Using time and stochastic models for software reliability forecasting and analysis

Tämä työ luo katsauksen ajallisiin ja stokastisiin ohjelmien luotettavuus malleihin sekä tutkii muutamia malleja käytännössä. Työn teoriaosuus sisältää ohjelmien luotettavuuden kuvauksessa ja arvioinnissa käytetyt keskeiset määritelmät ja metriikan sekä varsinaiset mallien kuvaukset. Työssä esitellään kaksi ohjelmien luotettavuusryhmää. Ensimmäinen ryhmä ovat riskiin perustuvat mallit. Toinen ryhmä käsittää virheiden ”kylvöön” ja merkitsevyyteen perustuvat mallit. Työn empiirinen osa sisältää kokeiden kuvaukset ja tulokset. Kokeet suoritettiin käyttämällä kolmea ensimmäiseen ryhmään kuuluvaa mallia: Jelinski-Moranda mallia, ensimmäistä geometrista mallia sekä yksinkertaista eksponenttimallia. Kokeiden tarkoituksena oli tutkia, kuinka syötetyn datan distribuutio vaikuttaa mallien toimivuuteen sekä kuinka herkkiä mallit ovat syötetyn datan määrän muutoksille. Jelinski-Moranda malli osoittautui herkimmäksi distribuutiolle konvergaatio-ongelmien vuoksi, ensimmäinen geometrinen malli herkimmäksi datan määrän muutoksille.

MCMC Analysis for Optimization of Stochastic Models

In any decision making under uncertainties, the goal is mostly to minimize the expected cost. The minimization of cost under uncertainties is usually done by optimization. For simple models, the optimization can easily be done using deterministic methods.However, many models practically contain some complex and varying parameters that can not easily be taken into account using usual deterministic methods of optimization. Thus, it is very important to look for other methods that can be used to get insight into such models. MCMC method is one of the practical methods that can be used for optimization of stochastic models under uncertainty. This method is based on simulation that provides a general methodology which can be applied in nonlinear and non-Gaussian state models. MCMC method is very important for practical applications because it is a uni ed estimation procedure which simultaneously estimates both parameters and state variables. MCMC computes the distribution of the state variables and parameters of the given data measurements. MCMC method is faster in terms of computing time when compared to other optimization methods. This thesis discusses the use of Markov chain Monte Carlo (MCMC) methods for optimization of Stochastic models under uncertainties .The thesis begins with a short discussion about Bayesian Inference, MCMC and Stochastic optimization methods. Then an example is given of how MCMC can be applied for maximizing production at a minimum cost in a chemical reaction process. It is observed that this method performs better in optimizing the given cost function with a very high certainty.

Public-private partnership investments in dry ports – Russian logistics markets and risks

