Locally Weighted Principal Component Analysis-Based Multimode Modeling for Complex Distributed Parameter Systems
Abstract : Global principal component analysis (PCA) has been successfully introduced for modeling distributed parameter systems (DPSs). In spite of the merits, this method is not feasible due to parameter variations and multiple operating domains. A novel multimode spatiotemporal modeling method based on the locally weighted PCA (LW-PCA) method is developed for large-scale highly nonlinear DPSs with parameter variations, by separating the original dataset into tractable subsets. This method implements the decomposition by making full use of the dependence among subset densities. First, the spatiotemporal snapshots are divided into multiple different Gaussian components by using a finite Gaussian mixture model (FGMM). Once the components are derived, a Bayesian inference strategy is then applied to calculate the posterior probabilities of each spatiotemporal snapshot belonging to each component, which will be regarded as the local weights of the LW-PCA method. Second, LW-PCA is adopted to calculate each locally weighted snapshot matrix, and the corresponding local spatial basis functions (SBFs) can be generated by the PCA method. Third, all the local temporal models are estimated using the extreme learning machine (ELM). Thus, the local spatiotemporal models can be produced with local SBFs and corresponding temporal model. Finally, the original system can be approximated using the sum form of each local spatiotemporal model. Unlike global PCA, which uses global SBFs to construct a global spatiotemporal model, LW-PCA approximates the original system by multiple local reduced SBFs. Numerical simulations verify the effectiveness of the developed multimode spatiotemporal model.
? The complex nonlinear dynamics exist in original space can be transformed into many simple local nonlinear spatiotemporal dynamics.
? Comparison with the other two existing methods demonstrate that the proposed method can lead to much higher modeling accuracy and efficiency.
? Each local model represents the same DPS and often has similar nonlinear dynamic characteristic, PCR is utilized to calculate the weights of each local spatiotemporal model to avoid the existence of multicollinearity.
? Compared with probabilistic PCA based multi-model, since each sub-model represents to the same system, multicollinearity will be existed with the increase of the number of sub-models.
? In theory, FGMM can be suited to any type of distribution, and it is usually used to solve the problem that snapshots in the same process driven by different operating modes.
? Moreover, since each subspace contains only a fraction of the snapshots, the computational complexity for calculating the BFs is reduced by solving a set of smaller eigenvalue problems.
? To overcome this problem, multi-modeling strategies have been investigated for DPSs by using probabilistic PCA based method.
? The parameters estimation problem can be solved by MLE method analytically by letting the first-order derivatives of the log-likelihood function equal to zero.
• The proposed method is a complex nonlinear spatiotemporal modeling method, which may not suit for weakly nonlinear DPSs.
• The proposed multi-model is a data-driven method, which strongly depends on the number of sensors.
• Multi-modeling can reduce the nonlinear complexity, the proposed model has strong ability to track and handle the complex nonlinear dynamics.
• The proposed model can track and handle the strong nonlinear spatiotemporal dynamics very well due to the multi-modeling mechanism.
• Rademacher complexity is developed for the evaluation of generalization bound of the proposed method.
? The modeling performance of the proposed method regarding the predicted output and the corresponding error distribution, respectively.
? It can be observed that the multimode modeling method obtains a well-performing modeling accuracy in the classical catalytic rod system.
? Therefore, a single set of global basis functions usually provides an unsatisfactory modeling performance for approximating complex DPSs with strong nonlinearities and time-varying dynamics.
? To further verify the model performance, the traditional global modeling and blind subspace modeling methods are adopted for comparison under the same configuration.
? In contrast, the proposed method shows a promising performance with smaller approximation errors.
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