Extended State Dependent Parameter modelling with a Data-Based Mechanistic approach to nonlinear model structure identification

Optimal PIP Control of a pH Neutralization Process Based on State-Dependent Parameter Model

This note compares and contrasts the non-linear parameter varying (NLPV) and state-dependent parameter (SDP) model classes. It shows that, while they have similarities, the two-stage SDP modelling procedure, involving non-parametric identification, followed by parametric estimation, is quite different from the single stage NLPV procedure. In particular, the SDP procedure allows for the identification of the model structure and the nature of the non-linearities, prior to the estimation of the parameters that characterize this identified model structure. In contrast to NLPV modelling, therefore, SDP estimation opens up the ‘black box’ and reveals the inner nature of the non-linear system.

<p>Recent investigations have shown it is possible to simultaneously calibrate model structures and model parameters to identify appropriate models for a given task (Spieler et al., 2020). However, this is computationally challenging, as different model structures may use a different number of parameters. While some parameters may be shared between model structures, others might be relevant for only a few structures, which theoretically requires the calibration of conditionally active parameters. Additionally, shared model parameters might cause different effects in different model structures, causing their optimal values to differ across structures. In this study, we tested how two current “of the shelf” mixed-integer optimization algorithms perform when having to handle these peculiarities during the automatic model structure identification (AMSI) process recently introduced by Spieler et al. (2020).</p><p>To validate the current performance of the AMSI approach, we conduct a benchmark experiment with a model space consisting of 6912 different model structures.  First, all model structures are independently calibrated and validated for three hydro-climatically differing catchments using the CMA-ES algorithm and KGE as the objective function. This is referred to as standard calibration procedure. We identify the best performing model structure(s) based on validation performance and analyze the range of performance as well as the number of structures performing in a similar range. Secondly, we run AMSI on all three catchments to automatically identify the most feasible model structure based on the KGE performance. Two different mixed-integer optimization algorithms are used – namely DDS and CMA-ES. Afterwards, we compare the results to the best performing models of the standard calibration of all 6912 model structures.</p><p>Within this experimental setup, we analyze if the best performing model structure(s) AMSI identifies are identical to the best performing structures of the standard calibration and if there are differences in performance when using different optimization algorithms for AMSI. We also validate if AMSI can identify the best performing model structures for a catchment at a fraction of the computational cost than the standard calibration procedure requires by using “off the shelf” mixed-integer optimization algorithms.</p><p> </p><p> </p><p> </p><p>Spieler, D., Mai, J., Craig, J. R., Tolson, B. A., & Schütze, N. (2020). Automatic Model Structure Identification for Conceptual Hydrologic Models. Water Resources Research, 56(9). https://doi.org/10.1029/2019WR027009</p>

<p>Recent studies have introduced methods to simultaneously calibrate model structure choices and parameter values to identify an appropriate (conceptual) model structure for a given catchment. This can be done through mixed-integer optimization to identify the graph structure that links dominant flow processes (Spieler et al., 2020) or, likewise, by continuous optimization of weights when blending multiple flux equations to describe flow processes within a model (Chlumsky et al., 2021). Here, we use the combination of the mixed-integer optimization algorithm DDS and the modular modelling framework RAVEN and refer to it as Automatic Model Structure Identification (AMSI) framework.</p><p>This study validates the AMSI framework by comparing the performance of the identified AMSI model structures to two different benchmark ensembles. The first ensemble consists of the best model structures from the brute force calibration of all possible structures included in the AMSI model space (7488+). The second ensemble consists of 35+ MARRMoT structures representing a structurally more divers set of models than currently implemented in the AMSI framework. These structures stem from the MARRMoT Toolbox introduced by Knoben et al. (2019) providing established conceptual model structures based on hydrologic literature.</p><p>We analyze if the model structure(s) AMSI identifies are identical to the best performing structures of the brute force calibration and comparable in their performance to the MARRMoT ensemble. We can conclude that model structures identified with the AMSI framework can compete with the structurally more divers MARRMoT ensemble. In fact, we were surprised to see how well we do with a simple two storage structure over the 12 tested MOPEX catchments (Duan et al.,2006). We aim to discuss several emerging questions, such as the selection of a robust model structure, Equifinality in model structures, and the role of structural complexity.</p><p> </p><p>Spieler et al. (2020). https://doi.org/10.1029/2019WR027009</p><p>Chlumsky et al. (2021). https://doi.org/10.1029/2020WR029229</p><p>Knoben et al. (2019). https://doi.org/10.5194/gmd-12-2463-2019</p><p>Duan et al. (2006). https://doi.org/10.1016/j.jhydrol.2005.07.031</p>

CAPTAIN is a MATLAB compatible toolbox for non-stationary time series analysis, system identification, signal processing and forecasting, using unobserved components models, time variable parameter models, state dependent parameter models and multiple input transfer function models. CAPTAIN also includes functions for true digital control.

