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Real-world data often exhibits sequential dependence, across diverse domains such as human behavior, medicine, finance, and climate modeling. Probabilistic methods capture the inherent uncertainty associated with prediction in these contexts, with autoregressive models being especially prominent. This dissertation focuses on using autoregressive models to answer complex probabilistic queries that go beyond single-step prediction, such as the timing of future events or the likelihood of a specific event occurring before another. In particular, we develop a broad class of novel and efficient approximation techniques for marginalization in sequential models that are model-agnostic. These techniques rely solely on access to and sampling from next-step conditional distributions of a pre-trained autoregressive model, including both traditional parametric models as well as more recent neural autoregressive models. Specific approaches are presented for discrete sequential models, for marked temporal point processes, and for stochastic jump processes, each tailored to a well-defined class of informative, long-range probabilistic queries.

Cellular signaling, crucial for biological processes like immune response and homeostasis, relies on specificity and fidelity in signal transduction to accurately respond to stimuli amidst biological noise. Kinetic proofreading (KPR) is a key mechanism enhancing signaling specificity through time-delayed steps, although its effectiveness is debated due to intrinsic noise potentially reducing signal fidelity. In this study, we reformulate the theory of kinetic proofreading (KPR) by convolving multiple intermediate states into a single state and then define an overall processing time required to traverse these states. This simplification allows us to succinctly describe kinetic proofreading in terms of a single waiting time parameter, facilitating a more direct evaluation and comparison of KPR performance across different biological contexts such as DNA replication and T cell receptor (TCR) signaling. We find that loss of fidelity for longer proofreading steps relies on the specific strategy of information extraction and show that in the first-passage time (FPT) discrimination strategy, longer proofreading steps can exponentially improve the accuracy of KPR at the cost of speed. Thus, KPR can still be an effective discrimination mechanism in the high noise regime. However, in a product concentration-based discrimination strategy, longer proofreading steps do not necessarily lead to an increase in performance. However, by introducing activation thresholds on product concentrations, can we decompose the product-based strategy into a series of FPT-based strategies to better resolve the subtleties of KPR-mediated product discrimination. Our findings underscore the importance of understanding KPR in the context of how information is extracted and processed in the cell.

Age and longevity are key parameters for demography and life-history evolution of organisms. In clonal species, a widespread life history among animals, plants, macroalgae and fungi, the sexually produced offspring (genet) grows indeterminately by producing iterative modules, or ramets, and so obscure their age. Here we present a novel molecular clock based on the accumulation of fixed somatic genetic variation that segregates among ramets. Using a stochastic model, we demonstrate that the accumulation of fixed somatic genetic variation will approach linearity after a lag phase, and is determined by the mitotic mutation rate, without direct dependence on asexual generation time. The lag phase decreased with lower stem cell population size, number of founder cells for the formation of new modules, and the ratio of symmetric versus asymmetric cell divisions. We calibrated the somatic genetic clock on cultivated eelgrass Zostera marina genets (4 and 17 years respectively). In a global data set of 20 eelgrass populations, genet ages were up to 1,403 years. The somatic genetic clock is applicable to any multicellular clonal species where the number of founder cells is small, opening novel research avenues to study longevity and, hence, demography and population dynamics of clonal species.

En els darrers anys, les tècniques generatives d'aprenentatge profund han vist un ampli ús, especialment en la generació d'imatges i text. Aquests mètodes també s'han mostrat prometedors en la generació de dades de sèries temporals, fins i tot en dominis continus. No obstant això, tenen dificultats per capturar certes distribucions de probabilitat, com les distribucions de cua pesada comúment observades en els rendiments d'actius financers. Aquesta limitació sorgeix de la continuïtat Lipschitz de les xarxes neuronals generadores, que impedeix la generació de distribucions amb cues més pesades que el soroll previ utilitzat en aquests mètodes. Utilitzar senzillament un soroll previ més pesat porta a problemes d'estabilitat i extrapolació. Per abordar aquest repte, el nostre treball proposa un enfocament innovador. Reduïm les dades d'entrenament de sèries temporals a l'espai de paràmetres d'una funció de distribució acumulativa de cua pesada, on els paràmetres evolucionen amb el temps. Després, s'utilitzen tècniques d'aprenentatge profund per generar aquesta evolució de paràmetres, aprofitant la distribució de probabilitat més favorable de les dades transformades. Finalment, es generen sèries temporals en l'espai original de les dades mitjançant la transformació inversa de l'espai de paràmetres utilitzant la funció quantil, el que resulta en un modelatge més precís de la distribució de la cua., En los últimos años, las técnicas generativas de aprendizaje profundo han visto un amplio uso, especialmente en la generación de imágenes y texto. Estos métodos también se han mostrado prometedores en la generación de datos de series temporales, incluso en dominios continuos. Sin embargo, tienen dificultades para capturar ciertas distribuciones de probabilidad, como las distribuciones de cola pesada comúnmente observadas en los rendimientos de activos financieros. Esta limitación surge de la continuidad Lipschitz de las redes neuronales generadoras, que impide la generación de distribuciones con colas más pesadas que el ruido previo utilizado en estos métodos. Simplemente usar un ruido previo más pesado conduce a problemas de estabilidad y extrapolación. Para abordar este desafío, nuestro trabajo propone un enfoque novedoso. Reducimos los datos de entrenamiento de series temporales al espacio de parámetros de una función de distribución acumulativa de cola pesada, donde los parámetros evolucionan con el tiempo. Luego, se emplean técnicas de aprendizaje profundo para generar esta evolución de parámetros, aprovechando la distribución de probabilidad más favorable de los datos transformados. Finalmente, se generan series temporales en el espacio original de los datos mediante la transformación inversa del espacio de parámetros utilizando la función cuantil, lo que resulta en un modelado más preciso de la distribución de la cola., In recent years, generative deep learning techniques have seen widespread use, particularly in image and text generation. These methods have also shown promise in generating time series data, even in continuous domains. However, they struggle to capture certain probability distributions, such as the heavy-tailed distributions commonly observed in financial asset returns. This limitation arises from the Lipschitz continuity of the generator neural networks, which prevents the generation of distributions with heavier tails than the prior noise used in these methods. Simply using a heavier prior noise leads to stability and extrapolation issues. To address this challenge, our work proposes a novel approach. We reduce time series training data to the parameter space of a heavy-tailed cumulative distribution function, where the parameters evolve over time. Deep learning techniques are then employed to generate this parameter evolution, leveraging the more favourable probability distribution of the transformed data. Finally, time series on the sample space are generated by inversely transforming the parameter space using the quantile function, resulting in a more accurate modelling of the tail distribution., Outgoing

