arXiv subjects

Simo Särkkä

Publications and source records attributed to Simo Särkkä.

At least 91 records · Page 5Linked to original sources

arXiv2020

Taylor Moment Expansion for Continuous-Discrete Gaussian Filtering and Smoothing

The paper is concerned with non-linear Gaussian filtering and smoothing in continuous-discrete state-space models, where the dynamic model is formulated as an Itô stochastic differential equation (SDE), and the measurements are obtained at discrete time instants. We propose novel Taylor moment expansion (TME) Gaussian filter and smoother which approximate the moments of the SDE with a temporal Taylor expansion. Differently from classical linearisation or Itô--Taylor approaches, the Taylor expansion is formed for the moment functions directly and in time variable, not by using a Taylor expansion on the non-linear functions in the model. We analyse the theoretical properties, including the positive definiteness of the covariance estimate and stability of the TME Gaussian filter and smoother. By numerical experiments, we demonstrate that the proposed TME Gaussian filter and smoother significantly outperform the state-of-the-art methods in terms of estimation accuracy and numerical stability.

stat.ME↗

arXiv2019

Automated Polysomnography Analysis for Detection of Non-Apneic and Non-Hypopneic Arousals using Feature Engineering and a Bidirectional LSTM Network

Objective: The aim of this study is to develop an automated classification algorithm for polysomnography (PSG) recordings to detect non-apneic and non-hypopneic arousals. Our particular focus is on detecting the respiratory effort-related arousals (RERAs) which are very subtle respiratory events that do not meet the criteria for apnea or hypopnea, and are more challenging to detect. Methods: The proposed algorithm is based on a bidirectional long short-term memory (BiLSTM) classifier and 465 multi-domain features, extracted from multimodal clinical time series. The features consist of a set of physiology-inspired features (n = 75), obtained by multiple steps of feature selection and expert analysis, and a set of physiology-agnostic features (n = 390), derived from scattering transform. Results: The proposed algorithm is validated on the 2018 PhysioNet challenge dataset. The overall performance in terms of the area under the precision-recall curve (AUPRC) is 0.50 on the hidden test dataset. This result is tied for the second-best score during the follow-up and official phases of the 2018 PhysioNet challenge. Conclusions: The results demonstrate that it is possible to automatically detect subtle non-apneic/non-hypopneic arousal events from PSG recordings. Significance: Automatic detection of subtle respiratory events such as RERAs together with other non-apneic/non-hypopneic arousals will allow detailed annotations of large PSG databases. This contributes to a better retrospective analysis of sleep data, which may also improve the quality of treatment.

eess.SP↗

arXiv2019

Dynamic 3D Tomographic X-ray Data of Ladybug

This is the documentation of the 3D dynamic tomographic X-ray (CT) data of a ladybug. The open data set is available https://zenodo.org/record/3375488#.XV_T3vxS9oA and can be freely used for scientific purposes with appropriate references to the data and to this document in http://arxiv.org/

physics.med-ph↗

arXiv2019

On the positivity and magnitudes of Bayesian quadrature weights

This article reviews and studies the properties of Bayesian quadrature weights, which strongly affect stability and robustness of the quadrature rule. Specifically, we investigate conditions that are needed to guarantee that the weights are positive or to bound their magnitudes. First, it is shown that the weights are positive in the univariate case if the design points locally minimise the posterior integral variance and the covariance kernel is totally positive (e.g., Gaussian and Hardy kernels). This suggests that gradient-based optimisation of design points may be effective in constructing stable and robust Bayesian quadrature rules. Secondly, we show that magnitudes of the weights admit an upper bound in terms of the fill distance and separation radius if the RKHS of the kernel is a Sobolev space (e.g., Matérn kernels), suggesting that quasi-uniform points should be used. A number of numerical examples demonstrate that significant generalisations and improvements appear to be possible, manifesting the need for further research.

math.ST↗

arXiv2019

Iterated Extended Kalman Smoother-based Variable Splitting for $L_1$-Regularized State Estimation

In this paper, we propose a new framework for solving state estimation problems with an additional sparsity-promoting $L_1$-regularizer term. We first formulate such problems as minimization of the sum of linear or nonlinear quadratic error terms and an extra regularizer, and then present novel algorithms which solve the linear and nonlinear cases. The methods are based on a combination of the iterated extended Kalman smoother and variable splitting techniques such as alternating direction method of multipliers (ADMM). We present a general algorithmic framework for variable splitting methods, where the iterative steps involving minimization of the nonlinear quadratic terms can be computed efficiently by iterated smoothing. Due to the use of state estimation algorithms, the proposed framework has a low per-iteration time complexity, which makes it suitable for solving a large-scale or high-dimensional state estimation problem. We also provide convergence results for the proposed algorithms. The experiments show the promising performance and speed-ups provided by the methods.

