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1

Improved detection of chemical threats by sensor data fusion

Kopp Norbert Ludwig, Koch-Eschweiler Helge

Security and Defence Quarterly

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2022

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Vol. 37, No. 1

70--93

EN

This paper presents some aspects of sensor data fusion that were derived from the EU-SENSE project of the European Commission (Horizon 2020, Grant Agreement No 787031). The aim of EU-SENSE was to develop a novel network of sensors for CBRNe applications through the exploitation of chemical detector technologies, advanced machine-learning and modelling algorithms. The high-level objectives of the project include improving the detection capabilities of the novel network of chemical sensors through the use of machine learning algorithms and reducing the impact of environmental noise. The focus in this paper is on the detection and data fusion aspects as well as the machine learning approaches that were used as part of the project. Detection (in the sense of detectto-warn) is a classification task and improvement of detection requires enhancing the discriminatory power of the classifier, that is reducing false alarms, false positives, and false negatives. This was achieved by a two-step procedure, that is a sensitive distance-based anomaly/change detection followed by downstream classification, identification and concentration estimation. Bayesian networks proved to be useful when fusing information from multiple sensors. For validation purposes, experimental data was gathered during the project and the developed approaches were applied successfully. Despite the development of several new, helpful tools within the project, the domain of chemical detection remains challenging, particularly regarding provisioning of the necessary prior-knowledge. It might make sense from a coverage point of view to look into integration of stand-off detection techniques into a sensor network, including data fusion too.

2

Wide estimation of dynamic properties of viscoelastic materials using Bayesian inference

Balbino Fernanda Oliveira, Préve Cíntia Teixeira, Munaro Marilda, Ribeiro Junior Paulo Justiniano, de Oliveira Lopes Eduardo Márcio

Journal of Theoretical and Applied Mechanics

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2021

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Vol. 59 nr 3

369–-384

EN

The dynamic behavior of a typical viscoelastic material in wide ranges of frequency and tem- perature is characterized. A four-parameter fractional derivative model was considered in the frequency domain along with the Arrhenius and WLF models, also for including tempera- ture as a source of variation. A Bayesian framework is adopted and inferences on parameters governing the model quantities of interest are based on samples from posterior distributions obtained by Monte Carlo Markov Chain (MCMC) methods. Posterior predictive checks were conducted to ensure the goodness-of-fit of the model. Based on the results we argue that the Bayesian framework allows more complete and suitable inference about dynamic properties of typical viscoelastic materials, as required for broad and sound vibration control actions.

3

On the use of viscoelastic materials characterized by Bayesian inference in vibration control

Préve Cíntia Teixeira, Balbino Fernanda Oliveira, Ribeiro Junior Paulo Justiniano, de Oliveira Lopes Eduardo Márcio

Journal of Theoretical and Applied Mechanics

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2021

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Vol. 59 nr 3

385–-399

EN

Viscoelastic materials are used to reduce vibrations in mechanical systems due to their con- trol efficacy. Considering that the dynamic behavior of those materials may be described by means of complex moduli, and experimental data may present ucertainties, an alternative is to use probabilistic methods, especially the Bayesian inference approach. By that approach, probability distribution functions are obtained for parameters of a model which describes the behavior of a given material. The present work employs a viscoelastic material modeled by the Bayesian approach in two vibration control actions, namely: a) use of vibration isolators; b) use of dynamic neutralizers. Transmissibility and receptance curves are displayed as well as dimensions of the control devices. Performance predictions are carried out in both cases. It is shown that the Bayesian approach can favourably reflect the presence of the uncertain- ties and advance their effects. Thus, more information can be provided for the designer of viscoelastic vibration control devices to anticipate eventual corrective measures.

4

A hierarchy of finite state machines as a scenario player in interactive training of pilots in flight simulators

Bach Małgorzata, Werner Aleksandra, Mrozik Magda, Cyran Krzysztof A.

International Journal of Applied Mathematics and Computer Science

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2021

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Vol. 31, no. 4

713--727

EN

The paper presents the concept of a control unit, i.e., a scenario player, for interactive training pilots in flight simulators. This scenario player is modelled as a hierarchy of finite state machines. Such an approach makes it possible to separate the details of an augmented reality display device which is used in training, from the core module of the system, responsible for contextual organization of the content. Therefore, the first contribution of this paper is the mathematical model of the scenario player as a universal formulation of the self-trained control unit for interactive learning systems, which is applicable in a variety of situations not limited solely to flight simulator related procedures. The second contribution is an experimental verification achieved by extensive simulations of the model, which proves that the proposed approach is capable to properly self-organize details of the context information by tracing preferences of the end users. For that latter purpose, the original algorithm is derived from statistical analysis, including Bayesian inference. The whole approach is illustrated by a real application of training the preflight procedure for the captain of the Boeing 737 aircraft in a flight simulator.

