SCOPE 35 - Scales and Global Change

 ABSTRACT

Our current ability to evaluate interactions of biological, chemical, and physical processes is limited by our difficulty of coping meaningfully with small-scale spatial and temporal heterogeneity in terrestrial, freshwater , marine, and atmospheric environments. This paper reviews the problems of measuring interactions in the presence of spatial and temporal heterogeneity from the viewpoint of mathematical statistics, and suggests that the difficulty stems as much from a general failure to plan environmental research adequately as from any supposed properties of the biosphere or geosphere.

Particular attention is focused on the problems of sampling in environmental research and on the measurement of interaction by the use of factorial experiments. It is stressed that the vast improvements which have taken place in data processing and in the modelling of ecological processes cannot be used  as a substitute for the detailed and careful planning necessary for the valid  estimation of population parameters and interactions. However, there are a number of exciting possibilities in the development of mathematical and statistical techniques which, combined with strict methods of experiment and  survey design, will enable us to:

 (a) integrate scales, conceptually and practically, between and among disciplines
 (b) aggregate local scales meaningfully into larger spatial scales
 (c) evaluate how global scale processes influence local processes.

INTRODUCTION

The general objective of this Workshop is to identify and describe the research needed to solve the scaling problems associated with quantifying and interpreting interactions within the biosphere and geosphere. This objective is part of a wider study of spatial and temporal variability of biosphere and geosphere processes, and it is suggested that our current ability to evaluate interactions of biological, chemical, and physical processes is limited by our difficulty of coping meaningfully with small-scale spatial and temporal heterogeneity in terrestrial, freshwater, marine, and atmospheric environments. It is this difficulty that is identified as 'the scaling problem' .

This chapter reviews the problems of measuring interactions in the presence of spatial and temporal heterogeneity from the viewpoint of mathematical statistics, though without embarking on any exercises of formal mathematics. It suggests, using examples drawn from current ecological research programmes, that the difficulty addressed by this Workshop stems as much from a general failure to plan environmental research adequately as from any supposed properties of the biosphere or geosphere. Repeated attempts to aggregate uncoordinated research programmes in the past have failed for reasons which are entirely explained by the logic of the scientific method, and 'the scaling problem' is a reflection of our scientific inadequacy in dealing with the complexity of environmental systems.

STATISTICAL CONCEPT OF HETEROGENEITY

If the biological, physical, and chemical processes of the biosphere all behaved in a perfectly orderly way, we would detect variation in space and time, but we would not regard such variation as 'heterogeneity' .The deterministic models of the Newtonian calculus could be used, with more or less difficulty, to model the changes that take place in time and space, or both together. The Lotka-Volterra equations might be used to describe, for example, the relationships between the numbers of a predatory animal and the numbers of its prey (Maynard Smith, 1974). While we might occasionally be surprised at the magnitude of the changes arising from relatively small induced perturbations in the system-because of the effects of feedback and the non-linearity of the relationships-the same changes would occur each time that the same starting conditions were input to the system. The presence of heterogeneity indicates either that there are parts of the process which are not adequately accounted for in our deterministic relationships or that the system itself is non-deterministic. Indeed, any system which includes living organisms is certain to show some degree of heterogeneity because of the genetic variation which occurs through sexual reproduction. Models of the reaction of some organism to persistent applications of a chemical which do not allow for the possible selection of resistant strains are mere caricatures.

 The statistician regards the measurement of variation as being of greater importance than the measurement of central tendency, i.e. means or algebraic relationships, and most of the now extensive theory of mathematical statistics, deal with the measurement of variation and its characterization. Where many scientists concentrate on the average values and regard any apparent variation from those average values as 'error' , the statistician accepts the variation as an essential part of the system being studied, and often uses that variation constructively to improve the modelling of the system. The variation can frequently be described by statistical distributions with well-known properties and parameters that are capable of independent estimation. Even where the variation does not follow exactly one of the better known statistical distributions, such a distribution may often be used as an approximation, sometimes by relatively simple transformations of the scale or dimensions of the original measurements. The dispersion of radio-nuclides in grassland ecosystems, for example, while showing extreme heterogeneity, can be regarded as approximately normal if the measure of radioactivity is first transformed to its reciprocal.

