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This chapter specifies the unified Bayesian approach of the recursive estimation to the dynamic mixture of normal regression models from Sect. 2.1, categorical components from Sect. 2.2 and state-space models from Sect. 2.3.

A mixture model considered in this book consists of a set of \(m_c\) components, which can be either regression models ( 2.1), categorical models ( 2.13) or state-space models ( 2.21)–( 2.22). In general form the component model is denoted by the pdf.

The paper describes application of data-based Bayesian approach to model identification and control problems in the field of fuel consumption optimization for conventional vehicles. The main contributions of the presented approach are: (i) analysis of data measured on a driven vehicle; (ii) data-based model construction, its real-time estimation and adaptation; (iii) control criterion using simultaneously setpoints for fuel consumption and speed; and (iv) universal recursive Bayesian algorithms of estimation and control implemented as semi-automatic eco-driving system. Experiments with real data report reduction in fuel consumption.

The paper proposes a recursive algorithm for estimation of mixtures with state-space components and a dynamic model of switching. Bayesian methodology is adopted. The main features of the presented approach are: (i) recursiveness that enables a real-time performance of the algorithm; (ii) one-pass elaboration of the data sample; (iii) dynamic nature of the model of switching active components; (iv) orientation at explicit solutions with exploitation of numerical procedures only in those parts which cannot be computed analytically; (v) systematic approach to the Bayesian mixture estimation theory.

Initialization is an extremely important part of the mixture estimation process. There exists a series of initialization approaches in the literature concerning the mixture initialization. However, the majority of them is directed at initialization of the expectation-maximization algorithm widely used in this area. This paper focuses on the initialization of the mixture estimation with normal components based on the recursive statistics update of involved distributions, where the mentioned methods are not suitable. Its key part is the choice of the initial statistics. The paper describes several relatively simple initialization techniques primarily based on processing the prior data. The experimental part of the paper represents results of validation on real data.

This paper deals with problem of analysis of traffic data. A traffic network has several types of roads: historical centre, peripherals, arterial roads, etc. They have specific properties. For a traffic analysis, large amounts of data are needed. Some traffic data are difficult to obtain due to their rare occurrence. Typical example is the investigation of traffic accidents. In these cases, data from other similar roads can be used. In such cases, an expert intervention added to the general analysis is very important. In this paper, the logistic regression with two types of expert intervention is briefly introduced. The performance of these methods is demonstrated on examples concerning seriousness of traffic accidents.

Classification is one of the frequently demanded tasks in data analysis. There exists a series of approaches in this area. This paper is oriented towards classification using the mixture model estimation, which is based on detection of density clusters in the data space and fitting the component models to them. A chosen function of proximity of the actually measured data to individual mixture components and the component shape play a significant role in solving the mixture-based classification task. This paper considers definitions of the proximity for several types of distributions describing the mixture components and compares their properties with respect to speed and quality of the resulting estimation interpreted as a classification task. Normal, exponential and uniform distributions as the most important models used for describing both Gaussian and non-Gaussian data are considered. Illustrative experiments with results of the comparison are provided.

The paper deals with a problem of fuel consumption optimization. Solutions existing in this field are mainly based on the various conceptual approaches such as hybrid and electric vehicles. However, it leads to high initial cost of a vehicle. The approach presented in this paper aims at conventional vehicles and is based on recursive algorithms of system identification and adaptive quadratic optimal control under Bayesian methodology. Experiments with real data measured on a driven vehicle are provided.

The paper deals with recursive state estimation for hybrid systems. An unobservable state of such systems is changed both in a continuous and a discrete way. Fast and efficient online estimation of hybrid system state is desired in many application areas. The presented paper proposes to look at this problem via Bayesian filtering in the factorized (decomposed) form. General recursive solution is proposed as the probability density function, updated entry-wise. The paper summarizes general factorized filter specialized for (i) normal state-space models; (ii) multinomial state-space models with discrete observations; and (iii) hybrid systems. Illustrative experiments and comparison with one of the counterparts are provided.

Online state prediction and fault detection are typical tasks in the chemical industry. In practice it often happens that some variables, important and critical for quality control, cannot be measured online due to such restrictions as cost and reliability. An uncertainty existing in real systems allows to use a probabilistic approach to online state estimation. Such an approach is proposed in this paper. Different types of information appearing in an online diagnostic system are processed via combination of algorithms subject to probability distributions. This combination of algorithms is presented as a decomposed version of Bayesian filtering. In this paper, the proposed solution is specialized for a system with mixed both continuous and discrete-valued measurements and unobserved variables.

