• The Report
• Appendix 1 (The beginning)
• Appendix 1 (Table of Contents)
• Appendix 2

Appendix 1. The Mathematical Sciences in New Zealand: Contributions to the Foresight Exercise

Table Of Contents

A1.1. Introduction
A1.2. Overview of Mathematical Sciences
A1.3. Algebra, Logic and Foundations
A1.4. Analysis, Geometry and Topology
A1.5. Biostatistics
A1.6. Econometrics
A1.7. Epidemiology
A1.8. Financial Mathematics
A1.9. Industrial Operations Research
A1.10. Mathematical Modelling
A1.11. Numerical Analysis and Scientific Computation
A1.12. Official Statistics
A1.13. Operations Research
A1.14. Probability and Stochastic Processes
A1.15. Statistics

Introduction

The terms of reference for the review included the requirement for a knowledge foresight exercise, identifying

• knowledge trends, and likely developments - where is the strength, where is it developing internationally and in New Zealand
• performance outside and inside New Zealand in identified "gaps"
• "breakthrough" areas of mathematics and its applications
• the implications of new technologies in computing, information and communications in the use of mathematics
• the opportunities for socio-economic sectors within New Zealand in mathematical developments and the supporting needs for mathematical services
• implications for PGSF priorities - any shift(s) in socio-economic science priorities
• the enabling science capability required to meet identified opportunities including infrastructure, human and other resources.

To this end, the authors of the knowledge base profiles in the MoRST report were asked to contribute to a discussion document by putting forward some preliminary views under the headings identified above.

A discussion document including these views was distributed at a meeting in Auckland on 9 July 1997, and criticisms sought. This appendix includes the subsequently modified profiles, together with a further series from mathematicians, working outside the university mathematical science departments, dealing with issues facing the mathematical sciences in industry, health, finance and government.

The views expressed are those of the individual authors, although there has, in the main, been considerable feedback from the mathematical community. The profiles have been subject to an absolute minimum of editing by the review team, enough only to present a common style and organisation.

We must warn that predicting knowledge trends over 25 years is fundamentally impossible. In a similar time frame, humankind has this century gone from biplanes to spaceflight; from dynamite to the atomic bomb; and created over 100 new countries.

In general, the contributors have concentrated on recent developments and trends, and their relation to present avenues of endeavour. One cannot say, in the mathematical sciences, whether proposed research will be successful since, if one knows the answer, one has done the research. This is a degree in which mathematical projects differ from many projects in the PGSF funding arena, in that such projects often have a foreseeable "result".

Overview of Mathematical Sciences

The mathematical sciences are universal. Having evolved out of early efforts to understand the natural and physical world, modern mathematical science consists of a body of knowledge which is a supreme creation of the human intellect, forms a basis and language for scientific investigation in many fields, and is now reaching far beyond the physical sciences into diverse areas of industry and commerce, medicine, life and social sciences, and every other area of knowledge that requires quantitative analysis.

The mathematical sciences are growing and diversifying, with several new areas coming to the fore in recent years of technological advance, and with increasing interaction between fields. A growing influence is the capacity of computers to solve problems of a discrete nature.

Outstanding strength exists in New Zealand in applied group theory, combinatorics, complex analysis, logic and computational complexity theory, experimental design theory, methods of mathematical modelling, numerical analysis, sample survey design and analysis, and spectral theory of operators in mathematical physics.

Developing as a parallel strength in New Zealand is the high level of applied research and experimental work, especially in the areas of mathematical and statistical modelling and operations research (and also more recently in mathematics education). A major feature is the interdisciplinary nature of this type of work, involving collaboration across institutions and with other areas of science and technology.

As is to be expected due to its size, there are several gaps in New Zealand's knowledge base in the mathematical sciences. Some of these include computational science, financial mathematics, theory of the finite element method, number theory, polyhedral combinatorics, stochastic differential equations, and stochastic optimisation.

All are important and active areas internationally, but it is difficult to assess the relative degree of their importance to New Zealand.

See submissions from area profile authors.

Mathematical and statistical software packages (and more generally the use of powerful computers) have enabled the solution of deeper and wider classes of problems, and have also impacted on the disciplines themselves, in all areas.

In addition, the use of electronic communications media (email, web browsers, news groups etc.) has dramatically improved international interaction between researchers and improved the speed of publication of research.

Opportunities exist for further collaboration in teaching and applied research with several other groups such as agriculture/horticulture, finance/risk management and health sciences. It is crucial that groups of mathematical scientists outside the universities be retained at a critical mass, and that interaction between universities and CRIs be fostered as much as possible (through collaborative research projects, graduate students and other joint ventures).

Gaining currency is the suggested formation and support of key centres of expertise in three or four broad application areas, each involving staff of universities and CRIs or research associations, together with representatives of the end users of the research. There is huge potential for development in this regard, with benefits possible both to science and to the New Zealand economy.

Recent structural reforms of the New Zealand science system have diffused mathematical scientists into different sectors, and created on the other hand an output-driven funding system (the PGSF) which does not cater for the type of methodological research of generic benefit carried out by mathematical and statistical modellers. In many cases these changes have caused a reduction in the size of (already small) research groups, replaced research by consulting duties, and discouraged collaboration between university researchers and CRI staff. Unless countered, this process will inevitably lead to a basic knowledge gap outside universities, as isolated researchers fall further behind in their knowledge of current research findings, and a serious decrease in the opportunities to make contact with New Zealand business and industry.

Continued strength in basic research depends on the ability of researchers to maintain a high level of contact with their international colleagues, and the recruitment and retention of young mathematical scientists of high calibre.

Teaching facilities in the mathematical sciences in New Zealand fall far below the standard of those in universities in comparable countries. Funding and space limitations are inhibiting the necessary development of applications-driven, computer-assisted and tutorial-based learning in mathematics and statistics, thus having a serious effect on New Zealand’s ability to provide adequate training for mathematical scientists and other scientists requiring quantitative skills.

What is required is a recognition within University funding that the mathematical sciences require support comparable to that of geography, psychology and computer science. Particular needs are teaching staff, computing equipment, and support for graduate students. Questions concerning the evolution of employment opportunities for graduates with specialist mathematical skills are considered elsewhere this study.

This was peer-reviewed by Alastair Scott, Alex McNabb, David Vere-Jones and Rod Downey, and incorporates corrections/improvements suggested by them.

Algebra, Logic and Foundations

The 1997 MoRST Knowledge-base Profile observed that New Zealand has about 40 researchers working in this area, mostly in university Mathematical Sciences departments, with others in Computer Science. The age profile is well spread but with a large proportion in the 40 to 55 age group.

Particularly notable is the work on computability and complexity, the combinatorics of phylogenetic trees, and computational investigations of groups, graphs, and networks.

The above areas of expertise have developed in an unplanned way:

Universities do not advertise vacancies at this level of specialism, and staff are generally free to decide their own research activities. Should this situation be left alone, or is there a case for targeting particular topics for support and funding? If so, what and why?

The future is of course unforeseeable, and predictions are likely to be wrong, but it is obvious that one significant influence in this area (as with others) is the emergence in the last few decades of computational science. This is evidenced by

• the emphasis on the study of discrete structures generally;
• the intensive activity in complexity theory;
• the extensive use of computer algebra packages as exploratory tools;
• the application of logic to computer science in logic programming, verification and synthesis systems etc.;
• the use of abstract algebra and category theory in developing the formal semantics of programming languages.

The latter is very much a foundational activity. A major interest in the future will continue to be in the mathematical foundations of computer science. For example, there appears to be a paradigm-shift going on from structured programming to object-oriented programming. Rigorous foundations for the latter, including a mathematical theory of "objects", is only just now being taken up. Tools developed in this area form a vital part of such studies.

What direction will this emphasis take? Some Computer Science departments are very practically oriented, with little interest in theory. On the other hand few Mathematics departments are interested in Logic. Theoretical Computer Science lies between the two. Will it simply become a mathematical science in its own right? Should it be targeted? Can it be linked to socio-economic priorities by explaining that practical applications depend on good theoretical foundations?