The investments have always been considered as an essential backbone and so-called ‘locomotive’ for the competitive economies. However, in various countries, the state has been put under tight budget constraints for the investments in capital intensive projects. In response to this situation, the cooperation between public and private sector has grown based on public-private mechanism. The promotion of favorable arrangement for collaboration between public and private sectors for the provision of policies, services, and infrastructure in Russia can help to address the problems of dry ports development that neither municipalities nor the private sector can solve alone. Especially, the stimulation of public-private collaboration is significant under the exposure to externalities that affect the magnitude of the risks during all phases of project realization. In these circumstances, the risk in the projects also is becoming increasingly a part of joint research and risk management practice, which is viewed as a key approach, aiming to take active actions on existing global and specific factors of uncertainties. Meanwhile, a relatively little progress has been made on the inclusion of the resilience aspects into the planning process of a dry ports construction that would instruct the capacity planner, on how to mitigate the occurrence of disruptions that may lead to million dollars of losses due to the deviation of the future cash flows from the expected financial flows on the project. The current experience shows that the existing methodological base is developed fragmentary within separate steps of supply chain risk management (SCRM) processes: risk identification, risk evaluation, risk mitigation, risk monitoring and control phases. The lack of the systematic approach hinders the solution of the problem of risk management processes of dry port implementation. Therefore, management of various risks during the investments phases of dry port projects still presents a considerable challenge from the practical and theoretical points of view. In this regard, the given research became a logical continuation of fundamental research, existing in the financial models and theories (e.g., capital asset pricing model and real option theory), as well as provided a complementation for the portfolio theory. The goal of the current study is in the design of methods and models for the facilitation of dry port implementation through the mechanism of public-private partnership on the national market that implies the necessity to mitigate, first and foremost, the shortage of the investments and consequences of risks. The problem of the research was formulated on the ground of the identified contradictions. They rose as a continuation of the trade-off between the opportunities that the investors can gain from the development of terminal business in Russia (i.e. dry port implementation) and risks. As a rule, the higher the investment risk, the greater should be their expected return. However, investors have a different tolerance for the risks. That is why it would be advisable to find an optimum investment. In the given study, the optimum relates to the search for the efficient portfolio, which can provide satisfaction to the investor, depending on its degree of risk aversion. There are many theories and methods in finance, concerning investment choices. Nevertheless, the appropriateness and effectiveness of particular methods should be considered with the allowance of the specifics of the investment projects. For example, the investments in dry ports imply not only the lump sum of financial inflows, but also the long-term payback periods. As a result, capital intensity and longevity of their construction determine the necessity from investors to ensure the return on investment (profitability), along with the rapid return on investment (liquidity), without precluding the fact that the stochastic nature of the project environment is hardly described by the formula-based approach. The current theoretical base for the economic appraisals of the dry port projects more often perceives net present value (NPV) as a technique superior to other decision-making criteria. For example, the portfolio theory, which considers different risk preference of an investor and structures of utility, defines net present value as a better criterion of project appraisal than discounted payback period (DPP). Meanwhile, in business practice, the DPP is more popular. Knowing that the NPV is based on the assumptions of certainty of project life, it cannot be an accurate appraisal approach alone to determine whether or not the project should be accepted for the approval in the environment that is not without of uncertainties. In order to reflect the period or the project’s useful life that is exposed to risks due to changes in political, operational, and financial factors, the second capital budgeting criterion – discounted payback period is profoundly important, particularly for the Russian environment. Those statements represent contradictions that exist in the theory and practice of the applied science. Therefore, it would be desirable to relax the assumptions of portfolio theory and regard DPP as not fewer relevant appraisal approach for the assessment of the investment and risk measure. At the same time, the rationality of the use of both project performance criteria depends on the methods and models, with the help of which these appraisal approaches are calculated in feasibility studies. The deterministic methods cannot ensure the required precision of the results, while the stochastic models guarantee the sufficient level of the accuracy and reliability of the obtained results, providing that the risks are properly identified, evaluated, and mitigated. Otherwise, the project performance indicators may not be confirmed during the phase of project realization. For instance, the economic and political instability can result in the undoing of hard-earned gains, leading to the need for the attraction of the additional finances for the project. The sources of the alternative investments, as well as supportive mitigation strategies, can be studied during the initial phases of project development. During this period, the effectiveness of the investments undertakings can also be improved by the inclusion of the various investors, e.g. Russian Railways’ enterprises and other private companies in the dry port projects. However, the evaluation of the effectiveness of the participation of different investors in the project lack the methods and models that would permit doing the particular feasibility study, foreseeing the quantitative characteristics of risks and their mitigation strategies, which can meet the tolerance of the investors to the risks. For this reason, the research proposes a combination of Monte Carlo method, discounted cash flow technique, the theory of real options, and portfolio theory via a system dynamics simulation approach. The use of this methodology allows for comprehensive risk management process of dry port development to cover all aspects of risk identification, risk evaluation, risk mitigation, risk monitoring, and control phases. A designed system dynamics model can be recommended for the decision-makers on the dry port projects that are financed via a public-private partnership. It permits investors to make a decision appraisal based on random variables of net present value and discounted payback period, depending on different risks factors, e.g. revenue risks, land acquisition risks, traffic volume risks, construction hazards, and political risks. In this case, the statistical mean is used for the explication of the expected value of the DPP and NPV; the standard deviation is proposed as a characteristic of risks, while the elasticity coefficient is applied for rating of risks. Additionally, the risk of failure of project investments and guaranteed recoupment of capital investment can be considered with the help of the model. On the whole, the application of these modern methods of simulation creates preconditions for the controlling of the process of dry port development, i.e. making managerial changes and identifying the most stable parameters that contribute to the optimal alternative scenarios of the project realization in the uncertain environment. System dynamics model allows analyzing the interactions in the most complex mechanism of risk management process of the dry ports development and making proposals for the improvement of the effectiveness of the investments via an estimation of different risk management strategies. For the comparison and ranking of these alternatives in their order of preference to the investor, the proposed indicators of the efficiency of the investments, concerning the NPV, DPP, and coefficient of variation, can be used. Thus, rational investors, who averse to taking increased risks unless they are compensated by the commensurate increase in the expected utility of a risky prospect of dry port development, can be guided by the deduced marginal utility of investments. It is computed on the ground of the results from the system dynamics model. In conclusion, the outlined theoretical and practical implications for the management of risks, which are the key characteristics of public-private partnerships, can help analysts and planning managers in budget decision-making, substantially alleviating the effect from various risks and avoiding unnecessary cost overruns in dry port projects.

