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• 1.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
A combined discontinuous Galerkin and finite volume scheme for multi–dimensional VPFP system2011In: AIP Conference Proceedings 1333, 2011, p. 57-62Conference paper (Refereed)

We construct a numerical scheme for the multi-dimensional Vlasov-Poisson-Fokker-Planck system based on a combined finite volume (FV) method for the Poisson equation in spatial domain and the streamline diffusion (SD) and discontinuous Galerkin (DG) finite element in time, phase-space variables for the Vlasov-Fokker-Planck equation.

• 2.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
A combined discontinuous Galerkin and finite volume scheme for multi–dimensional VPFP system2011In: AIP Conference Proceedings 1333 / [ed] Deborah A. Levin, Ingrid J. Wysong, Alejandro L. Garcia and Henry Abarbanel, American Institute of Physics (AIP), 2011, Vol. 1333, p. 57-62Conference paper (Refereed)

We construct a numerical scheme for the multi-dimensional Vlasov-Poisson-Fokker-Planck system based on a combined finite volume (FV) method for the Poisson equation in spatial domain and the streamline diffusion (SD) and discontinuous Galerkin (DG) finite element in time, phase-space variables for the Vlasov-Fokker-Planck equation.

• 3.
Gdansk University of Technology.
A Graph – String Model of Gene Assembly in Ciliates2006In: Zeszyty Naukowe Wydzialu ETI Politechniki Gdanskiej, 2006, Vol. 10, p. 521-534Conference paper (Refereed)

The ciliates are a family of unicellular organisms that characterize themselves by having two types of nuclei, micro - and macronuclei. During cell mating the genetic material must change from the micronuclei to the macronuclei form. The paper summarises a formal model for this change. The model, which is described in recent works, is based on strings and graphs. It shows that inside the cell complex computational operations have to take place.

• 4.
Gdansk University of Technology, Poland.
A Graph – String Model of Gene Assembly in Ciliates [Grafowo-tekstowy model rekombinacji DNA u orzęsek]2006In: Zeszyty Naukowe Wydzialu ETI Politechniki Gdanskiej, 2006, p. 521-534Conference paper (Refereed)

The ciliates are a family of unicellular organisms that characterize themselves by having two types of nuclei, micro - and macronuclei. During cell mating the genetic material must change from the micronuclei to the macronuclei form. The paper summarises a formal model for this change. The model, which is described in recent works, is based on strings and graphs. It shows that inside the cell complex computational operations have to take place.

• 5.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Exact and approximate limit behaviour of the Yule trees cophenetic index2018In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 303, p. 26-45Article in journal (Refereed)

In this work we study the limit distribution of an appropriately normalized cophenetic index of the pure-birth tree conditioned on n contemporary tips. We show that this normalized phylogenetic balance index is a sub-martingale that converges almost surely and in L-2. We link our work with studies on trees without branch lengths and show that in this case the limit distribution is a contraction-type distribution, similar to the Quicksort limit distribution. In the continuous branch case we suggest approximations to the limit distribution. We propose heuristic methods of simulating from these distributions and it may be observed that these algorithms result in reasonable tails. Therefore, we propose a way based on the quantiles of the derived distributions for hypothesis testing, whether an observed phylogenetic tree is consistent with the pure-birth process. Simulating a sample by the proposed heuristics is rapid, while exact simulation (simulating the tree and then calculating the index) is a time-consuming procedure. We conduct a power study to investigate how well the cophenetic indices detect deviations from the Yule tree and apply the methodology to empirical phylogenies.

• 6.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Limit distribution of the quartet balance index for Aldous’s $(\beta \ge 0)$-model2019In: Applicationes Mathematicae, ISSN 1233-7234, E-ISSN 1730-6280Article in journal (Refereed)

This paper builds on T. Martínez-Coronado, A. Mir, F. Rosselló and G. Valiente’s 2018 work, introducing a new balance index for trees. We show that this balance index, in the case of Aldous’s $(\beta \ge 0)$-model, converges weakly to a distribution that can be characterized as the fixed point of a contraction operator on a class of distributions.

• 7.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Phylogenetic effective sample size2016In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 407, p. 371-386Article in journal (Refereed)

In this paper I address the question—how large is a phylogenetic sample? I propose a definition of a phylogenetic effective sample size for Brownian motion and Ornstein-Uhlenbeck processes-the regression effective sample size. I discuss how mutual information can be used to define an effective sample size in the non-normal process case and compare these two definitions to an already present concept of effective sample size (the mean effective sample size). Through a simulation study I find that the AICc is robust if one corrects for the number of species or effective number of species. Lastly I discuss how the concept of the phylogenetic effective sample size can be useful for biodiversity quantification, identification of interesting clades and deciding on the importance of phylogenetic correlations.

