Modified multiple generalized regression neural network models using fuzzy Cmeans with principal component analysis for noise prediction of offshore platform
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Abstract
A modified multiple generalized regression neural network (GRNN) is proposed to predict the noise level of various compartments onboard of the offshore platform. With limited samples available during the initial design stage, GRNN can cause errors when it maps the available inputs to sound pressure level for the entire offshore platform. To obtain more relevant group for GRNNs training, fuzzy Cmean (FCM) is used. However, outliers in some group may interfere the prediction accuracy. The problem of selecting suitable inputs parameters (in each cluster) is often impeded by lack of accurate information. Principal component analysis (PCA) is used to ensure high relevance input variables in each cluster. By fusing multiple GRNNs by an optimal spread parameter, the proposed modeling scheme becomes quite effective for modeling multiple frequencydependent data set (ranging from 125 to 8000 Hz) with different input parameters. The performance of FCMPCAGRNNs has improved significantly as the results show a 25% improvement on the spatial sound pressure level (SPL) and 85% improvement on the spatial average SPL than just GRNNs alone. By comparing with data obtained from real engine room on a jackup rig, the FCMPCAGRNNs noise model performs better with around 16% less error than the empiricalbased acoustic models. Additionally, the results show comparable performance to statistical energy analysis that requires more time and resources to solve during the early stage of the offshore platform design.
Keywords
Fuzzy Cmean Principal component analysis Generalized regression neural network Noise prediction Offshore platform1 Introduction
Noise control is an important aspect which ensures the crew habitability onboard offshore platform. Implementing noise prediction is an effective way to identify the potential noise problem at the early stage of offshore platform design to avoid expensive retrofitting cost in the later stage of modification. Currently, excessive noise in the offshore and marine applications is identified mainly using the empirical formula or the computeraided design (CAD)based mathematical tools. For example, the finite element analysis (FEA) and the boundary element method (BEM) solve acoustic responses by considering wave propagation; the statistical energy analysis (SEA) and the energy finite element analysis (EFEA) determine the sound field based on power flow between subsystems. However, the accuracy of the results could not be guaranteed [1] if the empirical formulas are applied to different applications as some formulas are unable to meet the necessary assumptions such as room’s shape, room’s size and sound source. On the other hand, the CADbased numerical tool is proven to be quite accurate for certain frequency regime; however, using these tools for large scale system such as the offshore platform can be quite time and resource consuming.
For the past few decades, neural networks have been used to model complex systems. In machine learning, there are many methods available in the literature. In this study, a general regression neural network (GRNN) [2] is used. It is quite advantageous due to its ability to converge to the underlying function of the data after few training samples, and the results are quite consistent. A full knowledge of the system to be modeled is often not required. It makes GRNN a useful tool to perform prediction and comparison of system performance in practice. As a result, the noise engineers can spend more time on the noise analysis instead of creating an accurate CAD model that requires exact values of the model variables in the computerbased acoustic simulation.
Many applications including the noiserelated applications [3, 4, 5, 6] use GRNN. In the current literature, GRNN application on the offshore platform such as a jackup rig has not been discussed. In addition, the inherent use of steels for room construction in the jackup rig differs from most of the landbased industrial and acoustic rooms [7, 8] that increase the percentage of structureborne noise than airborne noise. Moreover, the problems of selecting the appropriate inputs from the design variables (e.g., actual position of the noise sources, room dimensions, and other acoustic variables) are often impeded by a lack of exact information during the early design stage of the offshore platform. The relevant inputs used for GRNN training are often quite subjective, and the types of input variables used for training can vary across different noise engineers due to their experience.
Hence, a modified multiple GRNN using fuzzy Cmeans (FCM) clustering and principal component analysis (PCA) is proposed to predict the noise level on the jackup rig with the least number of significant inputs. The training and test samples from 125 to 8000 Hz obtained from the computerbased statistical energy analysis (SEA) with direct field (SEADF) software approach validated by experimental data [9] will be used. These input data will be preprocessed by FCM and PCA to group the dominant samples together and reduce the dimensionality of the input variables before commencing the training using GRNN. With optimal spread variables obtained for each cluster at different frequencies, multiple GRNN can be fused to form an optimal GRNN. The proposed method enables noise engineers to predict the noise level on any similar offshore platform without repeating the SEA modeling that is often time and resource consuming.
The contributions of the paper are as follows. First, by fusing multiple GRNNs at different frequencies, the proposed modeling scheme is sufficient for modeling various frequencydependent data that contain several input variables (as compared to current acoustic room models in the literature that do not consider the frequency variation, room geometry, source power, and receiver position in a single formula). With more relevant variables used in each cluster after the FCMPCA, it consumes less computational time as compared to conventional GRNNs that applied to original data set with higher dimensions. Second, with multiple GRNNs training and FCMPCA, it enhances the input variables selection and thus delivers more reliability and robustness to the overall noise prediction model.
This paper has the following sections. Section 2 describes the proposed noise prediction using FCMPCAGRNNs. Section 3 illustrates the selection of input variables for FCMPCAGRNNS. Sections 4 and 5 introduce the real offshore structure case study and the data preprocessing using FCM and PCA, respectively. Section 6 describes the design of multiple FCMPCAGRNNs. Section 7 shows the results and discussion. Section 8 concludes the paper.
2 Proposed noise prediction using FCMPCAGRNNs
3 Selection of input parameters for FCMPCAGRNNs
Room types defined for compartment onboard
Room Type (1 to 8)  Descriptions  Compartments  Permitted noise level (dBA) 