Stochastic particle models: mean reversion and burgers dynamics. An application to commodity markets

The aim of this study is to propose a stochastic model for commodity markets linked with the Burgers equation from fluid dynamics. We construct a stochastic particles method for commodity markets, in which particles represent market participants. A discontinuity in the model is included through an interacting kernel equal to the Heaviside function and its link with the Burgers equation is given. The Burgers equation and the connection of this model with stochastic differential equations are also studied. Further, based on the law of large numbers, we prove the convergence, for large N, of a system of stochastic differential equations describing the evolution of the prices of N traders to a deterministic partial differential equation of Burgers type. Numerical experiments highlight the success of the new proposal in modeling some commodity markets, and this is confirmed by the ability of the model to reproduce price spikes when their effects occur in a sufficiently long period of time.

New directions in Stochastic Multicriteria Acceptability Analysis

Decisions taken in modern organizations are often multi-dimensional, involving multiple decision makers and several criteria measured on different scales. Multiple Criteria Decision Making (MCDM) methods are designed to analyze and to give recommendations in this kind of situations. Among the numerous MCDM methods, two large families of methods are the multi-attribute utility theory based methods and the outranking methods. Traditionally both method families require exact values for technical parameters and criteria measurements, as well as for preferences expressed as weights. Often it is hard, if not impossible, to obtain exact values. Stochastic Multicriteria Acceptability Analysis (SMAA) is a family of methods designed to help in this type of situations where exact values are not available. Different variants of SMAA allow handling all types of MCDM problems. They support defining the model through uncertain, imprecise, or completely missing values. The methods are based on simulation that is applied to obtain descriptive indices characterizing the problem. In this thesis we present new advances in the SMAA methodology. We present and analyze algorithms for the SMAA-2 method and its extension to handle ordinal preferences. We then present an application of SMAA-2 to an area where MCDM models have not been applied before: planning elevator groups for high-rise buildings. Following this, we introduce two new methods to the family: SMAA-TRI that extends ELECTRE TRI for sorting problems with uncertain parameter values, and SMAA-III that extends ELECTRE III in a similar way. An efficient software implementing these two methods has been developed in conjunction with this work, and is briefly presented in this thesis. The thesis is closed with a comprehensive survey of SMAA methodology including a definition of a unified framework.

Empirical Tests of Asset Pricing Models in Finnish Stock Market

Tämän tutkielman tavoitteena on selvittää mitkä riskitekijät vaikuttavat osakkeiden tuottoihin. Arvopapereina käytetään kuutta portfoliota, jotka ovat jaoteltu markkina-arvon mukaan. Aikaperiodi on vuoden 1987 alusta vuoden 2004 loppuun. Malleina käytetään pääomamarkkinoiden hinnoittelumallia, arbitraasihinnoitteluteoriaa sekä kulutuspohjaista pääomamarkkinoiden hinnoittelumallia. Riskifaktoreina kahteen ensimmäiseen malliin käytetään markkinariskiä sekä makrotaloudellisia riskitekijöitä. Kulutuspohjaiseen pääomamarkkinoiden hinnoinoittelumallissa keskitytään estimoimaan kuluttajien riskitottumuksia sekä diskonttaustekijää, jolla kuluttaja arvostavat tulevaisuuden kulutusta. Tämä työ esittelee momenttiteorian, jolla pystymme estimoimaan lineaarisia sekä epälineaarisia yhtälöitä. Käytämme tätä menetelmää testaamissamme malleissa. Yhteenvetona tuloksista voidaan sanoa, että markkinabeeta onedelleen tärkein riskitekijä, mutta löydämme myös tukea makrotaloudellisille riskitekijöille. Kulutuspohjainen mallimme toimii melko hyvin antaen teoreettisesti hyväksyttäviä arvoja.

