arXiv subjects

Oliver Kamps

Publications and source records attributed to Oliver Kamps.

At least 19 recordsLinked to original sources

arXiv2025

Hyperparameter Optimization in the Estimation of PDE and Delay-PDE models from data

We propose an improved method for estimating partial differential equations and delay partial differential equations from data, using Bayesian optimization and the Bayesian information criterion to automatically find suitable hyperparameters for the method itself or for the equations (such as a time-delay). We show that combining time integration into an established model estimation method increases robustness and yields predictive models. Allowing hyperparameters to be optimized as part of the model estimation results in a wider modelling scope. We demonstrate the method's performance on a number of synthetic benchmark problems of different complexity, representing different classes of physical behaviour. This includes the Allen-Cahn and Cahn-Hilliard models, as well as different reaction-diffusion systems without and with time-delay.

physics.comp-ph↗

arXiv2025

Sparse identification of evolution equations via Bayesian model selection

The quantitative formulation of evolution equations is the backbone for prediction, control, and understanding of dynamical systems across diverse scientific fields. Besides deriving differential equations for dynamical systems based on basic scientific reasoning or prior knowledge in recent times a growing interest emerged to infer these equations purely from data. In this article, we introduce a novel method for the sparse identification of nonlinear dynamical systems from observational data, based on the observation how the key challenges of the quality of time derivatives and sampling rates influence this problem. Our approach combines system identification based on thresholded least squares minimization with additional error measures that account for both the deviation between the model and the time derivative of the data, and the integrated performance of the model in forecasting dynamics. Specifically, we integrate a least squares error as well as the Wasserstein metric for estimated models and combine them within a Bayesian optimization framework to efficiently determine optimal hyperparameters for thresholding and weighting of the different error norms. Additionally, we employ distinct regularization parameters for each differential equation in the system, enhancing the method's precision and flexibility. We demonstrate the capabilities of our approach through applications to dynamical fMRI data and the prototypical example of a wake flow behind a cylinder. In the wake flow problem, our method identifies a sparse, accurate model that correctly captures transient dynamics, oscillation periods, and phase information, outperforming existing methods. In the fMRI example, we show how our approach extracts insights from a trained recurrent neural network, offering a novel avenue for explainable AI by inferring differential equations that capture potentially causal relationships.

physics.data-an↗

arXiv2024

Do Inner Greenland's Melt Rate Dynamics Approach Coastal Ones?

The Greenland Ice Sheet may be nearing a tipping point, transitioning to permanent melting. This article analyses two melt rate time series using the Bayesian Langevin estimation (BLE), providing further evidence for destabilizing melt dynamics, along with new insights from the method's nonlinear parameterisation. Comparing the results for Western Central Greenland and the Nuussuaq peninsula suggests that inland melt dynamics may approach less stable coastal dynamics on slow and fast scales. Both datasets show a significant increase in fast-scale amplitudes since the 1970s, possibly driven by large-scale atmospheric fluctuations. Additionally, the BLE's nonlinear drift proves crucial for obtaining these results, as a linear Ornstein-Uhlenbeck process fails to capture these dynamics due to the oversimplification of a strictly positive autocorrelation function.

physics.geo-ph↗

arXiv2024

Causal Hierarchy in the Financial Market Network -- Uncovered by the Helmholtz-Hodge-Kodaira Decomposition

Granger causality can uncover the cause and effect relationships in financial networks. However, such networks can be convoluted and difficult to interpret, but the Helmholtz-Hodge-Kodaira decomposition can split them into a rotational and gradient component which reveals the hierarchy of Granger causality flow. Using Kenneth French's business sector return time series, it is revealed that during the Covid crisis, precious metals and pharmaceutical products are causal drivers of the financial network. Moreover, the estimated Granger causality network shows a high connectivity during crisis which means that the research presented here can be especially useful to better understand crises in the market by revealing the dominant drivers of the crisis dynamics.

q-fin.ST↗

arXiv2024

Cooperation in a non-Ergodic World on a Network -- Insurance and Beyond

Cooperation between individuals is emergent in all parts of society, yet mechanistic reasons for this emergence is ill understood in the literature. A specific example of this is insurance. Recent work has, though, shown that assuming the risk individuals face is proportional to their wealth and optimising the time average growth rate rather than the ensemble average results in a non-zero-sum game, where both parties benefit from cooperation through insurance contracts. In a recent paper, Peters and Skjold present a simple agent-based model and show how, over time, agents that enter into such cooperatives outperform agents that do not. Here, we extend this work by restricting the possible connections between agents via a lattice network. Under these restrictions, we still find that all agents profit from cooperating through insurance. We, though, further find that clusters of poor and rich agents emerge endogenously on the two-dimensional map and that wealth inequalities persist for a long duration, consistent with the phenomenon known as the poverty trap. By tuning the parameters which control the risk levels, we simulate both highly advantageous and extremely risky gambles and show that despite the qualitative shift in the type of risk, the findings are consistent.

