Multidimensional Gaussian Process Research Articles

Trio of super-Earth candidates orbiting K-dwarf HD 48948: a new habitable zone candidate

ABSTRACT We present the discovery of three super-Earth candidates orbiting HD 48948, a bright K-dwarf star with an apparent magnitude of $m_V$ = 8.58 mag. As part of the HARPS-N Rocky Planet Search programme, we collect 189 high-precision radial velocity measurements using the HARPS-N spectrograph from 2013 October 6, to 2023 April 16. Various methodologies are applied to extract the radial velocities from the spectra, and we conduct a comprehensive comparative analysis of the outcomes obtained through these diverse extraction techniques. To ensure the robustness of our findings, we employ several methods to address stellar variability, with a focus on Gaussian Process regression. To account for the impact of stellar variability and correlated noise in the radial velocity data set, we include activity indicators, such as $\log R^{^{\prime }}_{\mathrm{HK}}$ and bisector span, in the multidimensional Gaussian Process regression. Our analysis reveals three planetary candidates with orbital periods of 7.3, 38, and 151 d, and minimum masses estimated at $4.88 \pm 0.21$ M$_{\oplus }$, $7.27 \pm 0.70$ M$_{\oplus }$, and $10.59 \pm 1.00$ M$_{\oplus }$, respectively. The outermost planet resides within the (temperate) habitable zone, positioned at a projected distance of $0.029\,{\rm arcsec}$ from its star. Given the close proximity of this planetary system, situated at a distance of 16.8 parsecs, HD 48498 emerges as a promising target (closest super-Earth around FGK stars) for future high-contrast direct imaging and high-resolution spectroscopic studies.

Smoothing in linear multicompartment biological processes subject to stochastic input.

Many physical and biological systems rely on the progression of material through multiple independent stages. In viral replication, for example, virions enter a cell to undergo a complex process comprising several disparate stages before the eventual accumulation and release of replicated virions. While such systems may have some control over the internal dynamics that make up this progression, a challenge for many is to regulate behavior under what are often highly variable external environments acting as system inputs. In this work, we study a simple analog of this problem through a linear multicompartment model subject to a stochastic input in the form of a mean-reverting Ornstein-Uhlenbeck process, a type of Gaussian process. By expressing the system as a multidimensional Gaussian process, we derive several closed-form analytical results relating to the covariances and autocorrelations of the system, quantifying the smoothing effect discrete compartments afford multicompartment systems. Semianalytical results demonstrate that feedback and feedforward loops can enhance system robustness, and simulation results probe the intractable problem of the first passage time distribution, which has specific relevance to eventual cell lysis in the viral replication cycle. Finally, we demonstrate that the smoothing seen in the process is a consequence of the discreteness of the system, and does not manifest in systems with continuous transport. While we make progress through analysis of a simple linear problem, many of our insights are applicable more generally, and our work enables future analysis into multicompartment processes subject to stochastic inputs.

Planets observed with CHEOPS

Context. M-dwarf stars are the most common of potential exoplanet host stars in the Galaxy. It is therefore very important to understand planetary systems orbiting such stars and to determine the physical parameters of such planets with high precision. Also with the launch of the James Webb Space Telescope (JWST) the observation of atmospheric parameters of planets orbiting these stars has begun. It is therefore required to determine properties of potential targets. Aims. Two planets around the red dwarf TOI-776 were detected by TESS. The objective of our study was to use transit observations obtained by the CHEOPS space mission to improve the current precision of the planetary radii, as well as additional radial velocity (RV) data in order to improve mass estimates of the two planets. Using these quantities, we wanted to derive the bulk densities of those planets, improving the precision in earlier results, and use this information to put them in context of other exoplanetary systems involving very low mass stars. Methods. Utilizing new transit data from the CHEOPS satellite and its photometric telescope, we obtained very high precision planetary transit measurements. Interpretation of these provides updated planetary radii, along with other system parameters. A concurrent ESO large observing program using the high precision spectrograph HARPS has doubled the available radial velocity data. Calculating the power spectrum of a number of stellar activity indices we update the previously estimated stellar rotation period to a lower value. Results. The CHEOPS data provide precise transit depths of 909 and 1177 ppm translating into radii of Rb = 1.798−0.077+0.078 R⊕ and Rc = 2.047−0.078+0.081 R⊕, respectively. Our interpretation of the radial velocities and activity indicator time series data estimates a stellar rotation period for this early M dwarf of ~21.1 days. A further multi-dimensional Gaussian process approach confirm this new estimate. By performing a Skew-Normal (SN) fit onto the Cross Correlation Functions we extracted the RV data and the activity indicators to estimate the planetary masses, obtaining Mb = 5.0−1.6+1.6 M⊕ and Mc = 6.9−2.5+2.6 M⊕. Conclusions. We improve the precision in planetary radius for TOI-776 b and c by a factor of more than two. Our data and modelling give us parameters of both bodies consistent with mini-Neptunes, albeit with a relatively high density. The stellar activity of TOI-776 is found to have increased by a factor larger than 2 since the last set of observations.

