In this paper, an algorithm of multiple signal classification (MUSIC) is proposed for two-dimensional (2-D) direction of- arrival (DOA) and polarization estimation of non-circular signal in three-dimensional (3-D) millimeter wave polarized largescale/ massive multiple-input-multiple-output (MIMO) systems. The traditional MUSIC-based algorithms can estimate either the DOA and polarization for circular signal or the DOA for non-circular signal by using spectrum search. By contrast, in the proposed algorithm only the DOA estimation needs spectrum search, and the polarization estimation has a closedform expression. First, a novel dimension-reduced MUSIC (DRMUSIC) is proposed for DOA and polarization estimation of circular signal with low computational complexity. Next, based on the quaternion theory, a novel algorithm named quaternion non-circular MUSIC (QNC-MUSIC) is proposed for parameter estimation of non-circular signal with high estimation accuracy. Then based on the DOA estimation result using QNC-MUSIC, the polarization estimation of non-circular signal is acquired by using the closed-form expression of the polarization estimation in DR-MUSIC. In addition, the computational complexity analysis shows that compared with the conventional DOA and polarization estimation algorithms, our proposed QNC-MUSIC and DRMUSIC have much lower computational complexity, especially when the source number is large. The stochastic Cramer-Rao Bound (CRB) for the estimation of the 2-D DOA and polarization parameters of the non-circular signals is derived as well. Finally, numerical examples are provided to demonstrate that the proposed algorithms can improve the parameter estimation performance when the large-scale/massive MIMO systems are employed.
To investigate the complex dynamics of a biological neuron that is subject to small random perturbations we can use stochastic neuron models. While many techniques have already been developed to study properties of such models, especially the analysis of the (expected) first-passage time or (E)FPT remains difficult. In this thesis I apply the large deviation theory (LDT), which is already well-established in physics and finance, to the problem of determining the EFPT of the mean-reverting Ornstein-Uhlenbeck (OU) process. The OU process instantiates the Stochastic Leaky Integrate and Fire model and thus serves as an example of a biologically inspired mathematical neuron model. I derive several classical results using much simpler mathematics than the original publications from neuroscience and I provide a few conceivable interpretations and perspectives on these derivations. Using these results I explore some possible applications for parameter estimation and I provide an additional mathematical justification for using a Poisson process as a small-noise approximation of the full model. Finally I perform several simulations to verify these results and to reveal systematic biases of this estimator.
We present a computational parameter estimation tool aimed at assisting molecule characterization applying nonlinear least-squares optimization to a specific closed-form solution of a capillary electrophoretic transport problem. Numerical estimates for the respective physicochemical parameters are generated. Among other features, the tool facilitates automated and user-controllable hypothesis testing and model selection. It can also be used to perform fully automated batch-processing in true concurrent mode.
The accuracy of least squares calibration using option premiums and particle filtering of price data to find model parameters is determined. Derivative models using exponential L\'evy processes are calibrated using regularized weighted least squares with respect to the minimal entropy martingale measure. Sequential importance resampling is used for the Bayesian inference problem of time series parameter estimation with proposal distribution determined using extended Kalman filter. The algorithms converge to their respective global optima using a highly parallelizable statistical optimization approach using a grid of initial positions. Each of these methods should produce the same parameters. We investigate this assertion.
In this paper, a concurrent learning based adaptive observer is developed for a class of second-order nonlinear time-invariant systems with uncertain dynamics. The developed technique results in simultaneous online state and parameter estimation. A Lyapunov-based analysis is used to show that the state and parameter estimation errors are uniformly ultimately bounded. As opposed to persistent excitation which is required for parameter estimation in traditional adaptive control methods, the developed technique only requires excitation over a finite time interval.
Optical biosensors are often used to measure kinetic rate constants associated with chemical reactions. Such instruments operate in the \textit{surface-volume} configuration, in which ligand molecules are convected through a fluid-filled volume over a surface to which receptors are confined. Currently, scientists are using optical biosenors to measure the kinetic rate constants associated with DNA translesion synthesis--a process critical to DNA damage repair. Biosensor experiments to study this process involve multiple interacting components on the sensor surface. This multiple-component biosensor experiment is modeled with a set of nonlinear integrodifferential equations (IDEs). It is shown that in physically relevant asymptotic limits these equations reduce to a much simpler set of Ordinary Differential Equations (ODEs). To verify the validity of our ODE approximation, a numerical method for the IDE system is developed and studied. Results from the ODE model agree with simulations of the IDE model, rendering our ODE model useful for parameter estimation.