The growth in this direction raises another issue: what is the likely future of more traditional branches of algebra? The period of the mid-20^th Century was characterised by great abstraction and generalisation (e.g. homological algebra), and has been followed by a period of focus on more concrete situations. It seems reasonable to predict that there will be less interest in the future in, say, the abstract study of rings, and continued development of computer-aided study of discrete structures as well as research on new concepts motivated by the needs of computer science.

The Profile noted that number theory represents a gap in the knowledge base. This is a high profile area internationally due to its applications in cryptanalysis and also the work of Wiles on Fermat's Last Theorem. Australia apparently has about 50 cryptomathematicians but New Zealand probably has only one or two. Is this lack of expertise in number theory an issue to be addressed, or is it irrelevant?

The emergence of Discrete Mathematics and Theoretical Computer Science impacts strongly on this area. In fact a future-looking approach might suggest the area be re-labelled "Discrete Mathematics, Theoretical Computer Science, and Logic". (However this would leave out more classical parts of algebra, such as Lie algebra and infinite group theory.)

More than 30 years ago, John McCarthy stated that "It is reasonable to hope that the relationship between computation and mathematical logic in the next century will be as fruitful as that between analysis and physics in the last. The development of this relationship demands a concern for both applications and mathematical elegance." It could be said that McCarthy's prophecy has already been realised, and that there has been a similar fruitful connection between computation and abstract algebra, leading to the creation of new structures (complete partial orders, continuous lattices, various types of category) required for the semantic modelling of programs. We can expect this trend to continue.

Any long-term prediction should take account of the fact that the life sciences are coming into ascendancy (witness the Human Genome Project, the great interest and progress in brain science, genetic engineering, the debate about the nature of consciousness, etc.). Impact on mathematics is evident in the applications of knot theory to DNA, the use of graph theory to study the structure of DNA and phylogenetic trees and the recent interest in "DNA computing". We can expect further algebraic and combinatorial work arising from the connection with biology. For any such observations about the progress of knowledge we can ask: is this just a trend to be observed, or should we be singling it out for emphasis and targeted support?

The creation of the Auckland/Waikato Centre for DMTCS has provided a focus for work in this area. Also the new Institute of Mathematics is considering hosting a workshop on complexity theory. Should these institutions be given more "official" support and recognition in the funding system? For example, would it be appropriate that proposals for Marsden/ FoRST funding could be submitted as a research programme of an institute, rather than attached to an individual or a team of individuals?

This was circulated to several people (Douglas Bridges, Cristian Calude, Marston Conder, Rod Downey, Derek Holton, and Geoff Whittle), and returned with a few minor comments which were incorporated in the final editing.

Analysis, Geometry and Topology

Analysis deals with limiting processes in mathematics; it permeates, and has great importance for, almost all branches of mathematics. Geometry, with ancient origins in the measurement of the features of the earth, now covers a wide range from planes with only a finite number of points to the differential geometry of manifolds, an abstraction of surfaces in space; geometry has strong links with analysis, topology, and algebra. Topology is the study of properties that do not change as geometric figures are distorted without tearing, and has strong interactions with analysis and algebra.

Some knowledge of analysis, in particular, is essential for all mathematicians, and for most physicists, economists, and others in increasingly mathematics-based disciplines. For example, quantum physics is founded upon Hilbert space theory, and, at its more theoretical end, requires a sound knowledge of the theory of operator algebras; on the other hand, relativists need the differential geometry of four-dimensional manifolds. Theoretical economists use functional analysis, measure theory, fixed-point theorems, and even nonstandard analysis (a subject at the frontier of logic and analysis) in their study of equilibria of large economies.

There are 26 researchers working in these areas in New Zealand. The main strengths here are in projective geometry, complex analysis, partial differential equations, operator theory applied to Mathematical Physics, constructive analysis, and set-theoretic topology. The University of Auckland is the only university in New Zealand with enough staff members active in analysis and related areas (such as topology) to count as a coherent research group; there are good analysts in the other universities, but they are essentially isolated, without day-to-day close contact with a group of like-minded researchers.

There are at least the following significant gaps in New Zealand.

• Analysis: operator algebra theory, measure theory, Banach algebras, the relation between set theory and analysis.
• Geometry and topology: differential geometry, algebraic topology, low-dimensional topology.

The research in New Zealand in the areas of analysis, geometry and topology in which there are strengths is excellent, at times outstanding. In the identified gaps there is little or no research in New Zealand at all. So, for example, there is very little contribution from New Zealand to the rapidly growing field of operator algebras and their applications in physics. The interaction of set theory and analysis (exemplified by the work of Dales (England) and Woodin (USA)) is an area that looks increasingly significant in analysis and in which there is no research whatsoever in New Zealand.

Internationally significant research findings include fundamental discoveries in knot theory and related areas by Vaughan Jones (Berkeley and Auckland), a New Zealander who is based overseas but spends part of each year in New Zealand; the discovery of a particular projective plane by Peter Lorimer (Auckland); and fundamental contributions by Gaven Martin (Auckland) to the theory of nonlinear partial differential equations.

As with most research in pure mathematics, it is difficult to predict where these developments will lead, or where they will find significant applications.

The increasing speed and power of desktop computers has had a major impact on research in a number of areas, for example, low-dimensional topology. There is little doubt that, as symbolic and numeric computation packages improve, computers will have a greater role to play in the formation of conjectures, the verification of routine, but lengthy, computations, and the graphical modelling of analytic and geometric phenomena.

Rapid communication via electronic mail has greatly facilitated international research collaboration, enabling mathematicians to interact at a distance almost as if they were together in the same room.

Research in the three branches of mathematics under discussion is, naturally, at the purer end of the mathematical spectrum. Nevertheless, the problems investigated often originate in the experimental or social sciences; and, in turn, the pure mathematical developments often have dramatic and unexpected applications within mathematics and in other disciplines. For example, the theory of Fourier analysis originated in Fourier's work in the theory of heat, developed into a major activity in analysis over the next 150 years, and has recently had significant applications in computer arithmetic (the Fast Fourier Transform).

There is a need to maintain strength in analysis in each university, since it is such a fundamental part of mathematics; but this need must be balanced by the reality that the New Zealand mathematical community cannot expect to have research groups in analysis in each university.

This is probably not applicable to the subjects under discussion; but see under the next heading.

Being perceived as one of the more difficult areas of mathematics, analysis is often shunned by students; so graduates in mathematics may be weak in analysis and related areas. It is imperative that universities do not yield, as some have done in the past, to student pressure to avoid difficult areas of mathematics in their undergraduate programmes. Some knowledge of the fundamentals of analysis and topology is an essential part of the education of any mathematics major; so each university must retain the capacity to teach modern analysis/topology courses up to graduate level.

As noted above, research in analysis, geometry, and topology is largely concentrated at Auckland, with isolated pockets of strength in the other universities. In the current funding climate, and with the relatively small size of the other New Zealand Mathematics departments, it seems unlikely that there will be a more even distribution of active researchers in these areas in the foreseeable future. One way to overcome the problems of associated with this uneven distribution might be to create a Centre for Analysis (and related subjects), facilitating and, where appropriate, coordinating research in analysis, geometry, and topology among universities. This arrangement, which would require funding from MoRST, the PGSF, or some other external source, would ensure the continuation of a major centre for analysis research, together with the distribution of at least some experts in that area around the universities. Its most likely location might be in Auckland, but there is also a reasonable case for strengthening mathematics outside of Auckland, and in particular for creating a centre in Canterbury, where there are researchers in aspects of analysis different to the strengths in Auckland.

Biostatistics

Bryan Manly's general comments on statistics also hold for biostatistics, which is the applied area of statistics concerned with living organisms, in particular health/medical research. Biostatisticians work in medical schools as academics, consultants, and members of research teams, and as academics in statistics departments.

As for statistics in general, the major theoretical and practical changes in the last two decades have come from increased computing power, which has enabled greater use of resampling methods such as the jackknife and the bootstrap, and Markov chain monte-carlo methods.