1  Unmanned machinery room  Engine room, fire pump room, emergency generator room, and thruster room  110 
2  Unmanned machinery room  AHU room  90 
3  Manned machinery room  Switchboard room, transformer room, drill floor, mud room, mixing area, pipe rack, general process and utility area, pump room, and cement room  85 
4  Unmanned instrument room  Local instrument room, electrical MCC room  75 
5  Store, workshop, and instrument room  Mechanical/electrical workshop, paint store, LQ stores, dish washing  70 
6  Living quarter public area  change room, LQ corridor, and toilets  60–65 
7  Living quarter public area, laboratory, and local control room  Local control room, laboratory, galley, mess room, workshop office, gymnasium, and lobby  50–60 
8  Cabin, hospital, and central control room  Cabin, hospital, and wheelhouse control room  45 

Compartments dominated by the airborne noise

Compartments influenced by the structureborne and transmission noise

Compartments influenced by airborne and structureborne noise

Compartments influenced by airborne and transmission noise

Compartments influenced by airborne, structureborne and transmission noise simultaneously
Based on the above sound analysis, several main parameters that determine the spatial and spatial average SPL of the room on the offshore platform can be obtained. These includes the following 13 inputs and two output parameters: (1) total interior source power level; (2) room type; (3) room surface area; (4) room volume; (5) first nearest source sound power level; (6) source/receiver distance from the first source; (7) second nearest source sound power level; (8) source/receiver distance from the second source; (9) room mean absorption coefficient; (10) maximum sound power level of adjacent rooms; (11) panel or insulation thickness; (12) room type of the adjacent room; (13) number of decks to the main deck; (14) spatial SPL; and (15) average spatial SPL.
4 Case study on real offshore structure
The hull dimensions of the jackup rig [9] involved in the study are approximately 88.8 m (length) × 115.1 m (width) × 12 m (height) as seen in Fig. 2a. There are four aspects of developing a SEA model: (a) the structure properties and configurations; (b) designed noise control treatment; (c) the source information; and lastly (d) the frequency range.
Input and output range for each input parameter
No.  Input variables and outputs  125 Hz  250 Hz  500 Hz  1000 Hz  2000 Hz  4000 Hz  8000 Hz  