cs.IT↗

arXiv2019

Hilbert Space Methods for Reduced-Rank Gaussian Process Regression

This paper proposes a novel scheme for reduced-rank Gaussian process regression. The method is based on an approximate series expansion of the covariance function in terms of an eigenfunction expansion of the Laplace operator in a compact subset of $\mathbb{R}^d$. On this approximate eigenbasis the eigenvalues of the covariance function can be expressed as simple functions of the spectral density of the Gaussian process, which allows the GP inference to be solved under a computational cost scaling as $\mathcal{O}(nm^2)$ (initial) and $\mathcal{O}(m^3)$ (hyperparameter learning) with $m$ basis functions and $n$ data points. Furthermore, the basis functions are independent of the parameters of the covariance function, which allows for very fast hyperparameter learning. The approach also allows for rigorous error analysis with Hilbert space theory, and we show that the approximation becomes exact when the size of the compact subset and the number of eigenfunctions go to infinity. We also show that the convergence rate of the truncation error is independent of the input dimensionality provided that the differentiability order of the covariance function is increases appropriately, and for the squared exponential covariance function it is always bounded by ${\sim}1/m$ regardless of the input dimensionality. The expansion generalizes to Hilbert spaces with an inner product which is defined as an integral over a specified input density. The method is compared to previously proposed methods theoretically and through empirical tests with simulated and real data.

stat.ML↗

arXiv2019

The Use of Gaussian Processes in System Identification

Gaussian processes are used in machine learning to learn input-output mappings from observed data. Gaussian process regression is based on imposing a Gaussian process prior on the unknown regressor function and statistically conditioning it on the observed data. In system identification, Gaussian processes are used to form time series prediction models such as non-linear finite-impulse response (NFIR) models as well as non-linear autoregressive (NARX) models. Gaussian process state-space models (GPSS) can be used to learn the dynamic and measurement models for a state-space representation of the input-output data. Temporal and spatio-temporal Gaussian processes can be directly used to form regressor on the data in the time domain. The aim of this article is to briefly outline the main directions in system identification methods using Gaussian processes.

stat.ML↗

arXiv2019

CT Data of a Pen-Spring: Application to Under-Sampled Dynamic X-ray Tomography

This is the documentation of Computed Tomography (CT) data of a pen-spring. The open data set is available at https://zenodo.org/record/3266936#.XRyMdCZS9oA and can be freely used for scientific purposes with appropriate references to the data and to this document in arxiv.org. The provided data set includes the X-ray sinograms ({\tt finalSino}) of a single 2D slice from a different height of the spring. The {\tt finalSino} was obtained from a measured 10-projection or 100-projection {\tt sinogram} using fan-beam geometry by down-sampling and taking logarithms. The data set includes also those original measured {\tt sinogram}s and corresponding measurement matrices.

physics.med-ph↗

arXiv2019

Probabilistic approach to limited-data computed tomography reconstruction

In this work, we consider the inverse problem of reconstructing the internal structure of an object from limited x-ray projections. We use a Gaussian process prior to model the target function and estimate its (hyper)parameters from measured data. In contrast to other established methods, this comes with the advantage of not requiring any manual parameter tuning, which usually arises in classical regularization strategies. Our method uses a basis function expansion technique for the Gaussian process which significantly reduces the computational complexity and avoids the need for numerical integration. The approach also allows for reformulation of come classical regularization methods as Laplacian and Tikhonov regularization as Gaussian process regression, and hence provides an efficient algorithm and principled means for their parameter tuning. Results from simulated and real data indicate that this approach is less sensitive to streak artifacts as compared to the commonly used method of filtered backprojection.

cs.CV↗

arXiv2019

Gaussian kernel quadrature at scaled Gauss-Hermite nodes

This article derives an accurate, explicit, and numerically stable approximation to the kernel quadrature weights in one dimension and on tensor product grids when the kernel and integration measure are Gaussian. The approximation is based on use of scaled Gauss-Hermite nodes and truncation of the Mercer eigendecomposition of the Gaussian kernel. Numerical evidence indicates that both the kernel quadrature and the approximate weights at these nodes are positive. An exponential rate of convergence for functions in the reproducing kernel Hilbert space induced by the Gaussian kernel is proved under an assumption on growth of the sum of absolute values of the approximate weights.

math.NA↗

arXiv2019

Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective

We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement sequence to consist of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP---which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a non-linear Bayesian filtering problem and all widely-used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers that are formulated in terms of generating synthetic measurements of the gradient field come out as specific approximations. Based on the non-linear Bayesian filtering problem posed in this paper, we develop novel Gaussian solvers for which we establish favourable stability properties. Additionally, non-Gaussian approximations to the filtering problem are derived by the particle filter approach. The resulting solvers are compared with other probabilistic solvers in illustrative experiments.

stat.ME↗

arXiv2019

Gaussian process classification using posterior linearisation

This paper proposes a new algorithm for Gaussian process classification based on posterior linearisation (PL). In PL, a Gaussian approximation to the posterior density is obtained iteratively using the best possible linearisation of the conditional mean of the labels and accounting for the linearisation error. PL has some theoretical advantages over expectation propagation (EP): all calculated covariance matrices are positive definite and there is a local convergence theorem. In experimental data, PL has better performance than EP with the noisy threshold likelihood and the parallel implementation of the algorithms.

