State filters on Dan MacKinlay
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Recent content in State filters on Dan MacKinlayHugo -- gohugo.ioen-usMon, 01 Mar 2021 17:08:40 +1100Random fields as stochastic differential equations
https://danmackinlay.name/notebook/random_fields_as_sdes.html
Mon, 01 Mar 2021 17:08:40 +1100https://danmackinlay.name/notebook/random_fields_as_sdes.htmlCreating a stationary Markov SDE with desired covariance Convolution representations Covariance representation Input measures \(\mu\) is a hypercube \(\mu\) is the unit sphere \(\mu\) is an isotropic Gaussian Without stationarity via Green’s functions References \(\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\sinc}{\operatorname{sinc}}\)
The representation of certain random fields, especially Gaussian random fields as stochastic differential equations. This is the engine that makes filtering Gaussian processes go, and is also a natural framing for probabilistic spectral analysis.Feynman-Kac formulae
https://danmackinlay.name/notebook/feynman_kac.html
Wed, 27 Jan 2021 11:55:19 +1100https://danmackinlay.name/notebook/feynman_kac.htmlReferences There is a mathematically rich theory about particle filters work. The notoriously abstruse Del Moral (2004); Doucet, Freitas, and Gordon (2001) are universally commended for unifying and making consistent the diffusion processes and Feynman-Kac formulae and “propagation of chaos”. I will get around to them eventually, maybe?
References Cérou, F., P. Del Moral, T. Furon, and A. Guyader. 2011. “Sequential Monte Carlo for Rare Event Estimation.Probabilistic spectral analysis
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Wed, 25 Nov 2020 11:33:34 +1100https://danmackinlay.name/notebook/probabilistic_spectral_analysis.htmlClassic: stochastic processes studied via correlation function Non-stationary spectral kernel Change point detection version Non-Gaussian approaches References Graphical introduction to nonstationary modelling of audio data. The input (bottom) is a sound recording of female speech. We seek to decompose the signal into Gaussian process carrier waveforms (blue block) multiplied by a spectrogram (green block). The spectrogram is learned from the data as a nonnegative matrix of weights times positive modulators (top).Hidden Markov Model inference for Gaussian Process regression
https://danmackinlay.name/notebook/gp_filtering.html
Wed, 25 Nov 2020 11:28:43 +1100https://danmackinlay.name/notebook/gp_filtering.htmlSpatio-temporal usage Miscellaneous notes towards implementation References Classic flavours together, Gaussian processes and state filters/ stochastic differential equations and random fields as stochastic differential equations.
I am interested here in the trick which makes certain Gaussian process regression problems soluble by making them local, i.e. Markov, with respect to some assumed hidden state, in the same way Kalman filtering does Wiener filtering. This means you get to solve a GP as an SDE.Particle filters
https://danmackinlay.name/notebook/particle_filters.html
Wed, 08 Apr 2020 10:50:05 +1000https://danmackinlay.name/notebook/particle_filters.htmlFeynman-Kac formulae Weird evolution equations Miscellaneous practical introductions Tooling References A field of study concerning certain kinds of stochastic processes. The easiest entry point is IMO to think about randomised generalisation of state filter models. This has nothing to to with filters for particulate matter as seen in respirators.
There is too much confusing and unhelpful terminology here, and I am only at the fringe of this field so I will not attempt to typologize.Gaussian processes on lattices
https://danmackinlay.name/notebook/gp_on_lattices.html
Wed, 30 Oct 2019 13:23:08 +1100https://danmackinlay.name/notebook/gp_on_lattices.htmlReferences Gaussian Processes with a stationary kernel are faster if you are working on a grid of points. The main tricks here seem to be circulant embeddings and circulant approximations, which enable one to leverage fast Fourier transforms. This complements, perhaps, the trick of filtering Gaussian processes.
References Chan, G., and A. T. A. Wood. 1999. “Simulation of Stationary Gaussian Vector Fields.” Statistics and Computing 9 (4): 265–68.State filtering parameters
https://danmackinlay.name/notebook/recursive_estimation.html
Tue, 01 Oct 2019 15:33:56 +1000https://danmackinlay.name/notebook/recursive_estimation.htmlClassic recursive estimation Iterated filtering Questions Basic Construction Awaiting filing Implementations References a.k.a. state space model calibration, recursive identification. Sometimes indistinguishable from online estimation.
State filters are cool for estimating time-varying hidden states given known fixed system parameters. How about learning those parameters of the model generating your states? Classic ways that you can do this in dynamical systems include basic linear system identification, and general system identification.Nonparametric state filters via Gaussian Processes
https://danmackinlay.name/notebook/gp_state_filters.html
Wed, 18 Sep 2019 10:21:15 +1000https://danmackinlay.name/notebook/gp_state_filters.htmlReferences Two classic flavours together, Gaussian Processes and state filters. There are other nonparametric state filters, e.g. Variational filters and particle filters.
This is a kind of a dual to using a state filter to calculate a Gaussian process regression as a computational shorthand.
Here we use Gaussian processes to define the filter, in particular to learn nonparametric transition, observation or state densities for a generalized Kalman filter.Variational state filtering
https://danmackinlay.name/notebook/state_filters_variational.html
Fri, 07 Dec 2018 12:39:45 +1100https://danmackinlay.name/notebook/state_filters_variational.htmlReferences A placeholder; State filtering and estimation where the unobserved state and/or process noise are variationally-learned distributions. For now the only version that is even peripherally related to my work is the Gaussian process state filter.
References Archer, Evan, Il Memming Park, Lars Buesing, John Cunningham, and Liam Paninski. 2015. “Black Box Variational Inference for State Space Models.” November 23, 2015. http://arxiv.org/abs/1511.07367. Bayer, Justin, and Christian Osendorfer.State filtering for hidden Markov models
https://danmackinlay.name/notebook/state_filters.html
Thu, 06 Jul 2017 10:57:30 +1000https://danmackinlay.name/notebook/state_filters.htmlLinear systems Non-linear dynamical systems As errors-in-variables models Discrete state Hidden Markov models Variational state filters Kalman filtering Gaussian processes State filter inference References Kalman-Bucy filter and variants, recursive estimation, predictive state models, Data assimilation. A particular sub-field of signal processing for models with hidden state.
In statistics terms, the state filters are a kind of online-updating hierarchical model for sequential observations of a dynamical system where the random state is unobserved, but you can get an optimal estimate of it based on incoming measurements and known parameters.