1,656 research outputs found
Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond
Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity
Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation
In this paper, we present the optimization formulation of the Kalman
filtering and smoothing problems, and use this perspective to develop a variety
of extensions and applications. We first formulate classic Kalman smoothing as
a least squares problem, highlight special structure, and show that the classic
filtering and smoothing algorithms are equivalent to a particular algorithm for
solving this problem. Once this equivalence is established, we present
extensions of Kalman smoothing to systems with nonlinear process and
measurement models, systems with linear and nonlinear inequality constraints,
systems with outliers in the measurements or sudden changes in the state, and
systems where the sparsity of the state sequence must be accounted for. All
extensions preserve the computational efficiency of the classic algorithms, and
most of the extensions are illustrated with numerical examples, which are part
of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure
A brief history of long memory: Hurst, Mandelbrot and the road to ARFIMA
Long memory plays an important role in many fields by determining the
behaviour and predictability of systems; for instance, climate, hydrology,
finance, networks and DNA sequencing. In particular, it is important to test if
a process is exhibiting long memory since that impacts the accuracy and
confidence with which one may predict future events on the basis of a small
amount of historical data. A major force in the development and study of long
memory was the late Benoit B. Mandelbrot. Here we discuss the original
motivation of the development of long memory and Mandelbrot's influence on this
fascinating field. We will also elucidate the sometimes contrasting approaches
to long memory in different scientific communitiesComment: 40 page
Practical High-Throughput, Non-Adaptive and Noise-Robust SARS-CoV-2 Testing
We propose a compressed sensing-based testing approach with a practical
measurement design and a tuning-free and noise-robust algorithm for detecting
infected persons. Compressed sensing results can be used to provably detect a
small number of infected persons among a possibly large number of people. There
are several advantages of this method compared to classical group testing.
Firstly, it is non-adaptive and thus possibly faster to perform than adaptive
methods which is crucial in exponentially growing pandemic phases. Secondly,
due to nonnegativity of measurements and an appropriate noise model, the
compressed sensing problem can be solved with the non-negative least absolute
deviation regression (NNLAD) algorithm. This convex tuning-free program
requires the same number of tests as current state of the art group testing
methods. Empirically it performs significantly better than theoretically
guaranteed, and thus the high-throughput, reducing the number of tests to a
fraction compared to other methods. Further, numerical evidence suggests that
our method can correct sparsely occurring errors.Comment: 8 Pages, 1 Figur
Statistically Evolving Fuzzy Inference System for Non-Gaussian Noises
Non-Gaussian noises always exist in the nonlinear system, which usually lead to inconsistency and divergence of the regression and identification applications. The conventional evolving fuzzy systems (EFSs) in common sense have succeeded to conquer the uncertainties and external disturbance employing the specific variable structure characteristic. However, non-Gaussian noises would trigger the frequent changes of structure under the transient criteria, which severely degrades performance. Statistical criterion provides an informed choice of the strategies of the structure evolution, utilizing the approximation uncertainty as the observation of model sufficiency. The approximation uncertainty can be always decomposed into model uncertainty term and noise term, and is suitable for the non-Gaussian noise condition, especially relaxing the traditional Gaussian assumption. In this paper, a novel incremental statistical evolving fuzzy inference system (SEFIS) is proposed, which has the capacity of updating the system parameters, and evolving the structure components to integrate new knowledge in the new process characteristic, system behavior, and operating conditions with non-Gaussian noises. The system generates a new rule based on the statistical model sufficiency which gives so insight into whether models are reliable and their approximations can be trusted. The nearest rule presents the inactive rule under the current data stream and further would be deleted without losing any information and accuracy of the subsequent trained models when the model sufficiency is satisfied. In our work, an adaptive maximum correntropy extend Kalman filter (AMCEKF) is derived to update the parameters of the evolving rules to cope with the non-Gaussian noises problems to further improve the robustness of parameter updating process. The parameter updating process shares an estimate of the uncertainty with the criteria of the structure evolving process to make the computation less of a burden dramatically. The simulation studies show that the proposed SEFIS has faster learning speed and is more accurate than the existing evolving fuzzy systems (EFSs) in the case of noise-free and noisy conditions
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