En els darrers anys s'ha observat un gran progrés experimental en les tecnologies quàntiques. La majoria dels sistemes d'interès són de naturalesa inherentment oberta i de molts cossos, i entendre els fenòmens col·lectius derivats d'aquestes característiques és crucial per a la metrologia, la informació quàntica, la computació quàntica i moltes altres aplicacions. No obstant, un tractament analític i simulacions exactes i eficients d'aquests sistemes segueixen sent difícils de dur a terme a causa de l'augment exponencial de la complexitat amb la mida del sistema. La representació en l'espai de fases, una formulació equivalent de la mecànica quàntica, proporciona una plataforma per explorar possibles solucions per evitar la complexitat exponencial. En el límit de la dinàmica semiclàssica, l'anomenada "truncated Wigner approximation" permet simulacions computacionalment eficients per a la dinàmica Hamiltoniana. Aquesta aproximació, inicialment destinada a sistemes bosònics, es pot generalitzar a sistemes d'espín amb dissipació. En aquest projecte, ens centrem principalment en aquest darrer cas i examinem diversos mètodes aproximats, ramificats de la truncated Wigner approximation original, proporcionant la comprensió física necessària de la validesa d'aquestes aproximacions. També generalitzem la representació en l'espai de fases per obtenir simulacions exactes de la dinàmica d'espins i caracteritzar l'estabilitat del mètode numèric., En los últimos años se han producido enormes avances experimentales en las tecnologías cuánticas. La mayoría de los sistemas de interés son inherentemente abiertos y de muchos cuerpos, y entender los fenómenos colectivos que surgen de estas características es crucial para la metrología, la información cuántica, la computación cuántica y muchas otras aplicaciones. Sin embargo, un tratamiento analítico y simulaciones exactas y eficientes de estos sistemas siguen siendo difíciles de alcanzar debido al aumento exponencial de la complejidad con el tamaño del sistema. La representación en el espacio de fases, una formulación equivalente de la mecánica cuántica, proporciona una plataforma para explorar posibles soluciones para sortear la complejidad exponencial. En el límite de la dinámica semiclásica, la "truncated Wigner approximation" permite realizar simulaciones computacionalmente eficientes de la dinámica Hamiltoniana. Esta aproximación, inicialmente destinada a sistemas bosónicos, puede generalizarse a sistemas de espín con disipación. En este proyecto, nos centramos principalmente en el último caso y examinamos varios métodos aproximados, partiendo de la truncated Wigner approximation original, proporcionando la comprensión física necesaria de la validez de estas aproximaciones. También generalizamos la representación en el espacio de fases para obtener simulaciones exactas de la dinámica del espín y caracterizamos la estabilidad del método numérico., Recent years have seen tremendous experimental progress in quantum technologies. Most of the systems of interest are inherently open and many-body in nature, and understanding the collective phenomena arising from these features is crucial for metrology, quantum information science, quantum computation, and many other applications. However, analytical treatments and efficient exact simulations of these systems remain elusive due to the exponential scaling of the complexity with system size. Phase space representation, an equivalent formulation of quantum mechanics, provides a platform to explore potential solutions to circumvent the exponential complexity. In the limit of semiclassical dynamics, the so-called truncated Wigner approximation allows for computationally inexpensive simulations for Hamiltonian dynamics. This approximation, initially intended for bosonic systems, can be generalized to dissipative spin systems. In this thesis, we primarily focus on the latter case and examine various approximate methods, branching from the original truncated Wigner approximation, and providing the necessary physical understanding of the validity of such approximations. We also generalize the phase space representation to obtain exact simulations of the spin dynamics and characterize the stability of the numerical method., Outgoing

Due to the influence of stochastic gradients, the existing algorithms suffer from slow convergence, noise explosion, and even failure to converge in practice, which motivates us to propose an accelerated algorithm to tackle these issues. Recognizing the potential of gradient, momentum, and conjugate gradient as promising search directions, we propose a 3-D acceleration algorithm, which uses a weighted combination of these three basis. Specifically, in order to analyze the dynamics of the discrete-time algorithm during the update process, we provide a general framework for approximating the discrete-time algorithm in the weak sense by a continuous-time stochastic differential equation. We exploit the continuous-time formulation together with Lyapunov drift optimization to derive novel adaptive step sizes, which effectively improve the performance of the algorithm in stabilizing noise and accelerating convergence. Extensive numerical experiments demonstrate the proposed algorithm’s superiority in convergence rate, computation complexity, and noise robustness compared to state-of-the-art baselines. IEEE

This paper explores distributed aggregative games in multi-agent systems. Current methods for finding distributed Nash equilibrium require players to send original messages to their neighbors, leading to communication burden and privacy issues. To jointly address these issues, we propose an algorithm that uses stochastic compression to save communication resources and conceal information through random errors induced by compression. Our theoretical analysis shows that the algorithm guarantees convergence accuracy, even with aggressive compression errors used to protect privacy. We prove that the algorithm achieves differential privacy through a stochastic quantization scheme. Simulation results for energy consumption games support the effectiveness of our approach. IEEE

