Inferring Latent States and Refining Force Estimates via Hierarchical Dirichlet Process Modeling in Single Particle Tracking Experiments

Optical microscopy provides rich spatio-temporal information characterizing in vivo molecular motion. However, effective forces and other parameters used to summarize molecular motion change over time in live cells due to latent state changes, e.g., changes induced by dynamic micro-environments, photobleaching, and other heterogeneity inherent in biological processes. This study focuses on techniques for analyzing Single Particle Tracking (SPT) data experiencing abrupt state changes. We demonstrate the approach on GFP tagged chromatids experiencing metaphase in yeast cells and probe the effective forces resulting from dynamic interactions that reflect the sum of a number of physical phenomena. State changes are induced by factors such as microtubule dynamics exerting force through the centromere, thermal polymer fluctuations, etc. Simulations are used to demonstrate the relevance of the approach in more general SPT data analyses. Refined force estimates are obtained by adopting and modifying a nonparametric Bayesian modeling technique, the Hierarchical Dirichlet Process Switching Linear Dynamical System (HDP-SLDS), for SPT applications. The HDP-SLDS method shows promise in systematically identifying dynamical regime changes induced by unobserved state changes when the number of underlying states is unknown in advance (a common problem in SPT applications). We expand on the relevance of the HDP-SLDS approach, review the relevant background of Hierarchical Dirichlet Processes, show how to map discrete time HDP-SLDS models to classic SPT models, and discuss limitations of the approach. In addition, we demonstrate new computational techniques for tuning hyperparameters and for checking the statistical consistency of model assumptions directly against individual experimental trajectories; the techniques circumvent the need for "ground-truth" and subjective information.Comment: 25 pages, 6 figures. Differs only typographically from PLoS One publication available freely as an open-access article at http://journals.plos.org/plosone/article?id=10.1371/journal.pone.013763

Modern Statistical/Machine Learning Techniques for Bio/Neuro-imaging Applications

Developments in modern bio-imaging techniques have allowed the routine collection of a vast amount of data from various techniques. The challenges lie in how to build accurate and efficient models to draw conclusions from the data and facilitate scientific discoveries. Fortunately, recent advances in statistics, machine learning, and deep learning provide valuable tools. This thesis describes some of our efforts to build scalable Bayesian models for four bio-imaging applications: (1) Stochastic Optical Reconstruction Microscopy (STORM) Imaging, (2) particle tracking, (3) voltage smoothing, (4) detect color-labeled neurons in c elegans and assign identity to the detections

Accelerator Mass Spectrometry and Bayesian Data Analysis

Accelerator mass spectrometry (AMS) is an ultrasensitive technique for measuring the concentration of a single isotope. The electric and magnetic fields of an electrostatic accelerator system are used to filter out other isotopes from the ion beam. The high velocity means that molecules can be destroyed and removed from the measurement background. As a result, concentrations down to one atom in 10^16 atoms are measurable. This thesis describes the construction of the new AMS system in the Accelerator Laboratory of the University of Helsinki. The system is described in detail along with the relevant ion optics. System performance and some of the 14C measurements done with the system are described. In a second part of the thesis, a novel statistical model for the analysis of AMS data is presented. Bayesian methods are used in order to make the best use of the available information. In the new model, instrumental drift is modelled with a continuous first-order autoregressive process. This enables rigorous normalization to standards measured at different times. The Poisson statistical nature of a 14C measurement is also taken into account properly, so that uncertainty estimates are much more stable. It is shown that, overall, the new model improves both the accuracy and the precision of AMS measurements. In particular, the results can be improved for samples with very low 14C concentrations or measured only a few times.Luonnossa useat prosessit muuttavat alkuaineen isotooppien runsaussuhteita. Näin ollen isotooppien määrän mittaaminen auttaa osaltaan ymmärtämään luonnon prosesseja ja tutkittavan kohteen historiaa. Yksi isotooppien runsaussuhteita muuttava prosessi on radioaktiivisuus. Esimerkiksi radioaktiivisen hiili-14 isotoopin määrä kertoo kuinka kauan eloperäinen näyte on ollut kuolleena tai eristettynä hiilen normaalista kiertokulusta. Luukappaleen, vuosikertaviinin tai vaikkapa ilmakehän aerosolien alkuperää voidaan siis ymmärtää paremmin mittaamalla kohteen hiili-14 pitoisuus. Koska saman alkuaineen eri isotoopit käyttäytyvät kemiallisesti samalla tavoin, harvinaisia isotooppeja voidaan myös käyttää leima-aineina tutkittaville prosesseille kuten lääkeaineiden kulkeutumiselle kehossa. Kiihdytinpohjaisessa massaspektrometriassa hiukkaskiihdytintä käytetään harvinaisten alkuaine-isotooppien runsaussuhteiden mittaamiseen. Kiihdyttimellä saavutettava hiukkasten suuri nopeus mahdollistaa häiritsevien molekyylien poistamisen hiukkassuihkusta törmäyksien avulla. Ei-halutut ionit erotetaan hiukkassuihkusta magneetti- ja sähkökentillä. Tavalliseen massaspektrometriin verrattuna kiihdytinmassaspektrometrillä voidaan mitata yli viisi kertaluokkaa pienempiä pitoisuuksia. Esimerkiksi hiili-14-isotoopin ilmaisukynnys on alle 1:10^15, joka vastaa yhtä milligrammaa hiili-14-isotooppia miljoonassa tonnissa tavallista hiiltä. Tässä väitöskirjatyössä esitellään Helsingin yliopiston fysikaalisten tieteiden laitoksen kiihdytinmassaspektrometri, joka on ensimmäinen laite Suomessa. Nykyinen laitteisto, ionioptiset laskut, kiihdyttimen vaatimat merkittävät muutostyöt sekä mittausten läpivienti kuvataan pääpiirteissään. Lisäksi käydään läpi laitteiston olennaiset ominaisuudet, suorituskyky sekä näiden toteamiseksi suoritetut mittaukset. Laitteen nykyinen tarkkuus on alle prosentin, ja pienin laitteella mitattu hiili-14 runsaussuhde on 4×10^-17, joka vastaa kansainvälisesti parasta tausta-arvoa. Kiihdytinmassaspektrometrian sovellusalueilla luotettavat tulokset ja mittausepävarmuudet ovat ensiarvoisen tärkeitä. Tässä työssä kehitetty bayeslainen tulostenkäsittelymenetelmä parantaa kiihdytinmassaspektrometrian tuloksia merkittävästi keskiarvoistukseen perustuviin menetelmiin verrattuna. Uudessa menetelmässä laitteesta itsestään johtuvaa virhettä kuvataan mukautuvalla autoregressiivisellä prosessilla. Mukautuva prosessi etsii parhaan tavan, jolla mittaustulokset normitetaan standardinäytteiden tuloksiin. Yksittäisten mittaustulosten tiedetään noudattavan Poisson-jakaumaa, joka myös huomioidaan uudessa menetelmässä. Siinä missä otos-pohjaisten menetelmien mittausepävarmuudet heittelevät aiheuttaen virherajojen moninkertaisia ylityksiä, uudessa menetelmässä esitieto virhelähteistä vakauttaa mittausepävarmuudet. Näin tulosten luotettavuus paranee merkittävästi. Myös varsinaisen tuloksen tarkkuus paranee hieman aikaisempiin menetelmiin verrattuna. Lisäksi menetelmä antaa erityisen tarkkoja ja luotettavia tuloksia 1) pienille konsentraatioille, 2) mittauksille joiden välillä on korrelaatioita sekä 3) tapauksille, joissa yhdestä näytteestä voidaan ottaa vain muutamia mittauksia