Regime Switching Models for Electricity Time Series that Capture Fat Tailed Distributions

In the power market, electricity prices play an important role at the economic level. The behavior of a price trend usually known as a structural break may change over time in terms of its mean value, its volatility, or it may change for a period of time before reverting back to its original behavior or switching to another style of behavior, and the latter is typically termed a regime shift or regime switch. Our task in this thesis is to develop an electricity price time series model that captures fat tailed distributions which can explain this behavior and analyze it for better understanding. For NordPool data used, the obtained Markov Regime-Switching model operates on two regimes: regular and non-regular. Three criteria have been considered price difference criterion, capacity/flow difference criterion and spikes in Finland criterion. The suitability of GARCH modeling to simulate multi-regime modeling is also studied.

Stochastic variability of output levels in a pulp mill

Quite often, in the construction of a pulp mill involves establishing the size of tanks which will accommodate the material from the various processes in which case estimating the right tank size a priori would be vital. Hence, simulation of the whole production process would be worthwhile. Therefore, there is need to develop mathematical models that would mimic the behavior of the output from the various production units of the pulp mill to work as simulators. Markov chain models, Autoregressive moving average (ARMA) model, Mean reversion models with ensemble interaction together with Markov regime switching models are proposed for that purpose.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Context-specific independence in graphical models

The theme of this thesis is context-speci c independence in graphical models. Considering a system of stochastic variables it is often the case that the variables are dependent of each other. This can, for instance, be seen by measuring the covariance between a pair of variables. Using graphical models, it is possible to visualize the dependence structure found in a set of stochastic variables. Using ordinary graphical models, such as Markov networks, Bayesian networks, and Gaussian graphical models, the type of dependencies that can be modeled is limited to marginal and conditional (in)dependencies. The models introduced in this thesis enable the graphical representation of context-speci c independencies, i.e. conditional independencies that hold only in a subset of the outcome space of the conditioning variables. In the articles included in this thesis, we introduce several types of graphical models that can represent context-speci c independencies. Models for both discrete variables and continuous variables are considered. A wide range of properties are examined for the introduced models, including identi ability, robustness, scoring, and optimization. In one article, a predictive classi er which utilizes context-speci c independence models is introduced. This classi er clearly demonstrates the potential bene ts of the introduced models. The purpose of the material included in the thesis prior to the articles is to provide the basic theory needed to understand the articles.

Adaptive Markov chain Monte Carlo and Bayesian filtering for state space models

This thesis is concerned with the state and parameter estimation in state space models. The estimation of states and parameters is an important task when mathematical modeling is applied to many different application areas such as the global positioning systems, target tracking, navigation, brain imaging, spread of infectious diseases, biological processes, telecommunications, audio signal processing, stochastic optimal control, machine learning, and physical systems. In Bayesian settings, the estimation of states or parameters amounts to computation of the posterior probability density function. Except for a very restricted number of models, it is impossible to compute this density function in a closed form. Hence, we need approximation methods. A state estimation problem involves estimating the states (latent variables) that are not directly observed in the output of the system. In this thesis, we use the Kalman filter, extended Kalman filter, Gauss–Hermite filters, and particle filters to estimate the states based on available measurements. Among these filters, particle filters are numerical methods for approximating the filtering distributions of non-linear non-Gaussian state space models via Monte Carlo. The performance of a particle filter heavily depends on the chosen importance distribution. For instance, inappropriate choice of the importance distribution can lead to the failure of convergence of the particle filter algorithm. In this thesis, we analyze the theoretical Lᵖ particle filter convergence with general importance distributions, where p ≥2 is an integer. A parameter estimation problem is considered with inferring the model parameters from measurements. For high-dimensional complex models, estimation of parameters can be done by Markov chain Monte Carlo (MCMC) methods. In its operation, the MCMC method requires the unnormalized posterior distribution of the parameters and a proposal distribution. In this thesis, we show how the posterior density function of the parameters of a state space model can be computed by filtering based methods, where the states are integrated out. This type of computation is then applied to estimate parameters of stochastic differential equations. Furthermore, we compute the partial derivatives of the log-posterior density function and use the hybrid Monte Carlo and scaled conjugate gradient methods to infer the parameters of stochastic differential equations. The computational efficiency of MCMC methods is highly depend on the chosen proposal distribution. A commonly used proposal distribution is Gaussian. In this kind of proposal, the covariance matrix must be well tuned. To tune it, adaptive MCMC methods can be used. In this thesis, we propose a new way of updating the covariance matrix using the variational Bayesian adaptive Kalman filter algorithm.