The proposed thesis complements generalized risk assessment frameworks for structural engineering tackling stochastic dynamic systems, by addressing multiple uncertainties, non-linearities, and full-time-dependent analyses. Two major complementary work packages have been pursued. Firstly, a novel point selection procedure for the Probability Density Evolution Method (PDEM) has been developed to enhance efficiency while retaining the full dynamic response of non-linear dynamic systems. This method aims to estimate time-dependent reliability and failure probabilities in dynamic systems under first-passage failure conditions. Leveraging features of the Subset simulation procedure, the Subset supported Point Selection (S-PS) method adaptively generates dependent sample sets, enhancing accuracy by incorporating performance function assessments as weighting factors. The proposed approach effectively identifies samples in the failure region, particularly benefiting dynamic systems under stochastic excitation. It offers a computationally efficient structural reliability estimation procedure by analyzing full-time-history responses, providing deeper insights into rare failure events and mechanisms through visualization of intermediate results. Secondly, to address the challenge of identifying patterns and underlying characteristics in natural or engineering time-varying phenomena, a data-driven stochastic representation of Evolutionary Power Spectral Density (EPSD) functions is introduced. The Relaxed Evolutionary Power Spectral Density (REPSD) function is derived from multiple similar data, accounting for uncertainties and providing a realistic representation of time data such as seismic ground motions or wind speed records in the time-frequency domain. Truncated normal distributions and kernel density estimates are used to determine a probability density function for each time-frequency component. The REPSD function enables the sampling of individual EPSD functions, facili, Die vorliegende Arbeit ergänzt Risikobewertungen für das Bauingenieurwesen, die sich mit stochastischen dynamischen Systemen befassen, indem sie sich mit mehreren Unsicherheiten, Nichtlinearitäten und vollzeitabhängigen Analysen befasst. Es wurden zwei große komplementäre Arbeitspakete verfolgt. Erstens wurde ein neuartiges Punktauswahlverfahren für die „Probability Density Evolution Method (PDEM)“ entwickelt, um die Effizienz zu steigern und gleichzeitig die volle dynamische Antwort nichtlinearer dynamischer Systeme beizubehalten. Diese Methode zielt darauf ab, zeitabhängige Zuverlässigkeit und Ausfallwahrscheinlichkeiten in dynamischen Systemen unter Bedingungen der ersten Durchgangszeit für kritische Systemantworten abzuschätzen. Unter Nutzung des Subset-Simulationsverfahrens generiert die vorgeschlagene Subset-unterstützte Punktauswahlmethode (S-PS) adaptiv abhängige Stichprobensätze und erhöht die Genauigkeit durch die Einbeziehung von Gewichtungsfaktoren. Der vorgeschlagene Ansatz identifiziert effektiv Proben im Fehlerbereich, was insbesondere dynamischen Systemen unter stochastischer Anregung zugute kommt. Es bietet ein rechnerisch effizientes Verfahren zur Schätzung der strukturellen Zuverlässigkeit durch die Analyse von Antworten über die Zeit und bietet durch die Visualisierung von Zwischenergebnissen tiefere Einblicke in seltene Fehlerereignisse und -mechanismen. Zweitens wird eine datengesteuerte stochastische Darstellung von EPSD-Funktionen (Evolutionary Power Spectral Density) eingeführt, um Muster und zugrunde liegende Merkmale in natürlichen oder technischen zeitlich variierenden Phänomenen zu identifizieren. Die „Relaxed Evolutionary Power Spectral Density“ (REPSD) Funktion wird aus mehreren ähnlichen Daten abgeleitet, berücksichtigt Unsicherheiten und bietet eine realistische Darstellung von Zeitdaten. Diese Daten können sich aus seismischen Bodenbewegungen oder Windgeschwindigkeitsaufzeichnungen im Zeit-Frequenz-Bereich zusammensetzen. Gestutzte

The identification of patterns and underlying characteristics of natural or engineering time-varying phenomena poses a challenging task, especially in the scope of simulation models and accompanying stochastic models. Because of their complex nature, time-varying processes such as wind speed, seismic ground motion, or vibrations of machinery in the presence of degradation oftentimes lack a closed-form description of their underlying Evolutionary Power Spectral Density (EPSD) function. To overcome this issue, a wide range of measurements exist for these types of processes. This opens up the path to a data-driven stochastic representation of EPSD functions. Rather than solely relying on time–frequency transform methods like the familiar short-time Fourier transform or wavelet transform for EPSD estimation, a probabilistic representation of the EPSD can provide valuable insights into the epistemic uncertainty associated with these processes. To address this problem, the evolutionary EPSD function is relaxed based on multiple similar data to account for these uncertainties and to provide a realistic representation of the time data in the time–frequency domain. This results is the so-called Relaxed Evolutionary Power Spectral Density (REPSD) function, which serves as a modular probabilistic representation of the time–frequency content of stochastic signals. For this purpose, truncated normal distributions and kernel density estimates are used to determine a probability density function for each time–frequency component. The REPSD function enables the sampling of individual EPSD functions, facilitating their direct application to the simulation model through stochastic simulation techniques like Monte Carlo simulation or other advanced methods. Even though the accuracy is highly dependant on the data available and the time–frequency transformation method used, the REPSD representation offers a stochastic representation of characteristics used to describe stochastic signal

Detailed longitudinal dynamics simulations may be used to predict the energy performance of road vehicles. However, including uncertainty in the operating conditions often implies high computational costs. Model-based formulations, in conjunction with statistical methods, may obviate this limitation by directly accounting for stochasticity, thus eliminating the need for simulating large populations of driving and operating cycles. To this end, leveraging directly the methods of stochastic calculus, this work presents a novel theory of longitudinal vehicle dynamics and energy consumption, where the vehicle’s speed varies stochastically depending on the characteristics of the operating environment. In particular, the proposed formulation, consisting of stochastic differential equations (SDEs) governing the longitudinal motion of road vehicles, inherently accounts for the statistical variation connected with uncertainties in the driver’s behavior and road properties, including, e.g., topography and legal speed. A Fokker-Planck partial differential equation (PDE) that describes the time evolution of the joint probability density function (PDF) of the vehicle’s speed, position, and road parameters is also derived from the SDEs established in the paper. The SDE and Fokker-Planck-based approaches enable statistical estimation of important quantities like speed fluctuations, instantaneous power requests, and energy consumption. The developed models may be used to assess the energy performance of road vehicles for different combinations of road transport missions. This is applicable at the early stages of the development, virtual testing, and certification processes, without the need to perform computationally expensive simulations, as corroborated by the virtual experiments conducted in the paper.

