H. Yoldaş
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This thesis proves a Cramér-type large deviation principle for drifted geodesic random walks on a complete Riemannian manifold. A geodesic random walk is the manifold analogue of a sum of independent increments: at each step a random tangent vector is drawn and the walk movesalong the geodesic that this vector defines. We add a deterministic drift, given by a smooth vector field V , so that each step follows the geodesic determined by the random increment together with the value of V at the current point. For the empirical average of such a walk we establish a large deviation principle at speed n, with an explicit good rate function obtained as the Legendre transform of the increment law, corrected by the drift. The proof rests on a single construction, the drift map, which sends a noise path to the path obtained by carrying it on top of the flow of V . We show that this map is continuous, and the large deviation principle then follows from the Riemannian version of Mogulskii’s theorem by two applications of the contraction principle. Setting the drift to zero recovers the undrifted theorem of Kraaij, Redig and Versendaal. A final chapter removes the bounded-increment hypothesis. It is replaced by Cram´er’s condition on the increment law together with a bounded-geometry assumption on the manifold, and the same rate function is obtained.
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This thesis proves a Cramér-type large deviation principle for drifted geodesic random walks on a complete Riemannian manifold. A geodesic random walk is the manifold analogue of a sum of independent increments: at each step a random tangent vector is drawn and the walk movesalong the geodesic that this vector defines. We add a deterministic drift, given by a smooth vector field V , so that each step follows the geodesic determined by the random increment together with the value of V at the current point. For the empirical average of such a walk we establish a large deviation principle at speed n, with an explicit good rate function obtained as the Legendre transform of the increment law, corrected by the drift. The proof rests on a single construction, the drift map, which sends a noise path to the path obtained by carrying it on top of the flow of V . We show that this map is continuous, and the large deviation principle then follows from the Riemannian version of Mogulskii’s theorem by two applications of the contraction principle. Setting the drift to zero recovers the undrifted theorem of Kraaij, Redig and Versendaal. A final chapter removes the bounded-increment hypothesis. It is replaced by Cram´er’s condition on the increment law together with a bounded-geometry assumption on the manifold, and the same rate function is obtained.
In many situations, when we have a group of people, they all formopinions on a subject. Everyone in a population influences each others opinion. Naturally, people how the opinion of children are influenced by other children differs from how adults influence their opinions. Between different age groups there are all kind of different interactions.
This thesis aims to model how such opinions evolve and influence each other over time. First We assume an individual can either have a negative or positive opinion on a subject. To model this, we use the Ising model, originally developed for the description of magnetism in metals. We model opinion changes as random processes influenced by the opinion of other individuals in the population. We divide the population into subgroups of people who interact similarly. In this thesis, we prove that if we make the total group of people larger and larger, this random process becomes a deterministic process. Just like when you flip a coin infinitely many times, you end up with heads 50% of the time. We then determine how the different populations influence each other’s opinions. Understanding group opinion dynamics can help explain the spread of misinformation on social media, the emergence and disappearance of political parties, or how companies can predict or start trends. ...
This thesis aims to model how such opinions evolve and influence each other over time. First We assume an individual can either have a negative or positive opinion on a subject. To model this, we use the Ising model, originally developed for the description of magnetism in metals. We model opinion changes as random processes influenced by the opinion of other individuals in the population. We divide the population into subgroups of people who interact similarly. In this thesis, we prove that if we make the total group of people larger and larger, this random process becomes a deterministic process. Just like when you flip a coin infinitely many times, you end up with heads 50% of the time. We then determine how the different populations influence each other’s opinions. Understanding group opinion dynamics can help explain the spread of misinformation on social media, the emergence and disappearance of political parties, or how companies can predict or start trends. ...
In many situations, when we have a group of people, they all formopinions on a subject. Everyone in a population influences each others opinion. Naturally, people how the opinion of children are influenced by other children differs from how adults influence their opinions. Between different age groups there are all kind of different interactions.
This thesis aims to model how such opinions evolve and influence each other over time. First We assume an individual can either have a negative or positive opinion on a subject. To model this, we use the Ising model, originally developed for the description of magnetism in metals. We model opinion changes as random processes influenced by the opinion of other individuals in the population. We divide the population into subgroups of people who interact similarly. In this thesis, we prove that if we make the total group of people larger and larger, this random process becomes a deterministic process. Just like when you flip a coin infinitely many times, you end up with heads 50% of the time. We then determine how the different populations influence each other’s opinions. Understanding group opinion dynamics can help explain the spread of misinformation on social media, the emergence and disappearance of political parties, or how companies can predict or start trends.