5

Learning causal theories with non-reversible MCMC methods

Krajewska Antonina

Control and Cybernetics

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2021

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Vol. 50, No. 3

323--361

EN

Causal laws are defined in terms of concepts and the causal relations between them. Following Kemp et al. (2010), we investigate the performance of the hierarchical Bayesian model, in which causal systems are represented by directed acyclic graphs (DAGs) with nodes divided into distinct categories. This paper presents two non-reversible search and score algorithms (Q1 and Q2) and their application to the causal learning system. The algorithms run through the pairs of class-assignment vectors and graph structures and choose the one which maximizes the probability of given observations. The model discovers latent classes in relational data and the number of these classes and predicts relations between objects belonging to them. We evaluate its performance on prediction tasks from the behavioural experiment about human cognition. Within the discussed approach, we solve a simplified prediction problem when object classification is known in advance. Finally, we describe the experimental procedure allowing in-depth analysis of the efficiency and scalability of both search and score algorithms.

6

Zastosowanie wnioskowania bayesowskiego w procesie diagnozowania systemów sterowania ruchem kolejowym

Nowakowski Waldemar

Zeszyty Naukowo-Techniczne Stowarzyszenia Inżynierów i Techników Komunikacji w Krakowie. Seria: Materiały Konferencyjne

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2019

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Nr 2(119)

159--168

PL

Systemy sterowania ruchem kolejowym odgrywają istotne znaczenie w zapewnieniu bezpieczeństwa przemieszczania osób i przewozu ładunków. Bardzo duża ilość urządzeń i systemów sterowania ruchem kolejowym, a także fakt, że wykonane są one w różnej technologii powoduje istotne utrudnienie w zapewnieniu, przez zarządcę infrastruktury kolejowej, wymaganego poziomu niezawodności. Konieczne jest więc wspieranie procesu ich utrzymania z użyciem metod diagnostyki technicznej. Oprogramowanie diagnostyczne CUiD przeznaczone jest głównie dla rozwiązań technicznych konkretnych producentów systemów sterowania ruchem kolejowym. Dlatego też autor artykułu zaproponował uniwersalną metodę diagnostyczną wykorzystującą wnioskowanie bayesowskie. Bazując na tej metodzie oraz protokole SNMP opracowano oprogramowanie komputerowe, które następnie użyto do diagnozowania uszkodzeń systemu SSP.

EN

Railway traffic control systems are essential to ensure the safety of passengers and freight transport. The very large number of controlling devices and systems, but also the fact that they are made in different technologies make it very difficult for the infrastructure manager to ensure the required level of reliability. Therefore, it is necessary to support the process of their maintenance with support of the application of technical diagnostic methods. The software M&DC is created mainly for technical solutions of specific manufacturers of railway traffic control systems. Therefore, the author of the article proposed a universal diagnostic method based on Bayesian inference. On the basis of this method and the SNMP protocol, computer software was developed, which is used to diagnose faults in the LCPS.

7

Zastosowanie wnioskowania Bayesa do oceny zagrożenia budynków wielkoblokowych na terenach górniczych

Rusek Janusz, Firek Karol

Przegląd Górniczy

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2019

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T. 75, nr 2

7--12

PL

W artykule przedstawiono przykłady wykorzystania modeli opartych na formalizmie wnioskowania Bayesa do analizy zagrożenia budynków zlokalizowanych na terenach górniczych. Przedmiotem badań była grupa 126 budynków wzniesionych w technologii wielkoblokowej. Przedstawiono metody wnioskowania wykorzystane w dotychczasowych badaniach ryzyka powstawania uszkodzeń w budynkach narażonych na negatywne skutki eksploatacji górniczej. Obejmowały one ocenę stanu technicznego (st), w ramach której do budowy modelu zastosowano naiwną klasyfikację Bayesa, a także analizę intensywności uszkodzeń elementów składowych budynku, z wykorzystaniem Bayesowskich sieci przekonań. W konkluzji przedstawiono koncepcję uszczegółowienia wyników wcześniejszych badań. Polega ona na samoistnym generowaniu struktury sieci Bayesa w oparciu o bazę danych o intensywności uszkodzeń istniejących budynków.

