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Looking Backwards to the Future

Tony Lawrance

Department of Statistics

University of Warwick

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2

First of all, sincere thanks for making this such a great day for me - (provisional remark…)

Especially – John Theodore

and thanks to the Statistics Department for ‘sponsoring’ the event

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Looking backwards to the future – what does it mean ?

An excuse to briefly look back on an enjoyable time in statistics with a wish to also look forward to some more time in statistics… Will try and pin the talk on some significant and not so significant events in my statistics life

Nearly 40 years of statistics before Warwick – so some reminiscing here for the first time here may be acceptable…

In Warwick for just less 10 years – but very enjoyable ones

Most of my publications are now on the site ‘researchgate.net’

Diary of Life Maths undergraduate in Leicester – graduated 1963 ‘Intimidated’ into statistics by Nageeb Rahman, a Cambridge PhD student of Henry Daniels – in that, I am the two-year elder ‘statistical brother’ of Phil Brown

Nageeb sent me in 1963 to Aberystwyth for an MSc (and then Phil Brown in 1965) because Dennis Lindley from Cambridge had started a Stats Department there in 1960, with David Bartholomew, Mervyn Stone and Ann Mitchell (Dennis was in Harvard for half my year, but taught frequentist inference in the second term)

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Carol? DonaldEast SylviaLutkins DavidBartholomew DennisLindley MervynStone AnnMitchell PeterKing Eileen?

GwynJones MikeSamworth PgslyGwynne GrahamPhipp ^ JeffWood ClivePayne ?Bambegye BasilSpringer ErylBasset RichdMorton

Department of Statistics, Aberystwyth 1963-64

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The IBM 1620 Electronic Computer, Aberystwyth Stats Dept 1963

Out of bounds to MSc students

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After MSc - Leicester October 1964 - started as a tutorial assistant 1 year -> assistant lecturer

Frank Downton, d 1986 ? Nageeb Rahman, d 90’s ? Mike Phillips – 1968-… Brian English – 1969-70?

Took 4 ‘summers’ to get a PhD, Stochastic Point Processes’, awarded in 1969. Started by Frank Downton giving me a sheet with a few references … To 7

Lightly supervised by Frank Downton, who almost immediately after my arrival back in Leicester moved to Birmingham, enticed by Henry Daniels

Never-the-less, Frank Downton had big influence encouraging me, research confidence building…

Another big influence in supporting my career was my external examiner David Cox

So this seems a good point to get a bit more technical

Diary of Life

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(back to 6)

PhD and Point Processes… Time series of point events on the line – mainly Poisson and renewal processes at the time – spatial or dependent interval versions had not been much considered

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My first issue was what was meant by a ‘typical event’ to start an interval in a stationary point process ?

I wrote to David Cox – good question, he said ! “We have avoided it in my just completed Methuen monograph with Peter Lewis” on ‘Series of Events’ – 1966

I went for dependent interval versions with stationarity and first studied Cox’s 1954 Biometrika paper on ‘superposition of renewal processes’ or ‘pooled processes’

What was the inter-point distribution and dependency of this process ?

So after a while I investigated two ideas…

time

Process 1

Process 2

Superposition

(I hope my memory is correct !)

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An Average Event – an interval beginning with an ‘average event’ in the stationary PP with intervals

1 2, ,...X X

has distribution 1

1lim n

i i

P X x P X xn n …a bit clunky

An Arbitrary Event - a more elegant approach follows from Khintchine’s (1955) work** on stationary input processes for queues**. This developed from ‘Palm distributions’, referencing Palm (1943) , who introduced the idea of an interval beginning with ‘at least one point’ in a telephone queuing context

( , )N t t Thus, with the counting variable in a stationary point process, the definition of the distribution of an interval beginning with an arbitrary event is

lim ( , ) 0 | (0, ) 1)0P X x P N x N It turned out that this definition mathematically connected the idea of an arbitrary event with that of an arbitrary time, and involved length-biased sampling and forward and backward recurrence times – previously informal concepts for a general stationary point process

My thesis work also contained work on this arbitrary event approach and on particular point processes…

(To 10)

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Khintchine (1894-1959). Mathematical Methods of Queuing, 1955, English Eds, 1960, 1969, Griffin

From the introduction…

(back to 9)

.

