Summary of research activities

My main interests lie in Applied Stochastic Modelling, within which broad field my work has been in four main areas.
Undoubtedly, the work that stands out as creating a new field of study, being the seminal work within that field, is the work with David Colquhoun FRS (Professor of Pharmacology at University College London), on the modelling of biological ion channels. I consider it to be a major achievement that will continue to be important for many years to come.
Earlier work in point process theory saw the introduction of a class of processes, which have become a standard feature of that branch of study and are known in the literature as ‘Hawkes Processes’. While these form a small part of a broad subject, they have some lasting significance theoretically and have continuing applications in various areas, including earthquake modelling and stock market transactions.
I have worked from time to time on problems in reliability theory, particularly the modelling of repairable systems. Currently work is going on, with colleagues in Beijing, that involves applying ion channel type models to systems reliability.
Early research was mainly in queueing theory, particularly applied to road traffic situations. The work done there used techniques that were later to be found useful in ion channel modelling and in systems reliability.
The above describes what I regard as my major interests and achievements.  There is the usual miscellaneous random assortment of bits and pieces that statisticians tend to collect. Applied statistical interest is mainly in medical, health care, engineering and general scientific problems.  Some of these are reflected in publications, but mostly it is consulting work; sometimes paid, more often unpaid statistical advice to research-minded local doctors, health workers and research students. There are also minor contributions to the literature of statistical computing using the APL language.

Ion channels
Working in conjunction with David Colquhoun FRS (Professor of Pharmacology at University College London), joined in recent years by Assad Jalali and Anton Merlushkin in Swansea, I have made a major contribution to the stochastic modelling of the activity of ion channels in biological membranes.  This is a very satisfying area in which some elegant mathematics sheds considerable light on physical mechanisms of fundamental importance to everyday life: ion channels are an essential link in the communication system carrying messages round the body, and are important sites of drug action.
In 1977 the classic paper [IC1] on macroscopic properties of ion currents (the sum of currents through a few thousand channels) described the relaxation following perturbations and also the noise properties which were at that time experimentally measurable (particularly the pioneering work of Sir Bernard Katz and co-workers).  At about this time Neher and Sakmann were developing the techniques necessary to enable measurement of currents through single ion channels, and which led to their Nobel prize in 1991.  In [IC2] and particularly [IC3], a classic foundation paper of single channel theory, we developed the theory needed to interpret observed single channel recordings. 
There have since grown up specialist groups around the World interested in the subject, which is now a recognised branch of stochastic modelling.  Notable contributors include Frank Ball (Nottingham), Geoff Yeo and Robin Milne (Western Australia), John Rice (Berkeley), Donald Fredkin (San Diego), Fred Sigworth (Yale) and Tony Auerbach (Buffalo). 
More importantly, the subject is of practical interest to Pharmacologists, Physiologists and Biophysicists.  In laboratories all over the world, they make daily use of the methods of analysis that follow from the theory.
A particularly significant achievement was to solve the problem of Time Interval Omission (TIO) which allows for the inability of the recording system to detect very short openings and shuttings of the channel.  The algebraic form and very accurate asymptotic approximations were obtained for the relevant distributions in a series of papers [IC8-11}: the Laplace transform had been published by Ball and Sansom.