EN

This research paper provides examples of the use of models based on the formalism of Bayesian inference for the analysis of the threats to building structures located in mining areas. The subject of the research study was a group of 126 buildings erected in the large-block technology. The authors presented the inference methods of the risk of the occurrence of damage to buildings exposed to the adverse effects of mining exploitation, which were used in the previous studies. They included the assessment of the technical condition (st), where the naive Bayes classification was used to build the model, as well as the analysis of the intensity of damage to the components of a building structure, using the Bayesian belief networks. The conclusion presents the concept of detailing the results of the previous research. It involves the Bayesian network structure being spontaneously generated, based on the database on the intensity of damage to the existing buildings.

8

A Bayesian model of group decision-making

Wibig T., Karbowiak M., Jaszczyk M.

Operations Research and Decisions

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2016

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Vol. 26, No. 1

95--110

EN

A change in the opinion of a group, treated as a network of communicating agents, caused by the accumulation of new information is expected to depend on communication within the group, cooperation and, possibly, a kind of conformity mechanism. We have developed a mathematical model of the creation of a group decision, including this effect. This is based on a Bayesian description of inference and can be used for both conscious and inattentive acts. This model can be used to study the effect of whether a leader exists or not and other group inhomogeneities, as well as establishing the (statistical) significance and quality of a group decision. The proposed evolution equations explain in a straightfor-ward, analytical way some general properties of the general phenomenon of conformity (groupthink). To illustrate this theoretical idea in practice, we created an information technology (IT) tool to study the effect of conformity in a small group. As an example, we present results of an experiment performed using a network of students’ tablets, which could not only measure group pressure, but also conduct and control collaborative thinking in the group.

9

Residual life prediction for highly reliable products with prior accelerated degradation data

Wang H.-W., Teng K.-N.

Eksploatacja i Niezawodność

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2016

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Vol. 18, no. 3

379--389

EN

To precisely predict the residual life for functioning products is a key of carrying out condition based maintenance. For highly reliable products, it is difficult to obtain abundant degradation data to precisely predict the residual life under normal stress levels. Thus, how to make use of historical degradation data to improve the accuracy of the residual life prediction is an interesting issue. Accelerated degradation testing, which has been widely used to evaluate the reliability of highly reliable products, can provide abundant accelerated degradation data. In this paper, a residual life prediction method based on Bayesian inference that takes accelerated degradation data as prior information was studied. A Wiener process with a time function was used to model degradation data. In order to apply the random effects of all the parameters of a Wiener process, the non-conjugate prior distributions were considered. Acceleration factors were introduced to convert the parameter estimates from accelerated stress levels to normal stress levels, so that the proper prior distribution types of the random parameters can be selected by the Anderson-Darling statistic. A Markov Chain Monte Carlo method with Gibbs sampling was used to evaluate the posterior means of the random parameters. An illustrative example of self-regulating heating cable was utilized to validate the proposed method.

PL

Precyzyjne przewidywanie trwałości resztkowej użytkowanego produktu stanowi klucz do prawidłowego utrzymania ruchu w oparciu o bieżący stan techniczny (condition-based maintenance).W przypadku produktów o wysokiej niezawodności, trudno jest uzyskać ilość danych degradacyjnych, która umożliwiałaby precyzyjne prognozowanie trwałości resztkowej przy normalnym poziomie obciążeń. Dlatego też bardzo ważnym zagadnieniem jest wykorzystanie historycznych danych degradacyjnych umożliwiających zwiększenie trafności prognozowania trwałości resztkowej. Przyspieszone badania degradacyjne, które powszechnie wykorzystuje się do oceny niezawodności wysoce niezawodnych produktów, mogą dostarczać bogatych danych o przyspieszonej degradacji. W przedstawionej pracy badano metodę prognozowania trwałości resztkowej opartą na wnioskowaniu bayesowskim, w którym jako uprzednie informacje wykorzystano dane z przyspieszonych badań degradacji. Dane degradacyjne modelowano za pomocą procesu Wienera z funkcją czasu. Aby móc zastosować efekty losowe wszystkich parametrów procesu Wienera, rozważano niesprzężone rozkłady a priori. Wprowadzono współczynniki przyspieszenia , które pozwoliły na przekształcenie szacowanych wartości parametrów z poziomu obciążeństosowanych w próbie przyspieszonej do poziomu obciążeń normalnych, co umożliwiło wybór odpowiednich typów parametrów losowych rozkładu a priori zwykorzystaniem statystyki testowej Andersona-Darlinga. Metodę Monte Carlo opartą na łańcuchach Markowa z próbnikiem Gibbsa wykorzystano do oceny średnich a posteriori parametrów losowych. Proponowaną metodę zweryfikowano na postawie przykładu samoregulującego przewodu grzejnego.

