Research

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My research has been defined by three ubiquitous features of the real world, namely that: (1)  life is essentially random, i.e. stochastic; (2) the vast majority of random processes have no memory, that is, the next step in their development depends only on the current state of the system and not on its previous history; and, (3) the universe and everything in it exhibits spatial relationships. The first two give rise to the study of Markov processes, my imagination being fired by an excellent undergraduate course at Imperial College. Whilst the third arose from a chance, and highly fortunate, encounter with Norman Bailey’s classic paper published in Biometrika in 1968 (vol. 55, 189-198).

The lack of memory property means that stochastic realizations are defined in terms of sequences of event-time pairs, whereupon the equations for the probability that a system is of size n at time t are straightforward to write down. So from a purely applied perspective the mathematical challenge lies in determining their solution in order to enhance understanding of the particular system under study. Now preceeding the Second World War, deep and insightful theoretical developments, pioneered by A.A. Markov and A.N. Kolmogorov, enabled many pure mathematically oriented results to be constructed. Whilst these have considerable intrinsic beauty, which stimulated great interest amongst pure probabilists, applied probabilists seeking solutions to systems oriented problems gained relatively little from them. Serious development over a rapidly widening spectrum of applications only really started to grow following a series of thought-provoking and innovative papers read to the Royal Statistical Society in 1949. Although studies in this new field swiftly prospered, with new stochastic results constantly being applied to fresh areas of application, as the associated level of mathematical difficulty increased, more and more researchers started to switch their attention back onto purer probability problems. For these are more likely to lead to intellectually appealing closed-form solutions. Unfortunately, success usually depended on making unrealistic, and hence detrimental, assumptions about the way in which particular scenarios develop. In recent years, however, there has been a resurgence of interest in constructing approximate theoretical solutions and simulation analyses. The latter has prospered greatly by the availability of fast computing power on inexpensive personal computers, which enables quite complex problems to be studied within a reasonable time frame.

Although my work initially focussed on my own interests in the nature of population spread, it swiftly became apparent that collaboration, both with other researchers in stochastic processes and applied practioners, was vital if real progress was to be made. I have been extremely fortunate in being able to work with many such people, all of whom are named as joint authors in the following list of selected publications and are easily Googled. As regards professional societies, I have enjoyed participating in the activities and administration of the Royal Statistical Society, the International Statistical Institute (elected 1984), the Royal Society of Edinburgh (elected 1995), the Bernoulli Society and the International Biometric Society.

Books:

1991     Modelling Biological Populations in Space and Time.  Cambridge University Press, pp. xvii + 403.
2010 Renshaw, E. Stochastic Population Processes: Analysis, Approximations and Simulations. Oxford University Press, pp. xii + 652.

Both appear in Google Books (select 1991 and 2010), whilst the PREFACE to the latter provides a good introduction to the overall content.

Selected Publications:

The following are a selection of my papers which cover the various research fields I have been involved in through the years. I have only just started to scan the hard copy in order to construct pdf files, so at present most of the pdf links cited below are not operational. However, if there is a particular paper that you would like to see a copy of please email me, and I will attempt to upload the associated file as soon as I can.

