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This article studies identification problems of multiple linear regression models, which may be described a class of multi-input multi-output systems (i.e. multivariable systems). Based on the coupling identification concept, a novel coupled-least-squares (C-LS) parameter identification algorithm is introduced for the purpose of avoiding the matrix inversion in the multivariable recursive least-squares (RLS) algorithm for estimating the parameters of the multiple linear regression models. The analysis indicates that the C-LS algorithm does not involve the matrix inversion and requires less computationally efforts than the multivariable RLS algorithm, and that the parameter estimates given by the C-LS algorithm converge to their true values. Simulation results confirm the presented convergence theorems.

This paper focuses on the identification of a multivariable controlled autoregressive-like (CAR-like) system. A joint identification algorithm of stochastic gradient and least squares is deduced for estimating the system parameters by decomposing the multivariable CAR-like system into two subsystems, which avoids the calculation of the matrix inversion. To further improve the parameter estimation accuracy, a joint identification algorithm of hierarchical multi-innovation stochastic gradient and least squares is proposed by using the multi-innovation identification theory. The simulation results confirm that these proposed algorithms are effective.

Mediation analysis seeks to explain the pathway(s) through which an exposure affects an outcome. Traditional, non-instrumental variable methods for mediation analysis experience a number of methodological difficulties, including bias due to confounding between an exposure, mediator and outcome and measurement error. Mendelian randomisation (MR) can be used to improve causal inference for mediation analysis. We describe two approaches that can be used for estimating mediation analysis with MR: multivariable MR (MVMR) and two-step MR. We outline the approaches and provide code to demonstrate how they can be used in mediation analysis. We review issues that can affect analyses, including confounding, measurement error, weak instrument bias, interactions between exposures and mediators and analysis of multiple mediators. Description of the methods is supplemented by simulated and real data examples. Although MR relies on large sample sizes and strong assumptions, such as having strong instruments and no horizontally pleiotropic pathways, our simulations demonstrate that these methods are unaffected by confounders of the exposure or mediator and the outcome and non-differential measurement error of the exposure or mediator. Both MVMR and two-step MR can be implemented in both individual-level MR and summary data MR. MR mediation methods require different assumptions to be made, compared with non-instrumental variable mediation methods. Where these assumptions are more plausible, MR can be used to improve causal inference in mediation analysis.

This paper is concerned with the identification problem for multivariable equation-error systems with autoregressive moving average noise using the hierarchical identification principle and the multi-innovation identification theory. We propose a hierarchical gradient-based iterative (HGI) identification algorithm and give a gradient-based iterative (GI) identification algorithm for comparison. Meanwhile, the multi-innovation theory is used to derive the hierarchical multi-innovation gradient-based iterative (HMIGI) identification algorithm. The analysis shows that the HGI algorithm has smaller computational burden and can give more accurate parameter estimates than the GI algorithm and the HMIGI algorithm can track time-varying parameters. Finally, a simulation example is provided to verify the effectiveness of the proposed algorithms.

For a multivariable system with moving average noise (i.e., a multivariable controlled autoregressive moving average system), this paper proposes a filtering based extended stochastic gradient (ESG) algorithm and a filtering based multi-innovation ESG algorithm for improving the parameter estimation accuracy. The key is using the filtering technique and the multi-innovation identification theory. The proposed algorithms can identify the parameters of the system model and the noise model. The filtering based multi-innovation ESG algorithm can give more accurate parameter estimates. The numerical simulation results demonstrate that the proposed algorithms work well.