Max.  Min.  Max.  Min.  Max.  Min.  Max.  Min.  Max.  Min.  Max.  Min.  Max.  Min.  
Inputs  
1  Total interior sound power level (dBA)  104.6  0.0  115.2  0.0  122.0  0.0  128.0  0.0  123.0  0.0  122.0  0.0  114.0  0.0 
2  Room type  8.0  1.0  8.0  1.0  8.0  1.0  8.0  1.0  8.0  1.0  8.0  1.0  8.0  1.0 
3  Room surface area (m^{2})  2052.0  39.2  2052.0  39.2  2052.0  39.2  2052.0  39.2  2052.0  39.2  2052.0  39.2  2052.0  39.2 
4  Room volume, V (m^{3})  2160.0  16.2  2160.0  16.2  2160.0  16.2  2160.0  16.2  2160.0  16.2  2160.0  16.2  2160.0  16.2 
5  First nearest source sound power levels (dBA)  101.0  0.0  112.0  0.0  119.0  0.0  125.0  0.0  120.0  0.0  119.0  0.0  111.0  0.0 
6  Source/receiver distance from the first source (m)  20.0  0.0  20.0  0.0  20.0  0.0  20.0  0.0  20.0  0.0  20.0  0.0  20.0  0.0 
7  Second nearest source sound power levels (dBA)  101.0  0.0  112.0  0.0  119.0  0.0  125.0  0.0  120.0  0.0  119.0  0.0  111.0  0.0 
8  Source/receiver distance from the second source (m)  20.2  0.0  20.2  0.0  20.2  0.0  20.2  0.0  20.2  0.0  20.2  0.0  20.2  0.0 
9  Room mean absorption coefficient  0.3  0.0  0.6  0.0  0.7  0.0  0.6  0.0  0.6  0.0  0.5  0.0  0.5  0.0 
10  Max sound power level of adjacent room (dBA)  104.6  0.0  115.2  0.0  122.0  0.0  128.0  0.0  123.0  0.0  122.0  0.0  114.0  0.0 
11  Room type of adjacent room  8.0  1.0  8.0  1.0  8.0  1.0  8.0  1.0  8.0  1.0  75.0  1.0  8.0  1.0 
12  Panel/insulation thickness between adjacent room (mm)  75.0  0.0  75.0  0.0  75.0  0.0  75.0  0.0  75.0  0.0  75.0  0.0  75.0  0.0 
13  Number of decks to main deck  6.0  −2.0  6.0  −2.0  6.0  −2.0  6.0  −2.0  6.0  −2.0  6.0  −2.0  6.0  −2.0 
Outputs  
14  Spatial l SPL (dBA)  90.5  20.4  97.2  21.0  103.3  16.2  109.4  12.9  104.5  9.9  103.9  0.0  95.9  0.0 
15  Spatial averaging SPL (dBA)  89.8  20.4  96.5  21.0  101.6  16.2  108.0  12.9  103.0  9.9  102.6  0.0  94.6  0.0 
5 Data preprocessing using FCMPCA
As discussed in Sect. 4, the sound transmission path in various compartments is different. By preprocessing the collected samples via data clustering can help to group samples into clusters of similar characteristics. The FCM algorithm [13, 14, 15] creates groups according to the distance between the data points and the cluster centers. Let x_{i} be input parameters at each frequency, e.g., 125, 250,…,8000 Hz. The input variables of ndimensional are denoted by \({\text{X}}_{i} = \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right) \in \Re^{n} ,\forall i = 1,2, \ldots ,N\) form the corresponding columns in the data matrix \({\text{X}} = [{\text{X}}_{1} ,{\text{X}}_{2} , \ldots ,{\text{X}}_{N} ]^{T} \in \Re^{N \times n}\) where N is the number of samples for each frequency as shown in Fig. 3.
After setting the number of clusters J = 5 and the maximum number of iterations as 200, the FCM algorithm is applied to all frequency samples. The clustering results are presented in Fig. 4a–g in the form of parallel coordinates plot to visualize and analyze multivariate data having different range and SI unit. The values of the thirteen input variables are polylines with vertices on the vertical axes. The numbers in the Xaxis represent the thirteen input variables as seen in Table 2. The position of the vertex on the ith axis corresponds to the ith coordinate of the sample [16]. For example, there exists a higher value in the sixth and eighth input within cluster 5. These high values can be contributed by the possible noise [17] within samples collected. The sound samples which are close to the cluster centers are considered as normal samples. However, they are assigned with very low or zero membership in the cluster group. As a result, the PCA is used to reduce the dimensionality through finding the high relevance input variables for each cluster at a particular frequency.
The correlations of input variables to the outputs are quite different in each cluster. The input variable selection is implemented on the data matrix \({\mathbf{X}}\) in jth cluster (denotes as \({\mathbf{X}}_{{}}^{j}\)) for each frequency to reduce the input dimension. Note that the superscript “j” will be used to define jth cluster and subscript “i” refers to the index for each sample. PCA uses the singular value decomposition (SVD) to rank the input variables in descending order of importance to least important. The most important variables are given a higher priority than the less significant ones.
 1.Subtract the mean of each data point in the data set \({\mathbf{X}}^{j}\) to produce a data set of zero means in the cluster j = 1, 2, …, J denotes aswhere the mean \({\bar{\mathbf{X}}}^{j} = \sum\nolimits_{i = 1}^{{N^{j} }} {{{{\text{X}}_{i}^{j} } \mathord{\left/ {\vphantom {{{\text{X}}_{i}^{j} } {N^{j} }}} \right. \kern0pt} {N^{j} }}}\), \({\text{X}}_{i}^{j}\) is the input samples, N^{j} is the number of samples in the jth cluster, respectively.$${\mathbf{X}}^{j}  {\bar{\mathbf{X}}}^{j}$$(9)
 2.
Compute the square covariance matrix \({\varvec{\Omega}}_{{}}^{j}\) of size l × l for jth cluster where l is the number of reduced input variables.
 3.
Perform singular value decomposition (SVD) on the covariance matrix \({\varvec{\Omega}}^{j}\).
where \({\bar{\mathbf{U}}}^{j}\) is a l × l matrix with columns being orthonormal eigenvectors or left singular vectors of \({\varvec{\Omega}}^{j} {\varvec{\Omega}}^{{j{\text{T}}}}\), \({\bar{\mathbf{V}}}^{{j{\text{T}}}}\) is a l × l matrix with columns being orthonormal eigenvectors or right singular vectors of \({\varvec{\Omega}}^{{j{\text{T}}}} {\varvec{\Omega}}^{j}\) and \({\mathbf{S}}^{j} = {\text{diag(}}s_{1}^{{}} , \ldots , {\text{s}}_{l}^{{}} )\) is a l × l diagonal matrix with the nonzero elements. It is also the singular values or the square roots of eigenvalues from \({\bar{\mathbf{U}}}^{j}\) or \({\bar{\mathbf{V}}}^{j}\) positioned in descending order.$${\varvec{\Omega}}^{j} = {\bar{\mathbf{U}}}^{j} {\mathbf{S}}^{j} {\bar{\mathbf{V}}}^{{j{\mathbf{T}}}}$$(10)  4.Apply \({\mathbf{U}}^{j}\), \({\mathbf{S}}^{j}\), and \({\mathbf{V}}^{j}\) to determine the inverse square root of the covariance matrix.where h is the number of eigenvectors for eigenvalues in \({\mathbf{S}}^{j}\).$${\varvec{\Omega}}^{j  1/2} = \sum\limits_{i = 1}^{h} {\frac{1}{{\sqrt {{\mathbf{S}}_{i}^{j} } }}} {\mathbf{U}}_{i}^{j} {\mathbf{V}}_{i}^{{j{\text{T}}}}$$(11)
 5.
Multiply the SVDcomputed inverse square root covariance matrix as shown to obtain the reduced dimensional data set.
$${\varvec{\Omega}}^{j  1/2} ({\mathbf{X}}^{j}  {\bar{\mathbf{X}}}^{j} )$$(12)
Based on the acoustic field behavior in Sect. 4, the samples are grouped into five clusters at different center frequencies using the FCM. The PCA is then applied to each cluster to determine the number of principal components. In this study, the cumulative percentage of variance criteria is applied to determine the number of principal components. According to this criterion, principal components are chosen based on their cumulative proportion of variance higher than a prescribed threshold value of 95%. The leverage scores for each dimension are obtained by calculating their two norms. Figure 5 shows the norm for the thirteen input parameters at each frequency. The different heights shown on the respective bar charts reflect the dominant input parameters used for each cluster. The dominant input parameters are only retained in each cluster thus reduces the problem dimension and eliminates the relativity between the input parameters.
Selection of input variables in clusters
Freq. (Hz)  Clusters  Total sound power level (dBA)  Room type  Room surface area (m^{2})  Room volume, V (111^{3})  Nearest source 1 SWL, dBA  Dist to nearest source 1  Nearest source 2 SWL, dBA 

