A Data Driven Prognostic Model for the Availability of Parking

Zusammenfassung In dieser Arbeit präsentieren wir ein zeitabhängiges nichthomogenes Markov Modell zur Vorhersage von freien Parkplätzen in urbanen Gebieten. Von speziellem Interesse ist die Schätzung der Parameter basierend auf einem gegebenen Datensatz. Dazu erweitern wir die Theorie zu Empirical Risk Minimization auf matrixwertige Funktionen und benutzen matrixwertige reproduzierende Kernhilberträume zum approximieren des zeitabhängigen Generators des Markovprozesses. Außerdem wird ein Überblick zu verwandten Themen und weiteren Forschungsmöglichkeiten gegeben. Abstract In this thesis, we propose a time-dependent nonhomogeneous Markov model for predicting free parking spaces in urban areas. Of particular interest is the parameter estimation based on a given data set. To this end, we extend the framework of empirical risk minimization to matrixvalued functions and use matrix-valued reproducing kernel Hilbert spaces to approximate the time-dependent generator of the Markov process. Also, an overview of related issues and further research opportunities is given. i Abstract Acknowledgements I want to use the opportunity to thank the people who have helped in the process of this thesis and supported me along the way. Above all, my thanks go to Dr. Frank Filbir for advising me while working on this thesis and for making it possible to choose this very interesting topic for my research. I especially want to thank him for the numerous insightful talks and discussions, his friendly ear for my questions and his advice and encouragement over the time working on this thesis and my studies in general. I could not have asked for a better guide throughout this time. I also thank Prof. Dr. Felix Krahmer for supervising this thesis. Furthermore, I would like to thank Dr. Daniel Kotzor for introducing me to this area of research and for o↵ering his expert opinion whenever needed. Moreover, I want to thank my fellow students, in particular Björn Häfner, David Melching, Florian Speicher, Rebecca Bürkle, David Matthäus, and Serghei Mogoreanu, for proofreading this work and the helpful discussions we had over the last months. Last but not least, I want to say thank you to my family for their neverending support during my studies and especially while writing this thesis. iiAcknowledgements I want to use the opportunity to thank the people who have helped in the process of this thesis and supported me along the way. Above all, my thanks go to Dr. Frank Filbir for advising me while working on this thesis and for making it possible to choose this very interesting topic for my research. I especially want to thank him for the numerous insightful talks and discussions, his friendly ear for my questions and his advice and encouragement over the time working on this thesis and my studies in general. I could not have asked for a better guide throughout this time. I also thank Prof. Dr. Felix Krahmer for supervising this thesis. Furthermore, I would like to thank Dr. Daniel Kotzor for introducing me to this area of research and for o↵ering his expert opinion whenever needed. Moreover, I want to thank my fellow students, in particular Björn Häfner, David Melching, Florian Speicher, Rebecca Bürkle, David Matthäus, and Serghei Mogoreanu, for proofreading this work and the helpful discussions we had over the last months. Last but not least, I want to say thank you to my family for their neverending support during my studies and especially while writing this thesis. ii Notation Symbol Use K number of parking segments E k parking segment k, k = 1, . ,K N k number of parking spots in segment E k , k = 1, . ,K P probability measure X(t) stochastic process S state space of the process E[·] expectation value P (t, s) matrix of transition probabilities Q(t) (time-dependent) generator matrix i birth rates, i = 0, . , N μ i death rates, i = 0, . , N X non-empty set Y closed subset of the space of matrices M closed subset of the space of matrices T (n) space of tridiagonal matrices of dimension n D data set, so subset of X⇥ Y M number of data points D empirical measure depending on the data set D L loss function R L,P risk functional w.r. t. distribution P and loss L ↵ regularization parameter R L,P,↵ regularized risk functional w.r. t distribution P, loss L and parameter ↵ > 0 L(X,M) space of measurable functions f : X ! M L(X,M, μ) space of p-integrable matrix-valued functions w.r. t. a measure μ Lp(X, μ) space of p-integrable real-valued functions w.r. t. a measure μ B(M,Z) space of bounded linear operators from M to Z B(M) space of bounded linear operators from M to M B + (M) space of non-negative bounded linear operators from M to M A ⇤ adjoint of an operator A K(x, t) matrix-valued kernel H reproducing kernel Hilbert space of M-valued functions iii