SUMMARY This paper concerns the Bayesian tracking of slowly varying parameters of a linear stochastic regression model. The modelled and predicted system output is assumed to possess time-varying mean value, whereas its dynamics are relatively stable. The proposed estimation method models the system output mean value by time-varying offset. It formulates three extreme hypotheses on model parameters' variability: (i) no parameter varies; (ii) all parameters vary; and (iii) the offset varies. The Bayesian paradigm then provides a mixture as posterior distribution, which is appropriately projected to a feasible class. Exponential forgetting at ‘second’ hypotheses level allows tracking of slow variations of respective hypotheses. The developed technique is an example of a general procedure called partial forgetting. Focus on a simple example allows to demonstrate essence of the approach. Moreover, it is important per se as it corresponds with a varying load of otherwise (almost) time-invariant dynamic system. Copyright © 2011 John Wiley & Sons, Ltd.

The paper introduces an algorithm for estimation of dynamic mixture models. A new feature of the proposed algorithm is the ability to consider a dynamic form not only for component models but also for the pointer model, which describes the activities of the mixture components in time. The pointer model is represented by a table of transition probabilities that stochastically control the switching between the active components in dependence on the last active one. This feature brings the mixture model closer to real multi-modal systems. It can also serve for a prediction of the future behavior of the modeled system. Copyright © 2011 John Wiley & Sons, Ltd.

The paper proposes a new estimating algorithm for linear parameter varying systems with slowly time-varying parameters when the rate of change of individual parameters is different. It introduces a true probability density function, describing ideally the behaviour of parameters. However, as it is unknown, we search for its best approximation. A convex combination of point estimates, defined by individual hypotheses about the true probability density function, is then approximated by a single density. That serves as the best available description of parameters’ behaviour and it is therefore suitable e.g. for prediction purposes.

The paper deals with detection of clusters in data measured on a driven vehicle. Such a clustering aims at distinguishing various driving styles for eco-driving and driver assistance systems. The task is solved with the help of the application of the recursive Bayesian mixture estimation theory. The main contribution of the paper is a demonstration that real measurements with non-linear relationships between them can be approximately described by the mixture model, which is known as the universal approximation. Validation experiments are shown.

This paper introduces a new concept of the state model of one traffic microregion based on a maximum utilization of information from all measured traffic variables. The aim of the model is to estimate length of queues that are formed on arms of junctions with traffic lights. This task is trivia in case of complete knowledge of all measured traffic quantities for all junction arms. Then the model only counts simply the queue length from input and output intensities. However, the net of all needed detectors is not usually complete and some significant traffic flows (parking cars, etc.) are not measurable in practice. The model estimates the queue length in this case. In the end of the paper, the model and estimation algorithm is tested for several types of disturbances which can arise in reality. At least partially, these experiments illustrate the functionality and effectiveness of the proposed model for estimating queue lengths on the junction arms in the real traffic.

SUMMARY Thispaperisconcernedwithdistributionsoftimeseries,which(i)arein∞uencedbyinitialconditions (ii) are stimulated by an exogenous signal or (iii) are obtained by recursive estimation of underlying parametersandthusundergoatransientperiod. In computer intensive applications, it is desirable to stop the processing when the transient period is practically over. This aspect is addressed here from a Bayesian perspective. Under an often met assumption that the model of a system’s time series is recursively estimated anyway, the computational overhead of the constructed stopping rule is negligible. Algorithmic details are presented for important normal ARX models (auto-regression with exogenous variable) and models ofdiscrete-valued,independent,identicallydistributeddata.Thelattercaseprovidesnon-parametric Bayesian estimation of credibility interval with sequential stopping. Copyright c ∞ 2004 John Wiley & Sons,Ltd.

Modelling of large transportation systems requires a reliable description of its elements that can be easily adapted to the specific situation. This paper offers mixture model as a flexible candidate for modelling of such element. The mixture model describes particular and possibly very different states of a specific system by its individual components. A hierarchical model built on such elements can describe complexes of big city communications as well as railway or highway networks. Bayesian paradigm is adopted for estimation of parameters and the actual component label of the mixture model as it serves well for the subsequent decision making. As a straightforward application of Bayesian method to mixture models leads to infeasible computations, an approximation is applied. For normal stochastic variations, the resulting estimation algorithm reduces to a simple recursive weighted least squares. The elementary modelling is demonstrated on a model of traffic flow state in a single point of a roadway. The examples for simulated as well as real data show excellent properties of the suggested model. They represent much wider set of extensive tests made. Copyright © 2003 John Wiley & Sons, Ltd.