For consulting biostatisticians the problem is one of being all things to all people and the major difficulty is to provide adequate up to date approaches to problems with little time to update let alone specialise. For the academic biostatisticians and for biostatisticians employed on specific projects or programmes there is more opportunity to specialise; for example the Christchurch Health and Development Study pioneered the use of structural equation modelling in health research in New Zealand. The dispersion of biostatisticians throughout the country is essential for consultation with researchers and for teaching but with the small numbers of biostatisticians there can be isolation. The continuing development of biostatistics courses in statistics departments is encouraging as it requires the appointment of more statisticians with expertise in biostatistics and results in more graduates with knowledge of the area.

Breakthrough areas in biostatistics, as in statistics generally, have been associated with computer-intensive methods. In addition, methods for dealing with correlated data and missing data have been particularly important for biostatistics. Some breakthroughs have resulted from attempts to provide solutions to new problems in health research such as the HIV/AIDS epidemic, and research in molecular biology and genetics.

Changes in computing have enabled new analyses, new research, and opened up worldwide communications.

Health is one of the major areas of expenditure for government and is also important for private spending. Within the health sector biostatistics is essential for research in aetiology, incidence and prevalence, need, effectiveness, and health systems delivery and performance, as well as being important in most areas of basic research.

Much of the work in biostatistics is applied and hence fits within the health outputs of PGSF. However there is need for the development of generic methods. These may arise out of specific applied projects but require further work to enable publication and application to other problems. Other generic work will arise out of the biostatistical literature itself. Without such generic work there will be a lack of methodological development in New Zealand and there will be problems in technology transfer, because of a lack of involvement with new methods for design and analysis. There is a danger that biostatistical methodological research might fall between PGSF (HRC) and the Marsden Fund.

As an interdisciplinary science, biostatistics exists to enable health research to take place. Most publications by biostatisticians are joint publications in medical/health journals. The increased emphasis on ethics committees, and their role in considering scientific validity as a pre-requisite to ethical research, has resulted in increased recognition of the important role biostatisticians can play in the design of studies as well as in their analysis. Collaboration requires some degree of mutual understanding. Biostatisticians need to learn something of the areas they are working in and this takes time, particularly for recent statistics graduates who may have no experience of health research. There is also a need for researchers to have a greater understanding of biostatistics, as few have even a basic knowledge of biostatistics. Many lack any understanding of the intellectual role of biostatistics and think of it as a bag of simple technical tricks rather than a tool to clarify thinking and to provide estimates of the scientifically relevant parameters.

Biostatisticians could benefit from greater collaboration among themselves for there are staff in university statistics departments who teach and do research in biostatistics, there are academic biostatisticians within medical schools, consulting biostatisticians who are in the same department within medical schools as the academic biostatisticians, and biostatisticians who are employed in research units, programmes and projects, usually within medical schools. Links between these groups are informal and sporadic. An important workforce issue concerns the employment of biostatisticians on research projects. These positions tend to be at junior level, which raises issues of training and supervision. A more major problem is that employment is short term, making these positions unattractive with no obvious career path.

International links are mainly individual through conferences or e-mail. A recent development has been the proliferation of workshops, some at basic level but with many offering opportunities for biostatisticians to learn new methods. there have been a small number in New Zealand and many more have been offered in Australia, as well as further afield. Support for attendance at such workshops, and some support for workshops to be held here, seems to be an increasingly useful way for biostatisticians to fill in the gaps in their knowledge and to keep up with international advances.

Econometrics

Econometrics "involves the unification of economic theory, statistics and mathematics," (Frisch, 1936). The Econometric Society has been in existence since 1930. As a subject it is a requirement for any academic economist and an increasing number of commercial economists.

The availability of powerful computing resources has changed both the access of the subject matter to a wider range of users, but also the nature of the tasks undertaken.

This has led to an increasing use of Monte Carlo simulation methods and large scale, cross-sectional, database usage in addition to time series-based methods. There is an increased interest in modelling generally and of the financial sector in particular.

Graduates with such knowledge bases are typically in high demand. Econometricians in New Zealand are generally well informed of world trends; many are members of the Econometric Society and regularly attend both world and regional, i.e., Australasian Meetings. Recently, the New Zealand Econometric Study Group was formed.

There are a number of gaps which exist, mainly because of the relatively small size of the group. These would involve panel data estimation; asymptotic theory; non-parametric methods and discrete choice modelling. Overseas visitors and exchange arrangements could help alleviate some of these gaps in addition to recruitment of suitably qualified individuals.

There is an increasing awareness of the importance of non-stationary processes within the time-series area of analysis. This involves work on testing for unit roots, cointegration and reduced-rank estimation. No other single issue has attracted the same level of attention in the last ten years in the subject area.

In addition the use of Monte Carlo simulation methods, aided by the increasing power of computers, has led to an increase in the use of bootstrap methods and simulation methods more widely defined.

Computing power, both local and networked, has had fundamental effects on what has been achieved and what can be achieved in the future. The internet is used increasingly both to acquire data and also use computer software online. Specialised user groups have arisen with online interaction as a consequence. The use of CD ROMS to disseminate large scale data sets cheaply and easily is an important development.

Traditional economic sectors and users, i.e., Treasury, Reserve Bank, etc., are clearly aware of the benefits of econometrics and regularly employ individuals on the basis of such expertise. There is also a growing tendency for financial sector firms to recruit similarly trained graduates. There appears to be a trend towards the employment and probable use of such skills in a wider range of enterprises. However, higher level analysis is still, in the main, undertaken via academic institution contacts.

Current output classes do not identify any separate category into which "econometrics" could bid. As such, the PGSF is not a good source for theoretically-based or applied-based research in econometrics. However, econometric analysis may be involved in a range of Social Science projects and potentially in other quantitatively orientated output classes.

It is essential that the teaching of econometric methods in universities continues and is encouraged and importantly that the mathematical and statistical skills required for such training be supported in the early years of social science and management degrees. Any tendency to reduce or remove compulsory introductory mathematics and statistics courses in undergraduate economics related degrees should be avoided.

There are two overriding impediments to research in econometrics at the highest level.

The first reflects the outside (university) options available to good graduates and established staff. Funding must be found to attract and retain the best researchers otherwise New Zealand will continue to look overseas for developments.

Secondly, New Zealand researchers need access to research funding to fund, in particular, theoretical research. The recently formed New Zealand Econometric Study Group is a significant development, but its lack of funding could jeopardise its long term success.

Epidemiology

Epidemiology can be defined briefly as the study of the distribution and determinants of disease in human populations. It is a very quantitative discipline, and makes extensive use of mathematical sciences, particularly biostatistics. A substantial part of biostatistics has been developed to address epidemiological research questions, and much of the current research in biostatistics is similarly motivated. Biostatistics is used in descriptive studies and studies of the causes, prevention and treatment of disease. Mathematical modelling is used to study the spread of epidemics. Computer methods are important for data bases and linkage, and epidemiological studies may also involve sample survey methods, official statistics, econometrics and operations research.

Epidemiological research in New Zealand is of a high standard. There are several research groups, the main ones being based in the medical schools and at ESR. Epidemiologists also work in the Ministry of Health, the Transitional Health Authority and the CHEs. Each group tends to have particular areas of expertise within epidemiology, and the research carried out addresses health issues particular to New Zealand as well as issues of international significance. A number of the research projects involve collaboration with overseas researchers and/or funding from overseas.

New Zealand epidemiologists make extensive use of the available biostatistical expertise. Each of the medical schools (Auckland, Wellington, Christchurch and Dunedin) employs biostatisticians, with the majority of positions funded by the Health Research Council, and some of the university departments of mathematics and statistics also have biostatisticians. Biostatisticians are also employed specifically by particular research groups. The main areas of biostatistics relevant to epidemiology are generalised linear models and survival analysis (with extensions to allow for more complex study designs and analyses such as the analysis of longitudinal data using random effects models) and study design (such as group sequential designs for clinical trials).

Expertise exists in New Zealand in many of these areas, including several statisticians researching statistical methods for epidemiology.

Mathematical modelling is also important for epidemiology, and there is some expertise in New Zealand in this area, particularly in deterministic epidemic models. Developments in computer software and hardware make possible the simulation of the effects of public health intervention strategies such as cancer screening programmes on desktop machines. Life history simulation methodology, as used in studies of cost-effectiveness of public health interventions, lies within operations research. Expertise is available at several, but not all, universities in New Zealand. Expertise in sample survey methods, official statistics and econometrics also exists.

