Parameter Identifiability Analysis Research Articles

A process-based model of nitrogen cycling in forest plantations: Part I. Structure, calibration and analysis of the decomposition model

We present a new decomposition model of C and N cycling in forest ecosystems that simulates N mineralisation from decomposing tree litter. It incorporates a mechanistic representation of the role of soil organisms in the N mineralisation-immobilisation turnover process during decomposition. We first calibrate the model using data from decomposition of 14C-labelled cellulose and lignin and 14C-labelled legume material and then calibrate and test it using mass loss and N loss data from decomposing Eucalyptus globulus residues. The model has been linked to the plant production submodel of the G’DAY ecosystem model, which previously used the CENTURY decomposition submodel for simulating C and N cycling. The key differences between this new decomposition model and the previous one, based on the CENTURY model, are: (1) growth of microbial biomass is the process that drives N mineralisation-immobilisation, and microbial succession is simulated; (2) decomposition of litter can be N-limited, depending on soil inorganic N availability relative to N requirements for microbial growth; (3) ‘quality’ of leaf and fine root litter is expressed in terms of biochemically measurable fractions; (4) the N:C ratio of microbial biomass active in decomposing litter is a function of litter quality and N availability; and (5) the N:C ratios of soil organic matter (SOM) pools are not prescribed but are instead simulated output variables defined by litter characteristics and soil inorganic N availability. With these modifications the model is able to provide reasonable estimates of both mass loss and N loss by decomposing E. globulus leaf and branch harvest residues in litterbag experiments. A sensitivity analysis of the decomposition model to selected parameters indicates that parameters regulating the stabilisation of organic C and N, as well as those describing incorporation of soil inorganic N in Young-SOM (biochemical immobilisation of N) are particularly critical for long-term applications of the model. A parameter identifiability analysis demonstrates that simulated short-term C and N loss from decomposing litter is highly sensitive to three model parameters that are identifiable from the E. globulus litterbag data.

A bio-physical coastal ecosystem model for assessing environmental effects of marine bivalve aquaculture

A simple lower trophic level, bio-physical marine ecosystem model is developed for the purpose of assessing the environmental effects of bivalve aquaculture in coastal embayments. The ecosystem box model includes pelagic and benthic components and describes the cycling of a most-limiting nutrient. The pelagic compartment is comprised of phytoplankton, zooplankton, nutrients and detritus. These populations interact following predator–prey dynamics and biogeochemical processes. Mixing processes within the bay, and exchange of waters with the adjacent open ocean, are included. The pelagic ecosystem is coupled to a simple benthos containing a dynamically active organic matter pool. Benthic–pelagic coupling includes episodic resuspension, remineralization, sinking, and permanent burial. A population of grazing bivalves is superimposed on this system as a diagnostic variable. The model is applied to a coastal bay and used to determine how bivalve populations affect nutrient cycling in the ecosystem. This is done by examining changes in the standing stock of the various populations, as well as associated nutrient (mass) fluxes, for cases both with and without intensive bivalve culture. It was demonstrated that bivalves divert production from the pelagic to benthic food webs. Phytoplankton and detritus are depleted from the water by bivalve filter feeding and biodeposited to the benthos as fecal matter. This organic loading causes order of magnitude changes in the benthic detrital pool and the associated benthic–pelagic fluxes. It was also shown that water motion and mixing is important in structuring the ecological dynamics in the bay. To faciliate future applications and observational studies, a retrospective analysis of parameter identifiability and uncertainty was also undertaken.

APPLICABILITY OF AN EXTANT BATCH RESPIROMETRIC ASSAY IN DESCRIBING DYNAMICS OF AMMONIA AND NITRITE OXIDATION IN A NITRIFYING BIOREACTOR

Several techniques have been proposed for biokinetic estimation of nitrification. Recently, an extant respirometric assay has been presented that yields kinetic parameters for both nitrification steps with minimal physiological change to the microorganisms during the assay. Herein, the ability of biokinetic parameter estimates from the extant respirometric assay to adequately describe concurrently obtained NH 4 + -N and NO 2 − -N substrate depletion profiles is evaluated. Based on our results, in general, the substrate depletion profiles resulted in a higher estimate of the maximum specific growth rate coefficient, μ max for both NH 4 + -N to NO 2 − -N oxidation and NO 2 − -N to NO 3 − -N oxidation compared to estimates from the extant respirograms. The trends in the kinetic parameter estimates from the different biokinetic estimation techniques are paralleled in the nature of substrate depletion profiles obtained from best-fit parameters. Based on a visual inspection, in general, best-fit parameters from optimally designed complete respirograms provided a better description of the substrate depletion profiles than estimates from isolated respirograms. Nevertheless, the sum of the squared errors for the best-fit respirometry based parameters was outside the 95% joint confidence interval computed for the best-fit substrate depletion based parameters. Notwithstanding the difference in kinetic parameter estimates determined in this study, the different biokinetic estimation techniques still are close to estimates reported in literature. Additional parameter identifiability and sensitivity analysis of parameters from substrate depletion assays revealed high precision of parameters and high parameter correlation. Although biokinetic estimation via automated extant respirometry is far more facile than via manual substrate depletion measurements, additional sensitivity analyses are needed to test the impact of differences in the resulting parameter values on continuous reactor performance.

