March 12, 2022

Follow–up to "Modern algebra and the Poisson point process".

Follow-up to Modern algebra and the Poisson point process from Random Curiosities

Here is a related question. Suppose I show you a network of intersecting lines, and I secretly choose a self-avoiding path on the line network. I tell you the total lengths of the intersections of this self-avoiding path with each line on the network. Can you then reconstruct the path?

For networks of lines in general position, the answer is "yes", using an argument based on the ideas in the "Modern algebra and the Poisson point process" post.

However this is not the case for all networks, and in particular it is not the case for a network determining a regular cartesian grid. Ed Kendall demonstrated this by exhibiting the following counterexample.


Illustration of counterexample


Modern algebra and the Poisson point process

Here is a trivial fact about the Poisson point process, which I find curious because it links directly to a concept from modern algebra (the notion of a free abelian group).

Recall that a Poisson point process on the real line can be viewed as a (locally finite) random pattern of points. We can define it in terms of its consecutive inter-point spacing lengths, Zn for n running through the integers. If Z0 corresponds to the spacing containing the origin, then the Zn are independent, with all but Z0 being of unit Exponential distribution and Z0 being of Exponential distribution of mean 2. (The discrepant behaviour of the Z0 distribution is due to size-biasing, and is associated with the famous Waiting Time Paradox.)

Suppose I secretly select finitely many different spacings, Zi1 , Zi2 , ..., Zik, and tell you only the total length of the spacings I have selected, namely z = Zi1 + Zi2 +...+ Zik.

If you are allowed to inspect closely the relevant realization of the unit-rate Poisson point process, then knowledge of the total length z is all you need to determine which are the spacings i1 < i2 < ... < ik that have been selected.

Indication of proof: the Zi's are independent with Exponential distributions (unit rate except for i=0, for which the rate is 2). But this means that almost surely the spacing lengths Zi generate a free abelian group under addition. In particular, almost surely

(Zi1 + Zi2 + ... + Zik) - (Zj1 + Zj2 + ... + Zjr)

must be non-zero unless the sequences i1 < i2 < ... < ik and j1 < j2 < ... < jr are identical. So (since there are only countably many finite integer-coefficient linear combinations) almost surely z = Zi1 + Zi2 +...+ Zik determines the sequence i1 < i2 < ... < ik.
End of proof
.

Notes:

  1. Free abelian groups are defined in the Encyclopaedia of Mathematics. I learnt about these in my second undergraduate year and then never thought about them again till I noticed this phenomenon just recently.
  2. So is the notion of a Poisson (point) process.
  3. For a child-friendly explanation of the Waiting Time paradox, see Masuda N and Porter MA (2021) The Waiting-Time Paradox. Frontiers for Young Minds 8:582433. DOI: 10.3389/frym.2020.582433.
  4. The result generalizes easily to stationary renewal processes for which the inter-point spacing has a probability density.
  5. I came across this trivial fact while working on generalizing my paper on random lines and metric spaces (Kendall, W. S. (2017). From random lines to metric spaces. Annals of Probability, 45(1), 469–517. https://doi.org/10.1214/14-AOP935); the Poisson process result is noted there (without the modern algebra adornments) in the process of proving that planar line-pattern-based geodesics between prescribed points are almost surely unique ...

September 18, 2020

Magic pencils

Writing about web page https://www.youtube.com/embed/IXNSavpJ3Bw

Statisticians learn to smile politely when people at parties roll out the so-called Mark Twain / Disraeli quote.


But the subject of statistics is like the magic pencils described in this charming story (less than 2.5 minutes); correct use will lead you to The Truth … .

The story is at the beginning of a full lecture on Youtube, and corresponds to Marie-Claire van Leunen's chapter 35 of Knuth, Donald E, Tracy L Larrabee, and Paul M Roberts. Mathematical Writing. Washington, D.C.: Mathematical Association of America, 1989.