In all of these areas of mathematical sciences the available expertise is often not sufficient to meet the need.

The gaps in support from mathematical scientists arise mainly from a lack of sufficient people, or lack of access to people with appropriate expertise in the same geographical region. Epidemiologists tend to have better access to statistical support than other health researchers partly because the biostatistics groups tend to be located in the same place, but also because epidemiologists tend to value biostatistical input very highly and develop good collaborative relationships with biostatisticians. Research in the mathematical sciences relevant to epidemiology is limited in New Zealand, which means that expertise for more mathematically sophisticated epidemiological research projects is not always available.

It is hard to predict the future developments in mathematical sciences which will be necessary for epidemiology since the mathematical developments are driven by emerging health issues. A recent example is the HIV/AIDS epidemic, which has been the impetus for a substantial amount of research in biostatistics and mathematical modelling. Current research areas include generalised linear mixed models, missing data problems, development of more sophisticated approaches to study design and stochastic epidemic models. Developments in computing power will impact on the statistical methods used in epidemiology, for example Markov chain monte-carlo methods. Initiatives such as the Cochrane Collaboration, an organisation which aims to prepare, maintain and distribute systematic reviews of the effectiveness of health care, require statistical and computing support. The increasing interest in genetic epidemiology will also have an impact. As more sophisticated statistical methods become available, the demands of epidemiology for mathematical science support are likely to increase.

As with all areas of mathematical sciences, developments in computing power enable the use of more sophisticated approaches. Use of the internet is important for collaborative research, for communication and for information transfer (such as randomisation and sharing data in multi-centre clinical trials).

The application of epidemiological research has potential to improve the standard of health in New Zealand. For example, New Zealand epidemiological studies of the causes of Sudden Infant Death Syndrome have lead to the development of important preventive measures, and their implementation has lead to a decrease in the incidence of SIDS. Good information is essential for making appropriate decisions regarding disease prevention and health care provision, and the collection and interpretation of this information is dependent upon adequate support from the mathematical sciences.

The Health Research Council is the major source of funding for epidemiological research. The HRC currently provides salary support for biostatisticians in each of the medical schools, as well as support for several more through research grants. The move of HRC to MoRST may impact on the nature of the support structure. Currently the level of support is not adequate to meet the needs of the health research community.

Much of the research into statistical methods for epidemiology is not directly output driven so, as with other applied mathematical sciences, tends to fall in the gap between output driven research (PGSF and HRC) and curiosity driven research (Marsden). The difficulty with obtaining funds has had an impact on the mathematical science workforce, with several very able people moving out of either epidemiological research or statistical research.

Epidemiology is heavily dependent on the mathematical sciences, and most research projects would include a biostatistician or a mathematical modeller as either a member of the research team or as a consultant. Lack of adequate support in these areas will hamper the efforts of epidemiologists to carry out research of high quality.

Access to biostatisticians or mathematical modellers with appropriate expertise will only be possible if the workforces in those disciplines are healthy. Currently they are not adequate to meet the needs. There is a lack of suitably qualified people, and there are problems with funding. These two issues are inter-related: the nature of much of the funding available and the attitudes of some of the universities to the profession of biostatistics have had a significant impact on the numbers of people seeking a career in biostatistics in New Zealand. Funding through specific research grants generally provides only short term partial support, which leads to patchy employment and the lack of a career structure. The current form of support provided by the Health Research Council is an excellent model, and extension of this would allow needs to be met.

Providing training opportunities for biostatisticians and entry level positions is essential for maintaining a workforce in the long term.

Efforts must be made to encourage more students into careers in the mathematical sciences. The education provided by the universities forms an excellent basis, but further training is then required specific to the field of biostatistics and epidemiology. The HRC has been active in trying to encourage statisticians into biostatistics by providing training grants, but their success has been limited by problems with career structures and funding support. In the past some of the HRC funded positions were reserved as training positions, but with the increase in the volume of work it has become necessary to employ experienced biostatisticians in these positions in an attempt to meet demands.

In addition to collaborating in epidemiological research, the mathematical community has a role in educating the public health workforce. This could take place through involvement in tertiary education, or through workshops and seminars. Some epidemiologists also have a background in biostatistics, so that providing education in biostatistics may also assist in training an epidemiology workforce.

Mathematical scientists working in the field of epidemiology should be encouraged to carry out research in methods applicable to epidemiology.

While this is often not seen as a priority, it is essential for maintaining a workforce whose knowledge is up to date, and has the capacity to develop methods when they are necessary for a particular epidemiological research project.

Development of links between centres would also help to match particular mathematical expertise with the specific requirements of a research project.

Financial Mathematics

The following comments are indicative of the author's opinion, and are believed to be accurate. Nevertheless, this paper has not been backed up by extensive research as to the numbers of students, and the precise nature of the institutions mentioned, and the courses that they run.

The label financial mathematics is a broad one, and it would seem reasonable to consider the subject under 3 headings, since the following comments ought to be differentiated depending on what one understands by financial mathematics.

At the first and most basic level financial mathematics comprises basic compound interest calculations, perhaps with discounted cash flow calculations. Incorporating the next level of difficulty, financial mathematics typically comprises topics such as the valuation of loans and annuities, calculating the duration of investments, and using probabilities applied to cash flows. At the third and highest level, financial mathematics comprises the use of stochastic processes to model security prices, mainly with the aim of pricing derivative securities.

In applied financial research, there are constantly more exotic derivatives, defined in more varied sorts of markets. There are constantly more means used to hedge risk, and more sophisticated ways of spreading both financial and generic risk.

In theoretical financial research, a pronounced trend in recent years is the greater use of heavy-tailed statistical distributions to model returns and share prices.

In financial and business practice there is increasing use being made of hedging, whether in currency, shares or commodity markets, and in risk management generally; and also within and outside firms.

On a personal level, there is increasing interest in DIY personal finance, especially in an era of likely decreasing government funding for old age pensions. With the advent of technology, more people desire to plan and track their own investments over time. But they are frequently held back by the absence of the most basic financial mathematical skills.

There are probably about 50 or 60 people in New Zealand who are comfortable at the third level of financial mathematical expertise, with about 10 graduates every year at that level. The rarity value for the graduates at that level is because of the combination of skills required, viz. about honours level mathematics, statistics and finance.

Most of the people with that third level of financial mathematics skills are in the finance sector. While there are some 30 finance specialists in New Zealand universities, most of them are at most at the second level of expertise mentioned above. There are perhaps 8 or so strongly quantitative finance specialists in universities in New Zealand.

Taking the number of actuaries as a rough proxy for the demand for financial mathematicians, the number of qualified actuaries in New Zealand is about 90, that in Australia is about 600. Student numbers are roughly the same as numbers of fellows, so that the numbers in those actuarial societies is about double the numbers given. The number of qualified actuaries in Australia has increased dramatically over the last 3 or 4 years, while the number in New Zealand has not changed much over the same period. A tentative conclusion is that New Zealand is lagging behind Australia in numbers of financial mathematicians, and the difference is increasing over time.

A strength in New Zealand is the development of the financial mathematics programme at Victoria University of Wellington. It is one of the few programmes in Australasia (in fact the author knows of no other such programme) which offers basic actuarial skills together with the opportunity to study finance and mathematics/statistics. It differs from similar programmes in Australia, especially at Melbourne and Macquarie universities, which offer a less flexible programme centred strongly on actuarial studies.

At a less exalted level, most commerce graduates in New Zealand would have some skills at the most basic level of financial mathematics. But there has been a strong tendency for commerce students to head into less conceptually demanding areas such as marketing, management and tourism; and far less tendency to go into quantitative areas. There is very little recognition within Commerce faculties of the need for anything more than the most basic of mathematical tools. This has severe ramifications at higher levels of study for those students without these basic tools.

It is only under exceptional circumstances that honours level skills of mathematics, statistics and finance can meet. The standard way in which this happens is through conjoint degrees, typically in Commerce and Science, as in Auckland. In Wellington, the main source of that mixture of skills is less through the conjoint degree than through the financial mathematics programme. In Otago, there is a melding of finance and statistics streams which is promising for the future supply of graduates at this level. The other universities in New Zealand apparently produce few graduates at this level.