Parametric identifiability of a structured kinetic model for mammalian cell cultures

Model parametric identifiability poses the question - "based on the mathematical behaviour of the model itself and assuming noise-free data - can all of the model parameters, in principle, be uniquely identified?" A class of parametric models, such as kinetic models, are used to capture the complex dynamics of cellular processes. However, increased model fidelity is often accompanied by an enlargement of the model parameter set and the expensive and time-consuming nature of mammalian cell culture experiments makes model parameter identifiability especially valuable as the development of unidentifiable models can be halted prior to experimental validation stages, in turn saving time and valuable resources. In this work, parameter identifiability analysis was performed on a state-of-the-art cell culture model described by a large differential algebraic equation system containing 200 equations and over 300 parameters. However, both the size and properties of this very large model has meant that a "practical identifiability" or estimability method had to be applied.

Identifiability and retrievability of unique parameters describing intrinsic Andrews kinetics

A key factor contributing to the variability in the microbial kinetic parameters reported from batch assays is parameter identifiability, i.e., the ability of the mathematical routine used for parameter estimation to provide unique estimates of the individual parameter values. This work encompassed a three-part evaluation of the parameter identifiability of intrinsic kinetic parameters describing the Andrews growth model that are obtained from batch assays. First, a parameter identifiability analysis was conducted by visually inspecting the sensitivity equations for the Andrews growth model. Second, the practical retrievability of the parameters in the presence of experimental error was evaluated for the parameter estimation routine used. Third, the results of these analyses were tested using an example data set from the literature for a self-inhibitory substrate. The general trends from these analyses were consistent and indicated that it is very difficult, if not impossible, to simultaneously obtain a unique set of estimates of intrinsic kinetic parameters for the Andrews growth model using data from a single batch experiment.

Toward Improved Identifiability of Soil Hydraulic Parameters: On the Selection of a Suitable Parametric Model

We present a thorough identifiability analysis of the soil hydraulic parameters in the parametric models of Brooks and Corey (BC; Brooks and Corey, 1964), Mualem–van Genuchten (VG; van Genuchten, 1980), and Kosugi (KC; Kosugi, 1996, 1999) using the recently developed Shuffled Complex Evolution Metropolis (SCEM‐UA) algorithm (Vrugt et al., 2002b, and unpublished data). Because the SCEM‐UA algorithm globally thoroughly exploits the parameter space and therefore explicitly accounts for parameter interdependence and nonlinearity of the employed parametric models, the algorithm is suited to generate a useful description of parameter uncertainty and its antithesis, parameter identifiability. A set of measured water retention characteristics of the UNSODA database (Leij et al., 1996) spanning a wide range of soil textures and three transient laboratory outflow experiments with decreasing flow rates were used to illustrate that a parameter identifiability analysis facilitates the selection of an adequate parametric model structure and provides useful information about the limitations of a model. Moreover, results suggest that one should be especially careful in establishing pedotransfer functions without knowledge of the underlying posterior uncertainty associated with the soil hydraulic parameters using direct estimation methods.

Toward Improved Identifiability of Soil Hydraulic Parameters: On the Selection of a Suitable Parametric Model

We present a thorough identifiability analysis of the soil hydraulic parameters in the parametric models of Brooks and Corey (BC; Brooks and Corey, 1964), Mualem-van Genuchten (VG; van Genuchten, 1980), and Kosugi (KC; Kosugi, 1996, 1999) using the recently developed Shuffled Complex Evolution Metropolis (SCEM-UA) algorithm (Vrugt et al., 2002b, and unpublished data). Because the SCEM-UA algorithm globally thoroughly exploits the parameter space and therefore explicitly accounts for parameter interdependence and nonlinearity of the employed parametric models, the algorithm is suited to generate a useful description of parameter uncertainty and its antithesis, parameter identifiability. A set of measured water retention characteristics of the UNSODA database (Leij et al., 1996) spanning a wide range of soil textures and three transient laboratory outflow experiments with decreasing flow rates were used to illustrate that a parameter identifiability analysis facilitates the selection of an adequate parametric model structure and provides useful information about the limitations of a model. Moreover, results suggest that one should be especially careful in establishing pedotransfer functions without knowledge of the underlying posterior uncertainty associated with the soil hydraulic parameters using direct estimation methods.