cs.LG↗

arXiv2019

1D Convolutional Neural Network Models for Sleep Arousal Detection

Sleep arousals transition the depth of sleep to a more superficial stage. The occurrence of such events is often considered as a protective mechanism to alert the body of harmful stimuli. Thus, accurate sleep arousal detection can lead to an enhanced understanding of the underlying causes and influencing the assessment of sleep quality. Previous studies and guidelines have suggested that sleep arousals are linked mainly to abrupt frequency shifts in EEG signals, but the proposed rules are shown to be insufficient for a comprehensive characterization of arousals. This study investigates the application of five recent convolutional neural networks (CNNs) for sleep arousal detection and performs comparative evaluations to determine the best model for this task. The investigated state-of-the-art CNN models have originally been designed for image or speech processing. A detailed set of evaluations is performed on the benchmark dataset provided by PhysioNet/Computing in Cardiology Challenge 2018, and the results show that the best 1D CNN model has achieved an average of 0.31 and 0.84 for the area under the precision-recall and area under the ROC curves, respectively.

eess.SP↗

arXiv2019

Symmetry Exploits for Bayesian Cubature Methods

Bayesian cubature provides a flexible framework for numerical integration, in which a priori knowledge on the integrand can be encoded and exploited. This additional flexibility, compared to many classical cubature methods, comes at a computational cost which is cubic in the number of evaluations of the integrand. It has been recently observed that fully symmetric point sets can be exploited in order to reduce - in some cases substantially - the computational cost of the standard Bayesian cubature method. This work identifies several additional symmetry exploits within the Bayesian cubature framework. In particular, we go beyond earlier work in considering non-symmetric measures and, in addition to the standard Bayesian cubature method, present exploits for the Bayes-Sard cubature method and the multi-output Bayesian cubature method.

stat.ME↗

arXiv2019

Iterative Statistical Linear Regression for Gaussian Smoothing in Continuous-Time Non-linear Stochastic Dynamic Systems

This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary Gaussian process. Two methods are developed based on 1) taking the limit of statistical linear regression of the discretised process and 2) minimising an upper bound to a cost functional. Their difference is manifested in the diffusion of the approximate processes. This in turn gives novel derivations of pre-existing Gaussian smoothers when Method 1 is used and a new class of Gaussian smoothers when Method 2 is used. Furthermore, based on the aforementioned development the iterative Gaussian smoothers in discrete-time are generalised to the continuous-time setting by iteratively re-linearising the stochastic differential equation with respect to the current Gaussian process approximation to the smoothed process. The method is verified in two challenging tracking problems, a reentry problem and a radar tracked coordinated turn model with state dependent diffusion. The results show that the method has better estimation accuracy than state-of-the-art smoothers.

stat.ME↗

arXiv2018

Numerical Integration as a Finite Matrix Approximation to Multiplication Operator

In this article, numerical integration is formulated as evaluation of a matrix function of a matrix that is obtained as a projection of the multiplication operator on a finite-dimensional basis. The idea is to approximate the continuous spectral representation of a multiplication operator on a Hilbert space with a discrete spectral representation of a Hermitian matrix. The Gaussian quadrature is shown to be a special case of the new method. The placement of the nodes of numerical integration and convergence of the new method are studied.

math.NA↗

arXiv2018

Kalman-based Spectro-Temporal ECG Analysis using Deep Convolutional Networks for Atrial Fibrillation Detection

In this article, we propose a novel ECG classification framework for atrial fibrillation (AF) detection using spectro-temporal representation (i.e., time varying spectrum) and deep convolutional networks. In the first step we use a Bayesian spectro-temporal representation based on the estimation of time-varying coefficients of Fourier series using Kalman filter and smoother. Next, we derive an alternative model based on a stochastic oscillator differential equation to accelerate the estimation of the spectro-temporal representation in lengthy signals. Finally, after comparative evaluations of different convolutional architectures, we propose an efficient deep convolutional neural network to classify the 2D spectro-temporal ECG data. The ECG spectro-temporal data are classified into four different classes: AF, non-AF normal rhythm (Normal), non-AF abnormal rhythm (Other), and noisy segments (Noisy). The performance of the proposed methods is evaluated and scored with the PhysioNet/Computing in Cardiology (CinC) 2017 dataset. The experimental results show that the proposed method achieves the overall F1 score of 80.2%, which is in line with the state-of-the-art algorithms.

eess.SP↗

arXiv2018

Gyroscope-Aided Motion Deblurring with Deep Networks

We propose a deblurring method that incorporates gyroscope measurements into a convolutional neural network (CNN). With the help of such measurements, it can handle extremely strong and spatially-variant motion blur. At the same time, the image data is used to overcome the limitations of gyro-based blur estimation. To train our network, we also introduce a novel way of generating realistic training data using the gyroscope. The evaluation shows a clear improvement in visual quality over the state-of-the-art while achieving real-time performance. Furthermore, the method is shown to improve the performance of existing feature detectors and descriptors against the motion blur.

cs.CV↗