The basic numeracy, let alone the ability at the lowest of the 3 financial maths levels mentioned above, of much of New Zealand’s workforce, is recognised to be poor. There is some in-house training within some large financial institutions to bring people up to say the first level of financial mathematics skills mentioned above. But these methods are ad-hoc, and un-coordinated between institutions for fear of giving away a competitive advantage.

There is the need for a better informed debate within circles of well informed lay people as well as academics of many issues regarding risk. Consider, for example, the aging problem; the debate about the future funding mechanism of ACC; the appropriate level of funding for earthquakes and other natural hazards which are liable to strike New Zealand. All of these problems demand quantification of risk on the one hand and financial mathematical tools for dealing with the financing of those risks on the other. Much of this work is done outside our shores if it is done at all at the moment.

There are new and improved ways of handling risk, and hedging risk, especially financial risk. Very recent developments in the financial markets include risk bonds, and futures markets in insurance.

It is possible to teach basic financial mathematics at night school and extra-mural studies. Such courses would need to be labelled attractively (possibly mentioning mortgages and personal finance), and the length would need to be carefully thought through. Teaching tools would contain lots of diagrams, and there could be provision for access to calculators or simple spread sheets during the course.

There is at the moment practically no effort being made in New Zealand to teach the sorts of courses mentioned under the last heading. The development of such courses would seem to be a worthy candidate for PGSF funding, and would not be an overly expensive item. After the initial development of these courses, they should be profitable on a stand alone basis.

The financial mathematics programme at Victoria University of Wellington has links with the Institute of Actuaries of Australia, and is the only university in New Zealand able to offer exemptions from professional actuarial exams.

While there is informal contact between academic financial mathematicians, and at least one umbrella organisation for those working in money and finance areas (mainly in Wellington), there is little coordination if any of the teaching of finance in New Zealand to ensure a stronger flow of graduates with the third skill level mentioned.

Industrial Operations Research

This appendix deals with operations research (OR) from a New Zealand industry perspective, in particular focussing on this activity in relation to the mathematical sciences. Although its theoretical basis is mathematical in nature, much of operations research in practice is not mathematical, especially in an industrial context. We will not discuss this here and confine our attention to the mathematical aspects of the subject as it is applied in New Zealand industry. The author is not employed by New Zealand industry, but has been involved in consulting work with New Zealand industries for the past 10 years. The opinions expressed here are those of the author, and do not reflect a corporate view of the ORSNZ.

Most industries in New Zealand do not have OR capability, and outsource their OR projects. There are a number of large management consulting firms (e.g. Putnam-Hayes-Bartlett, BCG, Coopers and Lybrand) who carry out OR consulting work with industrial clients. A considerable amount of OR consultancy is carried out in collaboration with University groups, either as sponsored student projects, or as Faculty consulting jobs. The mathematical end of OR consulting in New Zealand tends to focus on providing computing systems for clients to help them solve their problems, rather than written reports and recommendations. There are notable examples where reports have been written (e.g. the work of EMRG at Canterbury University) which contain considerable mathematical content.

It is possible that the explosive growth in management consultancy services seen worldwide is a passing phase. Certainly industry in the future will become more discerning in their acceptance of management consulting services, leading to an emphasis on quality. To guarantee successful OR implementation the education and training of the users of OR in the client organisation will become increasingly important. Indeed as the emphasis on nurturing knowledge workers increases into the next century, one might expect opportunities for a more literal OR consulting role (as a teacher) to replace the provision of glossy reports or turn-key solution boxes.

It is hard to be specific about gaps in OR in New Zealand industry, as it seems that there are many. The difficulty in comparison with larger countries such as the US is that the range of OR problems is not much less in a small country than a large one, so there are bound to be gaps because of a shortage of OR practitioners. (To keep this in perspective, OR in New Zealand has a much larger representation per capita—in terms of members of the respective OR Societies— than Australia does.) The existence of gaps presents great opportunities for the expansion of the discipline in New Zealand industry. The industries where these opportunities (from a mathematical perspective) are most evident are Finance, Agriculture, Fisheries, Manufacturing, and Telecommunications.

Recent breakthroughs in OR from a mathematical perspective have come from improvements in computer algorithms for optimisation, in particular interior-point methods and integer-programming techniques, and the development of modelling languages for mathematical programming. Work on metaheuristic techniques such as genetic algorithms, simulated annealing and tabu search, are leading to more widespread application of these ideas. Neural networks have also become very popular as a forecasting tool. The next decade will see a rapid growth in the development of models to help make decisions under uncertainty.

New technologies will have an impact on OR in three broad ways. First, information technology improvements will allow the solution of larger and more complex models more quickly. One might expect the routine application of optimisation procedures as subroutines for much more comprehensive models incorporating uncertainty, market behaviour, and complicated logical constraints. Secondly information technology will change the form of the models being studied. For example the decreasing cost of Global Positioning Systems (GPS) and Geographical Information Systems (GIS) will allow more sophisticated real-time vehicle routing and traffic engineering systems to be developed. The increase in bandwidth on the internet will lead to a continuation in the trend towards more decentralisation in decision making, which will require the development of new OR models. Finally new technologies will generate whole new classes of OR models and questions.

There are huge opportunities for Operations Research in all sectors of New Zealand Industry.

The recent move to make PGSF projects focussed on outcomes relevant to end users is a positive step for OR in New Zealand industry. New Zealand industry has historically been receptive to OR, and one hopes that this change will lead to increases in the number of OR research projects being funded by the PGSF. Like other parts of the mathematical sciences, OR methodologies span a number of output classes, and so there is a danger that it will receive funding from none. It is desirable that the output class, Information and Communication, be expanded to reflect this.

Operations Research is an interdisciplinary subject. It has strong links to Statistics, Mathematical Programming, Engineering, Commerce, and Economics. Many industries in New Zealand have links with New Zealand University OR Departments as well as local and overseas management consultancy firms (witness Air New Zealand and University of Auckland, Fletcher Challenge and MacKinseys, ECNZ and Putnam-Hayes-Bartlett). Some industries (e.g. Air New Zealand) send delegates to international OR conferences. Most industrial organisations, however, are relatively ignorant of the international state-of-the-art in operations research.

In this respect links with the New Zealand academic OR community (as a conduit to the world OR community) are vital.

Mathematical Modelling

Graeme Wake (University of Auckland) and Graham Weir (Industrial Research Ltd)

Mathematical modelling involves the use of mathematics to represent, either approximately or exactly, a physical, economic, biological, agricultural, or sociological process and many others.

Mathematical Modelling is a multidisciplinary, cross-output activity. It is rigorously based in mathematics and statistics but is problem-focused and driven. It provides a source of new ideas and methodologies for mathematics. It is a critical part of many science research programmes and underpins many investigations and processes. Mathematical Modelling can provide an auditing role on many processes and activities by providing cost benefit analysis, performance appraisal, and strategy development. Areas which benefit from this include energy, environment, agriculture, industry, health and finance. Indeed most PGSF areas should involve mathematical modelling and simulation.

New Zealand has strength in some specific areas:

• geophysical modelling,
• geothermal systems modelling,
• mathematical biology and agriculture,
• medical fluid dynamics,
• polar marine physics and modelling,
• industrial modelling (primary and secondary industries).

A likely trend is that teams will form and strengthen in response to the output structure. Cooperation should be encouraged but the present funding structure can work against this. The present tension between horizontally based disciplines and vertical output structures remains.

Internationally it is more pervasive with larger industrial participation, often in partnership with academics. We need more strength in computational mathematics as an adjunct to modelling and more depth in existing areas.

New Zealand is small and many groups cannot afford their own Mathematical Modelling group or even an individual. Therefore some teams are required - say in each main centre. These groups need to be very user oriented but also require excellent Mathematical Modelling skills.

The whole spectrum of mathematics from very pure to very application oriented is needed to support a good Mathematical Modelling effort but the balance in New Zealand is probably wrong at present. There is perhaps too much emphasis on pure mathematics and statistics at universities and high schools. A curriculum reform to include more Mathematical Modelling is necessary.