[IC1]  D. Colquhoun & A.G. Hawkes,  Relaxation and fluctuations of currents that flow through drug operated ion channels.  Proc. R. Soc. Lond., B 199, 1977, 231-262.
[IC2]  D. Colquhoun & A.G.  Hawkes,  On the stochastic properties of single ion channels. Proc. R. Soc. Lond., B 211, 1981, 205-235.
[IC3]  D. Colquhoun & A.G. Hawkes,  On the stochastic properties of bursts of single ion channel openings and of clusters of bursts.  Phil. Trans. R. Soc. Lond., B 300, 1982, 1-59.
[IC4]  D. Colquhoun & A.G. Hawkes,  The principles of the stochastic interpretation of ion channel mechanisms.  Chapter 9 of Single channel recording, eds. B. Sakmann and E. Neher, Plenum, 1983.
[IC5]  D. Colquhoun & A.G. Hawkes,  A note on correlations in single ion channel records. Proc. R. Soc. Lond., B 230, 1987, 15-52.
[IC6]  Y. Ebina, M. Mukuno, R. Shingai, K. Nakajima & A.G. Hawkes,  Power spectrum density equation of fluctuating membrane current based on discrete time Markov chain model - Analysis of ion channels with 2, 3 states, (in Japanese). Trans. Inst. Electronics, Information & Communication Eng., J72-D-II no.11, 1989, 1926-1934.
[IC7]  D. Colquhoun & A.G. Hawkes,  Stochastic properties of ion channel openings and bursts in a membrane patch that contains two channels: evidence concerning the number of channels present when a record containing only single openings is observed. Proc R. Soc. Lond., B 240, 1990, 453-477.
[IC8]  A.G. Hawkes, A. Jalali & D. Colquhoun,  The distributions of the apparent open times and shut times in a single ion channel when brief events can not be detected. Phil. Trans. R. Soc. Lond., A 332, 1990, 511-538.
[IC9]  A. Jalali & A.G. Hawkes,  The distribution of apparent occupancy times in a two-state Markov Process in which brief events cannot be detected.  Advances in Applied Probability, 24, 1992, 288-301.
[IC10] A. Jalali & A.G. Hawkes,  Generalised eigenproblems arising in aggregated Markov processes allowing for time interval omission.  Advances in Applied Probability, 24, 1992, 302-321.
[IC11] A.G. Hawkes, A. Jalali & D. Colquhoun,  Asymptotic distributions of apparent open times and shut times in a single channel record allowing for the omission of brief events.  Phil. Trans. R. Soc. Lond., B 337, 1992, 383-404.
[IC12] D. Colquhoun & A.G. Hawkes,  The interpretation of single channel recordings.  Microelectrode techniques: the Plymouth workshop handbook, D.C. Ogden (ed.), 1994, The Company of Biologists, Cambridge, 141-188.
[IC13] D. Colquhoun & A.G. Hawkes,  The principles of the stochastic interpretation of ion-channel mechanism.  Chapter in Single-Channel Recording 2nd. ed., B. Sakmann and E. Neher (eds.), 1995, Plenum Press, New  York, 397-482.
[IC14] D. Colquhoun & A.G. Hawkes,  A Q-matrix cookbook.  Chapter in Single-Channel Recording 2nd. ed., B. Sakmann and E. Neher (eds.), 1995, Plenum Press, New  York, 589-633.
[IC15] D. Colquhoun & A.G. Hawkes,  Desensitisation of N-methyl-D-aspartate receptors: a problem of interpretation.  Proc. Natl. Acad. Sci. USA, 92, 1995, 10327-10329.
[IC16]  D. Colquhoun, A.G. Hawkes & K. Srodzinski, Joint distributions of  apparent open times and shut times of single ion channels and the maximum likelihood fitting of mechanisms.  Phil. Trans. R. Soc. Lond., A 354, 1996, 2555-2590.
[IC17]  A. Merlushkin & A.G. Hawkes, Stochastic behaviour of ion channels in varying conditions.  IMA J. of Mathematics Applied in Medicine and Biology, 1996, 14, 1-26.
[IC18]  D. Colquhoun, A.G. Hawkes, A. Merlushkin & B. Edmonds,  Properties of single ion channel currents elicited by a pulse of agonist concentration or voltage. Phil. Trans. R. Soc. Lond., A, 1997, .
[IC19] D. Colquhoun,  C.J. Hatton and A.G. Hawkes, The quality of maximum likelihood estimation of ion channel rate constants. J. Physiol. Lond. 547, 2003, 699-728.
[IC20] A.G. Hawkes, Stochastic modelling of single ion channels, Computational Neuroscience: a comprehensive approach (Jianfeng Feng ed.), Chapman & Hall/CRC, 2004, 131-157.
[IC21] J.J. Celentano and A.G. Hawkes, Use of the covariance matrix in directly fitting kinetic parameters: application to GABAA receptors. Biophys. J. vol 87, July  2004,  276-94.
[IC22] A.G. Hawkes, Ion Channel Modeling, in Encyclopedia of Biostatistic 2nd Ed, Armitage, P. and Colton, T., (eds), vol 4, pp2625–2632,  Wiley, Chichester,
2005
Working papers:
[IC23] A. Merlushkin, A. Jalali & A.G. Hawkes, A stochastic model of a multilevel ion channel.  Working paper EBMS/1995/10, 1995.
[IC24] A. Merlushkin & A.G. Hawkes, Stochastic behaviour of an apparent aggregated Markov process following a jump in the generator of the underlying Markov process. Working paper EBMS/1995/12, 1995.
[IC25] A. Merlushkin & A.G. Hawkes, The effect of time interval omission on the apparent burst kinetics of ion channels.  Working paper EBMS/1996/1, 1996.

