Search Results
Solving linear parabolic rough partial differential equations
2018, Bayer, Christian, Belomestny, Denis, Redmann, Martin, Riedel, Sebastian, Schoenmakers, John
We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity with 1=3 < 1=2. Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented.
The degrees of freedom of partial least squares regression
2010, Krämer, Nicole, Sugiyama, Masashi
The derivation of statistical properties for Partial Least Squares regression can be a challenging task. The reason is that the construction of latent components from the predictor variables also depends on the response variable. While this typically leads to good performance and interpretable models in practice, it makes the statistical analysis more involved. In this work, we study the intrinsic complexity of Partial Least Squares Regression. Our contribution is an unbiased estimate of its Degrees of Freedom. It is defined as the trace of the first derivative of the fitted values, seen as a function of the response. We establish two equivalent representations that rely on the close connection of Partial Least Squares to matrix decompositions and Krylov subspace techniques. We show that the Degrees of Freedom depend on the collinearity of the predictor variables: The lower the collinearity is, the higher the Degrees of Freedom are. In particular, they are typically higher than the naive approach that defines the Degrees of Freedom as the number of components. Further, we illustrate that the Degrees of Freedom are useful for model selection. Our experiments indicate that the model complexity based on the Degrees of Freedom estimate is lower than the model complexity of the naive approach. In terms of prediction accuracy, both methods obtain the same accuracy as cross-validation
Robust optimal stopping
2015, Krätschmer, Volker, Ladkau, Marcel, Laeven, Roger J.A., Schoenmakers, John G.M., Stadje, Mitja
This paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a method to practically solve this problem in a general setting, allowing for general time-consistent ambiguity averse preferences and general payoff processes driven by jump-diffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backward-forward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We analyze the asymptotic behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples.
Robust multiple stopping -- A path-wise duality approach
2020, Laeven, Roger J. A., Schoenmakers, John G. M., Schweizer, Nikolaus F. F., Stadje, Mitja
In this paper we develop a solution method for general optimal stopping problems. Our general setting allows for multiple exercise rights, i.e., optimal multiple stopping, for a robust evaluation that accounts for model uncertainty, and for general reward processes driven by multi-dimensional jump-diffusions. Our approach relies on first establishing robust martingale dual representation results for the multiple stopping problem which satisfy appealing path-wise optimality (almost sure) properties. Next, we exploit these theoretical results to develop upper and lower bounds which, as we formally show, not only converge to the true solution asymptotically, but also constitute genuine upper and lower bounds. We illustrate the applicability of our general approach in a few examples and analyze the impact of model uncertainty on optimal multiple stopping strategies.
Nonparametric change point detection in regression
2020, Avanesov, Valeriy
This paper considers the prominent problem of change-point detection in regression. The study suggests a novel testing procedure featuring a fully data-driven calibration scheme. The method is essentially a black box, requiring no tuning from the practitioner. The approach is investigated from both theoretical and practical points of view. The theoretical study demonstrates proper control of first-type error rate under H0 and power approaching 1 under H1. The experiments conducted on synthetic data fully support the theoretical claims. In conclusion, the method is applied to financial data, where it detects sensible change-points. Techniques for change-point localization are also suggested and investigated
How the Selection of Training Data and Modeling Approach Affects the Estimation of Ammonia Emissions from a Naturally Ventilated Dairy Barn—Classical Statistics versus Machine Learning
2020, Hempel, Sabrina, Adolphs, Julian, Landwehr, Niels, Janke, David, Amon, Thomas
Environmental protection efforts can only be effective in the long term with a reliable quantification of pollutant gas emissions as a first step to mitigation. Measurement and analysis strategies must permit the accurate extrapolation of emission values. We systematically analyzed the added value of applying modern machine learning methods in the process of monitoring emissions from naturally ventilated livestock buildings to the atmosphere. We considered almost 40 weeks of hourly emission values from a naturally ventilated dairy cattle barn in Northern Germany. We compared model predictions using 27 different scenarios of temporal sampling, multiple measures of model accuracy, and eight different regression approaches. The error of the predicted emission values with the tested measurement protocols was, on average, well below 20%. The sensitivity of the prediction to the selected training dataset was worse for the ordinary multilinear regression. Gradient boosting and random forests provided the most accurate and robust emission value predictions, accompanied by the second-smallest model errors. Most of the highly ranked scenarios involved six measurement periods, while the scenario with the best overall performance was: One measurement period in summer and three in the transition periods, each lasting for 14 days.
Regression on particle systems connected to mean-field SDEs with applications
2017, Belomestny, Denis, Schoenmakers, John G.M.
In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKean-Vlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered.
How to gamble with non-stationary X-armed bandits and have no regrets
2020, Avanesov, Valeriy
In X-armed bandit problem an agent sequentially interacts with environment which yields a reward based on the vector input the agent provides. The agent's goal is to maximise the sum of these rewards across some number of time steps. The problem and its variations have been a subject of numerous studies, suggesting sub-linear and sometimes optimal strategies. The given paper introduces a new variation of the problem. We consider an environment, which can abruptly change its behaviour an unknown number of times. To that end we propose a novel strategy and prove it attains sub-linear cumulative regret. Moreover, the obtained regret bound matches the best known bound for GP-UCB for a stationary case, and approaches the minimax lower bound in case of highly smooth relation between an action and the corresponding reward. The theoretical result is supported by experimental study.