This paper considers the shaping of the amplitude spectra of perturbation signals for the identification of a thermostat system. The current approach in control engineering practice utilizes flat spectrum signals, which may not result in the highest possible accuracy. This research aims to investigate the effectiveness of optimal signals with amplitude spectra designed using two state-of-the-art software approaches, namely the model-based optimal signal excitation 2 (MOOSE2) design and the optimal excitation (optexcit) design, in improving estimation accuracy. Such a comparison on a real system is currently lacking. In particular, there exists a research gap on how the combined choice of signal and model structure affects performance measures. In this research, two model structures are used, which are the autoregressive with exogenous input (ARX) and the output error (OE) model structures. Four performance measures are compared, namely the determinant of the covariance matrix of the parameter estimates and the minimum error, mean error and maximum error in the frequency response. Results show that the optimal signals are effective in reducing the determinant of the covariance matrix and the maximum error in the frequency response for the thermostat system, when applied in combination with the ARX model structure. The flat spectrum signal remains very useful as a general broadband perturbation signal as it provides a good overall fit of the frequency response. The findings from this work highlight the benefits of applying optimal signals especially if the identification results are to be used for control, since these signals improve key performance measures which have direct implications on controller design.

A data based mechanistic approach to nonlinear flood routing and adaptive flood level forecasting

To overcome the disadvantage of traditional variance-based importance measures, i.e. the effects of different realizations of input variables on output response may mutually counteract each other, a modified variance-based importance measure is presented for importance analysis of the input variables. The proposed measure analyses the importance of the input variables comprehensively in terms of the expectation and variance of the output response. Compared with the traditional variance-based importance analysis method, the modified importance measure indices not only reflect the old one, but also provide a very useful supplement for it. Furthermore, combined with the advantages of the state dependent parameter model, a solution to the proposed measure indices is provided. Several examples are introduced to show that the modified importance measure is more comprehensive and reasonable, and the solution based on the state dependent parameter method can improve computational efficiency considerably with acceptable precision.

The problems of identifying the most appropriate model structure for a given problem and quantifying the uncertainty in model structure remain outstanding research challenges for the discipline of hydrology. Progress on these problems requires understanding of the nature of differences between models. This paper presents a methodology to diagnose differences in hydrological model structures: the Framework for Understanding Structural Errors (FUSE). FUSE was used to construct 79 unique model structures by combining components of 4 existing hydrological models. These new models were used to simulate streamflow in two of the basins used in the Model Parameter Estimation Experiment (MOPEX): the Guadalupe River (Texas) and the French Broad River (North Carolina). Results show that the new models produced simulations of streamflow that were at least as good as the simulations produced by the models that participated in the MOPEX experiment. Our initial application of the FUSE method for the Guadalupe River exposed relationships between model structure and model performance, suggesting that the choice of model structure is just as important as the choice of model parameters. However, further work is needed to evaluate model simulations using multiple criteria to diagnose the relative importance of model structural differences in various climate regimes and to assess the amount of independent information in each of the models. This work will be crucial to both identifying the most appropriate model structure for a given problem and quantifying the uncertainty in model structure. To facilitate research on these problems, the FORTRAN‐90 source code for FUSE is available upon request from the lead author.

This study presents a goal programming approach to determine the best self-drive tourism route in Terengganu. The goal programming was formulated based on the model of Traveling Salesman Problem (TSP). This study involved seven cities in Terengganu which can be connected by road network. The data of the tourist attractions for each city was used in this study. The model of self-drive tourism route was constructed so as to connect all these seven cities by minimizing the travel distance, minimizing accommodation costs and maximizing the total number of tourist attractions. To achieve the objectives of this study, we develop the goal programming model that considers three model structures with each of them consists three goal constraints which are subjected to four hard constraints. This model has been solved using LINGO12.0 software. The results obtained from each model structures will be compared in order to get the most optimum route for self-drive tourism. The result showed that the first model structure was suitable to be used because this model structure gave the optimum result compared to other model structures. It produced the minimum total traveling cost of RM967.70 compared to the model structure two of RM977.10 and model structure three of RM1233.10.

Although the nonlinear power form model structure is widely accepted by practitioners in the flood regionalization modelling, there is a lack of studies on whether there is a room for further improvement, and if the answer is yes, what should be done to explore alternative model structures. A framework is proposed in this study towards investigating this issue by the following steps: (i) a universal data‐driven model is utilized to see if there is a room for improvement compared with the conventional model, and (ii) if improvement is achieved, this means that there should exist more effective model structures than the current form. However, because the universal data‐driven models are usually opaque, more explicit model structures should be explored, which are convenient for practical usage. In this study, the proposed framework is applied in a case study using the catchment characteristics from the Flood Estimation Handbook in conjunction with the gamma test, support vector machine (SVM) and genetic programming (GP). First, the gamma test is used for the purpose of input variables selection where no model structure needs to be defined as a priori, and therefore, the result can be applied to any model structures for model building. Second, an SVM, which is a powerful data‐driven nonlinear model capable of modelling a variety of nonlinear systems, is applied to the index flood model for the first time. Once the best model is determined using those two data‐driven tools, GP is employed to find an alternative model structure. As the SVM is not formulated for producing explicit model functional form, the GP offers an advantage at this point where it can infer an explicit mathematical model functional form. Copyright © 2012 John Wiley & Sons, Ltd.