Point processes
Work on Point process theory was mainly done in the seventies.  Paper [P1] was motivated by the appearance of miniature end-plate potentials.  The problem involved gaps, and a technique used in this paper later found much use in the time interval omission problem in modelling ion channels.
In a series of papers [P2-4, P6] the ideas of self-exciting (and mutually exciting) point processes were developed, and have entered standard textbooks on point processes under the name of ‘Hawkes Processes’.  They were applied, along with other models, in a paper on earthquakes [P5] and continue to be used by others in the earthquake literature.  Recently they have been applied to stock market transactions. Paper [P5} also introduced the idea of spectral likelihood as a means of estimating parameters of a point process.  A 1994 PhD thesis at UMIST looked at this method in depth and showed it to be quite efficient as well as relatively simple to use. 
Further practical applications of point processes [P7-8], occurring in acoustic emissions from materials under stress (micro-earthquakes?), were made in the eighties.

[P1]  A.G. Hawkes,  Bunching in a semi-Markov process.  J. Appl. Prob., 7, 1970, 175-182.
[P2]  A.G. Hawkes,  Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58, 1971, 83-90.
[P3]  A.G. Hawkes,  Point spectra of some mutually exciting point processes.  J. Roy. Statist. Soc., B 33, 1971, 438-443.
[P4] A.G. Hawkes,  Spectra of some mutually exciting point processes with associated variables.   Stochastic Point Processes, P.A.W. Lewis (ed.), 1972, Wiley, New York, 261-271.
[P5] A.G. Hawkes & L. Adamopoulos,  Cluster models for earthquakes - regional comparisons.  Invited paper at the ISI conference, Vienna, 1973.
[P6] A.G. Hawkes & D. Oakes,  A cluster process representation of a self-exciting process. J. Appl. Prob., 11, 1974, 493-503.
[P7]  R.M. Bellchamber, D. Betteridge, T. Chow, A.G. Hawkes, M.E.A. Cudby & D.G.M. Wood, A study of acoustic emissions from stressed polypropelene-glass fibre compounds.  J. Composite Materials, 17, 1983, 420-434.
[P8]  R.M. Bellchamber, D. Betteridge, M.P. Collins, T. Lilley, C.Z. Marczowski & A.G. Hawkes, Time series analysis of acoustic emission signals from glass reinforced plastics.  Acoustic emission monitoring and analysis in manufacturing (ed. D.A. Cornfield), ASME, New  York, 1984.

