The kernel Kalman rule

Gebhardt, Gregor H. W.; Kupcsik, Andras; Neumann, Gerhard

doi:10.1007/s10994-019-05816-z

Cited by 9 publications

(21 citation statements)

References 16 publications

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“…Then, the posterior pdf is approximated by the KBR [21]. A variant of the KBR called the kernel Kalman rule (KKR) was formulated in [23] to overcome some of the instabilities that can be observed in using the KBR. These KME-based methods can effectively deal with problems that involve unknown measurement models or strong non-linear structures [6].…”

Section: A State Of the Art -Non-linear Filtersmentioning

confidence: 99%

“…Inspired by the KBR [21] and KKR [23], we introduce a full model based Bayesian filter called the adaptive kernel Kalman filter (AKKF). The main contributions of this paper can be summarized as:…”

Section: B Novelties and Contributionsmentioning

confidence: 99%

“…Here the weighted kernel mean vector and weighted kernel covariance matrix are predicted and updated using the KKR, i.e. the KKR is used to realize an unbiased update of the KME [23]. 3) Unlike most of the kernel based methods where the focus is on characteristic kernels [21], [23], we primarily focus on simple quadratic and quartic kernels which provide direct access to the mean and covariance of the hidden state.…”

Section: B Novelties and Contributionsmentioning

confidence: 99%

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Adaptive Kernel Kalman Filter

Sun¹,

Davies²,

Proudler³

et al. 2022

Preprint

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Sequential Bayesian filters in non-linear dynamic systems require the recursive estimation of the predictive and posterior distributions. This paper introduces a Bayesian filter called the adaptive kernel Kalman filter (AKKF). With this filter, the arbitrary predictive and posterior distributions of hidden states are approximated using the empirical kernel mean embeddings (KMEs) in reproducing kernel Hilbert spaces (RKHSs). In parallel with the KMEs, some particles, in the data space, are used to capture the properties of the dynamical system model. Specifically, particles are generated and updated in the data space, while the corresponding kernel weight mean vector and covariance matrix associated with the feature mappings of the particles are predicted and updated in the RKHSs based on the kernel Kalman rule (KKR). Simulation results are presented to confirm the improved performance of our approach with significantly reduced particle numbers, by comparing with the unscented Kalman filter (UKF), particle filter (PF) and Gaussian particle filter (GPF). For example, compared with the GPF, the proposed approach provides around 5% logarithmic mean square error (LMSE) tracking performance improvement in the bearing-only tracking (BOT) system when using 50 particles.

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Section: A State Of the Art -Non-linear Filtersmentioning

confidence: 99%

Section: B Novelties and Contributionsmentioning

confidence: 99%

Section: B Novelties and Contributionsmentioning

confidence: 99%

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Adaptive Kernel Kalman Filter

Sun¹,

Davies²,

Proudler³

et al. 2022

Preprint

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“…Applications of the conditional mean embedding in the context of sequential data include, among others, state-space models and filtering (Song et al, 2009;Fukumizu et al, 2013;Gebhardt et al, 2019), the embedding of transition probability models (Song et al, 2010;Grünewälder et al, 2012b;Nishiyama et al, 2012;Sun et al, 2019), predictive state representations (Boots et al, 2013), and reinforcement learning models (van Hoof et al, 2015(van Hoof et al, , 2017Stafford and Shawe-Taylor, 2018;Gebhardt et al, 2018).…”

Section: Conditional Mean Embedding Of Stationary Time Seriesmentioning

confidence: 99%

“…Recently, several applications for dependent data, sequence modeling, and time series analysis based on kernel mean embeddings have emerged. Popular approaches include state space models (Song et al, 2009), filtering (Fukumizu et al, 2013;Gebhardt et al, 2019), transition models (Sun et al, 2019;Grünewälder et al, 2012b) and reinforcement learning (van Hoof et al, 2015;Lever et al, 2016;van Hoof et al, 2017;Stafford and Shawe-Taylor, 2018;Gebhardt et al, 2018). A theoretical tool to understand these concepts is the kernel autocovariance operator, as its plays a role in the RKHS-based approximation of transition probabilities.…”

Section: Introductionmentioning

confidence: 99%

Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence

Mollenhauer¹,

Klus²,

Schütte³

et al. 2020

Preprint

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We consider autocovariance operators of a stationary stochastic process on a Polish space that is embedded into a reproducing kernel Hilbert space. We investigate how empirical estimates of these operators converge along realizations of the process under various conditions. In particular, we examine ergodic and strongly mixing processes and prove several asymptotic results as well as finite sample error bounds with a detailed analysis for the Gaussian kernel. We provide applications of our theory in terms of consistency results for kernel PCA with dependent data and the conditional mean embedding of transition probabilities. Finally, we use our approach to examine the nonparametric estimation of Markov transition operators and highlight how our theory can give a consistency analysis for a large family of spectral analysis methods including kernel-based dynamic mode decomposition.

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Scalable inference of cell differentiation networks in gene therapy clonal tracking studies of haematopoiesis

Del Core,

Pellin,

Wit

et al. 2023

Bioinformatics

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Motivation Investigating cell differentiation under a genetic disorder offers the potential for improving current gene therapy strategies. Clonal tracking provides a basis for mathematical modelling of population stem cell dynamics that sustain the blood cell formation, a process known as haematopoiesis. However, many clonal tracking protocols rely on a subset of cell types for the characterisation of the stem cell output, and the data generated are subject to measurement errors and noise. Results We propose a stochastic framework to infer dynamic models of cell differentiation from clonal tracking data. A state-space formulation combines a stochastic quasi-reaction network, describing cell differentiation, with a Gaussian measurement model accounting for data errors and noise. We developed an inference algorithm based on an extended Kalman filter, a nonlinear optimization, and a Rauch-Tung-Striebel smoother. Simulations show that our proposed method outperforms the state-of-the-art and scales to complex structures of cell differentiations in terms of nodes size and network depth. The application of our method to five in-vivo gene therapy studies reveals different dynamics of cell differentiation. Our tool can provide statistical support to biologists and clinicians to better understand cell differentiation and haematopoietic reconstitution after a gene therapy treatment. The equations of the state-space model can be modified to infer other dynamics besides cell differentiation. Availability The stochastic framework is implemented in the R package Karen which is available for download at https://cran.r-project.org/package=Karen. The code that supports the findings of this study is openly available at https://github.com/delcore-luca/CellDifferentiationNetworks. Supplementary information Supplementary data are available at Bioinformatics online.

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The kernel Kalman rule

Cited by 9 publications

References 16 publications

Adaptive Kernel Kalman Filter

Adaptive Kernel Kalman Filter

Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence

Scalable inference of cell differentiation networks in gene therapy clonal tracking studies of haematopoiesis

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