1972 Renshaw, E. Birth, death and migration processes. Biometrika, 59, 49-60. <1.pdf>
1973 Renshaw, E. Interconnected population processes. Journal of Applied Probability, 10, 1-14. <2.pdf>
1973 Renshaw, E. The effect of migration between two developing populations. Proceedings of the 39th Session of the International  Statistical Institute, 2, 294-298. <3.pdf>
1974 Renshaw, E. Stepping stone models for population growth. Journal of Applied Probability, 11, 16-31. <4.pdf>
1974 Renshaw, E. Spatial dynamics of biological populations. In Newcastle Conference on Spatial Processes 20-21 May, 1994. Advances in Applied Probability, 7, 449-460. <5.pdf>
1974 Mollison, D & Renshaw, E.  Conference on Stochastic Models in Space and Time, Edinburgh, 19-20 December 1974. Advances in Applied Probability, 7, 461-467. <6.pdf>
1977 Renshaw, E. Velocities of propagation for stepping-stone models of population growth. Journal of Applied Probability, 14, 591-597. <7.pdf>
1979 Renshaw, E. Waveforms and velocities for non-nearest-neighbour contact distributions. Journal of Applied Probability, 16, 1-11. <8.pdf>
1980 Henderson, R. & Renshaw, E.  Spatial stochastic models and computer simulation applied to the study of tree root systems. Compstat 80, 389-395, Physica Verlag: Vienna. <9.pdf>
1980 Renshaw, E. The spatial distribution of Tribolium confusum. Journal of Applied Probability, 17, 895-911. <10.pdf>
1981 Renshaw, E. Waveforms and velocities for models of spatial infection. Journal of Applied Probability, 18, 715-720.  <11.pdf>
1981 Renshaw, E. & Henderson, R.  The correlated random walk. Journal of Applied Probability, 18, 403-414. <12.pdf>
1982 Renshaw, E. The development of a spatial predator-prey process on interconnected sites. Journal of Theoretical Biology, 94, 355-365. <13.pdf>
1982 Renshaw, E. Change of genotype frequencies under selection in a self-pollinated population. Heredity, 48, 139-144. <14.pdf>
1983 Henderson, R., Ford, E.D., Renshaw, E. & Deans J.D.  Morphology of the structural root system of Sitka Spruce. 1. Analysis and quantitative description. Forestry, 56, 121-135. <15.pdf>
1983 Henderson, R., Ford, E.D. & Renshaw, E.  Morphology of the structural root system of Sitka Spruce. 2.  Computer simulation of rooting patterns. Forestry, 56, 137-153. <16.pdf>
1983 Henderson, R., Renshaw, E. & Ford, E.D.  A note on the recurrence of a correlated random walk. Journal of Applied Probability, 20, 696-699. <17.pdf>
1983 Renshaw. E. & Ford, E.D.  The interpretation of process from pattern using two-dimensional spectral analysis: methods and problems of interpretation. Applied Statistics, 32, 51-63. <18.pdf>
1984 Ford, E.D. & Renshaw, E.  The interpretation of process from pattern using two-dimensional spectral analysis: modelling single species patterns in vegetation. Vegetatio, 56, 113-123. <19.pdf>
1984 Henderson, R., Renshaw, E. & Ford, E.D.  A correlated random walk model for two-dimensional diffusion. Journal of Applied Probability, 21, 233-246. <20.pdf>
1984 Renshaw, E. SASP: a two-dimensional spectral analysis package for analysing spatial data. University of Edinburgh. <email>
1984 Renshaw, E. Competition experiments for light in a plant monoculture: an analysis based on two-dimensional spectra. Biometrics, 40, 717-728. <21.pdf>
1984 Renshaw. E. & Ford, E.D.  The description of spatial pattern using two-dimensional spectral analysis. Vegetatio, 56, 75-85. <22.pdf>
1985 Renshaw, E. Computer simulation of Sitka spruce: spatial branching models for canopy growth and root structure. Mathematics Applied in Medicine and Biology, 2, 183-200. <23.pdf>
1986 Newbery D. McC., Renshaw, E. & Brunig E.F. Spatial pattern of trees in Kerangas Forest, Sarawak. Vegetatio, 65, 77-89. <24.pdf>
1986 Renshaw, E. A survey of stepping-stone models in population dynamics. Advances in Applied Probability, 18, 581-627. <25.pdf>