We introduce a low dimensional function of the site frequency spectrum that is tailor-made for distinguishing coalescent models with multiple mergers from Kingman coalescent models with population growth, and use this function to construct a hypothesis test between these model classes. The null and alternative sampling distributions of the statistic are intractable, but its low dimensionality renders them amenable to Monte Carlo estimation. We construct kernel density estimates of the sampling distributions based on simulated data, and show that the resulting hypothesis test dramatically improves on the statistical power of a current state-of-the-art method. A key reason for this improvement is the use of multi-locus data, in particular averaging observed site frequency spectra across unlinked loci to reduce sampling variance. We also demonstrate the robustness of our method to nuisance and tuning parameters. Finally we show that the same kernel density estimates can be used to conduct parameter estimation, and argue that our method is readily generalisable for applications in model selection, parameter inference and experimental design.
Stochastic Boolean networks, or more generally, stochastic discrete networks, are an important class of computational models for molecular interaction networks. The stochasticity stems from the updating schedule. Standard updating schedules include the synchronous update, where all the nodes are updated at the same time, and the asynchronous update where a random node is updated at each time step. The former produces a deterministic dynamics while the latter a stochastic dynamics. A more general stochastic setting considers propensity parameters for updating each node. Stochastic Discrete Dynamical Systems (SDDS) is a modeling framework that considers two propensity parameters for updating each node and uses one when the update has a positive impact on the variable, that is, when the update causes the variable to increase its value, and uses the other when the update has a negative impact, that is, when the update causes it to decrease its value. This framework offers additional features for simulations but also adds a complexity in parameter estimation of the propensities. This paper presents a method for estimating the propensity parameters for SDDS. The method is based on adding noise to the system using the Google PageRank approach to make the system ergodic and thus guaranteeing the existence of a stationary distribution. Then with the use of a genetic algorithm, the propensity parameters are estimated. Approximation techniques that make the search algorithms efficient are also presented and Matlab/Octave code to test the algorithms are available at http://www.ms.uky.edu/~dmu228/GeneticAlg/Code.html.
This paper explores and develops alternative statistical representations and estimation approaches for dynamic mortality models. The framework we adopt is to reinterpret popular mortality models such as the Lee-Carter class of models in a general state-space modelling methodology, which allows modelling, estimation and forecasting of mortality under a unified framework. Furthermore, we propose an alternative class of model identification constraints which is more suited to statistical inference in filtering and parameter estimation settings based on maximization of the marginalized likelihood or in Bayesian inference. We then develop a novel class of Bayesian state-space models which incorporate apriori beliefs about the mortality model characteristics as well as for more flexible and appropriate assumptions relating to heteroscedasticity that present in observed mortality data. We show that multiple period and cohort effect can be cast under a state-space structure. To study long term mortality dynamics, we introduce stochastic volatility to the period effect. The estimation of the resulting stochastic volatility model of mortality is performed using a recent class of Monte Carlo procedure specifically designed for state and parameter estimation in Bayesian state-space models, known as the class of particle Markov chain Monte Carlo methods. We illustrate the framework we have developed using Danish male mortality data, and show that incorporating heteroscedasticity and stochastic volatility markedly improves model fit despite an increase of model complexity. Forecasting properties of the enhanced models are examined with long term and short term calibration periods on the reconstruction of life tables.
In cancer treatment, chemotherapy is administered according a constant schedule. The chronotherapy approach, considering chronobiological drug delivery, adapts the chemotherapy profile to the circadian rhythms of the human organism. This reduces toxicity effects and at the same time enhances efficiency of chemotherapy. To personalize cancer treatment, chemotherapy profiles have to be further adapted to individual patients. Therefore, we present a new model to represent cycle phenomena in circadian rhythms. The model enables a more precise modelling of the underlying circadian rhythms. In comparison with the standard model, our model delivers better results in all defined quality indices. The new model can be used to adapt the chemotherapy profile efficiently to individual patients. The adaption to individual patients contributes to the aim of personalizing cancer therapy.
Cardiac muscle tissue during relaxation is commonly modelled as a hyperelastic material with strongly nonlinear and anisotropic stress response. Adapting the behavior of such a model to experimental or patient data gives rise to a parameter estimation problem which involves a significant number of parameters. Gradient-based optimization algorithms provide a way to solve such nonlinear parameter estimation problems with relatively few iterations, but require the gradient of the objective functional with respect to the model parameters. This gradient has traditionally been obtained using finite differences, the calculation of which scales linearly with the number of model parameters, and introduces a differencing error. By using an automatically derived adjoint equation, we are able to calculate this gradient more efficiently, and with minimal implementation effort. We test this adjoint framework on a least squares fitting problem involving data from simple shear tests on cardiac tissue samples. A second challenge which arises in gradient-based optimization is the dependency of the algorithm on a suitable initial guess. We show how a multi-start procedure can alleviate this dependency. Finally, we provide estimates for the material parameters of the Holzapfel and Ogden strain energy law using finite element models together with experimental shear data.