physics.soc-ph↗

arXiv2023

Estimating Stable Fixed Points and Langevin Potentials for Financial Dynamics

The Geometric Brownian Motion (GBM) is a standard model in quantitative finance, but the potential function of its stochastic differential equation (SDE) cannot include stable nonzero prices. This article generalises the GBM to an SDE with polynomial drift of order q and shows via model selection that q=2 is most frequently the optimal model to describe the data. Moreover, Markov chain Monte Carlo ensembles of the accompanying potential functions show a clear and pronounced potential well, indicating the existence of a stable price.

q-fin.ST↗

arXiv2023

Anticipation of Oligocene's climate heartbeat by simplified eigenvalue estimation

The Eocene-Oligocene transition marks a watershed point of earth's climate history. The climate shifts from a greenhouse state to an icehouse state in which Antarctica glaciated for the first time and periodic dynamics arise which are still relevant for our current climate. We analyse a $CaCO_3$ concentration time series which covers the Eocene-Oligocene transition and which is obtained from a Pacific sediment core at site DSDP1218. Therefore, we introduce a simplified autoregression-based variant of the dominant eigenvalue (DEV) estimation procedure. The DEV works as leading indicator of bifurcation-induced transitions and enables us to identify the bifurcation type. We confirm its reliability in a methodological study and demonstrate the crucial importance of proper detrending to obtain unbiased results. As a remark, we discuss also possible pathways to estimate the stability of limit cycles based on the DEV and the alternative drift slope as a proof of principle. Finally, we present the DEV analysis results of the $CaCO_3$ concentration time series which are reproducible in a wide parameter range. Our findings demonstrate that the onset of Oligocene's periodic dynamics might be announced by a Neimark-Sacker/Hopf bifurcation in course of the Eocene-Oligocene transition 34 mya. (We follow the convention and use mya$\widehat{=}$"million years ago" and Ma$\widehat{=}$"million years" throughout the article.)

physics.ao-ph↗

arXiv2023

Non-linear, bivariate stochastic modelling of power-grid frequency applied to islands

Mitigating climate change requires a transition away from fossil fuels towards renewable energy. As a result, power generation becomes more volatile and options for microgrids and islanded power-grid operation are being broadly discussed. Therefore, studying the power grids of physical islands, as a model for islanded microgrids, is of particular interest when it comes to enhancing our understanding of power-grid stability. In the present paper, we investigate the statistical properties of the power-grid frequency of three island systems: Iceland, Ireland, and the Balearic Islands. We utilise a Fokker-Planck approach to construct stochastic differential equations that describe market activities, control, and noise acting on power-grid dynamics. Using the obtained parameters we create synthetic time series of the frequency dynamics. Our main contribution is to propose two extensions of stochastic power-grid frequency models and showcase the applicability of these new models to non-Gaussian statistics, as encountered in islands.

physics.data-an↗

arXiv2023

Efficient Multi-Change Point Analysis to decode Economic Crisis Information from the S&P500 Mean Market Correlation

Identifying macroeconomic events that are responsible for dramatic changes of economy is of particular relevance to understand the overall economic dynamics. We introduce an open-source available efficient Python implementation of a Bayesian multi-trend change point analysis which solves significant memory and computing time limitations to extract crisis information from a correlation metric. Therefore, we focus on the recently investigated S&P500 mean market correlation in a period of roughly 20 years that includes the dot-com bubble, the global financial crisis and the Euro crisis. The analysis is performed two-fold: first, in retrospect on the whole dataset and second, in an on-line adaptive manner in pre-crisis segments. The on-line sensitivity horizon is roughly determined to be 80 up to 100 trading days after a crisis onset. A detailed comparison to global economic events supports the interpretation of the mean market correlation as an informative macroeconomic measure by a rather good agreement of change point distributions and major crisis events. Furthermore, the results hint to the importance of the U.S. housing bubble as trigger of the global financial crisis, provide new evidence for the general reasoning of locally (meta)stable economic states and could work as a comparative impact rating of specific economic events.

q-fin.ST↗

arXiv2023

Memory Effects, Multiple Time Scales and Local Stability in Langevin Models of the S&P500 Market Correlation

The analysis of market correlations is crucial for optimal portfolio selection of correlated assets, but their memory effects have often been neglected. In this work, we analyse the mean market correlation of the S&P500 which corresponds to the main market mode in principle component analysis. We fit a generalised Langevin equation (GLE) to the data whose memory kernel implies that there is a significant memory effect in the market correlation ranging back at least three trading weeks. The memory kernel improves the forecasting accuracy of the GLE compared to models without memory and hence, such a memory effect has to be taken into account for optimal portfolio selection to minimise risk or for predicting future correlations. Moreover, a Bayesian resilience estimation provides further evidence for non-Markovianity in the data and suggests the existence of a hidden slow time scale that operates on much slower times than the observed daily market data. Assuming that such a slow time scale exists, our work supports previous research on the existence of locally stable market states.