125  1  O  X  O  O  O  X  O 
2  X  O  O  O  X  O  X  
3  O  X  O  O  O  X  O  
4  O  X  O  O  O  X  O  
5  O  O  O  O  O  X  O  
250  1  O  X  O  O  O  X  O 
2  O  X  O  O  O  X  O  
3  O  X  O  O  X  X  X  
4  X  O  O  O  X  O  X  
5  O  O  O  O  O  O  O  
500  1  O  O  O  O  O  X  O 
2  X  O  O  O  X  O  X  
3  O  X  O  O  O  X  O  
4  O  O  O  O  O  X  O  
5  X  X  O  O  X  X  O  
1000  1  O  X  O  O  O  X  O 
2  X  O  O  O  X  O  X  
3  O  O  O  O  O  X  O  
4  O  X  O  O  X  X  O  
5  O  X  O  O  O  X  O  
2000  1  O  O  O  O  O  X  O 
2  O  X  O  O  O  X  O  
3  O  X  O  O  O  X  O  
4  X  O  O  O  X  O  X  
5  O  X  O  O  O  X  O  
4000  1  O  X  O  O  O  X  O 
2  O  X  O  O  O  X  O  
3  X  O  O  O  X  O  X  
4  O  X  O  O  O  X  O  
5  X  X  O  O  O  X  O  
8000  1  O  X  O  O  O  X  O 
2  O  X  O  O  O  O  O  
3  O  X  O  O  O  X  O  
4  X  O  O  O  X  O  X  
5  O  O  O  O  O  X  O 
Freq.(Hz)  Dist. To nearest source 2  Mean absorption coefficient  Max sound power level of adjacent room (dBA)  Room type of adjacent room  Panel/insulation thickness between adjacent room (mm)  Number of decks to main deck  No. of dominant input variables  No. of samples (total 424 samples for each freq) 