• 8.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Phylogenetic effective sample size2016In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 407, p. 371-386Article in journal (Refereed)

In this paper I address the question—how large is a phylogenetic sample? I propose a definition of a phylogenetic effective sample size for Brownian motion and Ornstein-Uhlenbeck processes-the regression effective sample size. I discuss how mutual information can be used to define an effective sample size in the non-normal process case and compare these two definitions to an already present concept of effective sample size (the mean effective sample size). Through a simulation study I find that the AICc is robust if one corrects for the number of species or effective number of species. Lastly I discuss how the concept of the phylogenetic effective sample size can be useful for biodiversity quantification, identification of interesting clades and deciding on the importance of phylogenetic correlations.

• 9.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Quantifying the effects of anagenetic and cladogenetic evolution2014In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 254, p. 42-57Article in journal (Refereed)

An ongoing debate in evolutionary biology is whether phenotypic change occurs predominantly around the time of speciation or whether it instead accumulates gradually over time. In this work I propose a general framework incorporating both types of change, quantify the effects of speciational change via the correlation between species and attribute the proportion of change to each type. I discuss results of parameter estimation of Hominoid body size in this light. I derive mathematical formulae related to this problem, the probability generating functions of the number of speciation events along a randomly drawn lineage and from the most recent common ancestor of two randomly chosen tip species for a conditioned Yule tree. Additionally I obtain in closed form the variance of the distance from the root to the most recent common ancestor of two randomly chosen tip species.

• 10.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden.
Quantifying the effects of anagenetic and cladogenetic evolution2014In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 254, p. 42-57Article in journal (Refereed)

An ongoing debate in evolutionary biology is whether phenotypic change occurs predominantly around the time of speciation or whether it instead accumulates gradually over time. In this work I propose a general framework incorporating both types of change, quantify the effects of speciational change via the correlation between species and attribute the proportion of change to each type. I discuss results of parameter estimation of Hominoid body size in this light. I derive mathematical formulae related to this problem, the probability generating functions of the number of speciation events along a randomly drawn lineage and from the most recent common ancestor of two randomly chosen tip species for a conditioned Yule tree. Additionally I obtain in closed form the variance of the distance from the root to the most recent common ancestor of two randomly chosen tip species.

• 11.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Simulating an infinite mean waiting time2019In: Mathematica Applicanda, ISSN 1730-2668, Vol. 47, no 1, p. 93-102Article in journal (Refereed)

We consider a hybrid method to simulate the return time to the initial state in a critical-case birth-death process. The expected value of this return time is infinite, but its distribution asymptotically follows a power-law. Hence, the simulation approach is to directly simulate the process, unless the simulated time exceeds some threshold and if it does, draw the return time from the tail of the power law.

• 12.
Gdansk University of Technology.
The Bootstrap and Other Methods of Testing Phylogenetic Trees2007In: Zeszyty Naukowe Wydzialu ETI Politechniki Gdanskiej, 2007, Vol. 12, p. 103-108Conference paper (Refereed)

The final step of a phylogenetic analysis is the test of the generated tree. This is not a easy task for which there is an obvious methodology because we do not know the full probabilistic model of evolution. A number of methods have been proposed but there is a wide debate concerning the interpretations of the results they produce.

• 13.
Gdansk University of Technology, Poland.
The Bootstrap and Other Methods of Testing Phylogenetic Trees2007In: Zeszyty Naukowe Wydzialu ETI Politechniki Gdanskiej, 2007, p. 103-108Conference paper (Refereed)

The final step of a phylogenetic analysis is the test of the generated tree. This is not a easy task for which there is an obvious methodology because we do not know the full probabilistic model of evolution. A number of methods have been proposed but there is a wide debate concerning the interpretations of the results they produce.

• 14.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
The Laplace Motion in Phylogenetic Comparative Methods2012In: Proceedings of the XVIII National Conference on Applications of Mathematics in Biology and Medicine, 2012, p. 25-30Conference paper (Refereed)

The majority of current phylogenetic comparative methods assume that the stochastic evolutionaryprocess is homogeneous over the phylogeny or offer relaxations of this in rather limited and usually parameter expensive ways. Here we make a preliminary investigation, bymeans of a numerical experiment, whether the Laplace motion process can offer an alternative approach.

• 15.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
The phylogenetic effective sample size and jumps2018In: MATHEMATICA APPLICANDA (MATEMATYKA STOSOWANA), ISSN 1730-2668, Vol. 46, no 1, p. 25-33Article in journal (Refereed)

The phylogenetic effective sample size is a parameter that has as its goal the quantification of the amount of independent signal in a phylogenetically correlatedsample. It was studied for Brownian motion and Ornstein-Uhlenbeck models of trait evolution. Here, we study this composite parameter when the trait is allowedto jump at speciation points of the phylogeny. Our numerical study indicates thatthere is a non-trivial limit as the effect of jumps grows. The limit depends on thevalue of the drift parameter of the Ornstein-Uhlenbeck process.