1987 Jakeman, E. & Renshaw, E. Correlated random-walk model for scattering. Journal of the Optical Society of America, A, 4, 1206-1212. <26.pdf>
1987  Renshaw, E. The discrete Uhlenbech-Ornstein process. Journal of Applied Probability, 24, 908-917. <27.pdf>
1988 Renshaw, E. The high-order autocovariance structure of the telegraph wave. Journal of Applied Probability, 25, 744-751. <28.pdf>
1989 Renshaw, E. Discussion contribution to Haslett, J. & Raftery, A. Space-time modelling with long-memory dependence: assessing Ireland's wind power resource. Applied Statistics, 38, 24-25. <29.pdf>
1989 Gu, Z.Q., Duncan, C.N., Renshaw, E., Mugglestone, M.A., Cowan C.F.N. & Grant, P.M.  Comparison of techniques for measuring cloud texture in remotely-sensed satellite meteorological image data. Radar and Signal Processing, IEEE Proceedings-F, 136, 236-248. <30.pdf>
1990 Chen, A. & Renshaw, E.  Markov branching processes with instantaneous immigration. Probability Theory and Related Fields, 87, 209-240. <31.pdf>
1990 Jackson, D.R., Harwood, R.S. & Renshaw, E.  Tests of a scheme for regression retrieval and time-space interpolation of stratospheric temperature from satellite measurements. Quarterly Journal of the Royal Meteorological Society, 116, 1449-1470. <32.pdf>
1990 Renshaw, E. Discussion contribution to Bartlett, M.S. Chance or Chaos? Journal of the Royal Statistical Society, A, 153, 336.
1991 Gu, Z.Q., Duncan, C.N., Grant, P.M., Cowan, C.F.N., Renshaw, E. & Mugglestone, M.A.  Textural and spectral features as an aid to cloud classification. International Journal in Remote Sensing, 12, 953-968. <34.pdf>
1992 Chen, A. & Renshaw, E.  The Gillis-Domb-Fisher correlated random walk. Journal of Applied Probability, 29, 792-813. <35.pdf>
1992 Jackson, D.R. & Renshaw, E. A comparison of sequential estimation and time/space interpolation schemes for analysing retrieved stratospheric temperature. Annales Geophysicae, 10, 895-903. <36.pdf>
1992 Renshaw, E. Discussion contribution to RSS Chaos Meeting, London, Oct. 1991. Journal of the Royal Statistical Society, B, 54, 455.
1992 Renshaw, E. Lead Discussion Contribution to Joint Meeting of the RSS and LMS on The Future for Honours Degree Courses in Mathematics and Statistics, London, Oct. 1991. Journal of the Royal Statistical Society, A, 155, 199-200.
1993 Chen, A. & Renshaw, E. Recurrence of Markov branching processes with immigration. Stochastic Processes and Their Applications, 45, 231-242. <39.pdf>
1993 Chen, A. & Renshaw, E. Existence and uniqueness criteria for conservative uni-instantaneous denumerable Markov processes. Probability Theory and Related Fields, 94, 427-456. <40.pdf>
1993 Renshaw, E. Non-linear waves on the Turing ring. 49th Session of The International Statistical Institute. Firenze, Italy, 2, 339-340.
1993 Renshaw, E. Spatial reaction systems in a discrete environment. STATCHEM93, Venice, Italy, 1, 3-25.
1994 Chen, A. & Renshaw, E. The general correlated random walk. Journal of Applied Probability, 31, 869-884. <43.pdf>
1994 Renshaw, E. The linear spatial-temporal interaction process and its relation to 1/-noise. Journal of the Royal Statistical Society, B, 56, 75-91. <44.pdf>
1994 Renshaw, E. Chaos in Biometry. IMA Journal of Mathematics Applied in Medicine and Biology, 11, 17-44. <45.pdf>
1994 Renshaw, E. Non-linear waves on the Turing ring. The Mathematical Scientist, 19, 22-46. <46.pdf>
1994 Smith, M. & Renshaw, E. Parallel-prefix remapping for efficient data-parallel implementation of unbalanced simulations. Advances in Parallel Computing, 9, 215-222. <47.pdf>
1995 Chen, A. & Renshaw, E. Markov branching processes regulated by emigration and large immigration. Stochastic Processes and Their Applications, 57, 339-359. <48.pdf>