Bayesian analysis of SEIR epidemic models

This thesis concerns the analysis of epidemic models. We adopt the Bayesian paradigm and develop suitable Markov Chain Monte Carlo (MCMC) algorithms. This is done by considering an Ebola outbreak in the Democratic Republic of Congo, former Zaïre, 1995 as a case of SEIR epidemic models. We model the Ebola epidemic deterministically using ODEs and stochastically through SDEs to take into account a possible bias in each compartment. Since the model has unknown parameters, we use different methods to estimate them such as least squares, maximum likelihood and MCMC. The motivation behind choosing MCMC over other existing methods in this thesis is that it has the ability to tackle complicated nonlinear problems with large number of parameters. First, in a deterministic Ebola model, we compute the likelihood function by sum of square of residuals method and estimate parameters using the LSQ and MCMC methods. We sample parameters and then use them to calculate the basic reproduction number and to study the disease-free equilibrium. From the sampled chain from the posterior, we test the convergence diagnostic and confirm the viability of the model. The results show that the Ebola model fits the observed onset data with high precision, and all the unknown model parameters are well identified. Second, we convert the ODE model into a SDE Ebola model. We compute the likelihood function using extended Kalman filter (EKF) and estimate parameters again. The motivation of using the SDE formulation here is to consider the impact of modelling errors. Moreover, the EKF approach allows us to formulate a filtered likelihood for the parameters of such a stochastic model. We use the MCMC procedure to attain the posterior distributions of the parameters of the SDE Ebola model drift and diffusion parts. In this thesis, we analyse two cases: (1) the model error covariance matrix of the dynamic noise is close to zero , i.e. only small stochasticity added into the model. The results are then similar to the ones got from deterministic Ebola model, even if methods of computing the likelihood function are different (2) the model error covariance matrix is different from zero, i.e. a considerable stochasticity is introduced into the Ebola model. This accounts for the situation where we would know that the model is not exact. As a results, we obtain parameter posteriors with larger variances. Consequently, the model predictions then show larger uncertainties, in accordance with the assumption of an incomplete model.

Parameter estimation for Chaotic or Stochastic Dynamics

Since its discovery, chaos has been a very interesting and challenging topic of research. Many great minds spent their entire lives trying to give some rules to it. Nowadays, thanks to the research of last century and the advent of computers, it is possible to predict chaotic phenomena of nature for a certain limited amount of time. The aim of this study is to present a recently discovered method for the parameter estimation of the chaotic dynamical system models via the correlation integral likelihood, and give some hints for a more optimized use of it, together with a possible application to the industry. The main part of our study concerned two chaotic attractors whose general behaviour is diff erent, in order to capture eventual di fferences in the results. In the various simulations that we performed, the initial conditions have been changed in a quite exhaustive way. The results obtained show that, under certain conditions, this method works very well in all the case. In particular, it came out that the most important aspect is to be very careful while creating the training set and the empirical likelihood, since a lack of information in this part of the procedure leads to low quality results.

Structure learning of context-specific graphical models

Abstract The ultimate problem considered in this thesis is modeling a high-dimensional joint distribution over a set of discrete variables. For this purpose, we consider classes of context-specific graphical models and the main emphasis is on learning the structure of such models from data. Traditional graphical models compactly represent a joint distribution through a factorization justi ed by statements of conditional independence which are encoded by a graph structure. Context-speci c independence is a natural generalization of conditional independence that only holds in a certain context, speci ed by the conditioning variables. We introduce context-speci c generalizations of both Bayesian networks and Markov networks by including statements of context-specific independence which can be encoded as a part of the model structures. For the purpose of learning context-speci c model structures from data, we derive score functions, based on results from Bayesian statistics, by which the plausibility of a structure is assessed. To identify high-scoring structures, we construct stochastic and deterministic search algorithms designed to exploit the structural decomposition of our score functions. Numerical experiments on synthetic and real-world data show that the increased exibility of context-specific structures can more accurately emulate the dependence structure among the variables and thereby improve the predictive accuracy of the models.