Parameter Estimation In Stochastic Models Research Articles

Frequency Analysis of Hydrological Data for Urban Floods—Review of Traditional Methods and Recent Developments, Especially an Introduction of Japanese Proper Methods

Frequency analysis has long been an important theme of hydrology research. Although meteorological techniques (physical approaches) such as radar nowcasting, remote sensing, and forecasting heavy rainfall events using meteorological simulation models are quite effective for urban disaster prevention, statistical and stochastic theories that include frequency analysis, which are usually used in flood control plans, are also valuable for flood control plans for disaster prevention. Master plans for flood control projects in urban areas often use the concept of T-year hydrological values with a T-year return period. A flood control target is a “landside area that is safe against heavy rainfall or floods with a return period of T years”. This review emphasizes discussions of parameter estimation of stochastic models and selection of optimal statistical models, which include evaluation of goodness-of-fit techniques of statistical models. Based on those results, the authors criticize Japanese standard procedures recommended by the central government. Consistency between parameter estimation and evaluation of goodness-of-fit is necessary. From this perspective, we recommend using the maximum likelihood method and AIC, both of which are related to Kullback–Leibler divergence. If one prefers using SLSC, we recommend not SLSC itself but SLSC’s non-exceedance probability. One important purpose of this review is the introduction of well-used Japanese methods. Because some techniques that are slightly different from the international standard have been used for many years in Japan, we introduce those in the review article.

Parameter estimation of stochastic models based on limited data

Progress in experimental techniques enables a more accurate quantification of genes, mRNA, and proteins at the single cell level. Provided with limited time series data from single-cell measurements, this note proposes a new quasi-Newton optimization algorithm (QNSTOP) for parameter estimation of stochastic models. To capture the stochasticity inside models and data, the random objective function is constructed based on the maximum log-likelihood of transition probabilities rather than summary statistics, which relies heavily on stochastic simulations. Simple to use and efficient, QNSTOP can find the "best" parameter vector from far away starting points in just a few iterations. Results on a bistable model match well the bistable dynamics that can only be obtained from stochastic models.

MODELLING THE CANCER GROWTH PROCESS BY STOCHASTIC DELAY DIFFERENTIAL EQUATIONS UNDER VERHULTS AND GOMPERTZ’S LAW

In this paper, the uncontrolled environmental factors are perturbed into the intrinsic growth rate factor of deterministic equations of the growth process. The growth process under two different laws which are Verhults and Gompertz’s law are considered, thus leading to stochastic delay differential equations (SDDEs) of logistic and Gompertzian, respectively. Gompertzian deterministic model has been proved to fit well the clinical data of cancerous growth, however the performance of stochastic model towards clinical data is yet to be confirmed. The prediction quality of logistic and Gompertzian SDDEs are evaluating by comparing the simulated results with the clinical data of cervical cancer growth. The parameter estimation of stochastic models is computed by using simulated maximum likelihood method. We adopt 4-stage stochastic Runge-Kutta to simulate the solution of stochastic models.

Piecewise parameter estimation for stochastic models in COPASI.

Computational modeling is widely used for deepening the understanding of biological processes. Parameterizing models to experimental data needs computationally efficient techniques for parameter estimation. Challenges for parameter estimation include in general the high dimensionality of the parameter space with local minima and in specific for stochastic modeling the intrinsic stochasticity. We implemented the recently suggested multiple shooting for stochastic systems (MSS) objective function for parameter estimation in stochastic models into COPASI. This MSS objective function can be used for parameter estimation in stochastic models but also shows beneficial properties when used for ordinary differential equation models. The method can be applied with all of COPASI's optimization algorithms, and can be used for SBML models as well. The methodology is available in COPASI as of version 4.15.95 and can be downloaded from http://www.copasi.org frank.bergmann@bioquant.uni-heidelberg.de or fbergman@caltech.edu Supplementary data are available at Bioinformatics online.