The fitting or parameter estimation of complex ecological models is a challenging optimisation task, with a notable lack of tools for fitting complex, long runtime or stochastic models. calibrar is an R package that is dedicated to the fitting of complex models to data. It is a generic tool that can be used for any type of model, especially those with non-differentiable objective functions and long runtime, including Individual Based Models. calibrar supports multiple phases and constrained optimisation, includes 18 optimisation algorithms, including derivative-based and heuristic ones. It supports any type of parallelization, the restart of interrupted optimisations for long runtime models and the combination of different optimisation methods during the multiple phases of the calibration. User-level expertise in R is necessary to handle calibration experiments with calibrar, but there is no need to modify the model's code, which can be programmed in any language. It implements maximum likelihood estimation methods and automated construction of the objective function from simulated model outputs. For more experienced users, calibrar allows the implementation of user-defined objective functions. The package source code is fully accessible and can be installed directly from CRAN.
We explore the relationship among model fidelity, experimental design, and parameter estimation in sloppy models. We show that the approximate nature of mathematical models poses challenges for experimental design in sloppy models. In many models of complex biological processes it is unknown what are the relevant physics that must be included to explain collective behaviors. As a consequence, models are often overly complex, with many practically unidentifiable parameters. Furthermore, which details are relevant/irrelevant vary among potential experiments. By selecting complementary experiments, experimental design may inadvertently make details that were ommitted from the model become relevant. When this occurs, the model will fail to give a good fit to the data. We use a simple hyper-model of model error to quantify a model's inadequacy and apply it to two models of complex biological processes (EGFR signaling and DNA repair) with optimally selected experiments. We find that although parameters may be accurately estimated, the error in the model renders it less predictive than it was in the sloppy regime where model error is small. We introduce the concept of a \emph{sloppy system}--a sequence of models of increasing complexity that become sloppy in the limit of microscopic accuracy. We explore the limits of accurate parameter estimation in sloppy systems and argue that system identification better approached by considering a hierarchy of models of varying detail rather than focusing parameter estimation in a single model.
Ordinary differential equations (ODE) are widely used for modeling in Systems Biology. As most commonly only some of the kinetic parameters are measurable or precisely known, parameter estimation techniques are applied to parametrize the model to experimental data. A main challenge for the parameter estimation is the complexity of the parameter space, especially its high dimensionality and local minima. Parameter estimation techniques consist of an objective function, measuring how well a certain parameter set describes the experimental data, and an optimization algorithm that optimizes this objective function. A lot of effort has been spent on developing highly sophisticated optimization algorithms to cope with the complexity in the parameter space, but surprisingly few articles address the influence of the objective function on the computational complexity in finding global optima. We extend a recently developed multiple shooting for stochastic systems (MSS) objective function for parameter estimation of stochastic models and apply it to parameter estimation of ODE models. This MSS objective function treats the intervals between measurement points separately. This separate treatment allows the ODE trajectory to stay closer to the data and we show that it reduces the complexity of the parameter space. We use examples from Systems Biology, namely a Lotka-Volterra model, a FitzHugh-Nagumo oscillator and a Calcium oscillation model, to demonstrate the power of the MSS approach for reducing the complexity and the number of local minima in the parameter space. The approach is fully implemented in the COPASI software package and, therefore, easily accessible for a wide community of researchers.
Insurance and annuity products covering several lives require the modelling of the joint distribution of future lifetimes. In the interest of simplifying calculations, it is common in practice to assume that the future lifetimes among a group of people are independent. However, extensive research over the past decades suggests otherwise. In this paper, a copula approach is used to model the dependence between lifetimes within a married couple \eH{using data from a large Canadian insurance company}. As a novelty, the age difference and the \eH{gender} of the elder partner are introduced as an argument of the dependence parameter. \green{Maximum likelihood techniques are} thus implemented for the parameter estimation. Not only do the results make clear that the correlation decreases with age difference, but also the dependence between the lifetimes is higher when husband is older than wife. A goodness-of-fit procedure is applied in order to assess the validity of the model. Finally, considering several products available on the life insurance market, the paper concludes with practical illustrations.