q-fin.ST↗

arXiv2023

Identifying Dominant Industrial Sectors in Market States of the S&P 500 Financial Data

Understanding and forecasting changing market conditions in complex economic systems like the financial market is of great importance to various stakeholders such as financial institutions and regulatory agencies. Based on the finding that the dynamics of sector correlation matrices of the S&P 500 stock market can be described by a sequence of distinct states via a clustering algorithm, we try to identify the industrial sectors dominating the correlation structure of each state. For this purpose, we use a method from Explainable Artificial Intelligence (XAI) on daily S&P 500 stock market data from 1992 to 2012 to assign relevance scores to every feature of each data point. To compare the significance of the features for the entire data set we develop an aggregation procedure and apply a Bayesian change point analysis to identify the most significant sector correlations. We show that the correlation matrix of each state is dominated only by a few sector correlations. Especially the energy and IT sector are identified as key factors in determining the state of the economy. Additionally we show that a reduced surrogate model, using only the eight sector correlations with the highest XAI-relevance, can replicate 90% of the cluster assignments. In general our findings imply an additional dimension reduction of the dynamics of the financial market.

q-fin.ST↗

arXiv2022

Quantifying Tipping Risks in Power Grids and beyond

Critical transitions, ubiquitous in nature and technology, necessitate anticipation to avert adverse outcomes. While many studies focus on bifurcation-induced tipping, where a control parameter change leads to destabilization, alternative scenarios are conceivable, e.g. noise-induced tipping by an increasing noise level in a multi-stable system. Although the generating mechanisms can be different, the observed time series can exhibit similar characteristics. Therefore, we propose a Bayesian Langevin approach, implemented in an open-source tool, which is capable of quantifying both deterministic and intrinsic stochastic dynamics simultaneously. After a detailed proof of concept, we analyse two bus voltage frequency time series of the historic North America Western Interconnection blackout on 10th August 1996. Our results unveil the intricate interplay of changing resilience and noise influence. A comparison with the blackout's timeline supports our frequency dynamics' Langevin model, with the BL-estimation indicating a permanent grid state change already two minutes before the officially defined triggering event. A tree-related high impedance fault or sudden load increases may serve as earlier triggers during this event, as suggested by our findings. This study underscores the importance of distinguishing destabilizing factors for a reliable anticipation of critical transitions, offering a tool for better understanding such events across various disciplines.

physics.data-an↗

arXiv2022

Efficient Bayesian estimation of the generalized Langevin equation from data

Modeling non-Markovian time series is a recent topic of research in many fields such as climate modeling, biophysics, molecular dynamics, or finance. The generalized Langevin equation (GLE), given naturally by the Mori-Zwanzig projection formalism, is a frequently used model including memory effects. In applications, a specific form of the GLE is most often obtained on a data-driven basis. Here, Bayesian estimation has the advantage of providing both suitable model parameters and their credibility in a straightforward way. It can be implemented in the approximating case of white noise, which, far from thermodynamic equilibrium, is consistent with the fluctuation-dissipation theorem. However, the exploration of the posterior, which is done via Markov chain Monte Carlo sampling, is numerically expensive, which makes the analysis of large data sets unfeasible. In this work, we discuss an efficient implementation of Bayesian estimation of the GLE based on a piecewise constant approximation of the drift and diffusion functions of the model. In this case, the characteristics of the data are represented by only a few coefficients, so that the numerical cost of the procedure is significantly reduced and independent of the length of the data set. Further, we propose a modification of the memory term of the GLE, leading to an equivalent model with an emphasis on the impact of trends, which ensures that an estimate of the standard Langevin equation provides an effective initial guess for the GLE. We illustrate the capabilities of both the method and the model by an example from turbulence.

physics.data-an↗

arXiv2022

Efficient Bayesian estimation of a non-Markovian Langevin model driven by correlated noise

Data-driven modeling of non-Markovian dynamics is a recent topic of research with applications in many fields such as climate research, molecular dynamics, biophysics, or wind power modeling. In the frequently used standard Langevin equation, memory effects can be implemented through an additional hidden component which functions as correlated noise, thus resulting in a non-Markovian model. It can be seen as part of the model class of partially observed diffusions which are usually adapted to observed data via Bayesian estimation, whereby the difficulty of the unknown noise values is solved through a Gibbs sampler. However, when regarding large data sets with a length of $10^6$ or $10^7$ data points, sampling the distribution of the same amount of latent variables is unfeasible. For the model discussed in this work, we solve this issue through a direct derivation of the posterior distribution of the Euler-Maruyama approximation of the model via analytical marginalization of the latent variables. Yet, in the case of a nonlinear noise process, the inverse problem of model estimation proves to be ill-posed and still numerically expensive. We handle these complications by restricting the noise to an Ornstein-Uhlenbeck process, which considerably reduces the ambiguity of the estimation. Further, in this case, the estimation can be performed very efficiently if the drift and diffusion functions of the observed component are approximated in a piecewise constant manner. We illustrate the resulting procedure of efficient Bayesian estimation of the considered non-Markovian Langevin model by an example from turbulence.