July 03, 2020

Stats MSc student publishes in Early Medieval History

Follow-up to Perches, Post–holes and Grids from Random Curiosities

PEMLIt is not unheard-of for Stats MSc students to find their MSc dissertation leading to a publication, but rather less common for the publication to be a component of a book on early medieval history! But that's what happened to Clair Barnes' MSc dissertation ("Statistics in Anglo-Saxon Archaeology", Department of Statistics, Warwick, 2015); you can read all about it in:

Barnes, C., and W.S. Kendall. “Perches, Post-Holes and Grids.” In Planning in the Early Medieval English Landscape, edited by Blair, Rippon & Smart, Liverpool University Press, Appendix A, 213–31, 2020.

Clair started off studying English Literature as an undergraduate at UCL, but then took an OU degree in Math & Stats while working after graduation. That led to a Warwick MSc in Stats and most recently to a return to UCL, working for a PhD in statistical meteorology at UCL. Statistical science leads to all sorts of unexpected adventures ...


March 17, 2020

covid–19_instant_tracing

Writing about web page https://github.com/BDI-pathogens/covid-19_instant_tracing/blob/master/Manuscript%20-%20Modelling%20instantaneous%20digital%20contact%20tracing.pdf

My daughter-in-law Michelle is part of a team that has just published something rather interesting on controlling the epidemic. #VeryProudIndeed


December 17, 2018

Perches, Post–holes and Grids

Writing about web page https://arxiv.org/abs/1704.07342

I and my MSc student of two or three years back, Clair Barnes, produced an appendix, Perches, Post-holes and Grids, for a book being prepared by John Blair et al., arising from the project, Planning in the Early Medieval Landscape. You can find it on arXiv <https://arxiv.org/abs/1704.07342> of course! The appendix is aimed at demonstrating the application of statistical methods to the analysis of archeaological data, typically expressed in graphical form, with the objective of assessing the extent to which the spatial configuration exhibits planning by the original architects. Typical questions: did the builders use a common unit of measurement over a wide geographical region? to what extent is there evidence that they used a grid pattern when designing groups of buildings?

The name of the game is to contribute a statistical assessment to be mixed in with all sorts of other historical evidence. It's fun doing statistics in new areas like this: one learns a lot of stuff one didn't know before, and it provides a brilliant excuse to visit Anglo-Saxon Kingdoms <https://www.bl.uk/events/anglo-saxon-kingdoms> during the working week.


Crowd–sourcing data

Some really good ideas being implemented recently about and around the idea of crowdsourced data, For example:

Of course the cartoonists got there first:

Crowd-sourced data

Noise to Signal: Rob Cottingham
<https://www.robcottingham.ca/cartoon/archive/2007-08-07-crowdsourced/>


Gold Access and so on

Writing about web page https://www.xkcd.com/2085/

Lots gets said about the importance of open-access publishing. Researchers are under pressure to publish papers which are "Gold Access" (translation: they pay the publisher quite a lot of money so that the paper can be accessed freely by all and sundry). Many people discussing this, and/or making policy decisions, appear not to have noticed that in many research fields new work is invariably released as a freely available preprint using the wonderful arXiv <https://arxiv.org/>, for which the publication cost is extremely low (mostly met by academic institutions). For example virtually all of my work of the last 14 years can be found there using <https://arxiv.org/a/kendall_w_1.html>.

The web-comic xkcd makes the point well <https://www.xkcd.com/2085/>.


October 16, 2017

Password strength

Writing about web page https://xkcd.com/936/

Today in the ST116 group we discussed how to build strong passwords. A good exposition can be found in the xkcd cartoon referenced as the weblink for this article.


October 08, 2017

The media has a problem with uncertainty

Writing about web page https://fivethirtyeight.com/features/the-media-has-a-probability-problem/

Not just the media, but it's a fair point. Have a look at what Nate Silver has to say.


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