The most important of the statistical distributions is undoubtedly the normal distribution, not because it is 'normal' in the accepted sense of that word, but because its mathematical properties have made it the basis of a very wide range of statistical theory. Many of the other distributions can be shown to be a special case of the normal distribution, and the Central Limit Theory states the very remarkable fact that the distribution of the means of samples from any parent distribution approaches the normal distribution very closely as the size of the sample increases. While it is certainly true that the variability displayed by many environmental systems is markedly non-normal in the statistical sense, this distribution remains at the centre of the development of statistical theory. A relatively recent trend towards the use of the distribution- free methods in statistics has, if anything, served to emphasize the importance of the main core of statistical development (McNeil, 1977).

Nevertheless, many of the other statistical distributions have an important role to play in our description of environmental systems. The randomness of the Poisson distribution is often used as a first test to establish the presence of a spatial or temporal pattern, and there are several alternative distributions to describe 'clumping' in space or time, while retaining a random element in the heterogeneity. Statistical distributions may also be used to detect discontinuity in variability, and are especially useful when that variability is multivariate, so that the variation is represented by a cloud of points in multi-dimensional space. Where the discontinuities exhibit hysteresis, so that a system has an apparently delayed response to a changing stimulus, and the response follows one path when the stimulus increases and another when it decreases, the topological models of catastrophy theory may serve to describe the variation and its apparent heterogeneity (Poston and Stewart, 1978).

 In short, variation and heterogeneity are neither unexpected nor unwelcome to the statistician (Bartlett, 1975). Much of his craft has been developed in order to describe, measure and characterize heterogeneity. However, if we regard variation, and hence heterogeneity, as a property of environmental systems, it follows that what we measure can only be used to characterize that variation if our samples are fairly drawn from some defined population. It is the role of fair samples that forms the content of the next section of this chapter.

THE ROLE OF FAIR SAMPLES

The total set of individuals about which inferences are to be made is said by the statistician to constitute a 'population'. Those individuals may be organisms, communities, societies, or whole ecosystems, or indeed any measure or characteristic of these individuals. While such populations will usually be finite in both time and space, they will often be so large as to make it impossible for every member of the population to be investigated, measured, or counted. As scientists, we are usually forced to work with samples drawn from the population, and we will need to make the assumption that those samples are representative of the complete population. Only then can values calculated from the samples be regarded as estimates of the same values of the population. Characteristic values of the population are defined as parameters, in contrast to the corresponding values of samples which may, under certain conditions, be regarded as estimates of those parameters, possibly as constants or coefficients in model equations.

Much of the problem of 'scaling' arises from this distinction. As an example, we may be concerned to make inferences about tropical forests in the world. Ideally, we would like to make valid inferences about all tropical forest, but such forests exhibit marked heterogeneity of almost all of their properties in space or time, or both. Our access to tropical forests may be limited by political boundaries, by difficulties of travel and working in remote areas, by climate and topography, and by the sheer magnitude of the task of measuring, say, biological productivity of a tropical forest ecosystem. Inevitably, we have to make our measurements on sample areas of tropical forest, and  those samples have to be selected so that they are representative of the population about which we want to make inferences. Also, because the area we can measure will be limited, our measurement of the heterogeneity of biological production will be related to the scale of the sample areas. In what sense, then, can we regard the samples as justifying inferences about our defined population?