Real-world, multidimensional, dynamic, non-linear processes typically exhibit many distinct modes of operation. Mixtures of dynamic models improve greatly on traditional one-component linear models in this context. Improved prediction then points the way to effective adaptive control design. This paper presents the experience gained under the EU Project, ProDaCTool, in designing and implementing advisory systems, based on dynamic mixtures, in diverse domains: urban traffic regulation, therapy recommendations in nuclear medicine, and operator support for metal-strip rolling mills. Efficient, recursive estimation of the dynamic mixtures from archive data is accomplished using the quasi-Bayes (QB) algorithm, implemented with dedicated software developed within ProDaCTool. The advisory systems are designed using the probabilistic control design technique presented in the previous paper. Highly encouraging prediction and performance enhancements are reported for the applications considered. Copyright © 2003 John Wiley & Sons, Ltd.

Early recognition/isolation of a faulty behaviour of a dynamic system is the main task of a fault detection and isolation (FDI). FDI methods based on adaptive probabilistic models with multiple modes represent a theoretically well justified way of solution. Their use is severely restricted by an inherent computational complexity. The complexity problem is addressed here by employing an efficient quasi-Bayes estimation algorithm. It is directly applicable to the mixture of components created as products of factors belonging to the exponential family. It opens a novel way to deal adaptively with mixed continuous–discrete, dynamically related data. The presented theory and algorithmization are illustrated by a simple simulation example. Copyright © 2001 John Wiley & Sons, Ltd.

The paper deals with fuel consumption optimization under condition of keeping the recommended speed. The presented approach is based on data currently measured on a driven vehicle and on external observations. Using adaptive optimal control algorithms under Bayesian methodology, a compromise between fuel consumption minimization and keeping the recommended sceed is reached. Research is performed in collaboration with Skoda Auto (www.skoda-auto.com).

The presented paper deals with a problem of fuel consumption optimization. Today’s automotive industry solves this problem mainly via various conceptual approaches (hybrid and electric vehicles). However, it leads to high initial cost of a vehicle. This paper focuses on fuel economy for conventional vehicles. For this aim, recursive algorithms of adaptive optimal quadratic control under Bayesian methodology are used. A stochastic servo problem, including setpoint tracking, is a part of the considered adaptive control design. In this paper, fuel consumption and speed of a driven vehicle are the controlled variables, where the first one is to be optimized and the second one is pushed to track its set-point. This set-point is a recommended roaddependent speed. Experiments with real data measured on a driven vehicle are provided.

Many various algorithms are developed for state estimation of dynamic switching systems. It is not a straightforward task to choose the most suitable one. This paper deals with testing of state estimation via two well-known approaches: recursive estimation with finite mixtures and iterative technique with hidden Markov models. A discussion of comparison of these online and offline counterparts is of true interest. The paper describes experiments providing a comparison of these methods.

The paper describes a recent progress in searching for credible, well-grounded approximation of recursive Bayesian parameter estimation which would make the Bayesian paradigm feasible for a class of nonstandard (non-linear and/or non-Gaussian) models. The presented method is based on maximum-entropy approximation of the empirical distribution of data while just a reduced (non-sufficient) data statistic is available. The statistic is chosen so to induce an equivalence relation on the set of posterior probability distributions which is compatible with the Bayes-rule action. The approximating posterior density of unknown parameters is given by the standard Bayes-rule transformation of the approximating distribution of data. Numerical implementation of the general algorithm is considered using its discrete version or prior approximation of critical steps.

The paper deals with estimation of a state with discrete values. The proposed estimation technique is evolved as an application of Bayesian filtering to a state-space model with discrete distribution. The example of filtering is shown with Bernoulli distributions. The considered problem is one of the items aiming at filtering with mixed continuous and discrete state. Illustrative experiments demonstrate the filtering with discrete simulated data from the traffic control area, which is a potential application domain of the research.

In the light of today's high traffic volume, it is crucial to anticipate the traffic flow on traffic lights controlled intersections. Without this knowledge, it would be impossible to control the traffic in complicated hubs, such as, for instance, those in Prague. One of the characteristics, through which we can describe dynamics of vehicle movement on traffic lights controlled intersections, is the interval between departure of a vehicle from the space in front of the stop line and the arrival to the actual stop line. This is called a departure model.Our aim is, first, to create an appropriate mathematical application, which would take into account individual variables influencing departure of individual vehicles and determine the dependence among these variables. Second to create such a departure model, which would, on one hand, correspond as much as possible to the contemporary traffic situations, but on the other hand, be more exact for specific intersections.

A classical version of the EM algorithm is considered in the paper. Its numerical properties are improved using factorized algorithms for maximization in M step of the algorithm. The results are illustrated on simulated examples.