Modelling stellar variability in archival HARPS data: I - Rotation and activity properties with multidimensional Gaussian processes

ABSTRACT Although instruments for measuring the radial velocities (RVs) of stars now routinely reach sub-metre per second accuracy, the detection of low-mass planets is still very challenging. The rotational modulation and evolution of spots and/or faculae can induce variations in the RVs at the level of a few m s–1 in Sun-like stars. To overcome this, a multidimensional Gaussian Process framework has been developed to model the stellar activity signal using spectroscopic activity indicators together with the RVs. A recently published computationally efficient implementation of this framework, S + LEAF 2, enables the rapid analysis of large samples of targets with sizeable data sets. In this work, we apply this framework to HARPS observations of 268 well-observed targets with precisely determined stellar parameters. Our long-term goal is to quantify the effectiveness of this framework to model and mitigate activity signals for stars of different spectral types and activity levels. In this first paper in the series, we initially focus on the activity indicators (S-index and Bisector Inverse Slope), and use them to (a) measure rotation periods for 49 slow rotators in our sample, (b) explore the impact of these results on the spin-down of middle-aged late F, G, and K stars, and (c) explore indirectly how the spot to facular ratio varies across our sample. Our results should provide valuable clues for planning future RV planet surveys such as the Terra Hunting Experiment or the PLATO ground-based follow-up observations programme, and help fine-tune current stellar structure and evolution models.

TOI-733 b: A planet in the small-planet radius valley orbiting a Sun-like star

We report the discovery of a hot (Teq ≈ 1055 K) planet in the small-planet radius valley that transits the Sun-like star TOI-733. It was discovered as part of the KESPRINT follow-up program of TESS planets carried out with the HARPS spectrograph. TESS photometry from sectors 9 and 36 yields an orbital period of ${P_{{\rm{orb}}}} = 4.884765_{ - 2.4e - 5}^{ + 1.9e - 5}$ days and a radius of ${R_{\rm{p}}} = 1.992_{ - 0.090}^{ + 0.085}$ R⊕. Multi-dimensional Gaussian process modelling of the radial velocity measurements from HARPS and activity indicators gives a semi-amplitude of K = 2.23 ± 0.26 m s−1, translating into a planet mass of ${M_{\rm{p}}} = 5.72_{ - 0.68}^{ + 0.70}$ M⊕. These parameters imply that the planet is of moderate density (${\rho _{\rm{p}}} = 3.98_{ - 0.66}^{ + 0.77}$ g cm−3) and place it in the transition region between rocky and volatile-rich planets with H/He-dominated envelopes on the mass-radius diagram. Combining these with stellar parameters and abundances, we calculated planet interior and atmosphere models, which in turn suggest that TOI-733 b has a volatile-enriched, most likely secondary outer envelope, and may represent a highly irradiated ocean world. This is one of only a few such planets around G-type stars that are well characterised.