Toward Improved Identifiability of Soil Hydraulic Parameters

We present a thorough identifiability analysis of the soil hydraulic parameters in the parametric models of Brooks and Corey (BC; Brooks and Corey, 1964), Mualem–van Genuchten (VG; van Genuchten, 1980), and Kosugi (KC; Kosugi, 1996, 1999) using the recently developed Shuffled Complex Evolution Metropolis (SCEM-UA) algorithm (Vrugt et al., 2002b, and unpublished data). Because the SCEM-UA algorithm globally thoroughly exploits the parameter space and therefore explicitly accounts for parameter interdependence and nonlinearity of the employed parametric models, the algorithm is suited to generate a useful description of parameter uncertainty and its antithesis, parameter identifiability. A set of measured water retention characteristics of the UNSODA database (Leij et al., 1996) spanning a wide range of soil textures and three transient laboratory outflow experiments with decreasing flow rates were used to illustrate that a parameter identifiability analysis facilitates the selection of an adequate parametric model structure and provides useful information about the limitations of a model. Moreover, results suggest that one should be especially careful in establishing pedotransfer functions without knowledge of the underlying posterior uncertainty associated with the soil hydraulic parameters using direct estimation methods.

Practical identifiability of ASM2d parameters—systematic selection and tuning of parameter subsets

In many applications, some parameters of the Activated Sludge Model No. 2d (ASM2d) need calibration. Since ASM2d usually is overparameterized with respect to the available data, the subset of calibration parameters is not unique. In practice, calibration of ASM2d (and other ASMs) is often addressed by ad hoc selecting and tuning procedures. In this paper, a more systematic approach based on parameter identifiability analysis of parameter subsets is applied. The approach consists of a preliminary prior parameter analysis and a subsequent iterative parameter subset selection and tuning procedure. The former includes the choice of suitable prior parameter values and uncertainties and a pre-selection of parameters which are reasonably estimated from the data available. The latter is based on three diagnostic measures which are simple to calculate and easy to interpret. It is demonstrated as to how these measures can be used to identify the most important model parameters and to analyze their interdependencies. In addition, it is shown how these measures facilitate the analysis of the influence of fixed parameter values on parameter estimates.

Practical identifiability analysis of large environmental simulation models

Large environmental simulation models are usually overparameterized with respect to given sets of observations. This results in poorly identifiable or nonidentifiable model parameters. For small models, plots of sensitivity functions have proven to be useful for the analysis of parameter identifiability. For models with many parameters, however, near‐linear dependence of sensitivity functions can no longer be assessed graphically. In this paper a systematic approach for tackling the parameter identifiability problem of large models based on local sensitivity analysis is presented. The calculation of two identifiability measures that are easy to handle and interpret is suggested. The first accounts for the sensitivity of model results to single parameters, and the second accounts for the degree of near‐linear dependence of sensitivity functions of parameter subsets. It is shown how these measures provide identifiability diagnosis for parameter subsets, how they are able to guide the selection of identifiable parameter subsets for parameter estimation, and how they facilitate the interpretation of the correlation matrix of the parameter estimate with respect to parameter identifiability. In addition, we show how potential bias of the parameter estimates, due to a priori fixing of some of the parameters, can be analyzed. Finally, two case studies are presented in order to illustrate the suggested approach.

Applicability of two-step models in estimating nitrification kinetics from batch respirograms under different relative dynamics of ammonia and nitrite oxidation.

A mechanistically based nitrification model was formulated to facilitate determination of both NH(4)(+)-N to NO(2)(-)-N and NO(2)(-)-N to NO(3)(-)-N oxidation kinetics from a single NH(4)(+)-N to NO(3)(-)-N batch-oxidation profile by explicitly considering the kinetics of each oxidation step. The developed model incorporated a novel convention for expressing the concentrations of nitrogen species in terms of their nitrogenous oxygen demand (NOD). Stoichiometric coefficients relating nitrogen removal, oxygen uptake, and biomass synthesis were derived from an electron-balanced equation.%A parameter identifiability analysis of the developed two-step model revealed a decrease in correlation and an increase in the precision of the kinetic parameter estimates when NO(2)(-)-N oxidation kinetics became increasingly rate-limiting. These findings demonstrate that two-step models describe nitrification kinetics adequately only when NH(4)(+)-N to NO(3)(-)-N oxidation profiles contain sufficient information pertaining to both nitrification steps. Thus, the rate-determining step in overall nitrification must be identified before applying conventionally used models to describe batch nitrification respirograms.