Multiple models, neural networks, cluster analysis and probabilistic mixtures are prominent examples of situations when complex multi-modal models [1] are built using vast amount of data. Complexity and non-unicity of modified situation imply that resulting description depends heavily on the initial phase of search. The safest repetitive purely random search is mostly inhibited by computational complexity of the addressed task. For this reasons, various techniques have been designed. None of them, to our best knowledge, suits to cases when dynamic models are constructed. The paper describes a novel technique that fills this gap in a promising way. Essentially, the trial description is gradually split whenever there is possibility that a unimodal sub-model hides more modes.

A design of recursively implementable approximation of the optimal Bayesian estimation is addressed. The problem is imbedded into recursive least squares (RLS) framework. The imbedding is reached by formulating the approximation task at “upper” estimation level: a suitable transformation of a logarithmic likelihood function is approximated instead of attempting to find approximate estimator directly. In this way, the inherent recursivity of least squares (LS) determining a linear estimator of linearly evolving functions is exploited.

The present-day discrete LQ control synthesis is mostly based on ARUA-type models. The synthesis can be well algorithmized but it does not yield satisfactory results when sampling period is small with regard to dominant time-constants of the plant. In this case, the equations get ill-conditioned and the synthesis meets difficulties. Limiting of sampling period to zero cannot be made within the theory. In the paper an attempt has been made to overcome these problems by using difference-type discrete models. Such approach also tries to contribute to the trend to reduce a gap between the discrete control and the continuous one which is still felt in engineering practice. The algorithms of both discrete and continuous control synthesis are presented in a unified way and properties of the results are illustrated by means of a simple example.

A self-tuning controller with continuously operating control law updated in a discrete way is presented. The self-tuning is based on realtime identification of a stochastic continuous-process model and LQ-optimum control synthesis. The input and the output are filtered so that the derivatives of the filtered signals are available and their linear relation is estimated. From this relation and from the filter employed a stochastic model of the controlled process is reconstructed and used to update the control law. The controller can be realized purely digitally with high sampling rate for filtering and control and with a lower one for more complex algorithms of identification and control-law updating. The performance of the controller is demonstrated on simulations of a nontrivial case.

The algebraic theory of control of discrete linear time-invariant systems mostly utilizes polynomials in the delay operator.The presented paper generalizes this approach: instead of just one operator, a family of operators is used together with a group of transformations from one operator to another. As the group is the one-dimensional projective group, results of the projective geometry can be employed. The standard stochastic LQ control synthesis with polynomial equations and with algorithms for solving them are reformulated into a projectively covariant form. This leads to deeper understanding of equations and algorithms as well as to computational procedures with better numerical behaviour.

SummaryPsychotropic dopamine antagonists primarily stimulate prolactin release, and their subchronic administration does not influence the water-soluble protein concentration, the hexosemonophosphate shunt, and anaerobic glycolytic, oxidative, or lysosomal activity of the lactotrophs.

The activity of some glycolytic, oxidative, and degradative enzymes was studied in transplanted rat hormone-secreting pituitary tumors MtTW15 and 7315a and in the host pituitary gland. The elevated serum-hormone concentrations produced by 7315a tumor decreased the size of the host's pituitary gland, its hormone content, and G6P-DH, LDH, PK, and ICDH, but produced no changes in MDH, acid phosphatase, cathepsin-D, and LYSAR enzyme activities (mU/mg tissue). LDH and PK activities were greater in unit weight of pituitary tumors than in pituitary glands. Although more G6P-DH was found in MtTW15 tumor than in normal pituitary tissue, less of the enzyme was detected in 7315a pituitary tumor. It is concluded that elevated serum pituitary hormones selectively decrease hormone production and the activity of some enzymes in the pituitary gland, presumably through a feedback mechanism.

The effects of estradiol treatment on the synthesis and release of prolactin and GH in castrated male rats were studied in connection with the anterior pituitary enzymes that represent the hexosemonophosphate shunt, glycolytic, oxidative, and lysosomal activity. LDH and G6P-DH activities increased by 15%–30% at 12 hr and by 70% at 72 hr after estrogen administration. PK activity showed a statistically significant elevation of 20%–40% only after 48–72 hr. ICDH, MDH, acid phosphatase activities, and water-soluble protein concentrations were unchanged. Serum prolactin concentration increased about 400% 24 hr after estradiol injection, and the pituitary synthesized 1000%–1500% more radioactive prolactin in vitro than did control glands. However, no significant increase in prolactin synthesis was observed 12 hr after estradiol treatment. It is suggested that the primary effect of estradiol is on the synthesis of prolactin and that the increased rate of secretion is secondary. Radioimmunoassayable prolactin in the incubated gland tissue and its medium was greatly increased after estradiol treatment. A slight but statistically significant accumulation and decreased release of radioactive GH were also observed. The results suggest a correlation of pituitary prolactin production with the tissue's metabolic activity.