Reliability
Some work has been done on stochastic modelling, particularly concerning repairable systems [R2-R5], using techniques familiar from queueing theory. In so far as these models can be seen as collections of states which can be classified as either ‘system up’ or ‘system down’, they have some relation to ion channels, which have states classified as ‘open’ or ‘shut’.  Currently work is going on, with colleagues in Beijing, that involves applying ion channel type models to systems reliability, [R9, R14]. Some combinatorial arguments were used in looking at optimum properties in non-repairable systems, [R6], R[8] and [R11].

[R1] A.G. Hawkes,  A bivariate exponential distribution with applications to reliability.  J. Roy. Statist. Soc., B 34, 1972, 129-131.
[R2]  B.B. Fawzi & A.G. Hawkes,  Availability of Markov systems with spares and repair facilities.  Proc. Reliability '89, Brighton, 1989, 2B/4/1-19.
[R3]  B.B. Fawzi & A.G. Hawkes,  Availability of a series system with replacement and repair.  J. Appl. Prob., 27, 1990, 873-887.
[R4]  B.B. Fawzi & A.G. Hawkes,  Availability of an R-out-of-N system with spares and repairs.  J. Appl. Prob., 28, 1991, 397-408.
[R5]  L. Cui & A.G. Hawkes,  Availability of a series system  with warm spares. Microelectronics and Reliability, 34 no. 6, 1994, 1057-1069.
[R6]  L. Cui, A.G. Hawkes & A. Jalali,  The increasing failure rate property of consecutive-k-out-of-n systems. Probability in the Engineering and Informational Sciences, 9, 1995, 217-225.
[R7]  A.G. Hawkes and A. Jalali,  Analysis of censored lives and parameter estimation.  Building a better mousetrap: papers in honour of Don Leech, C. Jones and A. Watkins (eds.), 1995, Eurotrans Verlag,103-122.
[R8] A. Jalali, A.G. Hawkes, L.R.Cui, F.K. Hwang The optimal consecutive-k-out-of-n:G line for n ¿ 2k. J. Statistical Planning and Inference, 128, 2005, 281-287.
[R9] Z. Zheng, L. Cui and AG. Hawkes,  A study on a single-unit Markov repairable system with repair time omission, IEEE Trans. Reliability, Vol. 55 no. 2 June 2006, 182-188.
[R10] AG. Hawkes  and A. Jalali, Parameter estimation and censored lives, Communications in Statistics – Theory and Methods, 35, 2006, 1791-1802.
[R11] L. Cui & Alan G. Hawkes, A Note on the Proof for the Optimal Consecutive-k-out-of-n:G Line for n<=2k,  Journal of Statistical Planning and Inference, 2007, vol. 138 no.5, May 2008, p1516-1520 (SCI). {can be viewed online at the journal’s website before publication date}
Conference proceedings:
[R12] AG. Hawkes  and A. Jalali, Parameter estimation and censored lives, Proc .ICQR2005 Beijing Aug. 2005 (L. Cui, A, Tsang  & M.Xie eds.), Beijing Institute of Technology Press, 825-841.
[R13] Z. Zheng, L. Cui and AG. Hawkes,  A study on a single-unit Markov repairable system with repair time omission, Proc .ICQR2005 Beijing Aug. 2005 (L. Cui, A, Tsang  & M.Xie eds.), Beijing Institute of Technology Press, 907-912.
[R14] Zhihua Zheng, Lirong Cui and Alan G. Hawkes, A further study on a single-unit repairable system, International Conference on Reliability Maintainability and Safety, Beijing, China, Aug. 22-26, 2007, p151-155.