125  X  X  O  X  O  X  7  48 
O  O  X  O  X  O  8  205  
X  X  O  X  O  X  7  61  
X  X  O  X  O  X  7  66  
X  X  O  O  O  X  9  44  
250  X  X  O  X  O  X  7  25 
X  X  O  X  O  X  7  54  
X  X  O  X  O  X  5  83  
O  O  X  O  X  O  8  57  
X  O  X  O  O  O  8  205  
500  X  X  O  X  O  X  8  66 
O  O  X  O  X  O  8  205  
X  X  O  X  O  X  7  44  
X  X  O  O  O  X  9  48  
X  X  O  X  O  X  5  61  
1000  X  X  O  X  O  X  7  48 
O  O  X  O  X  O  8  66  
X  O  X  X  X  O  8  205  
X  X  O  X  O  X  6  61  
X  X  O  X  O  X  7  44  
2000  X  O  X  O  X  O  9  76 
X  X  O  X  O  X  7  61  
X  X  O  X  O  X  7  205  
O  O  X  O  X  O  8  48  
X  X  O  X  O  X  7  34  
4000  X  O  O  O  O  X  9  66 
X  X  O  X  O  X  7  44  
O  O  X  O  X  O  8  205  
X  X  O  X  O  X  7  61  
X  X  O  X  O  X  6  48  
8000  X  X  O  X  O  X  7  70 
X  X  O  X  O  X  8  38  
X  X  O  X  O  X  7  205  
O  O  X  O  X  O  8  76  
X  X  O  O  O  X  9  35 
6 Model of multiple GRNN after FCMPCA
The GRNN (see Fig. 6) is one type of radial basis function (RBF) networks based on the kernel regression [2] and is a robust regression tool for its strong nonlinear mapping capability and high training speed. Also, it overcomes the shortcoming of back propagation neural network which needs a large number of training samples. It is suitable for a problem with limited training samples, and GRNN has been proved to be a useful tool to perform prediction and comparison in many fields [6, 14, 18]. Briefly, the structure of GRNN is composed of four layers: an input layer, a pattern layer, summation layer, and output layer. The first input layer consists of reduced input variables from FCMPCA preprocess that connected to the second pattern layer. The neurons in the pattern layer can memorize the relationship between the neuron of entry and the proper response of pattern layer. The two summations \({\mathbf{S}}_{{\mathbf{s}}}\) and \({\mathbf{S}}_{{\mathbf{w}}}\) in the summation layer compute the arithmetic sum of the pattern outputs with the interconnection weight equals to one and compute the weighted sum of the pattern layer outputs with the interconnection weight, respectively. The neurons in the summation layer are then summed and fed into the output layer. The number of the neurons in the output layer equals to the dimension of the output vector. Since there are five clusters in each frequency, there are a total number of thirtyfive GRNN predictors for the seven frequencies.
The “spread” refers to the spread of radial basis functions which plays a significant role in FCMPCAGRNNs function approximation [2]. The larger spread gives a smoother function approximation while the smaller spread fits the data closely. The optimal spread variables can be selected based on prior knowledge or intelligent optimization algorithms [5]. In this study, a kfold crossvalidation method is used to find the corresponding spread parameter for each neuron based on the training samples in the clusters. The selected value of spread parameter is chosen once the error of the validation data starts to increase. It is the point where overtraining of the network may occur. The mean squared error (MSE) criteria measure the difference between the estimated and target. An updated spread parameter σ_{i+1} = σ_{i} + i × θ with θ is the adjustable learning factor and i is the current loop index.
7 Results and discussion
Summary of prediction errors between FCMPCAGRNNs and SEADF
Center frequency (Hz)  Error (dB)  

