Adaptive MCMC methods with applications in environmental and geophysical models

This work presents new, efficient Markov chain Monte Carlo (MCMC) simulation methods for statistical analysis in various modelling applications. When using MCMC methods, the model is simulated repeatedly to explore the probability distribution describing the uncertainties in model parameters and predictions. In adaptive MCMC methods based on the Metropolis-Hastings algorithm, the proposal distribution needed by the algorithm learns from the target distribution as the simulation proceeds. Adaptive MCMC methods have been subject of intensive research lately, as they open a way for essentially easier use of the methodology. The lack of user-friendly computer programs has been a main obstacle for wider acceptance of the methods. This work provides two new adaptive MCMC methods: DRAM and AARJ. The DRAM method has been built especially to work in high dimensional and non-linear problems. The AARJ method is an extension to DRAM for model selection problems, where the mathematical formulation of the model is uncertain and we want simultaneously to fit several different models to the same observations. The methods were developed while keeping in mind the needs of modelling applications typical in environmental sciences. The development work has been pursued while working with several application projects. The applications presented in this work are: a winter time oxygen concentration model for Lake Tuusulanjärvi and adaptive control of the aerator; a nutrition model for Lake Pyhäjärvi and lake management planning; validation of the algorithms of the GOMOS ozone remote sensing instrument on board the Envisat satellite of European Space Agency and the study of the effects of aerosol model selection on the GOMOS algorithm.

Ratiomuuttujat ja näennäinen korrelaatio

Eri tieteenalojen tutkijat ovat kiistelleet jo yli vuosisadan ajan ratiomuodossa olevien muuttujien käytön vaikutuksista korrelaatio- ja regressioanalyysien tuloksiin ja niiden oikeaan tulkintaan. Strategiatutkimuksen piirissä aiheeseen ei ole kuitenkaan kiinnitetty suuresti huomiota. Tämä on yllättävää, sillä ratiomuuttujat ovat hyvin yleisesti käytettyjä empiirisen strategiatutkimuksen piirissä. Tässä työssä luodaan katsaus ratiomuuttujien ympärillä käytyyn debattiin. Lisäksi selvitetään artikkelikatsauksen avulla niiden käytön yleisyyttä nykypäivän strategiatutkimuksessa. Työssä tutkitaan Monte Carlo –simulaatioiden avulla ratiomuuttujien ominaisuuksien vaikutuksia korrelaatio- ja regressioanalyysin tuloksiin erityisesti yhteisen nimittäjän tapauksissa.

Statistical Analysis of an SEIR Epidemic Model

This thesis was focussed on statistical analysis methods and proposes the use of Bayesian inference to extract information contained in experimental data by estimating Ebola model parameters. The model is a system of differential equations expressing the behavior and dynamics of Ebola. Two sets of data (onset and death data) were both used to estimate parameters, which has not been done by previous researchers in (Chowell, 2004). To be able to use both data, a new version of the model has been built. Model parameters have been estimated and then used to calculate the basic reproduction number and to study the disease-free equilibrium. Estimates of the parameters were useful to determine how well the model fits the data and how good estimates were, in terms of the information they provided about the possible relationship between variables. The solution showed that Ebola model fits the observed onset data at 98.95% and the observed death data at 93.6%. Since Bayesian inference can not be performed analytically, the Markov chain Monte Carlo approach has been used to generate samples from the posterior distribution over parameters. Samples have been used to check the accuracy of the model and other characteristics of the target posteriors.

Statistical variation of weld profiles and their expected influence on fatigue strength

The general objective of this study was to conduct astatistical analysis on the variation of the weld profiles and their influence on the fatigue strength of the joint. Weld quality with respect to its fatigue strength is of importance which is the main concept behind this thesis. The intention of this study was to establish the influence of weld geometric parameters on the weld quality and fatigue strength. The effect of local geometrical variations of non-load carrying cruciform fillet welded joint under tensile loading wasstudied in this thesis work. Linear Elastic Fracture Mechanics was used to calculate fatigue strength of the cruciform fillet welded joints in as-welded condition and under cyclic tensile loading, for a range of weld geometries. With extreme value statistical analysis and LEFM, an attempt was made to relate the variation of the cruciform weld profiles such as weld angle and weld toe radius to respective FAT classes.