Queueing theory
Research was done on queueing theory in the sixties, mainly in connection with road traffic, mostly in the course of my PhD studies under Maurice Bartlett.  It turns out that the work on gap acceptance, [Q2] and [Q5], is closely connected with the time interval omission problem in modelling ion channels which was done much later. In a more general sense, some of the ideas and techniques in queueing theory are also used in reliability of repairable systems.
[Q1]   A.G. Hawkes,  Queueing at traffic intersections.  Proc. 2nd Int. Symp. Traffic Flow, London, OECD, 1963, 190-199.
[Q2]  A.G. Hawkes,  Queueing for gaps in traffic.  Biometrika, 52, 1965, 79-85.
[Q3]  A.G.Hawkes,  Time-dependent solution of a priority queue with bulk arrival.  Opns. Res., 13, 1965, 586-595.
[Q4] A.G. Hawkes, Stochastic problems in traffic flow, Ph.D. Thesis, London University, 1965.
[Q5]  A.G. Hawkes,  Delay at traffic intersections.  J. Roy. Statist. Soc., B 28, 1966, 202-212.
[Q6]  A.G. Hawkes,  Gap acceptance in road traffic.  J. Appl. Prob., 5, 1968, 84-92.

APL
The Statistics Group at Swansea has made a contribution to the development of the use of the computer language APL in statistical research and teaching. The use of APL has significantly aided my main research activity although my contribution to the APL literature, per se, is modest.
[APL1] A.G. Hawkes,  The rôle of APL in statistics teaching. APL Business Technology, 1983, 123-132.
[APL2] A.G. Hawkes,  Complex numbers in APL. Vector, 1, 1984, 107-120, (reprinted also in Belgian APL-CAM Journal, vol. 6 No.4, 1984, 648-654.)
[APL3] A.G. Hawkes,  Statistical computing with APL. The Professional Statistician, Aug/Sept 1984, 21-22 (+49), (reprinted also in Vector, Vol. 1 No. 2, 1984, 113-116).
[APL4] A.G. Hawkes,  Numerical integration and probability distributions. Vector. Vol. 1 No. 2, 1984, 117-123.
[APL5] A.G. Hawkes & A.M. Sykes,  Teaching statistics through APL.  APL86 Tutorials (ed. A. Comacho), 1986, British Informatics Society, London, 175-197.
[APL6] A.M. Sykes & A.G. Hawkes,  Using APL2 in Statistics. APL'89, New  York, 1989, 346-354.
[APL7] A.G. Hawkes,  Markov processes in APL. APL'90, Copenhagen, 1990, APL Quote Quad, vol. 20 no. 4, 1990, 173-185.
[APL8]A.G. Hawkes & A.M. Sykes, Equilibrium distributions of finite-state Markov processes, IEEE Trans. Reliability, Vol. 29 no. 5, 1990, 592-595.

Miscellaneous
Finally, a collection of odd bits and pieces.
[M1] A.G. Hawkes, An approach to the analysis of electoral swing, J. Roy. Statist. Soc. A, 132, 1969,68-79.
[M2] A.G. Hawkes et al, Report on the joint University of Durham/Durham Constabulary research project on traffic tactics 1968-1969, 1970.
[M3]  A.G. Hawkes, A cross-over trial with categorized response, Br. J. Math. Statist. Psychol., 24, 1971, 93-100.
[M4] A.G. Hawkes, The development of statistical ideas and their applications, Inaugural lecture at University of Wales Swansea, 1975.
[M5] A.G. Hawkes, Teaching and examining applied statistics, The Statistician, 29, 1980, 81-90.
[M6] A.G. Hawkes, Random occurrence of stripes in a fabric, Applied Statistics, 29, 1980, 34-38.
[M7] A.G. Hawkes, Approximating the normal tail, The Statistician, 31, 1982, 231-236.
[M8]  A.G. Hawkes, Probability and the F. A. Cup, Teaching Statistics, 4, May 1982, 34-37.
[M9] Salahuddin & A.G. Hawkes,  Cross-validation in stepwise regression, Communications in Statistics, A20 no. 4, 1991, 1163-1181.
[M10] U.S. Pasaribu, A.G. Hawkes & S.J. Wainwright, Statistical assumptions underlying the fitting of the Michaelis-Menten equation, Journal of Applied Statistics, 26, 1999, 327-241.