Revisiting K2-233 spectroscopic time-series with multidimensional Gaussian processes

ABSTRACT Detecting planetary signatures in radial velocity time-series of young stars is challenging due to their inherently strong stellar activity. However, it is possible to learn information about the properties of the stellar signal by using activity indicators measured from the same stellar spectra used to extract radial velocities. In this manuscript, we present a reanalysis of spectroscopic High Accuracy Radial Velocity Planet Searcher data of the young star K2-233, which hosts three transiting planets. We perform a multidimensional Gaussian process regression on the radial velocity and the activity indicators to characterize the planetary Doppler signals. We demonstrate, for the first time on a real data set, that the use of a multidimensional Gaussian process can boost the precision with which we measure the planetary signals compared to a one-dimensional Gaussian process applied to the radial velocities alone. We measure the semi-amplitudes of K2-233 b, c, and d as $1.31_{-0.74}^{+0.81}$ , $1.81_{-0.67}^{+0.71}$ , and $2.72_{-0.70}^{+0.66}$${\rm m\, s^{-1}}$ , which translate into planetary masses of $2.4_{-1.3}^{+1.5}$ , $4.6_{-1.7}^{+1.8}$ , and $10.3_{-2.6}^{+2.4}$ M⊕, respectively. These new mass measurements make K2-233 d a valuable target for transmission spectroscopy observations with JWST. K2-233 is the only young system with two detected inner planets below the radius valley and a third outer planet above it. This makes it an excellent target to perform comparative studies, to inform our theories of planet evolution, formation, migration, and atmospheric evolution.

Modeling of uncertain geometry of cold formed steel members based on laser measurements and machine learning

This research focuses on modeling of as-true geometric uncertainty of cold-formed steel (CFS) members with a multidimensional Gaussian process, the coefficients of which are trained by a maximum-likelihood machine-learning method. The stochasticity and correlations of geometric features, including geometric imperfections, are embedded in the model. An optimization-based feature-recognition algorithm is devised to process general CFS members, by which the processed features are used as the benchmark data, and the entire geometry of a CFS member is reconstructed by the parameterized features. A multidimensional Gaussian process is used to model a joint distribution of feature parameters along the longitudinal direction, where some decorrelations in space and features are assumed to reduce the dimension of the process due to difficulties of acquiring massive samples. The Gaussian process model is trained by geometric features of 112 samples and tested by 48 samples, and results were satisfactory for the proposed method. Finite element analysis with different geometric uncertainty modeling methods applied are conducted and compared. The as-true geometric uncertainty provides more robust and accurate simulation, such as strength and failure modes, as a result. Thus, the proposed method provides a new way in geometric uncertainty modeling for advanced structural analysis and design, which thoroughly considers nonlinear geometric and material conditions.

Image denoising via neighborhood-based multidimensional Gaussian process regression

Image denoising via neighborhood-based multidimensional Gaussian process regression

Fast increased fidelity samplers for approximate Bayesian Gaussian process regression.

Gaussian processes (GPs) are common components in Bayesian non-parametric models having a rich methodological literature and strong theoretical grounding. The use of exact GPs in Bayesian models is limited to problems containing several thousand observations due to their prohibitive computational demands. We develop a posterior sampling algorithm using -matrix approximations that scales at . We show that this approximation's Kullback-Leibler divergence to the true posterior can be made arbitrarily small. Though multidimensional GPs could be used with our algorithm, d-dimensional surfaces are modeled as tensor products of univariate GPs to minimize the cost of matrix construction and maximize computational efficiency. We illustrate the performance of this fast increased fidelity approximate GP, FIFA-GP, using both simulated and non-synthetic data sets.

One year of AU Mic with HARPS – I. Measuring the masses of the two transiting planets

ABSTRACT The system of two transiting Neptune-sized planets around the bright, young M-dwarf AU Mic provides a unique opportunity to test models of planet formation, early evolution, and star–planet interaction. However, the intense magnetic activity of the host star makes measuring the masses of the planets via the radial velocity (RV) method very challenging. We report on a 1-yr, intensive monitoring campaign of the system using 91 observations with the HARPS spectrograph, allowing for detailed modelling of the ∼600 ${\rm m\, s^{-1}}$ peak-to-peak activity-induced RV variations. We used a multidimensional Gaussian Process framework to model these and the planetary signals simultaneously. We detect the latter with semi-amplitudes of Kb = 5.8 ± 2.5 ${\rm m\, s^{-1}}$ and Kc = 8.5 ± 2.5 ${\rm m\, s^{-1}}$, respectively. The resulting mass estimates, Mb = 11.7 ± 5.0 M⊕ and Mc = 22.2 ± 6.7 M⊕, suggest that planet b might be less dense, and planet c considerably denser than previously thought. These results are in tension with the current standard models of core-accretion. They suggest that both planets accreted a H/He envelope that is smaller than expected, and the trend between the two planets’ envelope fractions is the opposite of what is predicted by theory.