• 16.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Trait evolution with jumps: illusionary normality2017In: Proceedings of the XXIII National Conference on Applications of Mathematics in Biology and Medicine, 2017, p. 23-28Conference paper (Refereed)

Phylogenetic comparative methods for real-valued traits usually make use of stochastic process whose trajectories are continuous.This is despite biological intuition that evolution is rather punctuated thangradual. On the other hand, there has been a number of recent proposals of evolutionarymodels with jump components. However, as we are only beginning to understandthe behaviour of branching Ornstein-Uhlenbeck (OU) processes the asymptoticsof branching  OU processes with jumps is an even greater unknown. In thiswork we build up on a previous study concerning OU with jumps evolution on a pure birth tree.We introduce an extinction component and explore via simulations, its effects on the weak convergence of such a process.We furthermore, also use this work to illustrate the simulation and graphic generation possibilitiesof the mvSLOUCH package.

• 17.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Department of Probability and Biomathematics, Gdańsk University of Technology, Gdańsk, Poland.
A Noether theorem for stochastic operators on Schatten classes2017In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 452, no 2, p. 1395-1412Article in journal (Refereed)

We show that a stochastic (Markov) operator S acting on a Schatten class C-1 satisfies the Noether condition (i.e. S' (A) = A and S' (A(2)) = A(2), where A is an element of C-infinity is a Hermitian and bounded operator on a fixed separable and complex Hilbert space (H, <.,.>)), if and only if S(E-A(G)XEA(G)) = E-A (G)S(X)E-A (G) for any state X is an element of C-1 and all Borel sets G subset of R, where E-A (G) denotes the orthogonal projection coming from the spectral resolution A = integral(sigma(A)) zE(A)(dz). Similar results are obtained for stochastic one-parameter continuous semigroups.

• 18.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Gdansk Univ Technol, Dept Probabil & Biomath, Ul Narutowicza 11-12, PL-80233 Gdansk, Poland..
A Noether theorem for stochastic operators on Schatten classes2017In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 452, no 2, p. 1395-1412Article in journal (Refereed)

We show that a stochastic (Markov) operator S acting on a Schatten class C-1 satisfies the Noether condition (i.e. S' (A) = A and S' (A(2)) = A(2), where A is an element of C-infinity is a Hermitian and bounded operator on a fixed separable and complex Hilbert space (H, <.,.>)), if and only if S(E-A(G)XEA(G)) = E-A (G)S(X)E-A (G) for any state X is an element of C-1 and all Borel sets G subset of R, where E-A (G) denotes the orthogonal projection coming from the spectral resolution A = integral(sigma(A)) zE(A)(dz). Similar results are obtained for stochastic one-parameter continuous semigroups.

• 19.
Gdansk University of Technology.
Gdansk University of Technology.
On the Time Behaviour of Okazaki Fragments2006In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 43, p. 500-509Article in journal (Refereed)

We find explicit analytical formulae for the time dependence of the probability of the number of Okazaki fragments produced during the process of DNA replication. This extends a result of Cowan on the asymptotic probability distribution of these fragments.

• 20.
Gdansk University of Technology, Poland.
Gdansk University of Technology, Poland.
On the Time Behaviour of Okazaki Fragments2006In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 43, no 2, p. 500-509Article in journal (Refereed)

We find explicit analytical formulae for the time dependence of the probability of the number of Okazaki fragments produced during the process of DNA replication. This extends a result of Cowan on the asymptotic probability distribution of these fragments.

• 21.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Linkoping Univ, Dept Comp & Informat Sci, S-58183 Linkoping, Sweden.
State Univ Appl Sci Elblag, Krzysztof Brzeski Inst Appl Informat, Ul Wojska Polskiego 1, PL-82300 Elblag, Poland. Gdansk Univ Technol, Dept Probabil & Biomath, Ul Narutowicza 11-12, PL-80233 Gdansk, Poland.
Weak Stability of Centred Quadratic Stochastic Operators2019In: BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, ISSN 0126-6705, Vol. 42, no 4, p. 1813-1830Article in journal (Refereed)

We consider the weak convergence of iterates of so-called centred quadratic stochastic operators. These iterations allow us to study the discrete time evolution of probability distributions of vector-valued traits in populations of inbreeding or hermaphroditic species, whenever the offspring's trait is equal to an additively perturbed arithmetic mean of the parents' traits. It is shown that for the existence of a weak limit, it is sufficient that the distributions of the trait and the perturbation have a finite variance or have tails controlled by a suitable power function. In particular, probability distributions from the domain of attraction of stable distributions have found an application, although in general the limit is not stable.

