Periodic Covariance Functions Research Articles

From dawn till dusk: Time-adaptive bayesian optimization for neurostimulation.

Stimulation optimization has garnered considerable interest in recent years in order to efficiently parametrize neuromodulation-based therapies. To date, efforts focused on automatically identifying settings from parameter spaces that do not change over time. A limitation of these approaches, however, is that they lack consideration for time dependent factors that may influence therapy outcomes. Disease progression and biological rhythmicity are two sources of variation that may influence optimal stimulation settings over time. To account for this, we present a novel time-varying Bayesian optimization (TV-BayesOpt) for tracking the optimum parameter set for neuromodulation therapy. We evaluate the performance of TV-BayesOpt for tracking gradual and periodic slow variations over time. The algorithm was investigated within the context of a computational model of phase-locked deep brain stimulation for treating oscillopathies representative of common movement disorders such as Parkinson's disease and Essential Tremor. When the optimal stimulation settings changed due to gradual and periodic sources, TV-BayesOpt outperformed standard time-invariant techniques and was able to identify the appropriate stimulation setting. Through incorporation of both a gradual "forgetting" and periodic covariance functions, the algorithm maintained robust performance when a priori knowledge differed from observed variations. This algorithm presents a broad framework that can be leveraged for the treatment of a range of neurological and psychiatric conditions and can be used to track variations in optimal stimulation settings such as amplitude, pulse-width, frequency and phase for invasive and non-invasive neuromodulation strategies.

Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces

Optimal linear prediction (aka. kriging) of a random field {Z(x)}x∈X indexed by a compact metric space (X,dX) can be obtained if the mean value function m:X→R and the covariance function ϱ:X×X→R of Z are known. We consider the problem of predicting the value of Z(x∗) at some location x∗∈X based on observations at locations {xj}j=1n, which accumulate at x∗ as n→∞ (or, more generally, predicting φ(Z) based on {φj(Z)}j=1n for linear functionals φ,φ1,…,φn). Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure (m˜,ϱ˜), without any restrictive assumptions on ϱ, ϱ˜ such as stationarity. We, for the first time, provide necessary and sufficient conditions on (m˜,ϱ˜) for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to φ. These general results are illustrated by weakly stationary random fields on X⊂Rd with Matérn or periodic covariance functions, and on the sphere X= S2 for the case of two isotropic covariance functions.

State of health estimation of lithium-ion batteries based on a novel indirect health indicator

Accurately estimating the state of health (SOH) of lithium-ion batteries is necessary to ensure the battery system’s safe, stable, and efficient operation. It can be directly predicted by capacity, but the latter is difficult to measure online. Therefore, this paper first extracts new indirect health indicators from the voltage and current curves during charging and optimizes them using the Kalman filter. The Pearson correlation analysis method shows that the extracted HIs have a good correlation with the capacity. On this basis, Gaussian process regression (GPR) is modified into multi-kernel Gaussian process regression (MKGPR) by using the squared exponential covariance function and the periodic covariance function to refine the accurateness of SOH estimation. The hyper-parameters of the MKGPR model are solved employing particle swarm optimization (PSO) to reduce the errors caused by the artificial adjustment. Finally, the lithium-ion battery data set provided by NASA is used to evaluate the given method, and the experimental findings reveal that the proposed approach has high accuracy and stability.

Change-point detection in hierarchical circadian models

This paper addresses the problem of change-point detection in sequences of high-dimensional and heterogeneous observations, which also possess a periodic temporal structure. Due to the dimensionality problem, when the time between change points is of the order of the dimension of the model parameters, drifts in the underlying distribution can be misidentified as changes. To overcome this limitation, we assume that the observations lie in a lower-dimensional manifold that admits a latent variable representation. In particular, we propose a hierarchical model that is computationally feasible, widely applicable to heterogeneous data and robust to missing instances. Additionally, the observations’ periodic dependencies are captured by non-stationary periodic covariance functions. The proposed technique is particularly well suited to (and motivated by) the problem of detecting changes in human behavior using smartphones and its application to relapse detection in psychiatric patients. Finally, we validate the technique on synthetic examples and we demonstrate its utility in the detection of behavioral changes using real data acquired by smartphones.

Improved Gaussian Process Regression Inspired by Physical Optics for the Conducting Target's RCS Prediction

In this letter, we propose an improved Gaussian process regression (GPR) to accurately predict the monostatic radar cross section of conducting targets as a function of the incident angle and frequency. Inspired by physical optics, we assume the covariance function as the sum of linear periodic covariance functions. Experiments involving the simulated and measured data are carried out to assess the proposed method. Results show that our method has better prediction performance than GPR with a local periodic covariance function, with a consistent reduction, up to 39% on simulated data and 43% on measured data, of the predictive root mean square error.