Extrema of multi-dimensional Gaussian processes over random intervals

AbstractThis paper studies the joint tail asymptotics of extrema of the multi-dimensional Gaussian process over random intervals defined as $P(u)\;:\!=\; \mathbb{P}\{\cap_{i=1}^n (\sup_{t\in[0,\mathcal{T}_i]} ( X_{i}(t) +c_i t )>a_i u )\}$ , $u\rightarrow\infty$ , where $X_i(t)$ , $t\ge0$ , $i=1,2,\ldots,n$ , are independent centered Gaussian processes with stationary increments, $\boldsymbol{\mathcal{T}}=(\mathcal{T}_1, \ldots, \mathcal{T}_n)$ is a regularly varying random vector with positive components, which is independent of the Gaussian processes, and $c_i\in \mathbb{R}$ , $a_i>0$ , $i=1,2,\ldots,n$ . Our result shows that the structure of the asymptotics of P(u) is determined by the signs of the drifts $c_i$ . We also discuss a relevant multi-dimensional regenerative model and derive the corresponding ruin probability.

An application of Gaussian process modeling for high-order accurate adaptive mesh refinement prolongation

We present a new polynomial-free prolongation scheme for Adaptive Mesh Refinement (AMR) simulations of compressible and incompressible computational fluid dynamics. The new method is constructed using a multi-dimensional kernel-based Gaussian Process (GP) prolongation model. The formulation for this scheme was inspired by the GP methods introduced by A. Reyes et al. (A New Class of High-Order Methods for Fluid Dynamics Simulation using Gaussian Process Modeling, Journal of Scientific Computing, 76 (2017), 443-480; A variable high-order shock-capturing finite difference method with GP-WENO, Journal of Computational Physics, 381 (2019), 189-217). In this paper, we extend the previous GP interpolations and reconstructions to a new GP-based AMR prolongation method that delivers a high-order accurate prolongation of data from coarse to fine grids on AMR grid hierarchies. In compressible flow simulations special care is necessary to handle shocks and discontinuities in a stable manner. To meet this, we utilize the shock handling strategy using the GP-based smoothness indicators developed in the previous GP work by A. Reyes et al. We demonstrate the efficacy of the GP-AMR method in a series of testsuite problems using the AMReX library, in which the GP-AMR method has been implemented.

Detection of the tidal deformation of WASP-103b at 3 σ with CHEOPS

Context. Ultra-short period planets undergo strong tidal interactions with their host star which lead to planet deformation and orbital tidal decay. Aims. WASP-103b is the exoplanet with the highest expected deformation signature in its transit light curve and one of the shortest expected spiral-in times. Measuring the tidal deformation of the planet would allow us to estimate the second degree fluid Love number and gain insight into the planet’s internal structure. Moreover, measuring the tidal decay timescale would allow us to estimate the stellar tidal quality factor, which is key to constraining stellar physics. Methods. We obtained 12 transit light curves of WASP-103b with the CHaracterising ExOplanet Satellite (CHEOPS) to estimate the tidal deformation and tidal decay of this extreme system. We modelled the high-precision CHEOPS transit light curves together with systematic instrumental noise using multi-dimensional Gaussian process regression informed by a set of instrumental parameters. To model the tidal deformation, we used a parametrisation model which allowed us to determine the second degree fluid Love number of the planet. We combined our light curves with previously observed transits of WASP-103b with the Hubble Space Telescope (HST) and Spitzer to increase the signal-to-noise of the light curve and better distinguish the minute signal expected from the planetary deformation. Results. We estimate the radial Love number of WASP-103b to be hf = 1.59−0.53+0.45. This is the first time that the tidal deformation is directly detected (at 3 σ) from the transit light curve of an exoplanet. Combining the transit times derived from CHEOPS, HST, and Spitzer light curves with the other transit times available in the literature, we find no significant orbital period variation for WASP-103b. However, the data show a hint of an orbital period increase instead of a decrease, as is expected for tidal decay. This could be either due to a visual companion star if this star is bound, the Applegate effect, or a statistical artefact. Conclusions. The estimated Love number of WASP-103b is similar to Jupiter’s. This will allow us to constrain the internal structure and composition of WASP-103b, which could provide clues on the inflation of hot Jupiters. Future observations with James Webb Space Telescope can better constrain the radial Love number of WASP-103b due to their high signal-to-noise and the smaller signature of limb darkening in the infrared. A longer time baseline is needed to constrain the tidal decay in this system.