• 22.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
State Univ Appl Sci Elblag, Krzysztof Brzeski Inst Appl Informat, Ul Wojska Polskiego 1, PL-82300 Elblag, Poland. Gdansk Univ Technol, Dept Probabil & Biomath, Ul Narutowicza 11-12, PL-80233 Gdansk, Poland.
Weak Stability of Centred Quadratic Stochastic Operators2019In: BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, ISSN 0126-6705, Vol. 42, no 4, p. 1813-1830Article in journal (Refereed)

We consider the weak convergence of iterates of so-called centred quadratic stochastic operators. These iterations allow us to study the discrete time evolution of probability distributions of vector-valued traits in populations of inbreeding or hermaphroditic species, whenever the offsprings trait is equal to an additively perturbed arithmetic mean of the parents traits. It is shown that for the existence of a weak limit, it is sufficient that the distributions of the trait and the perturbation have a finite variance or have tails controlled by a suitable power function. In particular, probability distributions from the domain of attraction of stable distributions have found an application, although in general the limit is not stable.

• 23.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Ecology and Genetics, Plant Ecology and Evolution. CNRS Univ Montpellier IRD EPHE, UMR ISEM 5554, Montpellier, France.. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Ecology and Genetics, Plant Ecology and Evolution.
Using the Ornstein-Uhlenbeck process to model the evolution of interacting populations2017In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 429, p. 35-45Article in journal (Refereed)

The Ornstein-Uhlenbeck (OU) process plays a major role in the analysis of the evolution of phenotypic traits along phylogenies. The standard OU process includes random perturbations and stabilizing selection and assumes that species evolve independently. However, evolving species may interact through various ecological process and also exchange genes especially in plants. This is particularly true if we want to study phenotypic evolution among diverging populations within species. In this work we present a straightforward statistical approach with analytical solutions that allows for the inclusion of adaptation and migration in a common phylogenetic framework, which can also be useful for studying local adaptation among populations within the same species. We furthermore present a detailed simulation study that clearly indicates the adverse effects of ignoring migration. Similarity between species due to migration could be misinterpreted as very strong convergent evolution without proper correction for these additional dependencies. Finally, we show that our model can be interpreted in terms of ecological interactions between species, providing a general framework for the evolution of traits between "interacting" species or populations.

• 24.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences. Uppsala University, Sweden.
Uppsala University, Sweden; CNRS University of Montpellier IRD EPHE, France. Uppsala University, Sweden. Uppsala University, Sweden.
Using the Ornstein-Uhlenbeck process to model the evolution of interacting populations2017In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 429, p. 35-45Article in journal (Refereed)

The Ornstein-Uhlenbeck (OU) process plays a major role in the analysis of the evolution of phenotypic traits along phylogenies. The standard OU process includes random perturbations and stabilizing selection and assumes that species evolve independently. However, evolving species may interact through various ecological process and also exchange genes especially in plants. This is particularly true if we want to study phenotypic evolution among diverging populations within species. In this work we present a straightforward statistical approach with analytical solutions that allows for the inclusion of adaptation and migration in a common phylogenetic framework, which can also be useful for studying local adaptation among populations within the same species. We furthermore present a detailed simulation study that clearly indicates the adverse effects of ignoring migration. Similarity between species due to migration could be misinterpreted as very strong convergent evolution without proper correction for these additional dependencies. Finally, we show that our model can be interpreted in terms of ecological interactions between species, providing a general framework for the evolution of traits between "interacting" species or populations.(C) 2017 Elsevier Ltd. All rights reserved.

• 25.
Gdansk University of Technology.
Gdansk University of Technology. Gdansk University of Technology. Medical University of Gdansk. Gdansk University of Technology. Gdansk University of Technology.
Neural Network Breast Cancer Relapse Time Prognosis2006In: ASO Summer School 2006 Abstract Book, 2006, p. 8-10Conference paper (Other academic)

This paper is a result of a project at the Faculty of Electronics, Telecommunication and Computer Science (Technical University of Gdansk). The aim of the project was to create a neural network to predict the relapsetime of breast cancer. The neural network was to be trained on data collected over the past 20 years by dr. Jarosław Skokowski. The data includes 439 patient records described by about 40 parameters. For our neuralnetwork we only considered 6 medically most significant parameters the number of nodes showing evidence of cancer, size of tumour (in mm.), age, bloom score, estrogen receptors and proestrogen receptors and the relapsetime as the outcome. Our neural network was created in the MATLAB environment.