Monte Carlo Methods in Parameter Estimation of Nonlinear Models

Yksi keskeisimmistä tehtävistä matemaattisten mallien tilastollisessa analyysissä on mallien tuntemattomien parametrien estimointi. Tässä diplomityössä ollaan kiinnostuneita tuntemattomien parametrien jakaumista ja niiden muodostamiseen sopivista numeerisista menetelmistä, etenkin tapauksissa, joissa malli on epälineaarinen parametrien suhteen. Erilaisten numeeristen menetelmien osalta pääpaino on Markovin ketju Monte Carlo -menetelmissä (MCMC). Nämä laskentaintensiiviset menetelmät ovat viime aikoina kasvattaneet suosiotaan lähinnä kasvaneen laskentatehon vuoksi. Sekä Markovin ketjujen että Monte Carlo -simuloinnin teoriaa on esitelty työssä siinä määrin, että menetelmien toimivuus saadaan perusteltua. Viime aikoina kehitetyistä menetelmistä tarkastellaan etenkin adaptiivisia MCMC menetelmiä. Työn lähestymistapa on käytännönläheinen ja erilaisia MCMC -menetelmien toteutukseen liittyviä asioita korostetaan. Työn empiirisessä osuudessa tarkastellaan viiden esimerkkimallin tuntemattomien parametrien jakaumaa käyttäen hyväksi teoriaosassa esitettyjä menetelmiä. Mallit kuvaavat kemiallisia reaktioita ja kuvataan tavallisina differentiaaliyhtälöryhminä. Mallit on kerätty kemisteiltä Lappeenrannan teknillisestä yliopistosta ja Åbo Akademista, Turusta.

Development of data sheets for statistical evaluation of fatigue data

To enable a mathematically and physically sound execution of the fatigue test and a correct interpretation of its results, statistical evaluation methods are used to assist in the analysis of fatigue testing data. The main objective of this work is to develop step-by-stepinstructions for statistical analysis of the laboratory fatigue data. The scopeof this project is to provide practical cases about answering the several questions raised in the treatment of test data with application of the methods and formulae in the document IIW-XIII-2138-06 (Best Practice Guide on the Statistical Analysis of Fatigue Data). Generally, the questions in the data sheets involve some aspects: estimation of necessary sample size, verification of the statistical equivalence of the collated sets of data, and determination of characteristic curves in different cases. The series of comprehensive examples which are given in this thesis serve as a demonstration of the various statistical methods to develop a sound procedure to create reliable calculation rules for the fatigue analysis.

Statistical Analysis of Spectral Images

Tämä diplomityö liittyy Spektrikuvien tutkimiseen tilastollisen kuvamallin näkökulmasta. Diplomityön ensimmäisessä osassa tarkastellaan tilastollisten parametrien jakaumien vaikutusta väreihin ja korostumiin erilaisissa valaistusolosuhteissa. Havaittiin, että tilastollisten parametrien väliset suhteet eivät riipu valaistusolosuhteista, mutta riippuvat kuvan häiriöttömyydestä. Ilmeni myös, että korkea huipukkuus saattaa aiheutua värikylläisyydestä. Lisäksi työssä kehitettiin tilastolliseen spektrimalliin perustuvaa tekstuurinyhdistämisalgoritmia. Sillä saavutettiin hyviä tuloksia, kun tilastollisten parametrien väliset riippuvuussuhteet olivat voimassa. Työn toisessa osassa erilaisia spektrikuvia tutkittiin käyttäen itsenäistä komponenttien analyysia (ICA). Seuraavia itsenäiseen komponenttien analyysiin tarkoitettuja algoritmia tarkasteltiin: JADE, kiinteän pisteen ICA ja momenttikeskeinen ICA. Tutkimuksissa painotettiin erottelun laatua. Paras erottelu saavutettiin JADE- algoritmilla, joskin erot muiden algoritmien välillä eivät olleet merkittäviä. Algoritmi jakoi kuvan kahteen itsenäiseen, joko korostuneeseen ja korostumattomaan tai kromaattiseen ja akromaattiseen, komponenttiin. Lopuksi pohditaan huipukkuuden suhdetta kuvan ominaisuuksiin, kuten korostuneisuuteen ja värikylläisyyteen. Työn viimeisessä osassa ehdotetaan mahdollisia jatkotutkimuskohteita.