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My First Seminar was 25 Feb 1970 at UMIST, Manchester, on ‘selective interaction of point processes’, one of my PhD point processes

My Most Recent Seminar reconstructed part of my first seminar at the Maurice Priestley memorial meeting, 18 December 2013…

The selective interaction model was introduced by the Dutch neurophysiologists Ten Hoopen and Reuver (1965, 1967) to explain multi-modal inter-spike distributions for dark firing of lateral geniculate neurons, observed by Bishop et al (1964)

The process can be explained as follows - you can see that I was rather keen on graphics even in those distant days…

I explored it as an applied probability model. I really wish now that I had followed up on the statistical aspects, contacting the experimenters, analysing their data, attempting to collaborate, etc, and doing simulations – but there was little electronic computing and no internet, and Holland was a long way away

Diary of Life

(from my thesis)

The Selective Interaction Neuron Firing Point Process Model

Excitatory

Inhibitory

Observed Response

The model was justified empirically by a multi-modal distribution of times between the responses’, in the ‘spike trains’ of observed neuron firings – convolutions of excitatory intervals

Poisson excitatory results by very detailed calculation – in my thesis

General results by appealing to the compound distribution structure of the observed response count, resulting in

stnry Interval process

stnry stoc pnt count process

This image cannot currently be displayed.

, ( )i II y

This image cannot currently be displayed.Selective interaction process

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from Priestley meeting talk

( )

, , 1

( ) ( ) , 1 { ( ) 0} 1 { ( ) 0}, 0 IN t

i i R E E I E I E i E i

i

N t N t with prob P N I P N I otherwise

Continued, (J Appl Prob papers 1970-71 &1979)

Excitatory

Inhibitory

Response

It follows

and approximately (?) via compound distribution results

sdevs

Compounding the exciting process intervals using the inhibitory process to get the inter-response distribution is more difficult…but I used arbitrary events

For more detailed results when the excitatory process is Poisson, see my 4 JAP papers in the 70’s. No model fitting, no simulations – what a pity !

stationary interval process

stationary stoc pt count process

( )EN t

, ( )i II y

( )RN tSelective interaction process

0 { ( )} Pr{ ( ) 1} ( )R E I E Iy

E N t N y y dy t

,var{ ( )} [ ( ) var( )]R E I I E IN t E t

( )

, , 1

( ) ( ) , 1 { ( ) 0} 1 { ( ) 0}, 0 IN t

i i R E E I E I E i E i

i

N t N t with prob P N I P N I otherwise

,E I

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1970 - Next move - the year 970/71 at the ‘IBM Thomas J Watson Research Center’, New York, invited by Peter Lewis

Extended and consolidated PhD work by investigating branching Poisson process point models for computer failures, and co-organizing big point process conference

1972 – Returned to Leicester for 1 year – moved to Birmingham for years

1973-2004 My Birmingham Years

1970 – After PhD exam joined David Cox’s weekly PP journal club at IC from Leicester

met Valery Isham, Anthony Atkinson at IC

Diary Life

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Henry Daniels David Wishart Paul Davies Phil Bertram Roger Holder

Frank Downton Malcolm Faddy

Alan Girling John Copas Chris Jones

Richard Atkinson Frank Critchley Prakash Patil

Christmas Meal 1981/82 PhilB? FrankD ? Chris Gray AJL AnnieM ChrisJ TriciaC

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KamilaZ WolfgangB AlanG PrakashP SaidS MalcolmF RichardA

Birmingham Group (when MalcolmF moved back to NZ for second time, 2003)

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1973 – Farewell Point Processes

Found research opportunities in hydrology (from teaching with Nath Kottegoda in Civil Engineering) after devising a course in hydrological time series for Bham MSc in Hydrology

RSS Read Paper on the topic with