• 26.
Gdansk University of Technology.
Gdansk University of Technology. Gdansk University of Technology, Poland. Medical University of Gdansk, Poland. Gdansk University of Technology, Poland. Gdansk University of Technology, Poland.
Neural Network Breast Cancer Relapse Time Prognosis2006In: ASO Summer School 2006 abstract book Ostrzyce 30.06-2.07. 2006 / [ed] J. Skokowski and K. Drucis, 2006, p. 8-10Conference paper (Other academic)

This paper is a result of a project at the Faculty of Electronics, Telecommunication and Computer Science (Technical University of Gdansk). The aim of the project was to create a neural network to predict the relapsetime of breast cancer. The neural network was to be trained on data collected over the past 20 years by dr. Jarosław Skokowski. The data includes 439 patient records described by about 40 parameters. For our neuralnetwork we only considered 6 medically most significant parameters the number of nodes showing evidence of cancer, size of tumour (in mm.), age, bloom score, estrogen receptors and proestrogen receptors and the relapsetime as the outcome. Our neural network was created in the MATLAB environment.

• 27.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
University of Gothenburg. Chalmers University of Technology and the University of Gothenburg.
Time to a single hybridization event in a group of species with unknown ancestral history2013In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 322, p. 1-6Article in journal (Refereed)

We consider a stochastic process for the generation of species which combines a Yule process with a simple model for hybridization between pairs of co-existent species. We assume that the origin of the process, when there was one species, occurred at an unknown time in the past, and we condition the process on producing n species via the Yule process and a single hybridization event. We prove results about the distribution of the time of the hybridization event. In particular we calculate a formula for all moments, and show that under various conditions, the distribution tends to an exponential with rate twice that of the birth rate for the Yule process.

• 28.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden / Department of Biological and Environmental Science, University of Gothenburg, Gothenburg, Sweden. Department of Biological and Environmental Science, University of Gothenburg, Gothenburg, Sweden. Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden.
Time to a single hybridization event in a group of species with unknown ancestral history2013In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 322, p. 1-6Article in journal (Refereed)

We consider a stochastic process for the generation of species which combines a Yule process with a simple model for hybridization between pairs of co-existent species. We assume that the origin of the process, when there was one species, occurred at an unknown time in the past, and we condition the process on producing n species via the Yule process and a single hybridization event. We prove results about the distribution of the time of the hybridization event. In particular we calculate a formula for all moments, and show that under various conditions, the distribution tends to an exponential with rate twice that of the birth rate for the Yule process.

• 29.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Gdansk University of Technology.
Critical case stochastic phylogenetic tree model via the Laplace transform2014In: Demonstratio Matematicae, ISSN 0420-1213, Vol. 47, no 2, p. 474-481Article in journal (Refereed)

Birth-and-death models are now a common mathematical tool to describe branching patterns observed in real-world phylogenetic trees. Liggett and Schinazi (2009) is one such example. The authors propose a simple birth-and-death model that is compatible with phylogenetic trees of both in uenza and HIV, depending on the birth rate parameter. An interesting special case of this model is the critical case where the birth rate equals the death rate. This is a non-trivial situation and to study its asymptotic behaviour we employed the Laplace transform. With this we correct the proof of Liggett and Schinazi (2009) in the critical case.

• 30.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Sweden.
Gdansk University of Technology.
Critical case stochastic phylogenetic tree model via the Laplace transform2014In: Demonstratio Matematicae, ISSN 0420-1213, Vol. 47, no 2, p. 474-481Article in journal (Refereed)

Birth-and-death models are now a common mathematical tool to describe branching patterns observed in real-world phylogenetic trees. Liggett and Schinazi (2009) is one such example. The authors propose a simple birth-and-death model that is compatible with phylogenetic trees of both in uenza and HIV, depending on the birth rate parameter. An interesting special case of this model is the critical case where the birth rate equals the death rate. This is a non-trivial situation and to study its asymptotic behaviour we employed the Laplace transform. With this we correct the proof of Liggett and Schinazi (2009) in the critical case.

• 31.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Gdansk University of Technology. Medical University of Gdansk.
Survival time prognosis under a Markov model of cancer development2010In: Proceedings of the XVI National Conference on Applications of Mathematics in Biology and Medicine, 2010, p. 6-11Conference paper (Refereed)

In this study we look at a breast cancer data set of women from the Pomerania region collected in the year 1987- 1992 in the Medical University of Gdansk.We analyze the clinical risk factors in conjunction with a Markov model of cancer development. We evaluate Artificial Neural Network (ANN) survival time prediction (which was done on this data set in a previous study) via a simulation study.

• 32.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Gdansk University of Technology. Medical University of Gdansk.
Survival time prognosis under a Markov model of cancer development2010In: Proceedings of the XVI National Conference Applications of Mathematics to Biology and Medicine, Krynica, Poland, September 14–18, 2010 / [ed] M. Ziółko, M. Bodnar and E. Kutafina, 2010, p. 6-11Conference paper (Refereed)

In this study we look at a breast cancer data set of women from the Pomerania region collected in the year 1987- 1992 in the Medical University of Gdansk.We analyze the clinical risk factors in conjunction with a Markov model of cancer development. We evaluate Artificial Neural Network (ANN) survival time prediction (which was done on this data set in a previous study) via a simulation study.

