branching on The Dan MacKinlay family of variably-well-considered enterprises
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Recent content in branching 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.Contact tracing
https://danmackinlay.name/notebook/contact_tracing.html
Sun, 10 May 2020 07:21:14 +1000https://danmackinlay.name/notebook/contact_tracing.htmlDr. Evans, 1917, How to keep well
Privacy-respecting computing approaches are getting important in this time of epidemics.
A recent round up by Patrick Howell O'Neill, Tate Ryan-Mosley and Bobbie Johnson lists some of the apps in action.
John Langford:
For the following a key distinction to understand is between proximity and location approaches. In proximity approaches (such as DP3T, TCN, MIT PACT(*), Apple or one of the UW PACT(*) protocols which I am involved in) smartphones use Bluetooth low energy and possibly ultrasonics to discover other smartphones nearby.Epidemics
https://danmackinlay.name/notebook/epidemics.html
Fri, 03 Apr 2020 12:19:15 +1100https://danmackinlay.name/notebook/epidemics.htmlModeling Monitoring it Contact tracing Buy this from sam.
A grab-bag of links about disease spread in its messy glory.
Microbescope by David McCandless, Omid Kashan, Miriam Quick, Karl Webster, Dr Stephanie Starling
The spread of diseases in populations. A nitty-gritty messy empirical application for those abstract contagion models.
Connection with global trade networks: Cosma Shalizi on Ebola and Mongol Modernity.Cascade models
https://danmackinlay.name/notebook/cascade_models.html
Mon, 10 Feb 2020 09:28:49 +1100https://danmackinlay.name/notebook/cascade_models.html\(\newcommand{\rv}[1]{\mathsf{#1}}\)
Models for, loosely, the total population size arising from all generations the offspring of some progenitor.
Let us suppose that each individual \(i\) who catches a certain strain of influenza will go on to infect a further \(\rv{n}_i\sim F\) others. Assume the population is infinite, that no one catches influenza twice and that the number of transmission of the disease is distributed the same for everyone who catches it.Branching processes
https://danmackinlay.name/notebook/branching_processes.html
Fri, 07 Feb 2020 17:33:31 +1100https://danmackinlay.name/notebook/branching_processes.htmlTo learn We do not care about time Discrete index, discrete state, Markov: The Galton-Watson process Continuous index, discrete state: the Hawkes Process Continuous index, continuous state Parameter estimation Discrete index, continuous state Special issues for multivariate branching processes Classic data sets Implementations A diverse class of stochastic models that I am mildly obsessed with, where over some index set (usually time, space or both) there are distributed births of some kind, and we count the total population.Subordinators
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Wed, 05 Feb 2020 10:39:02 +1100https://danmackinlay.name/notebook/subordinators.htmlGamma processes Poisson processes Compound Poisson processes with non-negative increments Inverse Gaussian processes Positive linear combinations of other subordinators Generalized Gamma Convolutions via Kendall’s identity \[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\]
A subordinator is an a.s. non-decreasing Lévy process \(\{\rv{g}(t)\}, t \in \mathbb{R}\) with state space \(\mathbb{R}_+\equiv [0,\infty]\) such that
\[ \mathbb{P}(\rv{g}(t)-\rv{g}(s)\lt 0)=0, \,\forall t \geq s. \]
That is, it is an a.Hawkes processes
https://danmackinlay.name/notebook/hawkes_processes.html
Sun, 22 Dec 2019 16:36:44 +1100https://danmackinlay.name/notebook/hawkes_processes.htmlTime-inhomogeneous extension An intersection of point processes and branching processes is the Hawkes process. The classic is the univariate linear Hawkes process. For now we’ll assume it to be indexed by time.
Recall the log likelihood of a generic point process, with occurrence times \(\{t_i\}.\)
\[ \begin{aligned} L_\theta(t_{1:N}) &:= -\int_0^T\lambda^*_\theta(t)dt + \int_0^T\log \lambda^*_\theta(t) dN_t\\ &= -\int_0^T\lambda^*_\theta(t)dt + \sum_{j} \log \lambda^*_\theta(t_j) \end{aligned} \]
\(\lambda^*(t)\) is shorthand for \(\lambda^*(t|\mathcal{F}_t)\), and we call this the intensity.Generalized Galton-Watson processes
https://danmackinlay.name/notebook/discrete_hawkes.html
Fri, 11 Oct 2019 16:28:27 +1100https://danmackinlay.name/notebook/discrete_hawkes.htmlLong Memory Galton-Watson Autoregressive characterisation Estimation of parameters Influence kernels Endo-exo models This needs a better intro, but the Galton-Watson process is the archetype here.
There are many standard expositions. Two good ones:
Gesine Reinert’s Introduction to Branching Processes: Parts 1 and 2.
Steven Lalley’s intro.
Working through some generalisations of the Galton-Watson process as an INAR process. That is, this is something like the Galton-Watson process, butPoint processes
https://danmackinlay.name/notebook/point_processes.html
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.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.Contagion processes and their statistics
https://danmackinlay.name/notebook/contagion_processes.html
Fri, 28 Oct 2016 17:27:52 +1100https://danmackinlay.name/notebook/contagion_processes.htmlDirichlet Hawkes process The spread of quantities of things - earthquakes/diseases/innovations/credit defaults - between different georegions/populations/vertices/banks/variates.
In my own internal taxonomy growth in a single scalar value branching processes. Here I am concerned contagion between different variates. This distinction is arbitrary.
For now this is a mere collection of research links because I am hunting for data sets; no fancy analysis for the moment.
I’ll annotate a couple of useful models here, and hopefully talk about identifiability and noisy/incomplete data issues using a graphical model formalism.Fractional differential equations
https://danmackinlay.name/notebook/fractional_de.html
Thu, 05 May 2016 11:12:06 +1000https://danmackinlay.name/notebook/fractional_de.html“Super diffusive” systems, non-Markov processes… Classically, (stochastic or deterministic) ODEs are “memoryless” in the sense that the current state (and not the history) of the system determines the future states/distribution of states.
One way you can destroy this is by using fractional derivatives in the formulation of the equation. (Why this choice, as opposed to putting in explicit integrals over the history of the process, I have no idea.Earthquakes
https://danmackinlay.name/notebook/earthquakes.html
Tue, 14 Jul 2015 14:44:35 +0200https://danmackinlay.name/notebook/earthquakes.htmlKircher’s model of the seismic systems of the earth
A passing interest of mine, caught from Didier Sornette when we was my supervisor.
I’m mostly interested in the self-exciting process model of Ogata and Ozaki et al, but I’ll also accept notes on human tragedy and normal accidents.
KATHRYN SCHULZ at the New Yorker The Really Big One
To see the full scale of the devastation when that tsunami recedes, you would need to be in the international space station.