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.

Tilastotiede

Kartunta on vähäistä. Se johtunee osin siitä, että pääosa valtiotieteellisen tiedekunnan tilastotieteen tutkijoista työskentelee Kumpulan kampuksella, jonne he muuttivat vuonna 2004, jolloin HY:n matemaatikan laitos, tilastotieteen laitos ja Rolf Nevanlinna instituutti yhdistyivät matematiikan ja tilastotieteen laitokseksi. Tuolloin (tiedekunnan kirjastosta erillisen) tilastotieteen laitoksen kirjaston kokoelmia karsittiin ja jaettiin sekä sovittiin linjauksesta, jonka mukaan valtiotieteellisen tiedekunnan kirjaston tilastotieteen kokoelmaan hankitaan lähinnä tiedekunnan metodiopetuksen kannalta keskeistä tilastotieteellistä kirjallisuutta. Kirjastossa kokoelmassa on runsaat 700 nidettä, joista 15.8.2008 mennessä on Helkaan luetteloituina 703. Kokoelma sisältää pääosin (n. 80%) oppikirjoja tai sellaisiksi hyvin sopivia teoksia. Kokoelma tukeekin enemmän perustutkintoa suorittavien opintoja kuin tieteellistä tutkimusta. Nimekkeiden joukossa on myös jonkun verran matematiikan kirjoja. Kirjojen keski-ikä on varsin vanha. Pääosin kokoelma sisältää 1960-1989 ilmestyneitä teoksia (71%)mediaanin ollessa 1975. 2000-luvulla julkaistua kirjallisuutta on sangen vähän (9%). Kirjoista englanninkielisiä on 82%. Suomeksi julkaistuja on 9%. Tilastotiedettä sovelletaan monilla eri tieteenaloilla. Soveltavaa tilastotiedettä löytyy kirjaston muista kokoelmista etenkin kansantalouden ja sosiologian kokoelmista.

Hypoteettisesta universumista ja tilastollisista yhteyksistä johtolankoina

Vastine Pertti Tötön kirjoitukseen sosiologia-lehdessä 2/2000

Bayesian estimation of diameter distribution during harvesting

Abstract

Bayesian Estimation of Noise

In mathematical modeling the estimation of the model parameters is one of the most common problems. The goal is to seek parameters that fit to the measurements as well as possible. There is always error in the measurements which implies uncertainty to the model estimates. In Bayesian statistics all the unknown quantities are presented as probability distributions. If there is knowledge about parameters beforehand, it can be formulated as a prior distribution. The Bays’ rule combines the prior and the measurements to posterior distribution. Mathematical models are typically nonlinear, to produce statistics for them requires efficient sampling algorithms. In this thesis both Metropolis-Hastings (MH), Adaptive Metropolis (AM) algorithms and Gibbs sampling are introduced. In the thesis different ways to present prior distributions are introduced. The main issue is in the measurement error estimation and how to obtain prior knowledge for variance or covariance. Variance and covariance sampling is combined with the algorithms above. The examples of the hyperprior models are applied to estimation of model parameters and error in an outlier case.

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.