• 33.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences.
Univ Cambridge, England.
MODELLING TRAIT-DEPENDENT SPECIATION WITH APPROXIMATE BAYESIAN COMPUTATION2019In: ACTA PHYSICA POLONICA B PROCEEDINGS SUPPLEMENT, JAGIELLONIAN UNIV , 2019, Vol. 12, no 1, p. 25-47Conference paper (Refereed)

Phylogeny is the field of modelling the temporal discrete dynamics of speciation. Complex models can nowadays be studied using the Approximate Bayesian Computation approach which avoids likelihood calculations. The fields progression is hampered by the lack of robust software to estimate the numerous parameters of the speciation process. In this work, we present an R package, pcmabc, publicly available on CRAN, based on Approximate Bayesian Computations, that implements three novel phylogenetic algorithms for trait-dependent speciation modelling. Our phylogenetic comparative methodology takes into account both the simulated traits and phylogeny, attempting to estimate the parameters of the processes generating the phenotype and the trait. The user is not restricted to a predefined set of models and can specify a variety of evolutionary and branching models. We illustrate the software with a simulation-reestimation study focused around the branching Ornstein-Uhlenbeck process, where the branching rate depends non-linearly on the value of the driving Ornstein-Uhlenbeck process. Included in this work is a tutorial on how to use the software.

• 34.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
University of Cambridge. University of Cambridge.
Influenza differentiation and evolution2010In: Acta Physica Polonica B Proceedings Supplement, 2010, Vol. 3, p. 417-452Conference paper (Refereed)

The aim of the study is to do a very wide analysis of HA, NA and M influenza gene segments to find short nucleotide regions,which differentiate between strains (i.e. H1, H2, ... e.t.c.), hosts, geographic regions, time when sequence was found and combination of time and region using a simple methodology. Finding regions  differentiating between strains has as its goal the construction of a Luminex microarray which will allow quick and efficient strain recognition. Discovery for the other splitting factors could shed lighton structures significant for host specificity and on the history of influenza evolution. A large number of places in the HA, NA and M gene segments were found that can differentiate between hosts, regions, time and combination of time and region. Also very good differentiation between different Hx strains can be seen.We link one of our findings to a proposed stochastic model of creation of viral phylogenetic trees.

• 35.
Mathematical Statistics, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.
Computer Laboratory, University of Cambridge Cambridge, United Kingdom. Computer Laboratory, University of Cambridge Cambridge, United Kingdom.
Influenza differentiation and evolution2010In: Acta Physica Polonica B Proceedings Supplement, 2010, Vol. 3, p. 417-452, article id 2Conference paper (Refereed)

The aim of the study is to do a very wide analysis of HA, NA and M influenza gene segments to find short nucleotide regions,which differentiate between strains (i.e. H1, H2, ... e.t.c.), hosts, geographic regions, time when sequence was found and combination of time and region using a simple methodology. Finding regions  differentiating between strains has as its goal the construction of a Luminex microarray which will allow quick and efficient strain recognition. Discovery for the other splitting factors could shed lighton structures significant for host specificity and on the history of influenza evolution. A large number of places in the HA, NA and M gene segments were found that can differentiate between hosts, regions, time and combination of time and region. Also very good differentiation between different Hx strains can be seen.We link one of our findings to a proposed stochastic model of creation of viral phylogenetic trees.

• 36.
Linköping University, Department of Computer and Information Science, The Division of Statistics and Machine Learning. Linköping University, Faculty of Arts and Sciences. Uppsala Univ, Sweden.
Polish Acad Sci, Poland. Univ Lodz, Poland. Polish Acad Sci, Poland. Uppsala Univ, Sweden. Polish Acad Sci, Poland.
Predicting pathogenicity behavior in Escherichia coli population through a state dependent model and TRS profiling2018In: PloS Computational Biology, ISSN 1553-734X, E-ISSN 1553-7358, Vol. 14, no 1, article id e1005931Article in journal (Refereed)

The Binary State Speciation and Extinction (BiSSE) model is a branching process based model that allows the diversification rates to be controlled by a binary trait. We develop a general approach, based on the BiSSE model, for predicting pathogenicity in bacterial populations from microsatellites profiling data. A comprehensive approach for predicting pathogenicity in E. coli populations is proposed using the state-dependent branching process model combined with microsatellites TRS-PCR profiling. Additionally, we have evaluated the possibility of using the BiSSE model for estimating parameters from genetic data. We analyzed a real dataset (from 251 E. coli strains) and confirmed previous biological observations demonstrating a prevalence of some virulence traits in specific bacterial sub-groups. The method may be used to predict pathogenicity of other bacterial taxa.