This thesis aims to model how such opinions evolve and influence each other over time. First We assume an individual can either have a negative or positive opinion on a subject. To model this, we use the Ising model, originally developed for the description of magnetism in metals. We model opinion changes as random processes influenced by the opinion of other individuals in the population. We divide the population into subgroups of people who interact similarly. In this thesis, we prove that if we make the total group of people larger and larger, this random process becomes a deterministic process. Just like when you flip a coin infinitely many times, you end up with heads 50% of the time. We then determine how the different populations influence each other’s opinions. Understanding group opinion dynamics can help explain the spread of misinformation on social media, the emergence and disappearance of political parties, or how companies can predict or start trends.
How turtles find their way home
On the migration of loggerhead hatchlings: Using continuous and discrete methods to model their behaviour and assess recent survival rates
We investigate the migratory behaviour of juvenile loggerhead sea turtle (Caretta caretta) hatchlings within a region of the North Atlantic Gyre. To do so, we develop and compare an individual based (IB) model and a partial differential equation (PDE) model, specifically an advection-diffusion equation derived from a position-jump process. We compare their behaviour and show that they yield similar results, but that there are still some differences between the two. Using the IB model we assess survival probabilities of hatchlings in recent years (2016-2023), revealing that survival rates are statistically significantly (𝑝-value = 2.62 ⋅ 10^−6) not constant over time in this region. However, there could be global events that influence the survival probability, as years 2019 and 2023 have a large deviation in survival probability.
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We investigate the migratory behaviour of juvenile loggerhead sea turtle (Caretta caretta) hatchlings within a region of the North Atlantic Gyre. To do so, we develop and compare an individual based (IB) model and a partial differential equation (PDE) model, specifically an advection-diffusion equation derived from a position-jump process. We compare their behaviour and show that they yield similar results, but that there are still some differences between the two. Using the IB model we assess survival probabilities of hatchlings in recent years (2016-2023), revealing that survival rates are statistically significantly (𝑝-value = 2.62 ⋅ 10^−6) not constant over time in this region. However, there could be global events that influence the survival probability, as years 2019 and 2023 have a large deviation in survival probability.
Variational inference comprises a family of statistical methods to obtain the optimal approximation of a target probability distribution using some reference class of distributions and a cost function, commonly the Kullback-Leibler (KL) divergence. Recent work on variational inference has yielded a fast, stable set of mean and covariance evolutions which dynamically yield variational Gaussian approximations via a restriction to Gaussian measures of the well-known JKO scheme. The sequence of Gaussian measures thus generated converges towards the KL-optimal Gaussian approximation of the VI target: it may also be used to approximate the entire sequence of distributions generated by a JKO gradient flow directed at this same target, thereby supporting practical usage of Gaussian VI as well as fast, approximate modelling of the Fokker-Planck PDE. However, it is not immediately clear whether this Gaussian sequence offers valid, helpful approximations of the original JKO gradient flow. In this work, three upper bounds for the sequence of Wasserstein-2 distances between the two gradient flows are obtained by exploiting the Riemannian structure of the W2 manifold and the shared properties of the Gaussian and JKO evolutions. Numerical simulations support the validity of these bounds and test their performance in both ordinary and exceptional scenarios. One of the bounds may be computed solely using the Gaussian evolution and the target potential, thus offering a tractable estimator for the suitability of variational Gaussian approximations which retains the attractive properties of Wasserstein distances whilst avoiding their computational demands.
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Variational inference comprises a family of statistical methods to obtain the optimal approximation of a target probability distribution using some reference class of distributions and a cost function, commonly the Kullback-Leibler (KL) divergence. Recent work on variational inference has yielded a fast, stable set of mean and covariance evolutions which dynamically yield variational Gaussian approximations via a restriction to Gaussian measures of the well-known JKO scheme. The sequence of Gaussian measures thus generated converges towards the KL-optimal Gaussian approximation of the VI target: it may also be used to approximate the entire sequence of distributions generated by a JKO gradient flow directed at this same target, thereby supporting practical usage of Gaussian VI as well as fast, approximate modelling of the Fokker-Planck PDE. However, it is not immediately clear whether this Gaussian sequence offers valid, helpful approximations of the original JKO gradient flow. In this work, three upper bounds for the sequence of Wasserstein-2 distances between the two gradient flows are obtained by exploiting the Riemannian structure of the W2 manifold and the shared properties of the Gaussian and JKO evolutions. Numerical simulations support the validity of these bounds and test their performance in both ordinary and exceptional scenarios. One of the bounds may be computed solely using the Gaussian evolution and the target potential, thus offering a tractable estimator for the suitability of variational Gaussian approximations which retains the attractive properties of Wasserstein distances whilst avoiding their computational demands.