Detailed longitudinal dynamics simulations may be used to predict the energy performance of road vehicles. However, including uncertainty in the operating conditions often implies high computational costs. Model-based formulations, in conjunction with statistical methods, may obviate this limitation by directly accounting for stochasticity, thus eliminating the need for simulating large populations of driving and operating cycles. To this end, leveraging directly the methods of stochastic calculus, this work presents a novel theory of longitudinal vehicle dynamics and energy consumption, where the vehicle’s speed varies stochastically depending on the characteristics of the operating environment. In particular, the proposed formulation, consisting of stochastic differential equations (SDEs) governing the longitudinal motion of road vehicles, inherently accounts for the statistical variation connected with uncertainties in the driver’s behavior and road properties, including, e.g., topography and legal speed. A Fokker-Planck partial differential equation (PDE) that describes the time evolution of the joint probability density function (PDF) of the vehicle’s speed, position, and road parameters is also derived from the SDEs established in the paper. The SDE and Fokker-Planck-based approaches enable statistical estimation of important quantities like speed fluctuations, instantaneous power requests, and energy consumption. The developed models may be used to assess the energy performance of road vehicles for different combinations of road transport missions. This is applicable at the early stages of the development, virtual testing, and certification processes, without the need to perform computationally expensive simulations, as corroborated by the virtual experiments conducted in the paper.

The deterministic techniques in use for transmission-grid planning and operation have resulted in a high reliability, but they have certain limitations that could be removed by applying stochastic techniques. This paper presents a detailed review of one such stochastic technique: operational risk assessment. A general procedural model of operational risk assessment is introduced, based on one simple equation, highlighting the important elements. The research trends are presented for each of these elements: contingency definition and calculating the probability of a contingency case; contingency filtration methods; defining and calculating the severity factor. Next to an overview of the state-of-the-art, this paper contains a detailed discussion section, where the most important research gaps are identified. Emphasis is especially on bridging the gap between research and practical applications of operational risk assessment. The paper closes with a future outlook on operational risk assessment and its applications., Validerad;2024;Nivå 2;2024-05-28 (hanlid);Full text license: CC BY

We show how the classical Lagrangian approach to solving constrained optimization problems from standard calculus can be extended to solve continuous time stochastic optimal control problems. Connections to mainstream approaches such as the Hamilton-Jacobi-Bellman equation and the stochastic maximum principle are drawn. Our approach is linked to the stochastic maximum principle, but more direct and tied to the classical Lagrangian principle, avoiding the use of backward stochastic differential equations in its formulation. Using infinite dimensional functional analysis, we formalize and extend the approach first outlined in Chow (1992) within a rigorous mathematical setting using infinite dimensional functional analysis. We provide examples that demonstrate the usefulness and effectiveness of our approach in practice. Further, we demonstrate the potential for numerical applications facilitating some of our key equations in combination with Monte Carlo backward simulation and linear regression, therefore illustrating a completely different and new avenue for the numerical application of Chow's methods.

Detailed longitudinal dynamics simulations may be used to predict the energy performance of road vehicles. However, including uncertainty in the operating conditions often implies high computational costs. Model-based formulations, in conjunction with statistical methods, may obviate this limitation by directly accounting for stochasticity, thus eliminating the need for simulating large populations of driving and operating cycles. To this end, leveraging directly the methods of stochastic calculus, this work presents a novel theory of longitudinal vehicle dynamics and energy consumption, where the vehicle’s speed varies stochastically depending on the characteristics of the operating environment. In particular, the proposed formulation, consisting of stochastic differential equations (SDEs) governing the longitudinal motion of road vehicles, inherently accounts for the statistical variation connected with uncertainties in the driver’s behavior and road properties, including, e.g., topography and legal speed. A Fokker-Planck partial differential equation (PDE) that describes the time evolution of the joint probability density function (PDF) of the vehicle’s speed, position, and road parameters is also derived from the SDEs established in the paper. The SDE and Fokker-Planck-based approaches enable statistical estimation of important quantities like speed fluctuations, instantaneous power requests, and energy consumption. The developed models may be used to assess the energy performance of road vehicles for different combinations of road transport missions. This is applicable at the early stages of the development, virtual testing, and certification processes, without the need to perform computationally expensive simulations, as corroborated by the virtual experiments conducted in the paper.

We show how the classical Lagrangian approach to solving constrained optimization problems from standard calculus can be extended to solve continuous time stochastic optimal control problems. Connections to mainstream approaches such as the Hamilton-Jacobi-Bellman equation and the stochastic maximum principle are drawn. Our approach is linked to the stochastic maximum principle, but more direct and tied to the classical Lagrangian principle, avoiding the use of backward stochastic differential equations in its formulation. Using infinite dimensional functional analysis, we formalize and extend the approach first outlined in Chow (1992) within a rigorous mathematical setting using infinite dimensional functional analysis. We provide examples that demonstrate the usefulness and effectiveness of our approach in practice. Further, we demonstrate the potential for numerical applications facilitating some of our key equations in combination with Monte Carlo backward simulation and linear regression, therefore illustrating a completely different and new avenue for the numerical application of Chow's methods.

Detailed longitudinal dynamics simulations may be used to predict the energy performance of road vehicles. However, including uncertainty in the operating conditions often implies high computational costs. Model-based formulations, in conjunction with statistical methods, may obviate this limitation by directly accounting for stochasticity, thus eliminating the need for simulating large populations of driving and operating cycles. To this end, leveraging directly the methods of stochastic calculus, this work presents a novel theory of longitudinal vehicle dynamics and energy consumption, where the vehicle’s speed varies stochastically depending on the characteristics of the operating environment. In particular, the proposed formulation, consisting of stochastic differential equations (SDEs) governing the longitudinal motion of road vehicles, inherently accounts for the statistical variation connected with uncertainties in the driver’s behavior and road properties, including, e.g., topography and legal speed. A Fokker-Planck partial differential equation (PDE) that describes the time evolution of the joint probability density function (PDF) of the vehicle’s speed, position, and road parameters is also derived from the SDEs established in the paper. The SDE and Fokker-Planck-based approaches enable statistical estimation of important quantities like speed fluctuations, instantaneous power requests, and energy consumption. The developed models may be used to assess the energy performance of road vehicles for different combinations of road transport missions. This is applicable at the early stages of the development, virtual testing, and certification processes, without the need to perform computationally expensive simulations, as corroborated by the virtual experiments conducted in the paper.

The automatic projection filter is a recently developed numerical method for projection filtering that leverages sparse-grid integration and automatic differentiation. However, its accuracy is highly sensitive to the accuracy of the cumulant-generating function computed via the sparse-grid integration, which in turn is also sensitive to the choice of the bijection from the canonical hypercube to the state space. In this article, we propose two new adaptive parametric bijections for the automatic projection filter. The first bijection relies on the minimization of Kullback-Leibler divergence, whereas the second method employs the sparse-grid Gauss-Hermite quadrature. The two new bijections allow the sparse-grid nodes to adaptively move within the high-density region of the state space, resulting in a substantially improved approximation while using only a small number of quadrature nodes. The practical applicability of the methodology is illustrated in three simulated nonlinear filtering problems.