Recent studies has revealed that Microtubules (MTs) exhibit three transition states of growth, shrinkage and pause. In this paper, we first introduce a three states random evolution model as a framework for studying MTs dynamics in three transition states of growth, pause and shrinkage. Then, we introduce a non-traditional stack run encoding scheme with 5 symbols for detecting transition states as well as to encode MT experimental data. The peak detection is carried out in the wavelet domain to effectively detect these three transition states. One of the added advantages of including peak information while encoding being that it enables to detect the peaks efficiently and encodes them simultaneously in the wavelet domain without having the need to do further processing after the decoding stage. Experimental results show that using this form of non-traditional stack run encoding has better compression and reconstruction performance as opposed to traditional stack run encoding and run length encoding schemes. Parameters for MTs modeled in the three states are estimated and is shown to closely approximate original MT data for lower compression rates. As the compression rate increases, we may end up throwing away details that are required to detect transition states of MTs. Thus, choosing the right compression rate is a trade-off between admissible level of error in signal reconstruction, its parameter estimation and considerable rate of compression of MT data.
Interest in the study of in-host microbial communities has increased in recent years due to our improved understanding of the communities' significant role in host health. As a result, the ability to model these communities using differential equations, for example, and analyze the results has become increasingly relevant. The size of the models and limitations in data collection among many other considerations require that we develop new parameter estimation methods to address the challenges that arise when using traditional parameter estimation methods for models of these in-host microbial communities. In this work, we present the challenges that appear when applying traditional parameter estimation techniques to differential equation models of microbial communities, and we provide an original, alternative method to those techniques. We show the derivation of our method and how our method avoids the limitations of traditional techniques while including additional benefits. We also provide simulation studies to demonstrate our method's viability, the application of our method to a model of intestinal microbial communities to demonstrate the insights that can be gained from our method, and sample code to give readers the opportunity to apply our method to their own research.
Ecosystem approach to fisheries requires a thorough understanding of fishing impacts on ecosystem status and processes as well as predictive tools such as ecosystem models to provide useful information for management. The credibility of such models is essential when used as decision making tools, and model fitting to observed data is one major criterion to assess such credibility. However, more attention has been given to the exploration of model behavior than to a rigorous confrontation to observations, as the calibration of ecosystem models is challenging in many ways. First, ecosystem models can only be simulated numerically and are generally too complex for mathematical analysis and explicit parameter estimation; secondly, the complex dynamics represented in ecosystem models allow species-specific parameters to impact other species parameters through ecological interactions; thirdly, critical data about non-commercial species are often poor; lastly, technical aspects can be impediments to the calibration with regard to the high computational cost potentially involved and the scarce documentation published on fitting complex ecosystem models to data. This work highlights some issues related to the confrontation of complex ecosystem models to data and proposes a methodology for a sequential multi-phases calibration of ecosystem models. We propose criteria to classify the parameters of a model: model dependency and time variability of the parameters. These criteria and the availability of approximate initial estimates are used as decision rules to determine which parameters need to be estimated, and their precedence order in the sequential calibration process. The end-to-end ecosystem model ROMS-PISCES-OSMOSE applied to the Northern Humboldt Current Ecosystem is used as an illustrative case study.
In many chemical and biological applications, systems of differential equations containing unknown parameters are used to explain empirical observations and experimental data. The DEs are typically nonlinear and difficult to analyze, requiring numerical methods to approximate the solutions. Compounding this difficulty are the unknown parameters in the DE system, which must be given specific numerical values in order for simulations to be run. Estrogen receptor protein dimerization is used as an example to demonstrate model construction, reduction, simulation, and parameter estimation. Mathematical, computational, and statistical methods are applied to empirical data to deduce kinetic parameter estimates and guide decisions regarding future experiments and modeling. The process demonstrated serves as a pedagogical example of quantitative methods being used to extract parameter values from biochemical data models.
The correct use and interpretation of models depends on several steps, two of which being the calibration by parameter estimation and the analysis of uncertainty. In the biological literature, these steps are seldom discussed together, but they can be seen as fitting pieces of the same puzzle. In particular, analytical procedures for uncertainty estimation may be masking a high degree of uncertainty coming from a model with a stable structure, but insufficient data. Under a likelihoodist approach, the problem of uncertainty estimation is closely related to the problem of composite hypothesis. In this paper, we present a brief historical background on the statistical school of Likelihoodism, and examine the complex relations between the law of likelihood and the problem of composite hypothesis, together with the existing proposals for coping with it. Then, we propose a new integrative methodology for the uncertainty estimation of models using the information in the collected data. We argue that this methodology is intuitively appealing under a likelihood paradigm.