, with the better results for RMSE of [Formula: see text] , MAE of [Formula: see text] , [Formula: see text] of 0.99976, and MAPE of [Formula: see text].

J o u r n a l P r e -p r o o f Journal Pre-proof 1.1 

In recent years, the integration of Kalman ltering and fuzzy systems has 30 received increasing interest from the scientic community in several application domains [22, 23, 24, 25] . In [26] , an eight-layered neuro-fuzzy model is proposed to approximate nonane nonlinear dynamics with a state feedback quadratic stabilizing controller. In this approach, a constrained unscented Kalman lter is used for updating the parameters of both neuro-fuzzy model and the con- 35 troller. In [27] , a fuzzy adaptive error-state Kalman lter (FAESKF) is proposed and applied to attitude estimation problem of a small unmanned aerial vehicle (UAV). The eciency is illustrated from experimental ight results for situations with high disturbance and absence of Global Positioning System (GPS) measurements. In [28] , an active fault-tolerant control (FTC) scheme for robotic 40 manipulators, subject to actuator faults, is proposed. The approach combines an interval type-2 fuzzy satin bowerbird algorithm for parameters optimization and adaptive state-augmented extended Kalman lter as a real time fault detection and diagnosis module. In [29] , a strategy for optimizing the membership function parameters of an interval type-2 fuzzy system using Kalman lter is 45 presented. The performance of methodology was validated from its application to dierent benchmark functions, presenting better results when compared to other approaches from the literature.

Recently, several researchers have dedicated themselves for developing methodologies of modeling and forecasting using experimental epidemiological data 50 with applications to the novel Coronavirus 2019 [30, 31, 32, 33, 34] . In this In [35] , the rened instrumental variable method is applied for estimating hy- 60 brid BoxJenkins models of linear dynamic systems (RIVC). The mathematical model is applied to relate the daily death reports in the Italian and UK epidemics, and then provide 15-day-ahead forecasts of the UK daily death reports.

The approach is also used for modeling and forecasting the epidemic using daily reports of COVID-19 patients in UK hospitals. In [36] , a Bayesian structural 65 time series (BSTS) model is applied for investigating the temporal dynamics of COVID-19 in top ve aected countries around the world (United States, Brazil, Russia, India and United Kingdom), within the horizon of 30 days.

The spread of novel Coronavirus 2019 has motivated the analysis of epi-70 demiological data in order to contribute with the political/health authorities in decision-making and resource allocation [37, 38] . Thus, a large number of approaches have already been proposed in this context of forecasting the propagation dynamics of COVID-19 [39, 40, 41, 42] . However, the performance of The paper is organized as follows. In section 2, the description of the pro-90 posed methodology for design of interval type-2 fuzzy Kalman lters is presented. In section 3, experimental results for forecasting analysis of COVID-19 dynamic spread behavior is presented, as well as a comparative analysis between the proposed methodology and similar recent references from the literature. In Considering the initial experimental dataset referring to p time series under analysis, given by:

where y k ∈ R p , such that k = 1, . . . , N b . A set of ρ delayed vectors with dimension δ is dened, where δ is an integer number dened by user with 2 ≤ J o u r n a l P r e -p r o o f Journal Pre-proof

to construct a trajectory matrix H given by:

Then, a covariance matrix S is computed by:

The Singular Value Decomposition (SVD) procedure is applied to matrix S and a set of eigenvalues (σ 1 ≥ σ 2 ≥ · · · ≥ σ δ ≥ 0) and eigenvector (ϕ 1 , ϕ 2 , . . . , ϕ δ ) is

the SVD of H, is rewritten as follows:

where ∆ ς | ς=1,...,d is dened as:

The matrices ∆ ς | ς=1,...,d are regrouped into ξ independent matrices terms I j | j=1,...,ξ ,

where ξ ≤ d, as follows:

where ξ is the number of unobservable components extracted from experimental dataset. The unobservable spectral components α j | j=1,...,ξ are given by:

where δ * = min(δ, ρ), ρ * = max(δ, ρ) and N b = δ + ρ − 1 is the number of samples of initial experimental dataset. After the training step of spectral analysis algorithm, the next steps are repeated for k = N b + 1, N b + 2, . . . , as described in sequel. The values of ρ and S k are updated, respectively, as follows:

SVD method is applied to covariance matrix S k and the set of eigenvalues σ 1 k , σ 2 k , . . . , σ δ k and eigenvectors ϕ 1 k , ϕ 2 k , . . . , ϕ δ k are updated at k so that y k is rewritten as follows:

where h ς k = κ ς k ψ T k ϕ ς k , with ς = 1, . . . , d, where κ ς k is the last element of the 110 eigenvector ϕ ς k . The regrouping of the terms h ς k | ς=1,...,d in ξ disjoint terms I j k | j=1,...,ξ , results in

where I j k = α j k , with j = 1, . . . , ξ and k = N b + 1, N b + 2, . . . , corresponds to the extracted unobservable components at instant of time k. The Algorithm 1 presents the computational steps related to the pre-processing of experimental 115 data.

The rule-based structure of the interval type-2 fuzzy Kalman lter presents the i| [i=1,2,...,c] -th fuzzy rule, with n-th order, m inputs, p outputs, given by: Step 1: Compute ρ = N b − δ + 1;

Step 2: Construct the trajectory matrix H -Eq. (2);

Step 3: Calculate the covariance matrix S -Eq. (3);

Step 4: Apply the SVD method in the covariance matrix S and obtain the set of eigenvalues σ 1 ≥ σ 2 ≥ · · · ≥ σ δ ≥ 0 with their respective eigenvectors ϕ 1 , ϕ 2 , . . . , ϕ δ ;

Step 5: Rewrite the SVD of the matrix S in the form of Eq. (4);

Step 6: Regroup the matrices H ς | ς=1,...,d in ξ linearly independent matrix terms I j | j=1,...,ξ -Eq. (6);

Step 7: Compute the unobservable components α j k | j=1,...,ξ -Eq. (7); %Recursive step;

Step 1: Update ρ;

Step 2: Update the covariance matrix S k -Eq. (9);

Step 3: Update the set of eigenvalues with their respective eigenvectors applying the SVD method in S k ;

Step 4: Rewrite the sample y k in the form of Eq. (10);

Step

k ∈ R p×m are the state space matrices of the local submodel and K i k ∈ R n×p is the Kalman gain matrix. The residual error ϵ i k for i-th rule is dened as

The dynamics described by experimental dataset is approximated by the weighted sum of Kalman lters dened in the consequent proposition of interval type-2 fuzzy Kalman lter, considering the interval activation degrees µ i W i (Z k ), of each i-th rule, as follows:

are, respectively, the lower and upper activation degrees of i-th fuzzy rule, such that

, and c is the number of fuzzy rules dened for interval type-2 fuzzy Kalman lter. The rst step of interval type-2 fuzzy clustering algorithm is to compute the J o u r n a l P r e -p r o o f Journal Pre-proof centers of initial clusters dened by a random partition matrix U (0) as follows:

where Z k is the experimental dataset at instant k. After that, a covariance matrix F i is computed for each cluster, given by:

The covariance matrix is used to calculate the distances D i k F i between the sample Z k and the center v i (l) of the i-th cluster computed in the rst step, as follows:

The term det F i

in Eq. 17 corresponds to the norm-inducing matrix A i of i-th cluster, which are used as optimization parameters, allowing each cluster to adapt the distance norm to the local topological structure of the dataset [44]. The interval partition matrix U (l) , is updated as follows:

where

J o u r n a l P r e -p r o o f Journal Pre-proof 

nents that presents higher eigenvalue and are more signicant to represent the dynamics of experimental dataset. Initially, the appropriate number of Markov parameters q is chosen and a matrix of regressors is computed by:

The observer Markov parameters Y i are calculated as follows:

where

J o u r n a l P r e -p r o o f Journal Pre-proof corresponds to the diagonal weighting matrix related to the i-th fuzzy rule obtained by interval type-2 fuzzy clustering algorithm described in Section2.2.1,

165 are the interval observer Markov parameters of i-th rule and

corresponds to the coupled state matrix and coupled input matrix, since it has information about both the dynamics of experimental dataset and interval Kalman gain. From Eq. (22):

where y = [y 1 y 2 . . . y N b ] ∈ R p×N b corresponds to real experimental dataset. (27) is rewritten as:

Solving the Eq. (28) by QR factorization procedure [46] , it has:

where Q i is an orthogonal matrix, such that Q i 

where Y i j corresponds to the interval observer Markov parameters obtained from Eq. 29. Thus, the system Markov parameters vector Y i j , is given by:

and the observer gain Markov parameters Y i o j are obtained, as follows:

J o u r n a l P r e -p r o o f

A Hankel matrix H i (j − 1) ∈ R γp×βm is constructed from system Markov parameters, as follows:

where γ and β are arbitrary integers dened by user. Considering j = 1, the Hankel matrix H i (0) is decomposed using Singular Value Decomposition pro-180 cedure:

where Ξ i ∈ R γp×γp and Ψ i ∈ R βm×βm are orthogonal matrices and Σ i ∈ 

where

J o u r n a l P r e -p r o o f

Finally, the matrices that make up the consequent proposition of interval type-2 fuzzy Kalman lter inference system is computed, as follows:

The interval Kalman gain matrix K i k is calculated from the observer gain Markov parameters vector Y i o j , as follows:

where P i γ is the observability matrix computed from Eq. 42 and K i k is the interval Kalman gain matrix. From Eq. (50):

From Eq. (51) and considering

The Eq. (52) is solved by QR factorization procedure to obtain the interval Kalman gain matrix K i k .

After the initial estimation of the consequent proposition of interval type-2 fuzzy Kalman lter, the state space submodels are updated recursively at in- 

According to regressors vector dened in Eq. 53, the interval observer Markov parameters Y i k are updated at the instant of time k = N b + 1, k = N b + 2, . . . , by the following:

The QR factorization procedure is applied to the term U 

The Kalman gain matrices are updated through application of QR factorization to A Step 1: Construct the matrix of regressors Λ -Eq. (21);

Step 2: Calculate the observer Markov parameters Y i -Eq.

Step 3: Calculate the system Markov parameters Y i -Eq. (34)- (36) and the observer gain Markov parameters Y i o -Eq. (37)-(39) ;

Step 4: Construct the Hankel matrices H i (0) and H i (1) -Eq. (40);

Step 5: Decompose H i (0) using SVD method -Eq. (41);

Step 6: Calculate the observability matrix P Table 1 are shown the results for comparative analysis between the interval type-2 fuzzy Kalman lter and the approach in [47] . Once the approach in [47] consider the uncertainties inherent to COVID-19 propagation dynamics by using of fuzzy systems theory, it presents competitive results compared to interval type-2 fuzzy Kalman lter, but its performance is inferior due to its The approach in [48] is based on a rey algorithm for ensemble neural network optimization applied to COVID-19 time series prediction in Brazil using type-2 fuzzy logic, within the horizon of 30 days. In Table 2 are shown the results for comparative analysis between the interval type-2 fuzzy Kalman lter 285 and the approach in [48] . The approach in [48] , although uses a type-2 fuzzy structure for modeling the epidemiological experimental data, it presents slightly inferior results compared to interval type-2 fuzzy Kalman lter once it does not consider the variability in the dynamics of the experimental dataset to update the propagation forecasts of COVID-19 [50] . On the other hand, the 290 interval type-2 fuzzy Kalman lter presents greater eciency due to its recursive parameterizing mechanism for adaptive tracking and real time forecasting of experimental dataset.

The approach in [49] is based on a novel Deep Interval Type-2 Fuzzy LSTM Table 2 are shown the results for comparative analysis between the interval type-2 fuzzy Kalman lter and the approach in [49] . As it can be seen, although the approach in [49] presents satisfactory results once the LSTM model has the capability to learn higher-level features inherent to 300 dataset, its performance is inferior than the interval type-2 fuzzy Kalman lter due to its computing limitation from tuning the hyper-parameter model and determining the number of layers, the number of neural units by layer, the activation function, the learning rate, the loss function, and other parameters. 

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The authors are grateful to Coordination for the Improvement of Higher 

Dear Editor, The authors confirm that there is no conflict of interest in the submitted manuscript.