This letter explores distributed aggregative games in multi-agent systems. Current methods for finding distributed Nash equilibrium require players to send original messages to their neighbors, leading to communication burden and privacy issues. To jointly address these issues, we propose an algorithm that uses stochastic compression to save communication resources and conceal information through random errors induced by compression. Our theoretical analysis shows that the algorithm guarantees convergence accuracy, even with aggressive compression errors used to protect privacy. We prove that the algorithm achieves differential privacy through a stochastic quantization scheme. Simulation results for energy consumption games support the effectiveness of our approach.

This article proposes a method to design network-aware flexibility requests for local flexibility markets. These markets are becoming increasingly important for distribution system operators (DSOs) to ensure grid safety while minimizing costs and public opposition to new network investments. Despite extended recent literature on local flexibility markets, little attention has been paid to quantifying the flexibility required at each location, considering physical network constraints (e.g. line and voltage limits). The method introduced uses a chance-constrained optimization model and a LinDistFlow approximation to consider both physical network constraints and uncertainty caused by renewable production or demand fluctuations. Unlike other methods, it avoids sharing sensitive grid data with the market operator. We compare our approach against a stochastic market-clearing mechanism which serves as a benchmark, and we derive analytical conditions for the performance of our method to determine flexibility requests. We show on two case studies that our method outperforms the stochastic market-clearing benchmark in terms of computation time while achieving comparable social welfare and costs for the DSOs. One of the case studies is conducted on an actual German distribution grid, showing that the proposed method can scale well to real-sized networks.

The Hawkes process, also referred to as a self-exciting point process, is a class of point processes where the intensity is conditioned on previous events. More specifically, an event occurrence excites the process, temporarily increasing the probability of more events occurring. One application of Hawkes processes is to model occurrences of crimes, such as burglaries or shootings. The present thesis attempts to apply the theory of Hawkes processes to shooting occurrences in Sweden, by exploring whether a temporal Hawkes process model is suitable to model shooting occurrences in the Police regions of Stockholm, V¨ast (West) and Syd (South). The results indicate that although there is reason to believe that shooting occurrences in the regions exhibit self-exciting tendencies, a temporal Hawkes process model with time invariant parameters is not adequate. The present thesis suggests that the reason for this lie in the parameters not being constant over time. Furthermore, based on previous research it is likely that a spatial component is needed to adequately capture the underlying process.

The deterministic techniques in use for transmission-grid planning and operation have resulted in a high reliability, but they have certain limitations that could be removed by applying stochastic techniques. This paper presents a detailed review of one such stochastic technique: operational risk assessment. A general procedural model of operational risk assessment is introduced, based on one simple equation, highlighting the important elements. The research trends are presented for each of these elements: contingency definition and calculating the probability of a contingency case; contingency filtration methods; defining and calculating the severity factor. Next to an overview of the state-of-the-art, this paper contains a detailed discussion section, where the most important research gaps are identified. Emphasis is especially on bridging the gap between research and practical applications of operational risk assessment. The paper closes with a future outlook on operational risk assessment and its applications., Validerad;2024;Nivå 2;2024-05-28 (hanlid);Full text license: CC BY

We show how the classical Lagrangian approach to solving constrained optimization problems from standard calculus can be extended to solve continuous time stochastic optimal control problems. Connections to mainstream approaches such as the Hamilton-Jacobi-Bellman equation and the stochastic maximum principle are drawn. Our approach is linked to the stochastic maximum principle, but more direct and tied to the classical Lagrangian principle, avoiding the use of backward stochastic differential equations in its formulation. Using infinite dimensional functional analysis, we formalize and extend the approach first outlined in Chow (1992) within a rigorous mathematical setting using infinite dimensional functional analysis. We provide examples that demonstrate the usefulness and effectiveness of our approach in practice. Further, we demonstrate the potential for numerical applications facilitating some of our key equations in combination with Monte Carlo backward simulation and linear regression, therefore illustrating a completely different and new avenue for the numerical application of Chow's methods.

Technological and scientific advancements over the past 30-40 years make it possible to fabricate nanometer-sized systems where fluctuations and quantum effects are important features. These systems can be controlled and measured with high accuracy, making them suitable for exploring fundamental aspects of the microscopic world. This thesis makes contributions in this direction, focusing on the theory of continuous, Markovian feedback control and stochastic as well as quantum thermodynamics.In Paper I and III, we study an implementation of Maxwell’s demon in a double quantum dot. A feedback protocol, based on charge detection, transfers electrons against an external voltage bias, without performing any net work. The protocol is studied for both classical and quantum dynamics. We characterize the electronic transport properties, explore information-to-work conversion with realistic detector models, as well as the quantum-to-classical transition.Paper II introduces a novel master equation for continuous, Markovian feedback control. This equation describes the dynamics of a quantum system and a detector with finite bandwidth. For a fast detector, the equation reduces to a Markovian master equation for the system alone. This equation can describe protocols that depend linearly as well as nonlinearly on the measured signal. For linear protocols, this equation coincides with theWiseman-Milburn equation, illustrating a compelling connection to previous work in the field.Paper IV investigates a specific feedback protocol for increasing the entanglement generation in an autonomous thermal machine. Under optimal conditions, we find that the protocol significantly increases the entanglement. The entanglement violates the CHSH inequality and can uphold quantum teleportation.Paper V presents a setup for generating maximally entangled states in autonomous thermal machines. First, we show that maximal bipartite entanglement can be generated in a system of three qubits. This is lat

We show how the classical Lagrangian approach to solving constrained optimization problems from standard calculus can be extended to solve continuous time stochastic optimal control problems. Connections to mainstream approaches such as the Hamilton-Jacobi-Bellman equation and the stochastic maximum principle are drawn. Our approach is linked to the stochastic maximum principle, but more direct and tied to the classical Lagrangian principle, avoiding the use of backward stochastic differential equations in its formulation. Using infinite dimensional functional analysis, we formalize and extend the approach first outlined in Chow (1992) within a rigorous mathematical setting using infinite dimensional functional analysis. We provide examples that demonstrate the usefulness and effectiveness of our approach in practice. Further, we demonstrate the potential for numerical applications facilitating some of our key equations in combination with Monte Carlo backward simulation and linear regression, therefore illustrating a completely different and new avenue for the numerical application of Chow's methods.