10

Zastosowanie metod bayesowskich do prognozowania temperatury stali w kadzi odlewniczej i kadzi pośredniej

Miczka M.

Prace Instytutu Metalurgii Żelaza

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2015

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T. 67, nr 2

58--64

PL

W artykule przedstawiono podstawy metodyczne dla prognozowania i analizy ryzyka procesu produkcyjnego w hutnictwie żelaza i stali, ze szczególnym uwzględnieniem prognozowania temperatury w różnych fazach procesu stalowniczego. Metodyka oparta jest na reprezentacji systemu dynamicznego w przestrzeni stanu oraz wnioskowaniu bayesowskim. Pozwala to przede wszystkim uchylić założenie o stałości szacowanych parametrów, prowadzić analizę dla całości rozkładu statystycznego oraz uwzględnić tzw. informację a priori czyli pochodzącą spoza zbioru danych. Praca ma charakter przeglądowy i stanowi podstawę do dalszych badań, których ostatecznym celem jest wdrożenie systemu prognozowania i analizy ryzyka w jednej z polskich stalowni, a następnie opracowanie podobnych rozwiązań dla przypadku innych faz procesu hutniczego. Zaprezentowano zakres informacji na który zgodę wyraziło kierownictwo przedsiębiorstwa.

EN

The article presents the methodological basis for forecasting and risk analysis of the production process in the iron and steel industry, with particular emphasis on forecasting temperatures in the different stages of the steelmaking process. The methodology is based on the state space representation of a dynamic system and Bayesian inference. Above all it enables repeal the assumption of a constant estimated parameters, analyze the statistical distribution of the whole and take into account the so-called a priori information, from outside the dataset. Article is a review and provides a basis for further research, with the ultimate goal to implement a system for forecasting and risk analysis in one of the Polish steel mill, and then develop similar solutions for other phases of the metallurgical process. Presented range of information on which business executives expressed consent.

11

The advantages of bayesian methods over classical methods in the context of credible intervals

Grzenda W.

Information Systems in Management

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2015

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Vol. 4, No. 1

53--63

EN

The growing computational power of modern computer systems enables the efficient execution of algorithms. This is particularly important in Bayesian statistics, in which, nowadays, the key role is played by Markov Chain Monte Carlo methods. The primary objective of this work is to show the benefits arising from the use of Bayesian inference, especially confidence intervals in the context of logistic regression. The empirical analysis is based on "Household budgets" survey of Central Statistical Office. In this paper the unemployment among people over 55 will be investigated.

12

Efficient Markov chain Monte Carlo sampling for electrical impedance tomography

Ma E.

Computer Assisted Methods in Engineering and Science

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2014

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Vol. 21, no. 3-4

223--232

EN

This paper studies electrical impedance tomography (EIT) using Bayesian inference [1]. The resulting posterior distribution is sampled by Markov chain Monte Carlo (MCMC) [2]. This paper studies a toy model of EIT as the one presented in [3], and focuses on efficient MCMC sampling for this model. First, this paper analyses the computation of forward map of EIT which is the bottleneck of each MCMC update. The forward map is computed by the finite element method [4]. Here its exact computation was conducted up to five times more efficient, by updating the Cholesky factor of the stiffness matrix [5]. Since the forward map computation takes up nearly all the CPU time in each MCMC update, the overall efficiency of MCMC algorithms can be improved almost to the same amount. The forward map can also be computed approximately by local linearisation, and this approximate computation is much more efficient than the exact one. Without loss of efficiency, this approximate computation is more accurate here, after a log transformation is introduced into the local linearisation process. Later on, this improvement of accuracy will play an important role when the approximate computation of forward map will be employed for devising efficient MCMC algorithms. Second, the paper presents two novel MCMC algorithms for sampling the posterior distribution in the toy model of EIT. The two algorithms are made within the ‘multiple prior update’ [6] and the ‘delayed-acceptance Metropolis-Hastings’ [7] schemes respectively. Both of them have MCMC proposals that are made of localized updates, so that the forward map computation in each MCMC update can be made efficient by updating the Cholesky factor of the stiffness matrix. Both algorithms’ performances are compared to that of the standard single-site Metropolis [8], which is considered hard to surpass [3]. The algorithm of ‘multiple prior update’ is found to be six times more efficient, while the delayed-acceptance Metropolis-Hastings with single-site update is at least twice more efficient.

