spatial on The Dan MacKinlay family of variably-well-considered enterprises
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Recent content in spatial on The Dan MacKinlay family of variably-well-considered enterprisesHugo -- gohugo.ioen-usSat, 25 Jul 2020 10:27:21 +1000Lévy processes
https://danmackinlay.name/notebook/levy_processes.html
Sat, 25 Jul 2020 10:27:21 +1000https://danmackinlay.name/notebook/levy_processes.htmlGeneral form Intensity measure Subordinators Spectrally negative Martingales Sparsity properties Bridge processes Recommended readings \(\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\)
Stochastic processes with i.i.d. increments over disjoint intervals of the same length, i.e. which arise from divisible distributions. Specific examples of interest include Gamma processes, Brownian motions, certain branching processes, non-negative processes…
Let’s start with George Lowther:
Continuous-time stochastic processes with stationary independent increments are known as Lévy processes.Gaussian process regression
https://danmackinlay.name/notebook/gaussian_process_regression.html
Thu, 19 Mar 2020 10:23:56 +0800https://danmackinlay.name/notebook/gaussian_process_regression.htmlQuick intro Density estimation Kernels Using state filtering On lattice observations By variational inference With inducing variables Variational/inducing Approximation with dropout As dimension reduction Readings Implementations Chi Feng’s GP regression demo.
Gaussian processes are stochastic processes/fields with jointly Gaussian distributions of observations. In machine learning these models are used often as a means of regression or classification. They provide nonparametric method of inferring regression functions, with a conveniently Bayesian interpretation and reasonably elegant learning and inference steps.Morphogenesis
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Thu, 05 Mar 2020 14:50:13 +1100https://danmackinlay.name/notebook/morphogenesis.htmlimage from Growing Neural Cellular Automata
On instructing cells to grow into differentiated bodies. This notebook has been resurrected from the trash bin years after I deleted it because of my great enjoyment of Mordvintsev et al. (2020).
Mordvintsev et al. (2020) is a fun paper. They improve upon boring old school cellular automata in several ways (not all of which are completely novel, but are for sure a novel combination.Poisson processes
https://danmackinlay.name/notebook/poisson_processes.html
Wed, 29 Jan 2020 10:56:30 +1100https://danmackinlay.name/notebook/poisson_processes.htmlBasics Poisson distribution Moments Poisson bridge Hitting time Taylor expansion \[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]
Poisson processes are possibly the simplest subordinators, i.e. non-decreasing Lévy processes. They pop up everywhere, especially as a representation of point processes, and as the second continuous-time stochastic process anyone learns after the Brownian motion.
Basics A Poisson process \(\{\mathsf{n}\}\sim \operatorname{PoisP}(\lambda)\) is a stochastic process whose inter-occurrence times are identically and independently distributed such that \(\mathsf{t}_i-\mathsf{t}_{i-1}\sim\operatorname{Exp}(\lambda)\) (rate parameterization).Markov bridge processes
https://danmackinlay.name/notebook/bridge_processes.html
Mon, 20 Jan 2020 16:33:25 +1100https://danmackinlay.name/notebook/bridge_processes.html\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\]
A bridge process for some time-indexed Markov process \(\{\Lambda(t)\}_{t\in[S,T]}\) is obtained from that process by conditioning it to attain a fixed value \(\Lambda(T)=Y\) starting from \(\Lambda(S)=X\) on some interval \([S,T]\). We write that as \(\{\Lambda(t)\mid \Lambda(S)=X,\Lambda(T)=Y\}_{t\in[S,T]}.\) Put another way, given the starting and finishing values of a stochastic markov process, I would like to rewind time to find out the values of its path at a midpoint which is “compatible” with the endpoints.Spatial point process and their statistics
https://danmackinlay.name/notebook/point_processes_spatial.html
Wed, 04 Dec 2019 10:01:59 +1100https://danmackinlay.name/notebook/point_processes_spatial.htmlGibbs point processes Determinantal point processes Cox processes Log Gaussian point process Permanental point processes Spatio-temporal point processes Statistical theory Unconditional intensity estimation via pseudolikelihood Model estimation via GLM Implementations of methods The intersection of point processes and spatial processes. Popular in, e.g, earthquake modelling.
The references here are (Daley and Vere-Jones 2003, 2008; Møller and Waagepetersen 2003), especially the latter.Gaussian processes
https://danmackinlay.name/notebook/gaussian_processes.html
Tue, 03 Dec 2019 10:11:26 +1100https://danmackinlay.name/notebook/gaussian_processes.htmlRelationship between addition of covariance kernels and of processes “Gaussian Processes” are stochastic processes/fields with jointly Gaussian distributions of observations. The most familiar of these to many of us is the Gauss-Markov process, a.k.a. the Wiener process. These processes are convenient due to certain useful properties of the multivariate Gaussian distribution e.g. being uniquely specified by first and second moments, nice behaviour under various linear operations, kernel tricks….Non-Gaussian Bayesian functional regression
https://danmackinlay.name/notebook/stochastic_process_regression.html
Thu, 10 Oct 2019 10:20:07 +1100https://danmackinlay.name/notebook/stochastic_process_regression.htmlRandom fields with non-Gaussian marginals. Generalised Gaussian process regression.