Resilient software-defined Internet of Things (SD-IoT) networks are critical in offshore renewable energy supply (such as offshore wind farms) for wide-area monitoring, protection, automation, and control (WAMPAC). Offshore wind farms transmit time-sensitive data about the turbine performance, power generation, environmental conditions, and critical equipment status to the wind farm offshore landing point or the remote control room. As such, there is a need to guarantee real-time communication to coordinate protection and control actions that maximize production and minimize inadvertent wind farm downtime. This research designs a deep Q-Network (DQN) resilience model that quickly detects disruptions and applies optimal traffic engineering actions at the software-defined network (SDN) controller to guarantee high performance. This resilience model improves the quality of service of the SDN-enabled IoT networks in offshore wind farm communication networks.

Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this paper we develop theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady-state behavior. These theorems exploit structural properties of SCRNs, beyond those of general continuous-time Markov chains. Furthermore, we derive two theorems to compare stationary distributions and mean first passage times for SCRNs with different parameter values, or with the same parameters and different initial conditions. These tools are developed for SCRNs taking values in a generic (finite or countably infinite) state space and can also be applied for non-mass-action kinetics models. When propensity functions are bounded, our method of proof gives an explicit method for coupling two comparable SCRNs, which can be used to simultaneously simulate their sample paths in a comparable manner. We illustrate our results with applications to models of enzymatic kinetics and epigenetic regulation by chromatin modifications.

We make use of Borel resummation to extract the exact time dependence from the divergent series found in the context of stochastic inflation. Correlation functions of self-interacting scalar fields in de Sitter spacetime are known to develop secular IR divergences via loops, and the first terms of the divergent series have been consistently computed both with standard techniques for curved spacetime quantum field theory and within the framework of stochastic inflation. We show that Borel resummation can be used to interpret the divergent series and to correctly infer the time evolution of the correlation functions. In practice, we adopt a method called Borel-Padé resummation where we approximate the Borel transformation by a Padé approximant. We also discuss the singularity structures of Borel transformations and mention possible applications to cosmology.

This paper addresses an optimal control problem for one-way car-sharing systems under dynamic pricing. One-way car-sharing service allows customers to return vehicles to any available station. Although this service provides great convenience to customers, it has a serious drawback such that vehicles can be unevenly parked and some stations can be unavailable. To solve this problem, dynamic pricing is promising to distribute car parking, with which customers change their origins and destinations by walking according to prices. First, we develop a model of this system by using differential equations, including stochastic processes which represent uncertain human behavior such as the demand shift. Next, we design a uniform, distributed dynamic pricing policy by solving the optimization problem to minimize the unevenness of the vehicles. Then, we show that the effectiveness of the dynamic pricing depends on the network topology of stations. So, we propose an optimization method to relocate stations for effective dynamic pricing with PSO (Particle Swarm Optimization). Finally, numerical examples demonstrate the effectiveness of the developed methods.

Stochastic processes with reinforcement are the central theme of the present tandem workshop. We assembled a diverse group of international experts that worked on reinforcement dynamics from several different perspectives. We discussed progress and future strategies around a number of key open problems in the area of interacting urns with graph based interaction, preferential attachment, and reinforced random walks.

When a spatially localized stress is applied to a growing one-dimensional interface, the interface deforms. This deformation is described by the effective surface tension representing the stiffness of the interface. We present that the stiffness exhibits divergent behavior in the large system size limit for a growing interface with thermal noise, which has never been observed for equilibrium interfaces. Furthermore, by connecting the effective surface tension with a space-time correlation function, we elucidate the mechanism that anomalous dynamical fluctuations lead to divergent stiffness.

Remarkable achievements by deep neural networks stand on the development of excellent stochastic gradient descent methods. Deep-learning-based machine learning algorithms, however, have to find patterns between observations and supervised signals, even though they may include some noise that hides the true relationship between them, more or less especially in the robotics domain. To perform well even with such noise, we expect them to be able to detect outliers and discard them when needed. We, therefore, propose a new stochastic gradient optimization method, whose robustness is directly built in the algorithm, using the robust student-t distribution as its core idea. We integrate our method to some of the latest stochastic gradient algorithms, and in particular, Adam, the popular optimizer, is modified through our method. The resultant algorithm, called t-Adam, along with the other stochastic gradient methods integrated with our core idea is shown to effectively outperform Adam and their original versions in terms of robustness against noise on diverse tasks, ranging from regression and classification to reinforcement learning problems.

Large electrical transmission networks are susceptible to undergo very large blackouts due to cascading failures, with a very large associated economical cost. In this work we propose segmenting large power grids using controllable lines, such as high-voltage direct-current lines, to reduce the risk of blackouts. The method consists in modifying the power flowing through the lines interconnecting different zones during cascading failures in order to minimize the load shed. As a result, the segmented grids have a substantially lower risk of blackouts than the original network, with reductions up to 60% in some cases. The control method is shown to be specially efficient in reducing blackouts affecting more than one zone., Peer Reviewed, Postprint (published version)

We focus on planar Random Walks and some related stochastic processes. The discrete models are introduced and some of their core properties examined. We then turn to the question of continuous analogues, starting with the well-known convergence of the Random Walk to Brownian Motion. For the Harmonic Explorer and the Loop Erased Random Walk, we discuss the idea for convergence to SLE(\kappa) and carry out parts of the proof in the former case using a martingale observable to pin down the Loewner driving process.

We present a continuous-time probabilistic approach for estimating the chirp signal and its instantaneous frequency function when the true forms of these functions are not accessible. Our model represents these functions by non-linearly cascaded Gaussian processes represented as non-linear stochastic differential equations. The posterior distribution of the functions is then estimated with stochastic filters and smoothers. We compute a (posterior) Cramér-Rao lower bound for the Gaussian process model, and derive a theoretical upper bound for the estimation error in the mean squared sense. The experiments show that the proposed method outperforms a number of state-of-the-art methods on a synthetic data. We also show that the method works out-of-the-box for two real-world datasets.