13

Type A Standard Uncertainty of Long-Term Noise Indicators

Batko W. M., Stępień B.

Archives of Acoustics

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2014

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Vol. 39, No. 1

25--36

EN

The problem of estimation of the long-term environmental noise hazard indicators and their uncer- tainty is presented in the present paper. The type A standard uncertainty is defined by the standard deviation of the mean. The rules given in the ISO/IEC Guide 98 are used in the calculations. It is usually determined by means of the classic variance estimators, under the following assumptions: the normality of measurements results, adequate sample size, lack of correlation between elements of the sample and observation equivalence. However, such assumptions in relation to the acoustic measurements are rather questionable. This is the reason why the authors indicated the necessity of implementation of non-classical statistical solutions. An estimation idea of seeking density function of long-term noise indicators distri- bution by the kernel density estimation, bootstrap method and Bayesian inference have been formulated. These methods do not generate limitations for form and properties of analyzed statistics. The theoretical basis of the proposed methods is presented in this paper as well as an example of calculation process of expected value and variance of long-term noise indicators LDEN and LN. The illustration of indicated solutions and their usefulness analysis were constant due to monitoring results of traffic noise recorded in Cracow, Poland.

14

A study on the phylogeny and phylogeography of a marine cosmopolite diatom from the southern Black Sea

Baytut Ö.

Oceanological and Hydrobiological Studies

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2013

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Vol. 42, No. 4

406--411

EN

A single-cell PCR method was applied to Pseudo-nitzschia pungens strains from the southern Black Sea. Based on the aligment set of the LSU D1-D3 region, a Bayesian molecular phylogeny analysis and a parsimony network analysis were used to investigate phylogenetic clades (Clades I-III) in P. pungens and to determine the ancestral clades. The parsimony network analysis also demonstrated that ancestral haplotypes belonged to Clade II, residing around the northeastern Pacific, while Clade I was distributed globally but antitropicaly. According to the findings of this study, the Black Sea strain (Clade III) shows a global phylogeographic pattern.

15

Application of Monte Carlo filter for computer vision-based Bayesian updating of finite element model

Tekieli M., Słoński M.

Mechanics and Control

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2013

|

Vol. 32, no. 4

171--177

EN

In this paper we describe Bayesian inference-based approach to the solution of parametric identification problem in the context of updating of a finite element model of a structure. The proposed inverse solution is based on Monte Carlo filter and on the comparison of structure displacements extracted using digital image correlation method during a quasi-static loading and the corresponding displacements predicted by finite element method program. Our approach is applied to the problem of material model parameter identification of an aluminum laboratory-scale frame. The results are also verified by comparing the Monte Carlo filter-based solution with the analytical solution obtained using Kalman filter.

PL

Artykuł przedstawia zastosowanie podejścia opartego na wnioskowaniu bayesowskim do problemu identyfikacji parametrycznej w kontekście strojenia modelu MES konstrukcji. Proponowane rozwiązanie odwrotne opiera się na filtrze Monte Carlo oraz porównaniu przemieszczeń konstrukcji otrzymanych metodą korelacji obrazów cyfrowych podczas quasi statycznej próby obciążeniowej i odpowiadających im przemieszczeń przewidywanych przez program oparty na metodzie elementów skończonych. Nasze podejście zostało zastosowane do identyfikacji parametru modelu materiału aluminiowej ramki laboratoryjnej. Otrzymane wyniki porównano z wynikami otrzymanymi za pomocą filtru Kalmana.

16

Bayesian-Based Methods for the Estimation of the Unknown Model’s Parameters in the Case of the Localization of the Atmospheric Contamination Source

Borysiewicz M., Wawrzynczak A., Kopka P.