To take another example, there is currently much interest in the effect of acidic inputs to the environment on a range of ecological systems, including agricultural crops, forests, and freshwater. The acidic inputs are derived from point sources of sulphur, oxides of nitrogen, and ozone, widely scattered. In the atmosphere, the original emissions are subjected to complex chemical changes, as well as dispersion, mixing, and concentration by the atmospheric pressure systems. Our measurement of the effects of the resultant pollutant mix on actual ecosystems is necessarily based on sample sites, and on records taken over sample times, and these measurements display considerable heterogeneity. Here, the definition of the population about which we are seeking to make inferences needs particular care. It would be unrealistic to assume that the population was the totality of atmospheric pollution. More likely would be the assumption that our sample records of pollution episodes and their effects were representative of a particular type of forest or freshwater system over a defined period of time. Our ability to make inferences about 'episodes' would also depend on the frequency with which we had recorded the presence of pollutants during the period of observation. Changing our scale of definition requires a change of experimental procedure.

The statistician's requirement for a set of samples to be regarded as 'representative' of some defined population is uncompromising. The samples must be taken by a method which is objective and unbiased. Selection by some form of subjective choice, guided by whatever personal consideration of the representativeness of the samples, will not do-an unfortunate fact which eliminates much of the use of 'case-studies' by economists, sociologists, and ecologists from the statistician's definition of valid inference. Two methods of objective sampling have been traditionally used by statisticians. Systematic sampling has the appeal of simplicity, and, having selected the first individual or location at random, the remaining sample units are taken at a fixed interval. However, unless systematic sampling is repeated, severe problems occur in estimating the precision of estimates derived from the samples, and in characterizing the heterogeneity of the population. For these reasons, statisticians have given most emphasis to random sampling. The simple expedient of ensuring a genuinely random choice at an appropriate part of the sampling procedure guarantees the lack of bias, and also provides a methodology for estimating the heterogeneity of the sampled population, and the precision with which population parameters are estimated from the sample (Green, 1979).

Considerable refinements of methods of random sampling have been developed during the last 60 years-the major period of statistical development. The precision of sample estimates can be greatly increased by stratification of the population, ideally based on knowledge of major differences in heterogeneity of different parts of the population. Alternatively, stratification can be used as a kind of insurance, spreading the coverage of the sample units so that estimates can be made with a defined minimum precision for each part of the population. Multi-stage and cluster sampling provide further adaptations to particular requirements, while retaining the essential properties of randomness. There can be little excuse today for any investigation to be conducted with an inadequate sampling methodology, so diverse is the range of methods available to ensure that valid estimates can be made of population parameters. However, it still needs to be stressed that the design of valid sampling has to be done before any data are collected. It cannot be superimposed on a data collection that has already taken place. Even today, scarce resources are often wasted by inadequate design of the underlying sampling technique, leaving the research worker with a frequently impossible task of trying to unravel results which can never be reconciled with statistical theory. If anything, the use of computers, and the widespread belief that data processing techniques can be found to solve any problem, have only helped to encourage scientists to slide towards post-hoc inadequacies (Jeffers, 1979).

MEASUREMENT OF INTERACTION

The statistical definition of interaction is that it is a measure of the extent to which the effect of one factor varies with changes in the strength, grade, or level of other factors in an experiment. Thus, the response of an organism to the available supplies of some nutrient may be modified by the concentrations of other nutrients or pollutants to which it is exposed. Clearly, interactions may be either very simple, as when only two factors interact linearly, or extremely complex, as when many factors interact, and some of these interactions are non-linear. In both physical and biological systems, interactions may be greatly complicated by the existence of 'feedback', i.e. the carrying back of some effect to modify the factors causing that same effect, either positively or negatively.

As long ago as the mid-20s, R.A. Fisher (1925, 1935) pointed out that interactions could only be investigated experimentally if all of the factors were included in the same experiment. Together with his co-workers at the Rothamsted Experimental Station, he developed the concept of factorial experiments through which some or all of the combinations of factors could be used to determine the strength and character of the interactions. Combined with the analysis of variance, of which regression analysis is a special case, multifactorial experimentation provides an extremely powerful tool for the measurement and characterization of interaction. Like valid sampling,. how-ever, the use of factorial designs requires preliminary planning and a rigorous approach to experimentation. Estimation of the higher order interactions requires replication of treatments, together with randomization of treatment combinations in order to ensure valid inferences to the target population.