Synchronization phenomena is ubiquitous in nature, and in spite of having been studied for decades, it still attracts a lot of attention as is still challenging to detect and quantify, directly from the analysis of noisy signals. Semiconductor lasers are ideal for performing experiments because they are stochastic, nonlinear, and inexpensive and display different synchronization regimes that can be controlled by tuning the lasers’ parameters. Here, we analyze experiments done with two mutually optically coupled lasers. Due to the delay in the coupling (due to the finite time the light takes to travel between the lasers), the lasers synchronize with a lag: the intensity time traces show well-defined spikes, and a spike in the intensity of one laser may occur shortly before (or shortly after) a spike in the intensity of the other laser. Measures that quantify the degree of synchronization of the lasers from the analysis of the intensity signals do not fully quantify the synchronicity of the spikes because they also take into account the synchronization of fast irregular fluctuations that occur between spikes. By analyzing only the coincidence of the spike times, we show that event synchronization measures quantify spike synchronization remarkably well. We show that these measures allow us to quantify the degree of synchronization and, also, to identify the leading laser and the lagging one., C.M. acknowledges the support of Ministerio de Ciencia, Innovación y Universidades (No. PID2021-123994NB-C21) and Institució Catalana de Recerca i Estudis Avançats (Academia)., Peer Reviewed, Postprint (author's final draft)

Sick absence affects companies both operationally and economically and the matter has become increasingly prominent following the COVID-19 pandemic. MedHelp Care is an e-health company working towards increasing workplace wellness by offering insights into absence and rehabilitation matters. Forecasting future absence levels could help with matters such as staffing and garner more insight into which factors that impact absenteeism. Moreover, gauging how external factors affect the absence could help deepen the knowledge further. With absence data now being digital, it is able to be modeled using statistical tools. The dataset was aggregated to a daily basis and preprocessed to better represent the Swedish labour force. Then, by applying time series analysis and machine learning methods, predictions of sick absence were formed. Datasets of the Swedish stock market and the discourse around COVID-19 on Twitter, respectively, were processed to time series and their relation to the sick absence was explored using crosscorrelation. The sets were then individually incorporated into exogenous models using autoregression, and then their impact was evaluated. Results show that the SARIMA model is better for predicting short and medium length horizons while the machine learning methods used were more apt for long horizons. Results from the exogenous models were mixed; the OMX set improved short term prediction accuracy while other horizons were largely unaffected, and the Twitter set worsened performance for all horizon lengths. While the model results are promising, their current complexity hinder large-scale deployment and further research is needed to further verify the effect of exogenous datasets.

Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.

Stationary quantum information sources emit sequences of correlatedqudits---that is, structured quantum stochastic processes. If an observer performs identical measurements on a qudit sequence, the outcomes are a realization of a classical stochastic process. We introduce quantum-information-theoretic properties for separable qudit sequences that serve as bounds on the classical information properties of subsequent measured processes. For sources driven by hidden Markov dynamics we describe how an observer can temporarily or permanently synchronize to the source's internal state using specific positive operator-valued measures or adaptive measurement protocols. We introduce a method for approximating an information source with an independent and identically-distributed, Markov, or larger memory model through tomographic reconstruction. We identify broad classes of separable processes based on their quantum information properties and the complexity of measurements required to synchronize to and accurately reconstruct them.

Photosynthesis is generally assumed to be initiated by a single photon1-3 from the Sun, which, as a weak light source, delivers at most a few tens of photons per nanometre squared per second within a chlorophyll absorption band1. Yet much experimental and theoretical work over the past 40 years has explored the events during photosynthesis subsequent to absorption of light from intense, ultrashort laser pulses2-15. Here, we use single photons to excite under ambient conditions the light-harvesting 2 (LH2) complex of the purple bacterium Rhodobacter sphaeroides, comprising B800 and B850 rings that contain 9 and 18 bacteriochlorophyll molecules, respectively. Excitation of the B800 ring leads to electronic energy transfer to the B850 ring in approximately 0.7 ps, followed by rapid B850-to-B850 energy transfer on an approximately 100-fs timescale and light emission at 850-875 nm (refs. 16-19). Using a heralded single-photon source20,21 along with coincidence counting, we establish time correlation functions for B800 excitation and B850 fluorescence emission and demonstrate that both events involve single photons. We also find that the probability distribution of the number of heralds per detected fluorescence photon supports the view that a single photon can upon absorption drive the subsequent energy transfer and fluorescence emission and hence, by extension, the primary charge separation of photosynthesis. An analytical stochastic model and a Monte Carlo numerical model capture the data, further confirming that absorption of single photons is correlated with emission of single photons in a natural light-harvesting complex.

The application of classical Random Neural Networks (RNN) has been restricted to deterministic digital systems that generate probability distributions instead of the stochastic characteristics of random spiking signals. To optimize the utilization of the stochastic properties inherent in RNN, we propose a new category of supervised Random Quantum Neural Networks (RQNN) that incorporate a resilient training methodology. The RQNN under consideration utilizes a combination of classical and quantum algorithms, incorporating superposition state and angle encoding characteristics that draw inspiration from quantum information theory. Additionally, the model incorporates the stochastic random spiking property of neuron information encoding, which is known to exhibit spatial-temporal features similar to those observed in the brain. The proposed RQNN model has undergone thorough validation, relying on hybrid classical-quantum algorithms through a real IBM quantum processor. The proposed RQNN model is tested on the MNIST, Fashion-MNIST, and KMNIST datasets. The results indicate that it achieved an average classification accuracy of 59.6% when presented with unseen noisy test images. The experimental results demonstrate the efficacy and robustness of the proposed RQNN in classifying noisy images that have not been previously encountered.

The application of classical Random Neural Networks (RNN) has been restricted to deterministic digital systems that generate probability distributions instead of the stochastic characteristics of random spiking signals. To optimize the utilization of the stochastic properties inherent in RNN, we propose a new category of supervised Random Quantum Neural Networks (RQNN) that incorporate a resilient training methodology. The RQNN under consideration utilizes a combination of classical and quantum algorithms, incorporating superposition state and angle encoding characteristics that draw inspiration from quantum information theory. Additionally, the model incorporates the stochastic random spiking property of neuron information encoding, which is known to exhibit spatial-temporal features similar to those observed in the brain. The proposed RQNN model has undergone thorough validation, relying on hybrid classical-quantum algorithms through a real IBM quantum processor. The proposed RQNN model is tested on the MNIST, Fashion-MNIST, and KMNIST datasets. The results indicate that it achieved an average classification accuracy of 59.6% when presented with unseen noisy test images. The experimental results demonstrate the efficacy and robustness of the proposed RQNN in classifying noisy images that have not been previously encountered.