Is there ever an actual need for this? Or can we just use mostly-Gaussian process with some non-Gaussian distribution marginal and pretend? Presumably if we suspect higher moments than the second are important we might bother with this, but oh my there will be some bad scaling and ugly tensor mathematics.Spatial processes and statistics thereof
https://danmackinlay.name/notebook/spatial_statistics.html
Thu, 03 Oct 2019 09:46:31 +1000https://danmackinlay.name/notebook/spatial_statistics.htmlIntros Kriging Spatial point processes Implementations spatstat Pysal PASSaGE Why is there some kind of Catholic edict or something in the background? Image credit Jordan McDowall
Statistics on fields with index sets of more than one dimension of support and, frequently, an implicit 2-norm. Sometimes they are also time-indexed. Especially, for processes on a continuous index set with continuous state and undirected interaction. Sometimes over fancy manifolds, although often you can get away with plain old euclidean space, unless you if you are doing spatial statistics over the entire planet, which turns out to be curved.The interpretation of densities as intensities and vice versa
https://danmackinlay.name/notebook/densities_and_intensities.html
Mon, 23 Sep 2019 16:53:57 +1000https://danmackinlay.name/notebook/densities_and_intensities.htmlBasis function method for density Intensities Basis function method for intensity Count regression Probability over boxes Estimating densities by considering the observations drawn from that as a point process. In one dimension this gives us the particularly lovely trick of survival analysis, but the method is much more general, if not quite as nifty
Consider the problem of estimating the common density \(f(x)dx=dF(x)\) density of indexed i.Point processes
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Mon, 18 Feb 2019 09:47:08 +1100https://danmackinlay.name/notebook/point_processes.htmlTemporal point processes Spatial point processes Another intermittent obsession, tentatively placemarked. Discrete-state random fields/processes with a continuous index. In general I also assume they are non-lattice and simple, which terms I will define if I need them.
The most interesting class for me are the branching processes.
I’ve just spent 6 months thinking about nothing else, so I won’t write much here.
There are comprehensive introductions.Pattern formation
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Tue, 09 Oct 2018 16:44:25 +1100https://danmackinlay.name/notebook/pattern_formation.htmlDiffusion limited aggregation in cheap industrial hand soap
Reaction diffusion equations, and diffusion limited aggregation and other models for pretty blobs and swirls arising from a small number of parameters.
How the leopard got its spots and other special cases in morphogenesis.
To mention:
Reaction diffusion equations hallucinations Turing, the Brusselator etc. Can I file Lichtenberg figures here? Diffusion-limited aggregation? Lapalacian growth? …abolutely any such model involving PDEs?Linear and least-squares estimation of point processes
https://danmackinlay.name/notebook/point_processes_linear_estimation.html
Mon, 29 May 2017 13:23:15 +1000https://danmackinlay.name/notebook/point_processes_linear_estimation.htmlBerman-turner device. K-function. 🏗
Aalen, Odd. 1978. “Nonparametric Inference for a Family of Counting Processes.” The Annals of Statistics 6 (4): 701–26. https://doi.org/10.1214/aos/1176344247.
Aalen, Odd O. 1989. “A Linear Regression Model for the Analysis of Life Times.” Statistics in Medicine 8 (8): 907–25. https://doi.org/10.1002/sim.4780080803.
Adams, Ryan Prescott, Iain Murray, and David J. C. MacKay. 2009. “Tractable Nonparametric Bayesian Inference in Poisson Processes with Gaussian Process Intensities.” In, 1–8.Visualising spatial data
https://danmackinlay.name/notebook/spatial_data_visualisation.html
Mon, 04 Jul 2016 23:29:14 +1000https://danmackinlay.name/notebook/spatial_data_visualisation.html The particular pitfalls of data visualisation for spatial data.
John Krygier Perceptual Scaling of Map Symbols NYT’s Matthew Ericsson Beyond mapping Crowd-sourced science
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Mon, 10 Aug 2015 23:09:43 +0700https://danmackinlay.name/notebook/citizen_science.htmlGame theory of crowdsourced competitions Mapping the world from smartphones. Buzzwords: Citizen science
LIMN magazine on crowds and clouds Game theory of crowdsourced competitions Crowd sourcing competitions encourage malicious behaviour Game theory of crowdsource competition Charness, Gary, and Matthias Sutter. 2012. “Groups Make Better Self-Interested Decisions.” Journal of Economic Perspectives 26 (3): 157–76. https://doi.org/10.1257/jep.26.3.157.
Ghose, Anindya, and Panagiotis G. Ipeirotis. 2011.