Some 60 years later, it is sad to record that a significant proportion of the scientific community still does not understand what Fisher was saying in the 1920s. In every country in the world, developed or developing, major experiments are planned with neither replication nor randomization. The excuse is often that it would be too costly, or too difficult, to replicate treatments, and 'not commonsense' to randomize the application of treatments when it is obvious that one treatment would be more suitable for a particular experimental plot. The more costly the research, however, the more important it is that the experimental design should be efficient, and that the results that are obtained are capable of valid interpretation, including the correct identification of interactions (Jeffers, 1978).

Improvements in computing and automatic data collection are often used as an excuse for neglecting the provision of adequate design of sample surveys and experiments. Somehow, it seems to be felt that collecting and processing large numbers of observations can substitute for the features of design which are essential for the valid estimation of the effects of factors and their interactions. 'Within-plot' variation is commonly regarded as a perfectly adequate substitute for a valid estimate of 'experimental error'. Computational techniques, often poorly understood, are used to derive estimates of interactions from unplanned samples which do not cover the full range of admissible values of factors and their combinations. The effects of outliers to the main bulk of observations are frequently neither checked nor observed. 'Modelling' has sometimes been used as a substitute for careful thought about the logic of the experiment or survey which is being conducted (Jeffers, 1980).

MATHEMATICAL MODELS OF VARIATION AT DIFFERENT SCALES

However, I would not wish this paper to be regarded as being totally negative, and there are certainly some exciting possibilities for the understanding of biosphere or geosphere processes which exhibit variation at different scales. Indeed, it is these possibilities which underline the importance of the correct use of procedures of sampling and experimentation if the parameters of the heterogeneity exhibited by mathematical models are to be estimated efficiently.  A review of all the possible methods would make this paper too long, however, and it will be sufficient to mention a few which have especially interesting characteristics.

(a) Harmonic analysis 

Perhaps the best known of such methods is the harmonic analysis of time series data. Periodic variations of short duration can be superimposed on much longer term variations, with widely different frequencies and amplitudes being combined in the model. In recent years, some confusion has resulted from the competing claims and counterclaims of different schools of time series analysis, but there is now a reasonably unambiguous theoretical base for the analysis of consistently recorded data (Cliff and Ord, 1981). The fitting of harmonics to incomplete data remains both difficult and problematic, and is unlikely to yield useful results to the practical scientist, although theoreticians will undoubtedly continue to 'play' with the theory. As has been emphasized above, no useful statement can be made about variations which have not been sampled.

(b) Markov models

Although there is probably very little justification for the applicability of the Markov model to biosphere or geosphere processes, the model remains remarkably useful as a first approximation to changes taking place from one state to another when only the probabilities of the transitions can be estimated (Kemeny and Snell, 1960). Within major states, the variations in sub-states can also be modelled as a Markov process, resulting in a series of embedded Markov chains. Surprisingly, rather few attempts have so far been made to exploit the properties of Markov models possibly because most of the emphasis in the past has been on 'functional' models, using deterministic differential or difference equations (Usher, 1979).

(c) Fractal geometry

One of the most exciting developments of the last few years has been the use of fractal geometry to reproduce patterns at increasingly smaller scales. Some mathematicians have even hinted at the possibility that all of the heterogeneity which is evident in natural systems might be capable of being represented by this type of geometry (Mandelbrot, 1983). Most of us will probably remain somewhat sceptical until we have more evidence of the usefulness of the formulation in predicting change in environmental processes.