• 37.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Chalmers University of Technology and the University of Gothenburg. University of Gothenburg. Oslo University.
A phylogenetic comparative method for studying multivariate adaptation2012In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 314, p. 204-215Article in journal (Refereed)

Phylogenetic comparative methods have been limited in the way they model adaptation. Although some progress has been made, there are still no methods that can fully account for coadaptationbetween traits. Based on Ornstein-Uhlenbeck (OU) models of adaptive evolution, we present a method,with R implementation, in which multiple traits evolve both in response to each other and, as inprevious OU models, to fixed or randomly evolving predictor variables. We present the interpretation ofthe model parameters in terms of evolutionary and optimal regressions enabling the study of allometric and adaptive relationships between traits. To illustrate the method we reanalyze a data set of antlerand body-size evolution in deer (Cervidae).

• 38.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden.
Department of Genetics, University of Pretoria, Pretoria 0002, South Africa. Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden. Department of Biological and Environmental Sciences, University of Gothenburg, Gothenburg, Sweden. CEES, Department of Biology, University of Oslo, Oslo, Norway.
A phylogenetic comparative method for studying multivariate adaptation2012In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 314, p. 204-215Article in journal (Refereed)

Phylogenetic comparative methods have been limited in the way they model adaptation. Although some progress has been made, there are still no methods that can fully account for coadaptationbetween traits. Based on Ornstein-Uhlenbeck (OU) models of adaptive evolution, we present a method,with R implementation, in which multiple traits evolve both in response to each other and, as inprevious OU models, to fixed or randomly evolving predictor variables. We present the interpretation ofthe model parameters in terms of evolutionary and optimal regressions enabling the study of allometric and adaptive relationships between traits. To illustrate the method we reanalyze a data set of antlerand body-size evolution in deer (Cervidae).

• 39.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Cambridge University.
A novel algorithm to reconstruct phylogenies using gene sequences and expression data2014In: International Proceedings of Chemical, Biological & Environmental Engineering; Environment, Energy and Biotechnology III, 2014, p. 8-12Conference paper (Refereed)

Phylogenies based on single loci should be viewed with caution and the best approach for obtaining robust trees is to examine numerous loci across the genome. It often happens that for the same set of species trees derived from different genes are in conflict between each other. There are several methods that combine information from different genes in order to infer the species tree. One novel approach is to use informationfrom different -omics. Here we describe a phylogenetic method based on an Ornstein–Uhlenbeck process that combines sequence and gene expression data. We test our method on genes belonging to the histidine biosynthetic operon. We found that the method provides interesting insights into selection pressures and adaptive hypotheses concerning gene expression levels.

• 40.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Computer Laboratory , University of Cambridge, Cambridge, Un ited Kingdom.
A novel algorithm to reconstruct phylogenies using gene sequences and expression data2014In: International Proceedings of Chemical, Biological & Environmental Engineering; Environment, Energy and Biotechnology III, 2014, Vol. 70, p. 8-12Conference paper (Refereed)

Phylogenies based on single loci should be viewed with caution and the best approach for obtaining robust trees is to examine numerous loci across the genome. It often happens that for the same set of species trees derived from different genes are in conflict between each other. There are several methods that combine information from different genes in order to infer the species tree. One novel approach is to use informationfrom different -omics. Here we describe a phylogenetic method based on an Ornstein–Uhlenbeck process that combines sequence and gene expression data. We test our method on genes belonging to the histidine biosynthetic operon. We found that the method provides interesting insights into selection pressures and adaptive hypotheses concerning gene expression levels.

• 41.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.
Gdansk University of Technology.
Quadratic stochastic operators as a tool in modelling the dynamics of a distribution of a population trait2013In: Proceedings of the XIX National Conference on Applications of Mathematics in Biology and Medicine, 2013, p. 19-24Conference paper (Refereed)

Quadratic stochastic operators can exhibit a wide variety of asymptotic behaviours and these have been introducedand studied recently. In the present work we discuss biological interpretations that can be attributedto them. We also propose a computer simulation method to illustrate the behaviour of iterates of quadratic stochastic operators.

• 42.
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Sweden.
Department of Probability and Biomathematics, Gdánsk University of Technology, Gdánsk, Poland.
Quadratic stochastic operators as a tool in modelling the dynamics of a distribution of a population trait2013In: Proceedings of the 19th National Conference on Applications of Mathematics in Biology and Medicine / [ed] Katarzyna D. Lewandowska and Piotr Bogús, 2013, p. 19-24Conference paper (Refereed)

Quadratic stochastic operators can exhibit a wide variety of asymptotic behaviours and these have been introducedand studied recently. In the present work we discuss biological interpretations that can be attributedto them. We also propose a computer simulation method to illustrate the behaviour of iterates of quadratic stochastic operators.