Exploring Probability Measures with Markov Processes

In many domains where mathematical modelling is applied, a deterministic description of the system at hand is insufficient, and so it is useful to model systems as being in some way stochastic. This is often achieved by modeling the state of the system as being drawn from a probability measure, which is usually given algebraically, i.e. as a formula. While this representation can be useful for deriving certain characteristics of the system, it is by now well-appreciated that many questions about stochastic systems are best-answered by looking at samples from the associated probability measure. In this thesis, we seek to develop and analyse efficient techniques for generating samples from a given probability measure, with a focus on algorithms which simulate a Markov process with the desired invariant measure. The first work presented in this thesis considers the use of Piecewise-Deterministic Markov Processes (PDMPs) for generating samples. In contrast to usual approaches, PDMPs are i) defined as continuous-time processes, and ii) are typically non-reversible with respect to their invariant measure. These distinctions pose computational and theoretical challenges for the design, analysis, and implementation of PDMP-based samplers. The key contribution of this work is to develop a transparent characterisation of how one can construct a PDMP (within the class of trajectorially-reversible processes) which admits the desired invariant measure, and to offer actionable recommendations on how these processes should be designed in practice. The second work presented in this thesis considers the task of sampling from a probability measure on a discrete space. While work in recent years has made it possible to apply sampling algorithms to probability measures with differentiable densities on continuous spaces in a reasonably generic way, samplers on discrete spaces are still largely derived on a case-by-case basis. The contention of this work is that this is not necessary, and that one can in fact define quite generally-applicable algorithms which can sample efficiently from discrete probability measures. The contributions are then to propose a small collection of algorithms for this task, and verify their efficiency empirically. Building on the previous chapter’s work, our samplers are again defined in continuous time and non-reversible, each of which offer noticeable benefits in efficiency. The third work presented in this thesis concerns a theoretical study of a particular class of Markov Chain-based sampling algorithms which make use of parallel computing resources. The Markov Chains which are produced by this algorithm are mathematically equivalent to a standard Metropolis-Hastings chain, but their real-time convergence properties are affected nontrivially by the application of parallelism. The contribution of this work is to analyse the convergence behaviour of these chains, and to use the ‘optimal scaling’ framework (as developed by Roberts, Rosenthal, and others) to make recommendations concerning the tuning of such algorithms in practice. The introductory chapters provide a general overview on the task of generating samples from a probability measure, with particular focus on methods involving Markov processes. There is also an interlude on the relative benefits of i) continuous-time and ii) non-reversible Markov processes for sampling, which are intended to provide additional context for the reading of the first two works.PhD Studentship paid for by Cantab Capital Institute for the Mathematics of Informatio

Unraveling the Thousand Word Picture: An Introduction to Super-Resolution Data Analysis

Super-resolution microscopy provides direct insight into fundamental biological processes occurring at length scales smaller than light’s diffraction limit. The analysis of data at such scales has brought statistical and machine learning methods into the mainstream. Here we provide a survey of data analysis methods starting from an overview of basic statistical techniques underlying the analysis of super-resolution and, more broadly, imaging data. We subsequently break down the analysis of super-resolution data into four problems: the localization problem, the counting problem, the linking problem, and what we’ve termed the interpretation problem