Modelling is often dropped out of the vertically-structured applied sector programmes. We suggest each output has a formal scope (pool) for quantitative analysis and this be a pool which can be bid for and used, for modelling and nothing else.

A research base is needed which can adapt rapidly to urgent modelling needs. This base must be high quality since simple solutions found by non-modellers may be inaccurate and lead to economic disaster. Many physical systems behave in complicated non-intuitive ways and the mathematics is needed to understand these. Mathematics is the language of the physical, financial, and biological world.

Although these processes are critical to New Zealand’s development and economy, the application of modelling is under-utilised. There is the absence of critical mass and enough support to sustain it. Within mathematics there is insufficient support given to modelling. The need for inter-institutional links is clear and more cooperation needs to be encouraged and facilitated. The loss of Output 36 in PGSF was a pity for applied mathematics and we are now in the gap between PGSF and the Marsden Fund, with the latter being rather unsympathetic to problem-driven investigations.

Development of new mathematical modelling tools (such as leading finite element and computational fluid dynamical methods, numerical schemes for stochastic models, etc.) for generic use across outputs is now more difficult. For example the popular and useful web site on finite elements was initiated within a CRI under the now defunct output 36. Such initiatives are important and need fostering, if New Zealand is to keep abreast of developments in mathematical modelling.

The vertical output structures make the role of Mathematical Modelling less visible and hard to sustain.

• A critical mass as per the recommendation below. This would require administrative structures to change (employment opportunities etc) and need more personnel. A critical evaluation could be undertaken to identify which groups are doing well.
• New Zealand create a Centre for Applications of Mathematics and Statistics which links institutions, has a national focus, and has a movable programme around the country. This Centre to have a network of activity with regional convenors and minimal administrative overheads. At present the modelling community is too few in number and we are under-funded and over-stretched. We require a funding stream, otherwise it will be difficult to increase members and make further useful impacts.

This report has benefited from the comments of Associate-Professor Mike O'Sullivan (University of Auckland), Dr Mick Roberts (AgResearch) and Professor Vernon Squire (University of Otago) on an earlier draft.

Numerical Analysis and Scientific Computation

Numerical analysis deals with the systematic study of computational methods for the solution of scientific problems. Traditionally it considers several large problem types, such as linear algebra, ordinary differential equations, partial differential equations, and numerical quadrature. Each problem type has wide-ranging applications in many scientific disciplines. For each problem type, numerical methods are developed and studied from the points of view of accuracy, stability and computational efficiency.

Scientific computation refers to the closely-related practical implementation aspects of numerical methods.

New Zealand research in numerical methods and scientific computation has a long and distinguished history, dating from before the invention and general availability of modern computing equipment. However, both here and overseas, interest in and understanding of computational methods has grown most rapidly since the time that computers became fast, affordable and widely available. Traditional approaches in mathematical modelling and applications in many sciences have been affected by computational methods. There is today less emphasis on purely mathematical methods in favour of various types of computer simulation and modelling. With the rapid growth in new technologies, especially concerned with parallel and distributed computational techniques, these trends are likely to accelerate. Even though some traditional applications of mathematical analysis to applied mathematics have decreased in relative importance, others have grown to take their place. The reason for this can be found in the great complexity of many current computational schemes. These require new and deep analytical methodologies for their understanding and for their most effective computational application.

Because of its small size, New Zealand has to stand aside from many vigorous research efforts taking place throughout the world. However, it plays a full role in several key areas and it is essential that it continue to do so. This makes it possible for scientific links to be forged and maintained which give us a credibility in this strategically important area. It also enables us to retain the latest computational methodologies at our disposal, as these develop.

New Zealand has had more than its share of successes in many areas of computational science. However, it will be more difficult in the future to remain at the forefront of knowledge and research, simply because the questions being studied are more difficult and relate, in many cases, to computational tools to which we have very limited access. It is likely, for example, that methods will be found for solving many problems effectively, using, as yet unknown computer architectures, where such problems could not be handled at all in the past.

New technologies affect computational science in a fundamental way. In fact all mathematical sciences will be affected in much the same way but, for computation, the change will be more fundamental. The reason for this is that emerging technologies will simply transform other parts of mathematics to a more computational form. It is a matter of semantics what this does to the boundaries between the different mathematical sciences; but it is certain that links between the different areas will be stronger. Scientific computation will remain as ever a contributor to the development of other mathematical sciences and, at the same time, will continue to seek and receive inspiration from other sciences as sources of challenging and appropriate problems.

Many of the most intractable computational problems are in the area of partial differential equations and computational fluid mechanics. These have many applications to New Zealand industry and economic development. It is essential that strong links be formed between the numerical analysis community and the mathematicians and applied scientists and engineers who work most closely in these specialised application areas.

Every sector of the mathematical sciences is struggling for support and this is likely to continue to be the case. Both theoretical and applied aspects of numerical computation have a particularly difficult time in these struggles. This is partly a consequence of their growing significance because the needs of general approaches to computation are easy to confuse with particular application areas for which this need is specifically expressed. For example, technology changes will inevitably lead to the possibility of developing new algorithms and software for solving linear algebra problems. This is not only an appropriate activity for numerical analysis as a generally applicable science, but becomes a specific need of statistics, operations research, differential equations and other mathematical sciences which all use numerical linear algebra as part of their computational apparatus. Developing general techniques is often more difficult to justify but can be more cost effective than individual applications of the same technology. Funding decision-making processes will need to take this into account.

Numerical analysis has developed within universities as a teaching need, and research has flowed from this because of the enthusiasm of its practitioners. However, there has never been any national commitment to providing a basic cover. There are many areas which are vital to scientific applications for which there is no expertise in New Zealand at all. A centre of excellence, in at least one university, might be a way of overcoming such gaps.

Official Statistics

Statistics New Zealand has about 27 people whose involvement in Official Statistics involves applying "mathematical sciences". Across the public sector as a whole there would be perhaps another 20, mostly small numbers of individuals doing analysis in policy areas. Each of the universities has one or 2 staff whose research interests include topics central to Official Statistics (Sample Design, Time Series Analysis, Analysis of Data from Complex Surveys,...), and there are a handful of private consultants. These numbers are more than a little arbitrary; many more people are producing and using official statistics, at a lower level of sophistication.

The mathematical statisticians in Statistics New Zealand are predominantly young (as can be seen from the Statistics New Zealand return to the Mathematical Review survey), which is probably true of the other mathematical scientists working in official statistics in the public sector. The university experts are predominantly at the other end of the age scale. Obvious concerns arise about retaining the younger workers in the field, and ensuring older experts are replaced.

Unlike the situation some years ago (when there were groups like Applied Mathematics Division, the biometricians in Agriculture and Fisheries, etc.) Statistics New Zealand (henceforth SNZ) is now by far the largest single group of statisticians in New Zealand; other than perhaps Reserve Bank economists and Treasury analysts, other groups in the public sector would be only one or two individuals. (NZIER economists in the private sector might also be counted). Increasingly, statisticians in the universities are dispersed among various faculties, or absorbed into larger mathematical sciences groupings.

The resulting fragmentation is a serious constraint on the progress of Statistics in New Zealand.

There is an increasing tendency for students of Statistics to turn away from theoretically-oriented to practically oriented courses or what are simply service courses. Perhaps this is under the impression that this makes them more employable, but for employment in Official Statistics the opposite is true - only good statistical practice based on a sound theoretical understanding will enable the new techniques described in the journal literature to be adapted to the New Zealand situation.

Strengths in New Zealand Official Statistics include the soundly based core house-hold and business surveys run by SNZ, using up-to-date, but not necessarily state-of-the-art methods. There are some research strengths, some involving joint projects between SNZ and universities, such as improvements to seasonal adjustment techniques, confidentiality assurance techniques, and some aspects of analysis of complex survey data. On the whole, though, the major part of any New Zealand contribution to methodological research in Official Statistics is more likely to be through the testing we do to adapt new techniques to our conditions (which can be an important contribution, as our economy is sometimes rather different from the large economies in which the techniques are developed.)

An important recent breakthrough has been in use of computationally intensive methods in Statistics (e.g. Bootstrap, Gibbs sampling). These methods have not yet had a major impact on New Zealand Official Statistics.

