Stochastic Statistical Models

State that we cannot know all factors and processes that influence the values we observe for a certain wetland biogeochemical property. Simple statistical methods as a continuous response (regression analysis) or as a set of responses to discrete factors (analysis of variance, ANOVA) have been used extensively in wetland research, in order to relate elements in biogeochemical processes or determine the extent to which a particular factor may influence a process. 

Challenges in the future remain to model the high-order interactions among biogeochemical variables, to model complex linear and nonlinear processes, and to incorporate spatial and temporal autocorrelations into models. Scale is one of the most confounding factors when synthesizing and modeling wetland processes. Process-based models implicitly assume a specific scale at which 

For a more extensive overview of descriptive mnltivariate stochastic techniques it is referred to Hair et al. (1995), Berthouex and Brown (2002), and Schabenberger and Pierce (2002). For scaling methods in aquatic ecology it is referred to Seurant and Strntton (2004). 

---

The statistic models consider surface roughness as a stochastic process, and concern the averaged or statistic behavior of lubrication and contact. For instance, the average flow model, proposed by Patir and Cheng [2], combined with the Greenwood and Williamsons statistic model of asperity contact [3] has been one of widely accepted models for mixed lubrication in early times. 

The Stochastic and other Statistical Models of Long-Range Order. 324... 

Kousa et al. [20] classified exposure models as statistical, mathematical and mathematical-stochastic models. Statistical models are based on the historical data and capture the past statistical trend of pollutants [21]. The mathematical modelling, also called deterministic modelling, involves application of emission inventories, combined with air quality and population activity modelling. The stochastic approach attempts to include a treatment of the inherent uncertainties of the model [22],... 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

Matis, J., An introduction to stochastic compartmental models in pharmacokinetics, Pharmacokinetics-Mathematical and Statistical Approaches in Metabolism and Distribution of Chemicals and Drugs, edited by A. Pecile and A. Rescigno, Plenum Press, New York, 1988, pp. 113-128. 

Matis, J. and Kiffe, T., Stochastic Population Models. A Compartmental Perspective, Vol. 145 of Lecture Notes in Statistics, Springer-Verlag, New York, 2000. 

An estimator (or more specifically an optimal state estimator ) in this usage is an algorithm for obtaining approximate values of process variables which cannot be directly measured. It does this by using knowledge of the system and measurement dynamics, assumed statistics of measurement noise, and initial condition information to deduce a minimum error state estimate. The basic algorithm is usually some version of the Kalman filter.14 In extremely simple terms, a stochastic process model is compared to known process measurements, the difference is minimized in a least-squares sense, and then the model values are used for unmeasurable quantities. Estimators have been tested on a variety of processes, including mycelial fermentation and fed-batch penicillin production,13 and baker s yeast fermentation.15 The... 

The stochastic process models can be transformed by the use of specific theorems as well as various stochastic deformed models, more commonly called diffusion models (for more details see Chapter 4). In the case of statistical models, we can introduce other grouping criteria. We have a detailed discussion of this problem in Chapter 5. 

Stochastic mathematical modelling is, together with transfer phenomena and statistical approaches, a powerful technique, which can be used in order to have a good knowledge of a process without much tedious experimental work. The principles for establishing models, which were described in the preceding chapter, are still valuable. However, they will be particularized for each example presented below. 

The statistical modelling of a process can be applied in three different situations (i) the information about the investigated process is not complete and it is then not possible to produce a deterministic model (model based on transfer equations) (ii) the investigated process shows multiple and complex states and consequently the derived deterministic or stochastic model will be very complex (iii) the researcher s ability to develop a deterministic or stochastic model is limited. 

One of the goals of the experimental research is to analyze the systems in order to make them as widely applicable as possible. To achieve this, the concept of similitude is often used. For example, the measurements taken on one system (for example in a laboratory unit) could be used to describe the behaviour of other similar systems (e.g. industrial units). The laboratory systems are usually thought of as models and are used to study the phenomenon of interest under carefully controlled conditions. Empirical formulations can be developed, or specific predictions of one or more characteristics of some other similar systems can be made from the study of these models. The establishment of systematic and well-defined relationships between the laboratory model and the other systems is necessary to succeed with this approach. The correlation of experimental data based on dimensional analysis and similitude produces models, which have the same qualities as the transfer based, stochastic or statistical models described in the previous chapters. However, dimensional analysis and similitude do not have a theoretical basis, as is the case for the models studied previously. 