Spectral Color Appearance Modeling

Kuvien laatu on tutkituimpia ja käytetyimpiä aiheita. Tässä työssä tarkastellaan värin laatu ja spektrikuvia. Työssä annetaan yleiskuva olemassa olevista pakattujen ja erillisten kuvien laadunarviointimenetelmistä painottaen näiden menetelmien soveltaminen spektrikuviin. Tässä työssä esitellään spektriväriulkomuotomalli värikuvien laadunarvioinnille. Malli sovelletaan spektrikuvista jäljennettyihin värikuviin. Malli pohjautuu sekä tilastolliseen spektrikuvamalliin, joka muodostaa yhteyden spektrikuvien ja valokuvien parametrien välille, että kuvan yleiseen ulkomuotoon. Värikuvien tilastollisten spektriparametrien ja fyysisten parametrien välinen yhteys on varmennettu tietokone-pohjaisella kuvamallinnuksella. Mallin ominaisuuksien pohjalta on kehitetty koekäyttöön tarkoitettu menetelmä värikuvien laadunarvioinnille. On kehitetty asiantuntija-pohjainen kyselymenetelmä ja sumea päättelyjärjestelmä värikuvien laadunarvioinnille. Tutkimus osoittaa, että spektri-väri –yhteys ja sumea päättelyjärjestelmä soveltuvat tehokkaasti värikuvien laadunarviointiin.

Use of Statistical Methods in Demand Planning

In a very volatile industry of high technology it is of utmost importance to accurately forecast customers’ demand. However, statistical forecasting of sales, especially in heavily competitive electronics product business, has always been a challenging task due to very high variation in demand and very short product life cycles of products. The purpose of this thesis is to validate if statistical methods can be applied to forecasting sales of short life cycle electronics products and provide a feasible framework for implementing statistical forecasting in the environment of the case company. Two different approaches have been developed for forecasting on short and medium term and long term horizons. Both models are based on decomposition models, but differ in interpretation of the model residuals. For long term horizons residuals are assumed to represent white noise, whereas for short and medium term forecasting horizon residuals are modeled using statistical forecasting methods. Implementation of both approaches is performed in Matlab. Modeling results have shown that different markets exhibit different demand patterns and therefore different analytical approaches are appropriate for modeling demand in these markets. Moreover, the outcomes of modeling imply that statistical forecasting can not be handled separately from judgmental forecasting, but should be perceived only as a basis for judgmental forecasting activities. Based on modeling results recommendations for further deployment of statistical methods in sales forecasting of the case company are developed.

