point_processes on Dan MacKinlay
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Recent content in point_processes on Dan MacKinlayHugo -- gohugo.ioen-usTue, 16 Mar 2021 10:48:46 +1100Determinantal point processes
https://danmackinlay.name/notebook/point_processes_determinantal.html
Tue, 16 Mar 2021 10:48:46 +1100https://danmackinlay.name/notebook/point_processes_determinantal.htmlReferences Placeholder notes for a type of point process, with which I am unfamiliar, but about which I am incidentally curious.
Wikipedia says:
Let \(\Lambda\) be a locally compact Polish space and \(\mu\) be a Radon measure on \(\Lambda\). Also, consider a measurable function \(K:\Lambda^2\rightarrow \mathbb{C}\).
We say that \(X\) is a determinantal point process on \(\Lambda\) with kernel \(K\) if it is a simple point process on \(\Lambda\) with a joint intensity/Factorial_moment_densityorcorrelation function (which is the density of its factorial moment measure) given byStochastic processes which represent measures over the reals
https://danmackinlay.name/notebook/measure_priors.html
Mon, 08 Mar 2021 16:44:16 +1100https://danmackinlay.name/notebook/measure_priors.htmlSubordinators Other measure priors References Often I need to have a nonparametric representation for a measure over some non-finite index set. We might want to represent a probability, or mass, or a rate. I might want this representation to be something flexible and low-assumption, like a Gaussian process. If I want a nonparametric representation of functions this is not hard; I can simply use a Gaussian process.Convolutional subordinator processes
https://danmackinlay.name/notebook/subordinator_convolution.html
Mon, 08 Mar 2021 15:29:19 +1100https://danmackinlay.name/notebook/subordinator_convolution.htmlReferences Stochastic processes by convolution of noise with smoothing kernels, where the driving noise is a Lévy subordinator.
Why would we want this? One reason is that this gives us a way to create nonparametric distributions over measures.
References Barndorff-Nielsen, O. E., and J. Schmiegel. 2004. “Lévy-Based Spatial-Temporal Modelling, with Applications to Turbulence.” Russian Mathematical Surveys 59 (1): 65. https://doi.org/10.1070/RM2004v059n01ABEH000701. Çinlar, E. 1979. “On Increasing Continuous Processes.Subordinators
https://danmackinlay.name/notebook/subordinators.html
Thu, 08 Oct 2020 15:54:10 +1100https://danmackinlay.name/notebook/subordinators.htmlProperties Gamma processes Poisson processes Compound Poisson processes with non-negative increments Inverse Gaussian processes An increasing linear function is a subordinator Positive linear combinations of other subordinators Subordination of other subordinators Generalized Gamma Convolutions via Kendall’s identity Multivariate Subordinator-valued stochastic process References \[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\]
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 thatExtreme value theory
https://danmackinlay.name/notebook/extreme_value_theory.html
Fri, 25 Sep 2020 16:25:18 +1000https://danmackinlay.name/notebook/extreme_value_theory.htmlGeneralized Pareto Distribution Generalized Extreme Value distributions Burr distribution References In a satisfying way, it turns out that there are only so many shapes that probability densities can assume as they head off toward infinity. Extreme value theory makes this notion precise, and gives us some tools to work with them.
See also densities and intensities, survival analysis.
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Generalized Pareto Distribution Best intro from Hosking and Wallis (1987):Lévy 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 References \(\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\)
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.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 References 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.Survival analysis and reliability
https://danmackinlay.name/notebook/survival_analysis.html
Wed, 05 Feb 2020 14:08:55 +1100https://danmackinlay.name/notebook/survival_analysis.htmlEstimating survival rates Life table method Nelson-Aalen estimates Other reliability stuff tools References Estimating survival rates Here’s the set-up: looking at a data set of individuals’ lifespans you would like to infer the distributions—Analysing when people die, or things break etc. The statistical problem of estimating how long people’s lives are is complicated somewhat by the particular structure of the data — loosely, “every person dies at most one time”, and there are certain characteristic difficulties that arise, such as right-censorship.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 References \[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]
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.htmlReferences \[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\]
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.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 References 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.Permanental point processes
https://danmackinlay.name/notebook/point_processes_permanental.html
Wed, 04 Dec 2019 11:09:37 +1100https://danmackinlay.name/notebook/point_processes_permanental.htmlReferences Placeholder notes for a type of point process, with which I am unfamiliar, but about which I am incidentally curious.
This is, AFAICT, a point process whose intensity is a squared Gaussian process. The term permanental is because the matrix permanent arises somewhere in the model of this process although I know not where. (Walder and Bishop 2017) From some incidental comments at a seminar I presumed the permanental process was actually a Gibbs point process (i.(Discrete-measure)-valued stochastic processes
https://danmackinlay.name/notebook/discrete_measure_valued_processes.html
Thu, 10 Oct 2019 11:05:55 +1100https://danmackinlay.name/notebook/discrete_measure_valued_processes.htmlReferences \(\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\)
Stochastic processes indexed by time whose state is a discrete (possibly only countable) measure. Popular in, for example, mathematical models of alleles in biological evolution.
Population genetics keywords, in approximate order of generality and chronology: Fisher-Wright diffusion, Moran process, Viot-Fleming process. Obviously there are many other processes matching this broad description.
For one example, any element-wise-positive vector-valued stochastic process can be made into a discrete-measure-valued process by normalising the state vector to sum to 1.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 References 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
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 References 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.