Statistical Models. Due to the difficulties involved in calculating the composition distributions by purely deterministic techniques, statistical methods have been developed from which not only the CCD can be obtained but also the sequence length distribution. These methods view the chain growth as an stochastic process having possible states resulting from the kinetic mechanisms. Early work on this approach was reported by Merz, Alfrey and Goldfinger (4) who derived the copolymerization equation and the SLD for the ultimate effect case. Alfrey Bohrer and Mark (19) and Ham (9) formalized this approach. 

This means that larger scale motions can be explicitly resolved and deterministically forecasted. The smaller scale motions, namely turbulence, are not explicitly resolved. Rather, the effects of those sub-grid scales on the larger scales are approximated by turbulence models. These smaller size motions are said to be parameterized by sub-grid scale stochastic (statistical) approximations or modes. The referred experimental data analyzes of the flow in the atmospheric boundary layer determine the basis for the large eddy simulation (LES) approach developed by meteorologists like Deardorff [27] [29] [30]. Large-Eddy Simulation (LES) is thus a relatively new approach to the calculation of turbulent flows. 

Prior to model estimation the question that it will be used to answer and the specific manner in which it will be used should be explicitly stated. Using a model to answer a question is the act of simulation. There are two types of simulation deterministic and stochastic. In a deterministic simulation, the statistical model is ignored and no error is introduced into the model—the results are error-free. For example, given data from single-dose administration of a drug it may be of interest to predict the typical concentration-time profile at steady-state under a repeated dose administration regimen. A deterministic simulation would be useful in this case. 

Hidden Markov models (HMMs) are statistical models based on the Markov property [34]. A stochastic process has the Markov property if the conditional probability... 

HMMs are statistical models for systems that behave like a Markov chain, which a discrete-time stochastic process formed on the basis of the Markov property. 

Stochastic (Probabilistic) Models. One of the most significant advances in exposure estimation in the past 15 to 20 years has been the application of probabilistic statistical methods to many types of data analyses (Duan and Mage 1997 Finley and Paustenbach 1994 Morgan and Henrion 1990 US ERA 1995, 1997, 2000a). Stochastic or probabilistic techniques can help quantify variability and uncertainty in model inputs and outputs, can be used to better characterize the possible range of exposures for a particular scenario when measured data are minimal, and can be employed to better understand the uncertainty inherent in estimates developed from many different types of sources, whether quantitative or qualitative. 

Nonlinear dynamics of complex processes is an active research field with large numbers of publications in basic research and broad applications from diverse fields of science. Nonlinear dynamics as manifested by deterministic and stochastic evolution models of complex behaviour has entered statistical physics, physical chemistry, biophysics, geophysics, astrophysics, theoretical ecology, semiconductor physics and -optics etc. This research has induced a new terminology in science connected with new questions, problems, solutions and methods. New scenarios have emerged for spatio-temporal structures in dynamical systems far from equilibrium. Their analysis and possible control are intriguing and challenging aspects of the current research. 

A statistical model incorporates stochasticity in order to describe a biological phenomenon and will contain parameters that are of interest to the investigator. 

Hierarchical Bayesian models are commonly used in many biological apphcations because they incorporate both physical and statistical models with uncertainty. These models are used in pathway analysis because of their ability to manage multiple data sources, uncertain, physical models with stochastic parameters, and expert opinion. A summary of these models is given in WiMe (2004). In brief, all unknowns are treated as if they are random and evaluated probabilistically as follows ... 

Recent advances in statistical software have brought the use of hierarchical linear models and stochastic regression models into easy reach. PROC MIXED in the SAS software package (SAS Institute, Inc.) and BMDP 5V in the Biomedi-... 

Tervonen T, van Valkenhoef G, Buskens E, Hillege HL, Postmus D. A stochastic multicriteria model for evidence-based decision making in drug benefit-risk analysis. Statistics in Medicine 2011 30(12) 1419-1428. 

---