Max spatial  Mean spatial  Max spatial average  Mean spatial average  
125  0.9  −0.016  0.9  0.02 
250  0.7  0.01  0.7  −0.02 
500  1.4  −0.01  1.3  −0.02 
1000  1.8  0.03  1.75  0.01 
2000  1.1  −0.02  1.05  0.02 
4000  0.6  0.007  0.55  0.025 
8000  0.7  0.04  0.66  0 
Model performance with and without FCMPCA preprocessing
Frequency (Hz)  Description  SPL (dB)  Error in SPL (dB)  % of improvement using FCMPCAGRNNs  

Spatial  Spatial average  Spatial  Spatial average  Spatial  Spatial average  
125  GRNN  0.62  0.50  0.38  0.43  61  86 
FCMPCAGRNNs  0.24  0.07  
250  GRNN  0.54  0.29  0.19  0.25  35  86 
FCMPCAGRNNs  0.35  0.04  
500  GRNN  0.73  0.33  0.37  0.30  50  90 
FCMPCAGRNNs  0.36  0.03  
1000  GRNN  0.56  0.33  0.21  0.32  37  96 
FCMPCAGRNNs  0.35  0.01  
2000  GRNN  0.54  0.25  0.14  0.21  25  86 
FCMPCAGRNNs  0.40  0.04  
4000  GRNN  0.68  0.44  0.42  0.43  62  98 
FCMPCAGRNNs  0.26  0.01  
8000  GRNN  0.46  0.27  0.19  0.26  40  96 
FCMPCAGRNNs  0.27  0.01 
8 Conclusion
This paper proposed a modified multiple GRNN model with FCM and principal component analysis (PCA) before training to improve the performance of the GRNN models. The sound pressure level (SPL) on various compartments onboard of a jackup rig is influenced by many uncertain acoustical parameters. The implementation of the FCMPCA groups the data samples into clusters with less and more relevant input variables by removing the less correlated parameters from the clusters in each frequency. With the FCMPCA preprocessing, the FCMPCAGRNNs prediction accuracy has improved the spatial and spatial average SPL by approximately 0.14–0.42 dB and 0.21–0.43 dB, respectively. The spread parameters are identified by crossvalidation with minimum root mean squared error to ensure the FCMPCAGRNNs are an optimal and reliable predictor for the multiple frequencydependent data. In the engine room study, the FCMPCA on the fused multiple GRNN models exhibits less than 16% in the SPL error as compared to commercial acoustic software using statistical energy analysis (SEA) and empiricalbased acoustics models. The FCMPCAGRNNs are useful when the room arrangement tends to change too frequently due to different design requirements from owner and designers during the preliminary design stage. Hence, the proposed FCMPCAGRNNs model helps to predict the SPL of different compartments effectively at different frequencies as it consumes less time and resources when compared to the commercial acoustics software that requires approximately 2–3 months to build the functional acoustics model.
For future works, the proposed model will be further optimized and improved. More works will be done to improve the FCM partition and fuzzy membership functions for the multiple frequencydependent data set.
Notes
Acknowledgements
The authors would like to thank the Singapore Maritime Institute (ID: SMI2015MA11) for sponsoring and supporting the project.
Compliance with ethical standards
Conflict of interest
The authors declare that there is no conflict of interest.
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