Groundwater (GW) is the primary water source of socio‐economic development in water‐deficient regions, and long‐term overexploitation may cause GW depletion and deterioration. In this study, after analyzing the relationship between GW level and related factors, the main influencing factors were identified from the perspective of climate change and human activity. A novel and comprehensive prediction method for GW depth was developed by combining the multivariable controlled auto‐regressive model and copula functions. The capabilities of the proposed method extend beyond GW depth predictions in the plain area, as it also quantitatively assesses the probability of GW depth variation. The method was validated by using it to simulate GW depth in Beijing of China during 2019–2022, and the results indicate that the errors between simulated and observed GW depth are less than 1.5%. The Beijing's GW depth is likely to be gradually recovery with an increase in precipitation and cross‐regional water diversion in the future. The probability of Beijing's GW depth reaching 7.50 m by 2035 is 0.463 under annual average precipitation, ETa and inbound runoff. This study provides an effective method to predict GW depth variation and its probability in plain areas, and it also offers valuable insight for the protection and sustainable development of regional GW resources. Key Points A novel and comprehensive prediction method for groundwater (GW) depth in the plain region is established by combining the multivariable controlled auto‐regressive model and copula functions The developed method can not only predict the yearly variation of GW depth in the future, but also quantitatively provide its probability The application results reveal that the GW depth in Beijing is likely to be recovery with an increase in precipitation in the future

In power systems, although the inertia energy in power sources can partly cover power unbalances caused by load disturbance or renewable energy fluctuation, it is still hard to maintain the frequency deviation within acceptable ranges. However, with the vehicle-to-grid (V2G) technique, electric vehicles (EVs) can act as mobile energy storage units, which could be a solution for load frequency control (LFC) in an isolated grid. In this paper, a LFC model of an isolated micro-grid with EVs, distributed generations and their constraints is developed. In addition, a controller based on multivariable generalized predictive control (MGPC) theory is proposed for LFC in the isolated micro-grid, where EVs and diesel generator (DG) are coordinated to achieve a satisfied performance on load frequency. A benchmark isolated micro-grid with EVs, DG, and wind farm is modeled in the Matlab/Simulink environment to demonstrate the effectiveness of the proposed method. Simulation results demonstrate that with MGPC, the energy stored in EVs can be managed intelligently according to LFC requirement. This improves the system frequency stability with complex operation situations including the random renewable energy resource and the continuous load disturbances.

Regression models have been in use for decades to explore and quantify the association between a dependent response and several independent variables in environmental sciences, epidemiology and public health. However, researchers often encounter situations in which some independent variables exhibit high bivariate correlation, or may even be collinear. Improper statistical handling of this situation will most certainly generate models of little or no practical use and misleading interpretations. By means of two example studies, we demonstrate how diagnostic tools for collinearity or near-collinearity may fail in guiding the analyst. Instead, the most appropriate way of handling collinearity should be driven by the research question at hand and, in particular, by the distinction between predictive or explanatory aims.

This paper proposes an adaptive stabilization control scheme for a class of multivariable interconnected nonlinear systems with nonlinear parametrization. It is a systematic result in a sense that the proposed control scheme is a general one that also applies to systems with linear parametrization, without changing the controller structure. A novel integrated framework is built by means of a functional bounding technique for handling nonlinear parameter/structure uncertainties, a modified backstepping method for designing continuous state-feedback controllers, and Lyapunov stability analysis for stabilizing interconnected system states and parameter estimates. A commonly physical simulation and a representative numerical simulation are presented, and their results demonstrate the effectiveness of the proposed control scheme.

This paper investigates the multivariable control of wastewater treatment processes (WWTP). This paper integrates deep reinforcement learning (DRL) with PID control and proposes a multivariable adaptive PID control strategy based on multi‐agent DRL (MADRL) for WWTP. The approach begins with the construction of a MADRL‐PID controller structure, consisting of an agent and a PID controller module. The agent adjusts the PID controller values while the PID module calculates the control signal. To enhance the agent's ability to cooperatively tune multiple PID controllers, the algorithm's components–reward function, action space, environment, and state space–are designed according to the BSM1 simulation platform principles and the MADRL framework requirements. Additionally, to handle WWTP's non‐linearities, uncertainties, and parameter coupling, the multi‐agent deep deterministic policy gradient algorithm is selected as the foundation for training the agents. Experimental results demonstrate that the proposed algorithm exhibits greater adaptability than traditional PID control and achieves superior control performance. This paper investigates the multivariable control of wastewater treatment processes. This paper integrates deep reinforcement learning (DRL) with PID control and proposes a multivariable adaptive PID control strategy based on multi‐agent DRL for wastewater treatment processes.