• 43.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Gdansk University of Technology.
Asymptotic properties of quadratic stochastic operators acting on the L1 space2015In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 114, p. 26-39Article in journal (Refereed)

Quadratic stochastic operators can exhibit a wide variety of asymptotic behaviours andthese have been introduced and studied recently in the l1 space. It turns out that inprinciple most of the results can be carried over to the L1 space. However, due to topologicalproperties of this space one has to restrict in some situations to kernel quadratic stochasticoperators. In this article we study the uniform and strong asymptotic stability of quadratic stochastic operators acting on the L1 space in terms of convergence of the associated (linear)nonhomogeneous Markov chains.

• 44.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Department of Probability and Biomathematics, Gdańsk University of Technology, Gdańsk, Poland.
Asymptotic properties of quadratic stochastic operators acting on the L1 space2015In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 114, p. 26-39Article in journal (Refereed)

Quadratic stochastic operators can exhibit a wide variety of asymptotic behaviours andthese have been introduced and studied recently in the l1 space. It turns out that inprinciple most of the results can be carried over to the L1 space. However, due to topologicalproperties of this space one has to restrict in some situations to kernel quadratic stochasticoperators. In this article we study the uniform and strong asymptotic stability of quadratic stochastic operators acting on the L1 space in terms of convergence of the associated (linear)nonhomogeneous Markov chains.

• 45.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Gdansk University of Technology.
Prevalence Problem in the Set of Quadratic Stochastic Operators Acting on L12018In: Bulletin of the Malaysian Mathematical Sciences Society, ISSN 0126-6705, Vol. 41, no 1, p. 159-173Article in journal (Refereed)

This paper is devoted to the study of the problem of prevalence in the classof quadratic stochastic operators acting on the L1 space for the uniform topology.We obtain that the set of norm quasi-mixing quadratic stochastic operators is a denseand open set in the topology induced by a very natural metric. This shows the typicallong-term behaviour of iterates of quadratic stochastic operators.

• 46.
Department of Mathematics, Uppsala University, Uppsala, Sweden.
Department of Probability and Biomathematics, Gdańsk University of Technology, Gdańsk, Poland.
Prevalence Problem in the Set of Quadratic StochasticOperators Acting on L12018In: Bulletin of the Malaysian Mathematical Sciences Society, ISSN 0126-6705, Vol. 41, no 1, p. 159-173Article in journal (Refereed)

This paper is devoted to the study of the problem of prevalence in the classof quadratic stochastic operators acting on the L1 space for the uniform topology.We obtain that the set of norm quasi-mixing quadratic stochastic operators is a denseand open set in the topology induced by a very natural metric. This shows the typicallong-term behaviour of iterates of quadratic stochastic operators.

• 47.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
A consistent estimator of the evolutionary rate2015In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 371, p. 69-78Article in journal (Refereed)

We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate 2, conditioned on the observed number of particles to be equal to n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance sigma(2). In this paper we treat n particles as a sample of related species. The spatial Brownian motion of a particle describes the development of a trait value of interest (e.g. log-body-size). We propose an unbiased estimator 4 of the evolutionary rate rho(2) - sigma(2)/lambda. The estimator R-n(2) is proportional to the sample variance S-n(2) computed from n trait values. We find an approximate formula for the standard error of R-n(2), based on a neat asymptotic relation for the variance of S-n(2). (C) 2015 Elsevier Ltd. All rights reserved.

• 48.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Chalmers University of Technology and the University of Gothenburg.
A consistent estimator of the evolutionary rate2015In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 371, p. 69-78Article in journal (Refereed)

We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate λ, conditioned on the observed number of particles to be equal to n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance σ2. In this paper we treat n particles as a sample of related species. The spatial Brownian motion of a particle describes the development of a trait value of interest (e.g. log-body-size). We propose an unbiased estimator Rn2 of the evolutionary rate ρ22/λ. The estimator Rn2 is proportional to the sample variance Sn2 computed from n trait values. We find an approximate formula for the standard error of Rn2 based on a neat asymptotic relation for the variance of Sn2.

• 49.
Uppsala universitet, Tillämpad matematik och statistik.
A consistent estimator of the evolutionary rate2015In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 371, p. 69-78Article in journal (Refereed)

We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate 2, conditioned on the observed number of particles to be equal to n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance sigma(2). In this paper we treat n particles as a sample of related species. The spatial Brownian motion of a particle describes the development of a trait value of interest (e.g. log-body-size). We propose an unbiased estimator 4 of the evolutionary rate rho(2) - sigma(2)/lambda. The estimator R-n(2) is proportional to the sample variance S-n(2) computed from n trait values. We find an approximate formula for the standard error of R-n(2), based on a neat asymptotic relation for the variance of S-n(2). (C) 2015 Elsevier Ltd. All rights reserved.

• 50.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Chalmers University of Technology and the Unversity of Gothenburg.
Phylogenetic confidence intervals for the optimal trait value2015In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 52, no 4, p. 1115-1132Article in journal (Refereed)

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree ismodelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein–Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the samplemean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean isapproximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

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