Spectral density estimation for random fields via periodic embeddings

We introduce methods for estimating the spectral density of a random field on a [Formula: see text]-dimensional lattice from incomplete gridded data. Data are iteratively imputed onto an expanded lattice according to a model with a periodic covariance function. The imputations are convenient computationally, in that circulant embedding and preconditioned conjugate gradient methods can produce imputations in [Formula: see text] time and [Formula: see text] memory. However, these so-called periodic imputations are motivated mainly by their ability to produce accurate spectral density estimates. In addition, we introduce a parametric filtering method that is designed to reduce periodogram smoothing bias. The paper contains theoretical results on properties of the imputed-data periodogram and numerical and simulation studies comparing the performance of the proposed methods to existing approaches in a number of scenarios. We present an application to a gridded satellite surface temperature dataset with missing values.

Modeling of subway indoor air quality using Gaussian process regression

Soft sensor modeling of indoor air quality (IAQ) in subway stations is essential for public health. Gaussian process regression (GPR), as an efficient nonlinear modeling method, can effectively interpret the complicated features of industrial data by using composite covariance functions derived from base kernels. In this work, an accurate GPR soft sensor with the sum of squared-exponential covariance function and periodic covariance function is proposed to capture the time varying and periodic characteristics in the subway IAQ data. The results demonstrate that the prediction performance of the proposed GPR model is superior to that of the traditional soft sensors consisting of partial least squares, back propagation artificial neural networks, and least squares support vector regression (LSSVR). More specifically, the values of root mean square error, mean absolute percentage error, and coefficient of determination are improved by 12.35%, 9.53%, and 40.05%, respectively, in comparison with LSSVR.

Short-term Wave Forecasting using Gaussian Process for Optimal Control of Wave Energy Converters

Wave forecasting is an important prerequisite for implementing optimal control of wave energy converters since the control of power take-off problem is dependent on a reference strategy that must be calculated from a time-varying wave excitation force value. The real-time computation of the wave force excitation is a considerable challenge, depending on predicted values of wave elevation and short-term frequency. In this study, the performance of a Gaussian Process (GP) procedure for short-term wave forecasting is investigated and compared with Neural Network (NN) and Autoregressive modelling (AR) methods. The GP strategy is not only capable of forecasting the mean value of wave elevations but also provides the uncertainty of forecasting, which is beneficial to the safe operation, robust and optimal control of the wave energy converter devices. According to the non-linear and stochastic nature of wave motion, 3 different covariance functions including spectral mixture (SM), quasi-periodic spectral mixture (QPSM) and Matérn (MA) periodic covariance functions are adopted and analyzed in the work. Results indicate the feasibility of using the GP approach in a real application from the viewpoints of high forecasting accuracy and acceptable computational complexity. The forecasting performance comparisons between the AR and NN models consider two dissimilar real wave validation datasets. The overall outcome is that the GP outperforms its counterparts. However, prediction of the uncertainty is a strong contribution compared with other approaches.

Gaussian Process-Based Predictive Control for Periodic Error Correction

Many controlled systems suffer from unmodeled nonlinear effects that recur periodically over time. Model-free controllers generally cannot compensate these effects, and good physical models for such periodic dynamics are challenging to construct. We investigate nonparametric system identification for periodically recurring nonlinear effects. Within a Gaussian process (GP) regression framework, we use a locally periodic covariance function to shape the hypothesis space, which allows for a structured extrapolation that is not possible with more widely used covariance functions. We show that hyperparameter estimation can be performed online using the maximum a posteriori point estimate, which provides an accuracy comparable with sampling methods as soon as enough data to cover the periodic structure has been collected. It is also shown how the periodic structure can be exploited in the hyperparameter optimization. The predictions obtained from the GP model are then used in a model predictive control framework to correct the external effect. The availability of good continuous predictions allows control at a higher rate than that of the measurements. We show that the proposed approach is particularly beneficial for sampling times that are smaller than, but of the same order of magnitude as, the period length of the external effect. In experiments on a physical system, an electrically actuated telescope mount, this approach achieves a reduction of about 20% in root mean square tracking error.

Estimation for a class of nonstationary processes

Random processes with almost periodic covariance function are considered from a spectral outlook. Given suitable conditions, spectral estimation problems are discussed for Gaussian processes of this type that are neither stationary nor locally stationary. Spectral mass is concentrated on lines parallel to the main diagonal in the spectral plane. A method of estimation of the support of spectral mass under appropriate restraints is considered. Some open questions are discussed. Extension of the methods for a class of nonGaussian nonstationary processes with mean value function a trigonometric regression is given. Consistent estimates for frequency, amplitude and phase of the regression are noted when the residual process is zero mean almost periodic. The resulting estimation of the spectral mass of the residual is also considered.

Estimation for almost periodic processes

Processes with almost periodic covariance functions have spectral mass on lines parallel to the diagonal in the two-dimensional spectral plane. Methods have been given for estimation of spectral mass on the lines of spectral concentration if the locations of the lines are known. Here methods for estimating the intercepts of the lines of spectral concentration in the Gaussian case are given under appropriate conditions. The methods determine rates of convergence sufficiently fast as the sample size n→∞ so that the spectral estimation on the estimated lines can then proceed effectively. This task involves bounding the maximum of an interesting class of non-Gaussian possibly nonstationary processes.