physics.data-an↗

arXiv2022

Quantifying resilience and the risk of regime shifts under strong correlated noise

Early warning indicators often suffer from the shortness and coarse-graining of real-world time series. Furthermore, the typically strong and correlated noise contributions in real applications are severe drawbacks for statistical measures. Even under favourable simulation conditions the measures are of limited capacity due to their qualitative nature and sometimes ambiguous trend-to-noise ratio. In order to solve these shortcomings, we analyse the stability of the system via the slope of the deterministic term of a Langevin equation, which is hypothesized to underlie the system dynamics close to the fixed point. The open-source available method is applied to a previously studied seasonal ecological model under noise levels and correlation scenarios commonly observed in real world data. We compare the results to autocorrelation, standard deviation, skewness and kurtosis as leading indicator candidates by a Bayesian model comparison with a linear and a constant model. We show that the slope of the deterministic term is a promising alternative due to its quantitative nature and high robustness against noise levels and types. The commonly computed indicators apart from the autocorrelation with deseasonalization fail to provide reliable insights into the stability of the system in contrast to a previously performed study in which the standard deviation was found to perform best. In addition, we discuss the significant influence of the seasonal nature of the data to the robust computation of the various indicators, before we determine approximately the minimal amount of data per time window that leads to significant trends for the drift slope estimations.

physics.data-an↗

arXiv2021

Bayesian on-line anticipation of critical transitions

The design of reliable indicators to anticipate critical transitions in complex systems is an im portant task in order to detect a coming sudden regime shift and to take action in order to either prevent it or mitigate its consequences. We present a data-driven method based on the estimation of a parameterized nonlinear stochastic differential equation that allows for a robust anticipation of critical transitions even in the presence of strong noise levels like they are present in many real world systems. Since the parameter estimation is done by a Markov Chain Monte Carlo approach we have access to credibility bands allowing for a better interpretation of the reliability of the results. By introducing a Bayesian linear segment fit it is possible to give an estimate for the time horizon in which the transition will probably occur based on the current state of information. This approach is also able to handle nonlinear time dependencies of the parameter controlling the transition. In general the method could be used as a tool for on-line analysis to detect changes in the resilience of the system and to provide information on the probability of the occurrence of a critical transition in future.

physics.data-an↗

arXiv2021

Non-parametric estimation of a Langevin model driven by correlated noise

Langevin models are frequently used to model various stochastic processes in different fields of natural and social sciences. They are adapted to measured data by estimation techniques such as maximum likelihood estimation, Markov chain Monte Carlo methods, or the non-parametric direct estimation method introduced by Friedrich et al. The latter has the distinction of being very effective in the context of large data sets. Due to their $δ$-correlated noise, standard Langevin models are limited to Markovian dynamics. A non-Markovian Langevin model can be formulated by introducing a hidden component that realizes correlated noise. For the estimation of such a partially observed diffusion a different version of the direct estimation method was introduced by Lehle et al. However, this procedure includes the limitation that the correlation length of the noise component is small compared to that of the measured component. In this work we propose another version of the direct estimation method that does not include this restriction. Via this method it is possible to deal with large data sets of a wider range of examples in an effective way. We discuss the abilities of the proposed procedure using several synthetic examples.

physics.data-an↗

arXiv2020

Adaptive stochastic continuation with a modified lifting procedure applied to complex systems

Many complex systems occurring in the natural or social sciences or economics are frequently described on a microscopic level, e.g., by lattice- or agent-based models. To analyze the states of such systems and their bifurcation structure on the level of macroscopic observables, one has to rely on equation-free methods like stochastic continuation. Here, we investigate how to improve stochastic continuation techniques by adaptively choosing the parameters of the algorithm. This allows one to obtain bifurcation diagrams quite accurately, especially near bifurcation points. We introduce lifting techniques which generate microscopic states with a naturally grown structure, which can be crucial for a reliable evaluation of macroscopic quantities. We show how to calculate fixed points of fluctuating functions by employing suitable linear fits. This procedure offers a simple measure of the statistical error. We demonstrate these improvements by applying the approach in analyses of (i) the Ising model in two dimensions, (ii) an active Ising model, and (iii) a stochastic Swift-Hohenberg model. We conclude by discussing the abilities and remaining problems of the technique.

nlin.AO↗