Statistical Methods for Conservation and Alignment Quality in Proteins

Construction of multiple sequence alignments is a fundamental task in Bioinformatics. Multiple sequence alignments are used as a prerequisite in many Bioinformatics methods, and subsequently the quality of such methods can be critically dependent on the quality of the alignment. However, automatic construction of a multiple sequence alignment for a set of remotely related sequences does not always provide biologically relevant alignments.Therefore, there is a need for an objective approach for evaluating the quality of automatically aligned sequences. The profile hidden Markov model is a powerful approach in comparative genomics. In the profile hidden Markov model, the symbol probabilities are estimated at each conserved alignment position. This can increase the dimension of parameter space and cause an overfitting problem. These two research problems are both related to conservation. We have developed statistical measures for quantifying the conservation of multiple sequence alignments. Two types of methods are considered, those identifying conserved residues in an alignment position, and those calculating positional conservation scores. The positional conservation score was exploited in a statistical prediction model for assessing the quality of multiple sequence alignments. The residue conservation score was used as part of the emission probability estimation method proposed for profile hidden Markov models. The results of the predicted alignment quality score highly correlated with the correct alignment quality scores, indicating that our method is reliable for assessing the quality of any multiple sequence alignment. The comparison of the emission probability estimation method with the maximum likelihood method showed that the number of estimated parameters in the model was dramatically decreased, while the same level of accuracy was maintained. To conclude, we have shown that conservation can be successfully used in the statistical model for alignment quality assessment and in the estimation of emission probabilities in the profile hidden Markov models.

Statistical Inversion Theory in X-ray Tomography

In this thesis the X-ray tomography is discussed from the Bayesian statistical viewpoint. The unknown parameters are assumed random variables and as opposite to traditional methods the solution is obtained as a large sample of the distribution of all possible solutions. As an introduction to tomography an inversion formula for Radon transform is presented on a plane. The vastly used filtered backprojection algorithm is derived. The traditional regularization methods are presented sufficiently to ground the Bayesian approach. The measurements are foton counts at the detector pixels. Thus the assumption of a Poisson distributed measurement error is justified. Often the error is assumed Gaussian, altough the electronic noise caused by the measurement device can change the error structure. The assumption of Gaussian measurement error is discussed. In the thesis the use of different prior distributions in X-ray tomography is discussed. Especially in severely ill-posed problems the use of a suitable prior is the main part of the whole solution process. In the empirical part the presented prior distributions are tested using simulated measurements. The effect of different prior distributions produce are shown in the empirical part of the thesis. The use of prior is shown obligatory in case of severely ill-posed problem.

Statistical Optimum Design of Heat Exchangers

The optimal design of a heat exchanger system is based on given model parameters together with given standard ranges for machine design variables. The goals set for minimizing the Life Cycle Cost (LCC) function which represents the price of the saved energy, for maximizing the momentary heat recovery output with given constraints satisfied and taking into account the uncertainty in the models were successfully done. Nondominated Sorting Genetic Algorithm II (NSGA-II) for the design optimization of a system is presented and implemented inMatlab environment. Markov ChainMonte Carlo (MCMC) methods are also used to take into account the uncertainty in themodels. Results show that the price of saved energy can be optimized. A wet heat exchanger is found to be more efficient and beneficial than a dry heat exchanger even though its construction is expensive (160 EUR/m2) compared to the construction of a dry heat exchanger (50 EUR/m2). It has been found that the longer lifetime weights higher CAPEX and lower OPEX and vice versa, and the effect of the uncertainty in the models has been identified in a simplified case of minimizing the area of a dry heat exchanger.

Statistical Analysis and Numerics of Heat Exchanger Models

The identifiability of the parameters of a heat exchanger model without phase change was studied in this Master’s thesis using synthetically made data. A fast, two-step Markov chain Monte Carlo method (MCMC) was tested with a couple of case studies and a heat exchanger model. The two-step MCMC-method worked well and decreased the computation time compared to the traditional MCMC-method. The effect of measurement accuracy of certain control variables to the identifiability of parameters was also studied. The accuracy used did not seem to have a remarkable effect to the identifiability of parameters. The use of the posterior distribution of parameters in different heat exchanger geometries was studied. It would be computationally most efficient to use the same posterior distribution among different geometries in the optimisation of heat exchanger networks. According to the results, this was possible in the case when the frontal surface areas were the same among different geometries. In the other cases the same posterior distribution can be used for optimisation too, but that will give a wider predictive distribution as a result. For condensing surface heat exchangers the numerical stability of the simulation model was studied. As a result, a stable algorithm was developed.