Also, important overseas work on the effect of imputation on data (such as Rubin's work on multiple imputation) has not been followed up here. Improvements in techniques for integrating data from different administrative and survey sources is also an area in which not much work is done here, and which would bring dividends.

Methodologies used in Official Statistics are under wider scrutiny than in the past, and there is more emphasis on measuring non sampling error.

Changes in technology have revolutionised Official Statistics in the last 30 years, and will continue to do so. This has been primarily computing technology; increasing power and storage capacity of computers has lead to an exponential increase in data collection in one form or another, and has totally changed the way we analyse and visualise data.

Allied technologies are also important. One such technology, starting to be applied overseas, but not yet here, is survey interviewing through hand-held computing. Data input technology is changing (e.g., scanning for the 1996 Census) and it is likely this will be radically different in 10 years time. Data dissemination may also be radically different by then - e.g., access to output through the internet. Methodological research overseas will increasingly start from use of these newer collection technologies as standard; unless New Zealand remains not too far behind other countries of our type in use of this technology, we will no longer be able to use such overseas work here, in a way we have relied on in the past.

As stated above, technology will also make it possible to use and integrate administrative records of government and business, in completely new ways. There is increasing interest in derived measures of difficult concepts such as competitiveness, productivity and poverty.

There are substantial gaps in existing statistics on areas important for policy e.g., statistics on Maori, longitudinal data.

There is considerable opportunity, through the use of better data collection and more sophisticated statistical analysis, to raise the level of policy advice and debate. (e.g., on superannuation policy). The 1995 report to MoRST "Drawing on the Evidence" by Professor Hawke noted that the Social Science sector is underdeveloped in relation to comparable research areas in the physical and biological sciences.

Sound statistical methodology and "best statistical practice" are at the heart of the scientific method, and necessary inputs to the enhanced decision making and planning that underpin New Zealand's competitive economy and contribute to a more complete understanding of its natural environment. It is thus essential that proposals using statistics and mathematics are assessed by professionals in these areas as well as by experts in the subject matter.

Statistics straddles the existing PGSF output classes, and it would be good to have an output for "methodological research with an applied focus"

Statistics New Zealand has strong links with major overseas statistical agencies (the Australian Bureau of Statistics, Statistics Canada, Netherlands Bureau, Statistics Sweden, US Bureau of the Census, ...). Particularly through the ABS link, it has been possible, in conjunction with the universities, to bring over to New Zealand such recent statistical visitors to Australia as Danny Pfefferman, Jim Durbin, and (in 1998) Estella Dagum.

Contact of this kind is essential; while there have been improvements over the last 10 years (e.g., publication of journals such as "Survey Methodology" and "Journal of Official Statistics"), a lot of methodological detail does not make into the major statistical journals, or is excluded from discussion of the results in subject matter journals.

It is also essential to improve linkages between statisticians inside New Zealand, and also between the official statistical community and applied social science and policy analysts. The setting up of a Center for Research into Official Statistics, as recently proposed, is one initiative which could help. Initiatives aimed to give researchers better access to data, such as the Data Laboratory SNZ has set up, could also help.

This has been circulated to a number of people in Statistics New Zealand, and has been altered to account for 4 replies from workers in Official Statistics outside Statistics New Zealand.

Operations Research

Operations Research (OR) is the application of scientific method and mathematical models to assist in the solution of decision problems.

The area profile identified several areas of strength in Operations Research in New Zealand, classified by application area rather than mathematical technique; this approach is guided by the fact that Operations Research should not be viewed as a collection of mathematical techniques but a discipline in which these techniques are applied to solve practical real world problems. The subject's generality makes knowledge trends difficult to forecast, although one can safely assume that they will definitely be driven by market need. This means that the knowledge trends in New Zealand will rely on the extent to which operations researchers can educate potential users as to the value of the OR approach.

The advent of spreadsheets and their inbuilt solvers has led to a greater awareness of optimisation models as useful decision tools. Modelling languages like GAMS and AMPL now make optimisation techniques easily available to users wishing to solve large scale mathematical programming models with continuous variables. On the other hand methods for solving discrete programs to optimality are much more problem dependent. One might hope that the development of integer programming modelling languages will overcome this, but this is unlikely in the short term. Big challenges remain in solving the seemingly intractable models which arise when uncertainty is incorporated into mathematical programming, and one might hope that some of these will be overcome. The growth of mathematical programming will continue, driven by applications in new areas such as engineering design, chemical engineering, finance, and systems control. In manufacturing and transport the application of optimisation technologies to small parts of the enterprise (e.g. trim loss) will grow as tools become more widespread. The use of mathematical programming as a strategic tool for companies has little widespread acceptance, because of difficulties in modelling uncertainty and competitive market behaviour. In manufacturing there is a need to address flexibility in production planning as well as cost. There is no accepted modelling methodology for a world with rapidly changing technology.

Research on OR in New Zealand is primarily the domain of university departments. There is a lot of work being done in the US and Europe by mathematicians and operations research people in financial engineering, especially in models and techniques for evaluating financial derivatives. This is lacking here, and is critically important for issues such as valuation of forests, natural gas reserves, etc. New Zealand is also weak in the application of polyhedral combinatorics to the solution of combinatorial optimisation problems arising in OR applications. New Zealand is also lacking expertise in the proper application (using e.g. variance reduction techniques) of Monte-Carlo simulation methodology in industry.

What breakthroughs are currently occurring in the subject? Linear and nonlinear programming has achieved a classical status, and new developments in methodology are focussed on new and novel applications. There are exciting developments in the modelling and optimisation of systems subject to uncertainty, with applications in many areas. Forecasting using neural network techniques has become very popular with practitioners. What breakthroughs would cause a transformation in major parts of the discipline? Breakthroughs in OR, like the announcement of Karmarkar's algorithm in 1984, tend to result in furious changes of direction of research activity, especially in the US, with a somewhat attenuated impact on the end users. Some (unlikely) discoveries which would have a major impact are the resolution of the P = NP question, and the discovery of efficient computational schemes for general contingent-claims pricing.

In practical terms, geographical information systems (GIS) and global positioning systems (GPS) will lead to more sophisticated vehicle routing and scheduling techniques. They will also aid in solving facility location problems, and traffic congestion problems. The availability of cheap portable computer power will increase the opportunities for the discipline. The internet will change the way many companies market their products and services, and so it is bound to generate many interesting problems for operations research as this service grows. Developing methods of refining useful information from a deluge of net data will also be a fertile source of operations research problems.

Operations Research by definition endeavours to solve decision problems relating to situations arising in real organisations and industries. Many sectors in New Zealand currently benefit from this discipline, through the adoption of OR models by individual organisations. However there are a number of areas such as Agriculture, Horticulture, Transportation Planning, and Environmental Planning in which OR is under-utilised. Operations Research could also contribute a lot more to public policy decision making.

The recent indication that one of the criteria in judging PGSF grant applications will be the endorsement by end users of the research is good news for the field of Operations Research, which is essentially driven by the needs of end users. Of course enthusiastic endorsement of a project by any group of end users raises the possibility that FRST will recommend that the industry funds the research rather than the taxpayer. However, although this was a perceived problem for Operations Research in the early days of the PGSF, FRST seem to be willing to contemplate supporting very applied work, as evidenced by the industry commitment required to get a GRIF approved.

The main barrier to the capability of research in Operations Research is the availability of funds for postgraduate and post-doctoral fellowships. In general, the GRIF and TBG awards have been welcomed by the academic Operations Research community. However, there is still a need for a postgraduate award at the same funding level as the GRIF but with no requirement for the fellow to spend 50% of their time on the premises of the industrial partner. Many organisations do not have the infrastructure to support extra personnel, nor the expertise to guide the student in appropriate directions, and so they are unable to benefit from University research under the GRIF scheme. A common criticism of the awards is that GRIF students also spend less time with their peers, which detracts from the overall educational experience of being a graduate student, as well as making it harder for academic research groups to form a critical mass of research activity.