1995 Jakeman, E., Phayre, S. & Renshaw, E. The evolution and measurement of a population of pairs. Journal of Applied Probability, 32, 1048-1062. <49.pdf>
1996 Mugglestone, M.A. & Renshaw, E. A practical guide to the spectral analysis of spatial point processes. Computational Statistics and Data Analysis, 21, 43-65. <50.pdf>
1996 Mugglestone, M.A. & Renshaw, E. The exploratory analysis of bivariate spatial point patterns using cross-spectra. Environmetrics, 7, 361-377. <51.pdf>
1997 Chen, A. & Renshaw, E. The M/M/1 queue with mass exodus and mass arrivals when empty. Journal of Applied Probability, 34, 192-207. <52.pdf>
1997 Renshaw, E. Spectral techniques in spatial analysis. Forestry Ecology and Management, 94, 165-174. <53.pdf>
1997 Renshaw, E. & Chen, A.Y. Birth-death processes with mass annihilation and state-dependent immigration. Stochastic Models, 13, 239-254. <54.pdf>
1997 Renshaw, E. & Dai, Y. Regularity and reversibility results for birth-death-migration processes. Journal of Applied Probability, 34, 192-207. <55.pdf>
1998 Capobianco, R. & Renshaw, E. The autocovariance function for marked point processes: a comparison between two different approaches. Biometrical Journal, 40, 1-16. <56.pdf>
1998 Gibson, G.J. & Renshaw, E. Estimating parameters in stochastic compartmental models using Markov chain methods. IMA Journal of Mathematics Applied in Medicine and Biology, 15, 19-40. <57.pdf>
1998 Marion, G., Renshaw, E. & Gibson, G. Stochastic effects in a model of nematode infection in ruminants. IMA Journal of Mathematics Applied in Medicine and Biology, 15, 97-116. <58.pdf >
1998 Mugglestone, M.A. & Renshaw, E. Detection of geological lineations on aerial photographs using two-dimensional spectral analysis. Computers and Geosciences, 24, 771-784. <59.pdf>
1998 Renshaw, E. Epidemic Models, Spatial. In Encyclopaedia of Biostatistics, eds. P. Armitage & T. Colton, 1341-1345, Wiley. <60.pdf>
1998 Renshaw, E. Saddlepoint approximations for stochastic processes with truncated cumulant generating functions. IMA Journal of Mathematics Applied in Medicine and Biology, 15, 1-12. <61.pdf>
1998 Renshaw, E. & Gibson, G.J. Can Markov Chain Monte Carlo be usefully applied to stochastic processes with hidden birth times? Inverse Problems, 14, 1581-1606. <62.pdf>
1999 Renshaw, E. Pearl-Verhulst logistic process. In Encyclopaedia of Mathematics Supplement Vol. 1 (ed. M. Hazewinkel),  Kluwer, 404. <63.pdf>
1999 Renshaw, E. Stochastic Effects in Population Models. Chapter 2 of Advanced Theoretical Ecology: Advances in Principles and Applications (ed. J. McGlade), 23-63. Blackwell Science: Oxford. <64.pdf>
2000 Chen, A. & Renshaw, E. Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration. Stochastic Processes and Their Applications, 88, 177-193. <65.pdf>
2000 Dai, Y. & Renshaw, E. The Markov oscillation problem in discrete time. Journal of the London Mathematical Society, 61, 301-314. <66.pdf>
2000 Dalgleish, A.N., Boulton, G.S. & Renshaw, E. The ice age cycle and the deglaciations: an application of nonlinear regression modelling. Quaternary Science Reviews, 19, 687-697. <67.pdf>
2000 Marion, G., Renshaw, E. & Gibson, G. Stochastic modelling of environmental variation for biological populations. Theoretical Population Biology, 57, 197-217. <68.pdf>
2000 Renshaw, E. Epidemic Models, Spatial. In Encyclopaedia of Epidemiologic Methods, (eds. M. Gail & J. Benichou), Wiley, 362-366. <69.pdf>
2000 Renshaw, E. Applying the saddlepoint approximation to bivariate stochastic processes. Mathematical Biosciences, 168, 57-75. <70.pdf>
2000 Renshaw, E., Phayre, S. & Jakeman, E. The development of space-time interaction processes with given spectral structure. Inverse Problems. 16, 877-890. <71.pdf>
2001 Gibson, G.J. & Renshaw, E. Inference for immigration-death processes with single and paired immigrants. Inverse Problems, 17, 455-466. <72.pdf>