(d) Expert systems

Our experience of modelling biosphere processes has been almost entirely based on the use of procedural algorithms, principally because it was this type of algorithm which was emphasized by the development of the early. computers. The more recent development of intelligent knowledge-based systems and declarative programming heuristics has opened up some alternative approaches to the modelling of processes. There are already some examples of data sets which have proved difficult to intrepret by the more conventional statistical models, but which have responded to analysis by rule-based methods (Jeffers, 1985). Computer languages like LISP and PROLOG are already available for the exploration of models incorporating widely varying scales of heterogeneity, but, so far, relatively few ecologists have made the transition from the better-known procedural languages (Conlon, 1985).

CONCLUSIONS

I have argued that much of our difficulty with spatial and temporal scales within the biosphere/geosphere arises from a failure to apply the necessarily strict rules of the logic of the scientific method. That method demands the clear definition of the population about which we will seek to make inferences, and the adoption of valid methods of sampling and experimental design in order to estimate the parameters-including the heterogeneity-of the target population. If my argument is correct, it will, therefore, be useless for us to seek for computational methods which bypass the necessarily laborious and methodologically strict formulation of hypotheses and collection of data in order to:

   (a) integrate scales, conceptually and practically, between and among disciplines
   (b) aggregate local scales meaningfully into larger spatial scales
   (c) evaluate how global scale processes influence local processes.

Each of these goals, separately or in combination, requires the formulation of a priori hypotheses, and the collection of data in carefully prescribed ways so as to test those hypotheses explicitly. The search for a method, computational or conceptual, which will convert a haphazard collection of data into a useful working hypothesis is a delusion!

Elsewhere, I have provided checklists for sampling, experimental design and modelling (Jeffers, 1978, 1979, 1980). The difficulty that the scientist confronts in making a correct use of the scientific method is reflected in the fact that each of those checklists contains some 70 questions which need to be asked of any piece of research. Improvement in our ability to quantify and interpret the interactions of the biosphere and geosphere with a changing global environment depends on our convincing the working scientist-and his administrators-that time spent considering the logic of what he intends to do is not wasted. Indeed, taking sufficient time to plan the investigation with due regard to the difficulties of estimating population parameters, before any data are collected, is the essential  pre-requisite for the efficient use of the scarce resource of scientific expertise and equipment.

REFERENCES

Bartlett, M. S. (1975). The Statistical Analysis of Spatial Pattern. Cambridge University Press, Cambridge.

Cliff, A. D. and Ord, J. K. (1981). SpatialAutocorrelation. Pion, London.

Conlon, T. (1985). Start Problem Solving with PROLOG Addison-Wesley, London.

Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver & Boyd, Edinburgh.

Fisher, R. A. (1935). The Design of Experiments. Oliver & Boyd, Edinburgh.

Green, R. H. (1979). Sampling Design and Statistical Methods for Environmental Biologists. John Wiley & Sons, New York.

Jeffers, J. N. R. (1978). Design of Experiments. Statistical checklist No.1. Institute of Terrestrial Ecology, Huntingdon.

Jeffers, J. N. R. (1979). Sampling. Statistical checklist No.2. Institute of Terrestrial Ecology, Huntingdon.

Jeffers, J. N. R. (1980). Modelling. Statistical checklist No.3. Institute of Terrestrial Ecology, Huntingdon.

Jeffers, J. N. R. (1985). Ecological Advice through Expert Systems. Proceedings of First International Expert System Conference. Learned Information, Oxford.

Kemeny, J. G. and Snell, J. L. (1960). Finire Markov chains. Van Nostrand, New York.

Mandelbrot, B. B. (1983). The Fractal Geometry of Nature. W. H. Freeman & Co., New York.

Maynard Smith, J. (1974). Models in Ecology. Cambridge University Press, Cambridge.

McNeil, D. R. (1977). Interactive Data Analysis. John Wiley & Sons, New York, London.

Poston, T. and Stewart, I. (1978). Catastrophe Theory and its Applications. Pitman Publishing Ltd, London.

Usher, M. B. (1979). Markovian approaches to ecological succession, J. Anim. Ecol., 48, 413-426.