Challenges in Applying Calibration Methods to Stochastic Traffic Models

This paper evaluates calibration and validation as a means to understand traffic flow models better. The paper concentrates on the car-following part of these models and demonstrates that the calibration of stochastic models under certain circumstances can become difficult. Three types of stochasticity are distinguished for microscopic traffic flow models: the stochasticity coming from noisy data, the stochasticity coming from the distribution of the parameters describing the driver’s behavior, and the stochasticity coming from the model itself when a noise component is added to a deterministic differential equation governing the vehicle’s movements. By using four submodels with four noise terms and an identical deterministic part, this paper shows that a calibration with synthetic and, therefore, reproducible data can lead to results that are awry. The parameters fitted by the calibration procedure were significantly different for the deterministic and stochastic models. It was concluded that stochasticity was the reason why the parameter estimation of stochastic models sometimes failed. Up to now, the authors have not been able to propose a solution to cope with this intrinsic pitfall of genuine stochastic models.

Comparison of approaches for parameter estimation on stochastic models: Generic least squares versus specialized approaches

Parameter estimation for models with intrinsic stochasticity poses specific challenges that do not exist for deterministic models. Therefore, specialized numerical methods for parameter estimation in stochastic models have been developed. Here, we study whether dedicated algorithms for stochastic models are indeed superior to the naive approach of applying the readily available least squares algorithm designed for deterministic models.We compare the performance of the recently developed multiple shooting for stochastic systems (MSS) method designed for parameter estimation in stochastic models, a stochastic differential equations based Bayesian approach and a chemical master equation based techniques with the least squares approach for parameter estimation in models of ordinary differential equations (ODE). As test data, 1000 realizations of the stochastic models are simulated. For each realization an estimation is performed with each method, resulting in 1000 estimates for each approach. These are compared with respect to their deviation to the true parameter and, for the genetic toggle switch, also their ability to reproduce the symmetry of the switching behavior. Results are shown for different set of parameter values of a genetic toggle switch leading to symmetric and asymmetric switching behavior as well as an immigration-death and a susceptible-infected-recovered model. This comparison shows that it is important to choose a parameter estimation technique that can treat intrinsic stochasticity and that the specific choice of this algorithm shows only minor performance differences.

English

In this paper, the uncontrolled environmental factors are perturbed into the growth rate deceleration factor of the Gompertzian deterministic model. The growth process under Gompertz’s law is considered, thus lead to stochastic differential equations of Gompertzian with time delay. The Gompertzian deterministic model has proven to fit well with the clinical data of cancerous growth, however the performance of stochastic model towards clinical data is yet to be confirmed. The prediction quality of stochastic model is evaluated by comparing the simulated results with the clinical data of cervical cancer growth. The parameter estimation of stochastic models is computed by using simulated maximum likelihood method. 4-stage stochastic Runge-Kutta is applied to simulate the solution of stochastic model. Low values of root mean-square error (RMSE) of Gompertzian model with random effect indicate good fits.

Parameter Estimation for Stochastic Models of Biochemical Reactions

Parameter estimation is central for the analysis of models in Systems Biology. Stochastic models are of increasing importance. However, parameter estimation for stochastic models is still in the early phase of development and there is need for efficient methods to estimate model parameters from time course data which is intrinsically stochastic, only partially observed and has measurement noise. The thesis investigates methods for parameter estimation for stochastic models presenting one efficient method based on integration of ordinary differential equations (ODE) which allows parameter estimation even for models which have qualitatively different behavior in stochastic modeling compared to modeling with ODEs. Further methods proposed in the thesis are based on stochastic simulations. One of the methods uses the stochastic simulations for an estimation of the transition probabilities in the likelihood function. This method is suggested as an addition to the ODE-based method and should be used in systems with few reactions and small state spaces. The resulting stochastic optimization problem can be solved with a Particle Swarm algorithm. To this goal a stopping criterion suited to the stochasticity is proposed. Another approach is a transformation to a deterministic optimization problem. Therefore the polynomial chaos expansion is extended to stochastic functions in this thesis and then used for the transformation. The ODE-based method is motivated from a fast and efficient method for parameter estimation for systems of ODEs. A multiple shooting procedure is used in which the continuity constraints are omitted to allow for stochasticity. Unobserved states are treated by enlarging the optimization vector or using resulting values from the forward integration. To see how well the method covers the stochastic dynamics some test functions will be suggested. It is demonstrated that the method works well even in systems which have qualitatively different behavior in stochastic modeling than in modeling with ODEs. From a computational point of view, this method allows to tackle systems as large as those tackled in deterministic modeling.