2001 Gibson, G.J. & Renshaw, E. Likelihood estimation for stochastic compartmental models using Markov chain methods.  Statistics and Computing, 11, 347-358. <73.pdf>
2001 Mugglestone, M.A. & Renshaw, E. (2001) Spectral tests for randomness for spatial point patterns. Environmental and Ecological Statistics, 8, 237-251. <74.pdf>
2001 Renshaw, E. & Särkkä, A. Gibbs point processes for studying the development of spatial-temporal stochastic processes. Computational Statistics and Data Analysis, 36, 85-105. <75.pdf>
2002 Mao, X., Marion, G. & Renshaw, E. Environmental Brownian noise suppresses explosions in population dynamics.  Stochastic Processes and Their Applications, 97, 95-110. <76.pdf>
2002 Marion, G., Mao, X. & Renshaw, E.  Convergence of the Euler scheme for a class of stochastic differential equation. International Journal of Mathematics, 1, 9-22. <77.pdf>
2002 Renshaw, E. Epidemic models. In Encyclopaedia of Environmetrics. (eds. A.H. El-Shaarawi & W.W. Piegorsch), volume 2, 703-708. Chichester: Wiley. <78.pdf>
2002 Renshaw, E. Two-dimensional spectral analysis for marked point processes. Biometrical Journal, 44, 718-745. <79.pdf>
2002 Marion, G., Mao, X., Renshaw, E. & Liu, J. Spatial heterogeneity and the stability of reaction states in autocatalysis. Physical Review E, 66, 051915.1-9. <80.pdf>
2003 Mao, X., Sabanis, S. & Renshaw, E. Asymptotic behaviour of the stochastic Lotka-Volterra model. Journal of Mathematical Analysis and Applications, 287, 141-156. <81.pdf>
2003 Marion, G., Gibson, G.J. & Renshaw, E. (2003) Estimating likelihoods for spatio-temporal models using importance sampling. Statistics and Computing, 13, 111-119. <82.pdf>
2003 Matis, J.H., Kiffe, T.R., Renshaw, E. & Hassan, J. A simple saddlepoint approximation for the equilibrium distribution of the stochastic logistic model of population growth. Ecological Modelling, 161, 239-248. <83.pdf>
2004 Chen, A. & Renshaw, E. Markovian bulk-arriving queues with state-dependent control at idle time.  Advances in Applied Probability, 36, 499-524. <84.pdf>
2004 Renshaw, E. Metropolis-Hastings from a stochastic population dynamics perspective. Computational Statistics & Data Analysis, 45, 765-786. <85.pdf>
2005 Gillespie, C. & Renshaw, E. The evolution of a batch-immigration death process subject to counts. Proceedings of the Royal Society, A 461, 1563-1581. <86.pdf>
2005 Renshaw, E. Epidemic Models, Spatial. In Encyclopaedia of Biostatistics (2nd edition), eds. P. Armitage & T. Colton, 1693-1699, Wiley. <87.pdf>
2006 Särkkä, A. & Renshaw, E. The analysis of marked point patterns evolving through space and time. Computational Statistics and Data Analysis, 51, 1698-1718. <88.pdf>
2007 Gillespie, C. & Renshaw, E. An Improved Saddlepoint Approximation.  Mathematical Biosciences, 208, 359-374. <89.pdf>
2007 Renshaw, E., Mateu, J. & Saura, F. Disentangling mark/point interaction in marked point processes. Computational Statistics and Data Analysis, 51, 3123-3144. <90.pdf>
2008 Gillespie C.S. & Renshaw, E. The evolution of a single-paired immigration death process. Journal of Physics, A 41, 1-20. <91.pdf>
2009 Renshaw, E. & Comas, C. Space-time generation of high intensity patterns using growth-interaction processes. Statistics and Computing, 19, 423-437. <92.pdf>
2009 Renshaw, E., Comas, C. & Mateu, J. Analysis of forest thinning strategies through the development of space-time growth-interaction simulation models. Stochastic Environmental and Research Risk Assessment, 23, 275-288. <93.pdf>
2009 Starr, J.M., Campbell, A., Renshaw, E. Poxton, I.R. & Gibson, G.J. (2009) Spatio-temporal stochastic modelling of Clostridium difficile. Journal of Hospital Infection, 71, 49-56. <94.pdf>
2009 Renshaw, E. Spatial-temporal marked point processes: a spectrum of stochastic models. Environmetrics, 20, 1-17. <95.pdf>