<p>Lumped karst hydrological models often suffer from the over-simplified structures. In recent days, hydrochemical has been used as an auxiliary information to define realistic karst model structures. For karst aquifers, the hydrochemical dynamics of karst springs contain important information about the internal behavior of the karst aquifer that can be conducive to model identification and calibration. In this presentation, electrical conductivity (EC), as a substitution for Ca<sup>2+</sup> concentration of the spring discharge, is evaluated for its potential for model structure identification and reduction of simulation uncertainty. A new framework to integrate EC in karst modeling is developed at a small, well-instrumented karst catchment near Guilin City, China. A set of different hydrologic models were enabled to consider the linear dissolution process of Ca<sup>2+</sup> and its transport to investigate the models’ abilities to reproduce the behavior of spring discharge and EC. We found that most hydrologic models obtain similar performances concerning spring discharge, however, their performances in simulating spring EC show obvious differences. The combination of observed spring discharge and EC could identify a more realistic model structure that was in accordance with the observed perception of this karst aquifer’s functioning. Using the identified “most realistic” model, we use sensitivity analysis to show that spring EC only improves the identifiability of one hydrologic parameter mostly due to too complex EC dynamics during recharge events and therefore has limited potential to reduce discharge simulation uncertainty. Consequently, our new framework to include EC in karst models opens new doors for more realistic simulation but an explicit treatment of uncertainties remains necessary due to EC’s limited potential to reduce simulation uncertainty. The next step of our work is to integrate appropriate nonlinear dissolution process of carbonate rock in the model to further improve the simulation of spring EC (Ca<sup>2+</sup>).</p>

The paper presents a new recursive estimation algorithm designed expressly for the purpose of model structure identification (not for state estimation or primarily for parameter estimation) and discusses two applications thereof, one to a motivating, hypothetical example and one to data from whole‐pond manipulations designed to explore sediment‐nutrient‐phytoplankton dynamics. The algorithm is the current culmination of a long‐term technical development from state estimation using a Kalman filter, through state parameter estimation using an extended Kalman filter, through a recursive prediction error (RPE) algorithm for parameter estimation cast in the state space and recently modified for estimating time‐varying model parameters, to an RPE algorithm for estimating time‐varying parameters but cast in a parameter space formulation. It is concluded that the algorithm performs well, in the sense of being robust and indeed in revealing specifically where (but less so exactly how) a prior candidate model's structure may be in error.

Multi-model studies are widespread in large-sample hydrology. However, significant challenges remain in identifying interpretable connections between high-performing model structures and catchment characteristics, and thus in developing a coherent strategy for developing tailored multi-model ensembles. Here, we assess the importance of selecting model structures that are consistent with the expected hydrological variability across the study domain. We compare results of two modular modelling frameworks across 998 catchments in Great Britain. The RRMT framework includes model structures historically evolved in the UK, while the FUSE framework employs model structures from diverse global origins. While both groups of model structures contain high-performing members, the historically evolved group members separate between catchments in line with our expectation of hydrologic differences. We find that four hydrologic signatures organize these distinctions. Our results emphasize (1) the importance of model structure selection based on explicit perceptual models, and (2) the need to look beyond statistical performance alone.

Simulations of human behavior in water resources systems are challenged by uncertainty in model structure and parameters. The increasing availability of observations describing these systems provides the opportunity to infer a set of plausible model structures using data‐driven approaches. This study develops a three‐phase approach to the inference of model structures and parameterizations from data: problem definition, model generation, and model evaluation, illustrated on a case study of land use decisions in the Tulare Basin, California. We encode the generalized decision problem as an arbitrary mapping from a high‐dimensional data space to the action of interest and use multiobjective genetic programming to search over a family of functions that perform this mapping for both regression and classification tasks. To facilitate the discovery of models that are both realistic and interpretable, the algorithm selects model structures based on multiobjective optimization of (1) their performance on a training set and (2) complexity, measured by the number of variables, constants, and operations composing the model. After training, optimal model structures are further evaluated according to their ability to generalize to held‐out test data and clustered based on their performance, complexity, and generalization properties. Finally, we diagnose the causes of good and bad generalization by performing sensitivity analysis across model inputs and within model clusters. This study serves as a template to inform and automate the problem‐dependent task of constructing robust data‐driven model structures to describe human behavior in water resources systems.