Kalman filtering in groundwater flow modelling: problems and prospects

The popularity of applying filtering theory in the environmental and hydrological sciences passed its first climax in the 1970s. Like so many other new mathematical methods it was simply the fashion at the time. The study of groundwater systems was not immune to this fashion, but neither was it by any means a prominent area of application. The spatial-temporal characteristics of groundwater flow are customarily described by analytical or, more frequently, numerical, physics-based models. Consequently, the state-space representations associated with filtering must be of a high order, with an immediately apparent computational over-burden. And therein lies part of the reason for the but modest interest there has been in applying Kalman filtering to groundwater systems, as reviewed critically in this paper. Filtering theory may be used to address a variety of problems, such as: state estimation and reconstruction, parameter estimation (including the study of uncertainty and its propagation), combined state-parameter estimation, input estimation, estimation of the variance-covariance properties of stochastic disturbances, the design of observation networks, and the analysis of parameter identifiability. A large proportion of previous studies has dealt with the problem of parameter estimation in one form or another. This may well not remain the focus of attention in the future. Instead, filtering theory may find wider application in the context of data assimilation, that is, in reconstructing fields of flow and the migration of sub-surface contaminant plumes from relatively sparse observations.

Parameter estimation of a monod-type model. Part II: Use of a neural network with associative memories

On the basis of a previous study related with parameter identifiability and sensitivity analysis of a Monod-type model, a parameter estimation method based on Artificial Neural Networks (ANN) with Associative Memories (AMs) is presented. A combination of an iterative procedure and a convergence index given by AMs allows to confirm the nature of relations existing between state variables and parameters which were found in the first part of the study. The convergence criterion is particularly well adapted to showing various influences of state variables on parameter estimation of such a model.

Parameter identifiability problem in signal processing

The authors give a broader perspective on the problem addressed by Choate and Tsui (1993). The parameter identifiability analysis in that paper is significantly simplified and extended. In addition, it is pointed out that the 'new' Prony's method discussed in that paper is nothing but the well-known forward-backward Prony's approach.

A Model of Leucine Kinetics: A Priori Identifiability and Minimal Input-output Configuration

The minimal input-output configuration necessary to identify a model of the kinetics of the aminoacid leucine from tracer data in vivo is studied. The experimental configuration originally designed for model identification consisted of two separate tracer experiments, a single input-single output and a two input-five output respectively. The proposed compartmental model is linear, time invariant and has ten compartments. A priori parameter identifiability analysis shows that the model can be identified with a reduced experimental effort: the single input-single output experiment is not necessary and, in addition, only four outputs instead of five are necessary and sufficient in the other experiment.

Identifiability analysis of an in vivo receptor-binding radiopharmacokinetic system.

A five-state, nonlinear model describing the pharmacokinetics of a receptor-binding radiopharmaceutical is presented. Local parameter identifiability analysis was performed preexperimentally to investigate the chemical and observational conditions required for adequate estimates of hepatic blood flow, receptor-ligand affinity, and receptor population. The ability to estimate the above parameters from externally observed kinetic data depended upon receptor-ligand affinity, the molar dose of the radiopharmaceutical, the number and coupling of the observers, and independent knowledge of the coupling coefficients or ligand-receptor affinity.

Identification of phosphorus dynamics in Kootenay Lake, Canada

This paper regards the use of system theory techniques in the development of models describing the phosphorus kinetics in fresh waters with the objective to understand the observed kinetics better in order to obtain indications on the design of future field experiments. Identification techniques have been applied to the data regarding 49 experiments on phosphorus cycling in Kootenay Lake, British Columbia, Canada, in order to determine the minimal number of compartments required to model the phosphorus dynamical behaviour. Canonical state-space models have then been identified both from raw and smoothed data. The parameters in these models do not have a direct physical meaning; they can, however, be identified from the measured data and the obtained models describe with good accuracy the considered kinetics. The identification of physical models has also been performed; in this case, however, only two compartments have been considered according to the results of the identifiability analysis of the complete model parameters. Results give further insight on the phosphorus processes and show that the complete phosphorus cycle through the four compartments described by Lean can be identified only when phosphorus cycles rapidly through the system.

Structural identifiability in linear time-invariant systems

A matrix-operator representation of parameter sensitivities is used to provide an algebraic "structural" analysis of local parameter identifiability in linear time-invariant ordinary differential equation systems. Necessary conditions for identifiability depend only upon the system matrices, no integrals must be computed, and arbitrary parametrization may be used. Relations to insensitivity are discussed, and design techniques are suggested which use the nonidentifiable (insensitive) subspace to systematically reduce the number of exciting parameters in individually designed parameter identification experiments. Finally, sufficiency conditions for zero-state identifiability are examined in terms of control inputs which continually excite the natural modes of the parameter sensitivities.