Probability and Stochastic Processes

Stochastic processes deals with modelling incompletely (partially) understood systems, typically evolving over time and/or space. Stochastic or random components are substituted for the unknown, or uncontrollable factors, and the resulting system analysed in order to evaluate proposed changes, better understand the underlying system, or to make predictions. The remarks made under the heading "Mathematical Modelling - a Definition" are apposite here.

Probability is the tool used in this endeavour, much as calculus underlies mathematical modelling, and a research topic in its own right.

The nature of the subject is changing, with greater reliance on computation and graphics going hand in hand with increasing emphasis on the modelling of large, complex, interacting systems.

Computational methods and discrete event simulation/scenario generation are thus strong research areas. There is more interest in modelling generally, particularly in the financial field, where many overseas financial institutions employ numbers of PhDs in the field.

Other developing research areas in this vein are stochastic geometry (models for spatial processes) and inference for stochastic processes (which is vital to modelling using stochastic processes).

The earlier knowledge base exercise identified the following gaps in New Zealand:

Stochastic Differential Equations, Stochastic Geometry, Stochastic Optimisation, Computational Methods, and the Theory of Stochastic Processes generally.

While these are "hot" research areas overseas (see the paragraph above), New Zealand lacks the practitioners to be able to do much more than be conversant with overseas developments.

The lack of theoretical practitioners weakens the field generally, and the use of computational methods is retarded by the lack of facilities.

The most important of these is Time Series Analysis (and Stochastic Differential Equations), because of the ubiquitous and increasingly important use of forecasting methods for better management and control of resources, economics, biological systems, etc. Telecommunications modelling is also a hot topic, with the onset of the "information age". All telecommunications models, for bandwidth, encoding, routing etc., are stochastic in nature, due to the random demands on the resource. Queuing theory has recently seen a renaissance with the development of fluid flow models, and work on stability of queuing systems. This has implications for the design and management of large telecommunications and computer networks. A final area that may lead to great things, particularly with the increasing power of computation, is that of Interacting Particle Systems (an area small in size at present).

This has already been alluded to above. As far as this field is concerned, this is a two-way street. Increasing computing power, electronic mail, and the use of the world wide web for increased (graphical) interaction, all enrich the field. One way of utilising the internet would be the establishment of a "virtual" centre, with links to people and organisations etc. On the other hand, models for the new technologies, such as communication, network design, dynamic routing etc. are one of the important research areas in the field.

Important opportunities exist in the financial industry, in hazard management through geophysical modelling, in resource management, and in Telecommunications. DNA profiling in Forensic science is an important probabilistic problem. A vast range of management decisions can benefit from consultation in this field.

It is important that there be some recognition of the underlying role of mathematics. Any proposal that relies on the application of mathematical and/or statistical techniques should be reviewed not only by people in the output field, but also by a professional statistician or mathematician. Further, since professional mathematical/statistical advice is often the first item dispensed with under a reduced budget, we suggest a post report critique on these issues.

Only in this way can quality of the output be assured.

An important issue is the age structure of the current body of researchers, with insufficient numbers being trained for replacement. Funding is needed for post graduate study, and the GRIF schemes etc., are not readily available to the mathematical sciences because of difficulty in finding an industrial partner. Professional development would be greatly enhanced by increased New Zealand involvement in overseas (particularly in Australia if it is established) institutes such as the Fields institute and the Isaac Newton centre. International linkages generally need more funding, due to New Zealand's geographical position. The maintenance of a "critical mass" in the field, supported by service teaching, is vital, particularly since there is no backup since the dissolution of the Applied Maths Division. Finally, the increased reliance of the mathematical sciences in general on computing requires the establishment of a centre for high performance computing, as a national resource. Funding should be sought from outside the Universities and CRIs.

The final version has benefited from the comments and suggestions of Peter Thomson and Ilze Ziedins.

Statistics

There seems little doubt that the factor that has had most impact on theoretical and practical developments of statistics in recent years has been the increasing availability of powerful computers. Simulation and other computer-intensive methods such as bootstrapping are being used more and more widely, and it seems inevitable that this trend will continue in the foreseeable future. One of the results of improved computer capabilities has also been a renewed interest in Bayesian inference where calculations which were once difficult to do can now be accomplished using new approaches such as Monte-Carlo Markov chain simulation.

Statisticians in New Zealand are generally keeping up with these world-wide trends, and this is likely to continue, providing that funding continues to be available to support visits to the country by leading overseas statisticians and for New Zealand statisticians to attend international conferences and to collaborate with overseas colleagues.

Statistical theory and methods develop in so many different areas of application that it is difficult to identify specific gaps. However, the wide range of application areas does itself raise some concerns. One of these is that there is insufficient communication between practitioners in different subject areas. This leads to innovative new approaches sometimes being rediscovered several times by researchers who are completely unaware of earlier work. Another concern is the very real danger that statistics as a subject will lose its central core entirely, with no communication at all between biometricians, econometricians, environmental statisticians, medical statisticians, etc. This can only be avoided by maintaining adequate funding for statisticians who are developing the subject without being specifically tied to any particular application area.

One gap that can be identified is the lack of research at the interface between the subjects of information science and statistics. Information scientists have developed new approaches such as neural networks for solving what are essentially classical statistical problems. As yet these methods have been largely ignored by statisticians. Another related gap is in the analysis of the huge data sets coming from geographical information systems. In the past much of the effort in statistics has involved extracting the maximum amount of information from small or moderate sized data sets. More attention is now needed on methods for dealing with very large sets of correlated data.

In the recent past, breakthrough areas have been associated with the improved computer technology already noted. Bootstrapping and other resampling methods have received a great deal of attention by researchers since the early 1970s, and the development and uses of Monte-Carlo Markov chain algorithms for inferences associated with models with many parameters is a more recent innovation.

The impact of new computing technology has already been noted. Improved communications are having an effect on the practice of statistics as it is in every other subject. It is now much easier to collaborate on a daily basis with colleagues on the other side of the world, and generally interactions between statisticians are becoming more international.

Although statistical methods are widely used in many socio-economic sectors in the country, it seems very likely that there are important opportunities that are still untapped.

A common complaint from statisticians about current PGSF priorities is that they fail to recognise the need for support for the development of generic statistical methods that can be applied in a wide range of applications, rather than being specific for a particular problem. Some recognition of the value of supporting research in applied statistics in general would be of great benefit to the subject.

By its nature statistics links in easily with other sciences, and much of statistical research is published in journals in applications areas. Indeed, individual applied statisticians often work with colleagues in a very wide range of different sciences. Traditionally there has been particularly strong links between statistics and the biological and medical sciences, both globally and in New Zealand.

There are a number of groups in New Zealand where the interaction between statisticians and other scientists is particularly noteworthy. For example:

• Work on the design and analysis of agricultural experiments is carried out by staff of AgResearch, Crops and Foods, HortResearch and universities.
• Staff from AgResearch, the National Institute of Water and Atmospheric Research, the University of Auckland, and the University of Otago are working on various problems related to environmental statistics.
• Inference methods related to forensic science are studied by staff at the University of Auckland.
• Methods for industrial quality improvement are being developed by groups working for Massey University, the University of Waikato, Industrial Research Limited, and HortResearch.
• Staff at Industrial Research Limited, Massey University and Victoria University are working on models in seismology, including those for quantifying and predicting earthquake hazards.
• Groups at AgResearch, Landcare, the University of Auckland, Victoria University, the University of Canterbury, and the University of Otago are working on problems related to animal and plant ecology.
• AgResearch has an active programme in the development of statistical methods related to animal genetics and linkage studies while some staff at the Universities of Auckland and Otago are involved with similar methods for human genetics.

International links in statistics are usually at an informal level, with individual statisticians or research groups collaborating with overseas colleagues via e-mail, with occasional travel by individuals in either direction. These links are crucial to the health of the subject in New Zealand. In addition, many New Zealand statisticians are members of international statistical societies and are involved in the management of these societies.

Although data are lacking on the number of links with other countries, it seems likely that these are mainly with Australia, Canada, the United Kingdom, and the United States.

Last modified 29.5.98. Final Version
Edith.Hodgen@vuw.ac.nz

• The Report
• Appendix 1 (The beginning)
• Appendix 1 (Table of Contents)
• Appendix 2