Path Loss Model Based on Machine Learning Using Multi-Dimensional Gaussian Process Regression

For beyond the fifth-generation (5G) and future wireless communications, spatial consistency that represents the correlation between propagation channel characteristics in close proximity has become one of the major issues in channel modeling to describe channels more realistically in emerging scenarios, such as device-to-device (D2D). In this study, we propose a novel path loss model based on multi-dimensional Gaussian process regression (GPR) that provides spatial consistency to channels in propagation environment by predicting local shadow fading while fitting large-scale path loss from measured data. The proposed model has a special structure consisting of a radial mean function and local shadow fading term, which are modeled by independent Gaussian Processes. In contrast to the log-distance path loss model and other regression-based models, the proposed model trains two functions simultaneously; thus, predicts path loss well by capturing both global tendency and local correlation. Moreover, because the proposed model is based on GPR, it provides uncertainty of the predicted path loss. We validated the performance of the proposed model in terms of prediction accuracy with the measurement datasets from two different indoor environments. Our experiments showed that the proposed model predicts better than the log-distance path loss model, especially when spatial correlation becomes significant. The proposed model can be used to simulate path loss in a general environment after training the measurement data.

Quantifying Experimental Edge Plasma Evolution Via Multidimensional Adaptive Gaussian Process Regression

The edge density and temperature of tokamak plasmas are strongly correlated with energy and particle confinement and their quantification is fundamental to understanding edge dynamics. These quantities exhibit behaviours ranging from sharp plasma gradients and fast transient phenomena (e.g. transitions between low and high confinement regimes) to nominal stationary phases. Analysis of experimental edge measurements therefore require robust fitting techniques to capture potentially stiff spatiotemporal evolution. Additionally, fusion plasma diagnostics inevitably involve measurement errors and data analysis requires a statistical framework to accurately quantify uncertainties. This paper outlines a generalized multidimensional adaptive Gaussian process routine capable of automatically handling noisy data and spatiotemporal correlations. We focus on the edge-pedestal region in order to underline advancements in quantifying time-dependent plasma profiles including transport barrier formation on the Alcator C-Mod tokamak.

pyaneti – II. A multidimensional Gaussian process approach to analysing spectroscopic time-series

ABSTRACT The two most successful methods for exoplanet detection rely on the detection of planetary signals in photometric and radial velocity time-series. This depends on numerical techniques that exploit the synergy between data and theory to estimate planetary, orbital, and/or stellar parameters. In this work, we present a new version of the exoplanet modelling code pyaneti. This new release has a special emphasis on the modelling of stellar signals in radial velocity time-series. The code has a built-in multidimensional Gaussian process approach to modelling radial velocity and activity indicator time-series with different underlying covariance functions. This new version of the code also allows multiband and single transit modelling; it runs on Python 3, and features overall improvements in performance. We describe the new implementation and provide tests to validate the new routines that have direct application to exoplanet detection and characterization. We have made the code public and freely available at https://github.com/oscaribv/pyaneti. We also present the codes citlalicue and citlalatonac that allow one to create synthetic photometric and spectroscopic time-series, respectively, with planetary and stellar-like signals.