Foundations of Computing and Decision Sciences

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2012

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Vol. 37, No. 4

256--272

EN

In many areas of application it is important to estimate unknown model parameters in order to model precisely the underlying dynamics of a physical system. In this context the Bayesian approach is a powerful tool to combine observed data along with prior knowledge to gain a current (probabilistic) understanding of unknown model parameters. We have applied the methodology combining Bayesian inference with Markov chain Monte Carlo (MCMC) to the problem of the atmospheric contaminant source localization. The algorithm input data are the on-line arriving information about concentration of given substance registered by distributed sensor network. We have examined different version of the MCMC algorithms in effectiveness to estimate the probabilistic distributions of atmospheric release parameters. The results indicate the probability of a source to occur at a particular location with a particular release rate.

17

Mathematical Foundations of Cognitive Radios

Couillet R., Debbah M.

Journal of Telecommunications and Information Technology

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2009

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nr 4

108-117

EN

Recently, much interest has been directed towards software defined radios and embedded intelligence in telecommunication devices. However, no fundamental basis for cognitive radios has ever been proposed. In this paper, we introduce a fundamental vision of cognitive radios from a physical layer viewpoint. Specifically, our motivation in this work is to embed human-like intelligence in mobile wireless devices, following the three century-old work on Bayesian probability theory, the maximum entropy principle and minimal probability update. This allows us to partially answer such questions as, what are the signal detection capabilities of a wireless device, when facing a situation in which most parameters are missing, how to react and so on. As an introductory example, we will present previous works from the same authors following the cognitive framework, and especially the multi-antenna channel modeling and signal sensing.

18

HPC strength prediction using Bayesian neural networks

Słoński M.

Computer Assisted Mechanics and Engineering Sciences

|

2007

|

Vol. 14, No. 2

345-352

EN

The objective of this paper is to investigate the efficiency of nonlinear Bayesian regression for modelling and predicting strength properties of high-performance concrete (HPC). A multilayer perceptron neural network (MLP) model is used. Two statistical approaches to learning and prediction for MLP based on the likelihood function maximization and Bayesian inference are applied and compared. Results of experimental data sets show that Bayesian approach for MLP offers some advantages over classical one.

19

Bayesian and empirical Bayesian approach to weighted averaging of ECG signal

Mamot A., Mamot M., Łęski J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2007

|

Vol. 55, nr 4

341-350

EN

One of the prime tool in non-invasive cardiac electrophysiology is the recording of an electrocardiographic signal (ECG) which analysis is greatly useful in the screening and diagnosis of cardiovascular diseases. However, one of the greatest problems is that usually recording an electrical activity of the heart is performed in the presence of noise. The paper presents Bayesian and empirical Bayesian approach to problem of weighted signal averaging in time domain which is commonly used to extract a useful signal distorted by a noise. The averaging is especially useful for biomedical signal such as ECG signal, where the spectra of the signal and noise significantly overlap. Using the methods of weighted averaging are motivated by variability of noise power from cycle to cycle, often observed in reality. It is demonstrated that exploiting a probabilistic Bayesian learning framework leads to accurate prediction models. Additionally, even in the presence of nuisance parameters the empirical Bayesian approach offers the method of theirs automatic estimation which reduces number of preset parameters. Performance of the new method is experimentally compared to the traditional averaging by using arithmetic mean and weighted averaging method based on criterion function minimization.

20

Empirical Bayesian averaging method and its application to noise reduction in ECG signal

Momot A., Momot M., Łęski J.

Journal of Medical Informatics & Technologies

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2006

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Vol. 10

93--101

EN

An electrocardiogram (ECG) is the prime tool in non-invasive cardiac electrophysiology and has a prime function in the screening and diagnosis of cardiovascular diseases. However one of the greatest problems is that usually recording an electrical activity of the heart is performed in the presence of noise. The paper presents empirical Bayesian approach to problem of signal averaging which is commonly used to extract a useful signal distorted by a noise. The averaging is especially useful for biomedical signal such as ECG signal, where the spectra of the signal and noise significantly overlap. In reality the variability of noise can be observed, with power from cycle to cycle, which is motivation for weighted averaging methods usage. It is demonstrated that by exploiting a probabilistic Bayesian learning framework, it can be derived accurate prediction models offering significant additional advantage, namely automatic estimation of 'nuisance' parameters. Performance of the new method is experimentally compared to the traditional averaging by using arithmetic mean and weighted averaging method based on criterion function minimization.