Slowing down occurs in dynamical systems close to bifurcations or phase transitions. The time length (t) of ghost transients close to a saddle-node bifurcation in deterministic systems follows t~|¿-¿c|-1/2, ¿ being the bifurcation parameter and ¿c its critical value. Recently, we numerically investigated how intrinsic noise affected the deterministic picture, finding a more complicated scaling law. We here provide a theoretical basis for this new law with two models of cooperation using a Wentzel–Kramers–Brillouin asymptotic approximation of the Master Equation. A study of the phase space of the Hamiltonian derived from the Hamilton–Jacobi equation shows that the statistically significant orbits (paths) reproduce the scaling function observed in the stochastic simulations. The flight times tied to these orbits underpin the scaling law of the stochastic system, and the same properties should extend in a universal way to all stochasticsystems whose associated Hamiltonian exhibits the same behaviour. Our approach allows to make useful theoretical predictions of transient times in stochastic systems close to a bifurcation., Peer Reviewed, Postprint (published version)

This paper studies the assembly line balancing problem with collaborative robots in light of recent efforts to implement collaborative robots in industrial production systems under random processing time. A stochastic version with uncertain human processing time is considered for the first time. The issue is defined by the potential for simultaneous human and robot task execution at the same workpiece, either in parallel or in collaboration. We provide stochastic mixed-integer programming based on Monte Carlo sampling approach for the balancing and scheduling of collaborative robot assembly lines for this novel issue type. In order to minimise the line cost including fixed workstation operating costs and resource costs caused by exceeding cycle time, the model determines both the placement of collaborative robots at stations and the distribution of work among humans and robots.

In this paper, we formulate feedback control problems involving stochastic systems and stochastic signals and obtain stability conditions based on the notions called gains and angles. The family of systems under investigation is represented by a discrete-time linear time-varying stochastic (LTVS) model which generalizes the intensively studied multiplicative stochastic model. Such formulation may incorporate widely encountered networked control problems involving fading and packet-drop communication channels. For resolving feedback stability problems of LTVS systems, small gain and small angle conditions are proposed, and illustrative examples are presented. IEEE

The Gumbel-max trick is a method to draw a sample from a categorical distribution, given by its unnormalized (log-)probabilities. Over the past years, the machine learning community has proposed several extensions of this trick to facilitate, e.g., drawing multiple samples, sampling from structured domains, or gradient estimation for error backpropagation in neural network optimization. The goal of this survey article is to present background about the Gumbel-max trick, and to provide a structured overview of its extensions to ease algorithm selection. Moreover, it presents a comprehensive outline of (machine learning) literature in which Gumbel-based algorithms have been leveraged, reviews commonly-made design choices, and sketches a future perspective.

To alleviate the local computation demands from the ever-increasing computation-intensive mobile applications, Mobile Edge Computing (MEC) has proved promising. Especially, by opportunistically offloading these computation tasks to the MEC server, the delay of computing could be significantly improved through communication. In this paper, we develop an analytical framework for joint communication and computation resources allocation for multi-user MEC systems. Specifically, to retrieve the combined effect of communication and computation capabilities, we establish a dual queue system, including a data queue sub-system and a computation queue sub-system. To address the associated stochastic resource optimization problem, we propose a low-complexity resource allocation algorithm by Lyapunov optimization to stabilize all the sub-queue systems. As the practical buffers are finite, the conventional delay analysis of Lyapunov optimization becomes inaccurate. Alternatively, we model the stochastic queue lengthes as discrete time controlled random walk processes, which are transformed to continuous time Stochastic Differential Equations (SDEs) with reflections by strong approximation. According to the steady state analysis on the SDEs, we derive closed-form steady state distributions of the queue lengths, and then obtain the average delay performance with finite buffers. Finally, the accuracy of the proposed delay analysis is verified through simulation. IEEE

Automatically synthesizing controllers for continuous-state nonlinear stochastic systems, while giving guarantees on the probability of satisfying (infinite-horizon) temporal logic specifications crucially depends on abstractions with a quantified accuracy. For this similarity quantification, approximate stochastic simulation relations are often used. To handle the nonlinearity of the system effectively, we use finite-state abstractions based on piecewise-affine approximations together with tailored simulation relations that leverage the local affine structure. In the end, we synthesize a robust controller for a nonlinear stochastic Van der Pol oscillator.

We focus on planar Random Walks and some related stochastic processes. The discrete models are introduced and some of their core properties examined. We then turn to the question of continuous analogues, starting with the well-known convergence of the Random Walk to Brownian Motion. For the Harmonic Explorer and the Loop Erased Random Walk, we discuss the idea for convergence to SLE(\kappa) and carry out parts of the proof in the former case using a martingale observable to pin down the Loewner driving process.

This paper jointly optimizes the zoning, scheduling, and pricing of a zonal-based flexible bus service (ZBFBS). ZBFBS categorizes and groups the origin-destination (OD) pairs of ride requests by geographical zones to provide door-to-door transit services while considering dynamic stochastic elastic demand volume, stochastic ride request locations, and time window constraints. The integrated problem can be formulated in two ways, either by designing the bus service routing in response to the realized demand, referred to as the demand-responsive method, or by a stochastic programming approach that accounts for the demand distribution. The former first clusters the realized requests into zones through the k-means algorithm, and then solves the joint vehicle scheduling and passenger assignment problem. As for the latter, a bi-level framework is proposed: the upper level optimizes the zoning, and the lower level maximizes the profit by scheduling and pricing the ZBFBS service. For this purpose, hexagonal and rectangular zonings are considered, whose positions and dimensions are optimized by a line search algorithm, with a deterministic approximation approach to significantly reduce the solution time. This framework is applied to a scenario based on actual ride-hailing data in Chengdu, China. The results demonstrate the benefit of the bi-level framework in producing an optimal zonal design. Also, the proposed bi-level stochastic zonal design framework outperforms the demand-responsive method in a tight planning time. IEEE