On asymptotic merging of nodes set for multichannel stochastic networks

In this paper, a multichannel queueing network and the problem of asymptotic merging for its set of nodes are considered. Rate of input flow arriving to the network depends on time. Service times in the network nodes are generally distributed. We introduce a multivariate service process as the number of calls being processed in the nodes. Under heavy traffic conditions, functional limit theorems of Gaussian approximation type are proved for the normalized service process of both original and merged networks. Limit Gaussian processes are constructed, and their correlation characteristics are written via the network parameters. In the case of the merging of the network nodes set, the limit process is a multidimensional diffusion Gaussian process. Moreover, due to the approach with merging, dimension of the limit process is reduced.

Two-Dimensional Sampling-Recovery Algorithm of a Realization of Gaussian Processes on the Input and Output of Linear Systems.

Based on the application of the conditional mean rule, a sampling-recovery algorithm is studied for a Gaussian two-dimensional process. The components of such a process are the input and output processes of an arbitrary linear system, which are characterized by their statistical relationships. Realizations are sampled in both processes, and the number and location of samples in the general case are arbitrary for each component. As a result, general expressions are found that determine the optimal structure of the recovery devices, as well as evaluate the quality of recovery of each component of the two-dimensional process. The main feature of the obtained algorithm is that the realizations of both components or one of them is recovered based on two sets of samples related to the input and output processes. This means that the recovery involves not only its own samples of the restored realization, but also the samples of the realization of another component, statistically related to the first one. This type of general algorithm is characterized by a significantly improved recovery quality, as evidenced by the results of six non-trivial examples with different versions of the algorithms. The research method used and the proposed general algorithm for the reconstruction of multidimensional Gaussian processes have not been discussed in the literature.

Radial velocity confirmation of K2-100b: a young, highly irradiated, and low-density transiting hot Neptune

ABSTRACT We present a detailed analysis of HARPS-N radial velocity observations of K2-100, a young and active star in the Praesepe cluster, which hosts a transiting planet with a period of 1.7 d. We model the activity-induced radial velocity variations of the host star with a multidimensional Gaussian Process framework and detect a planetary signal of 10.6 ± 3.0 ${\rm m\, s^{-1}}$, which matches the transit ephemeris, and translates to a planet mass of 21.8 ± 6.2 M⊕. We perform a suite of validation tests to confirm that our detected signal is genuine. This is the first mass measurement for a transiting planet in a young open cluster. The relatively low density of the planet, $2.04 _{ - 0.61 } ^ { + 0.66 }$ ${\rm g\, cm^{-3}}$, implies that K2-100b retains a significant volatile envelope. We estimate that the planet is losing its atmosphere at a rate of $10^{11}\!-\!10^{12}\, {\rm g\, s^{-1}}$ due to the high level of radiation it receives from its host star.

Bayesian multi-task learning methodology for reconstruction of structural health monitoring data

Reconstruction of structural health monitoring data is a challenging task, since it involves time series data forecasting especially in the case with a large block of missing data. In this study, we propose a novel methodology for structural health monitoring data recovery in the context of Bayesian multi-task learning with multi-dimensional Gaussian process prior. The proposed methodology stands to model a series of tasks simultaneously rather than modeling each task independently while explicitly encoding the correlations among tasks that can be learnt efficiently from data. The primary advantage of Bayesian multi-task learning for data reconstruction is that it makes more efficient use of the data available and gives rise to enhanced reconstruction capability by making use of the underlying task relatedness. Since the modeling performance of the Gaussian process–based Bayesian approach heavily relies on the selected covariance function, particular focus has been laid on the influences of various kinds of covariance functions including the unblended and composite (hybrid) ones on reconstruction performance. The instrumented Canton Tower of 600 m high is used as a test bed to illustrate the effectiveness of the proposed method in reconstruction of structural health monitoring data. The traditional Bayesian single-task learning approach is also implemented for comparison purpose. The reconstruction results of the structural health monitoring data show that the proposed Bayesian multi-task learning methodology affords an excellent performance, while the Bayesian single-task learning method is unreliable in certain cases; yet, the selection of covariance function has a significant impact on the reconstruction performance of the proposed methodology. The work presented in this study also gains insight into how to choose an appropriate covariance function for reconstruction of missing structural health monitoring data.