STATISTICAL METHODS FOR ENVIRONMENTAL SCIENCE 1135
Stochastic models of environmental interest are often
multivariate. Mathematical models applied to air pollu-
tion may deal with the concentrations of a number of pol-
lutants, as well as such variables as temperature, pressure,
precipitation, and wind direction. Special problems arise in
the evaluation of such models because of the large numbers
of variables involved, the large amounts of data which must
be processed for each variable, and the fact that the distri-
butions of the variables are often nonnormal, or not well
known. Instead of using analytic methods to obtain solu-
tions, it may be necessary to seek approximate solutions; for
example, by extensive tabulation of data for selected sets of
conditions, as has been one in connection with models for
urban air pollution.
The development of computer technology to deal with
the very large amounts of data processing often required has
made such approaches feasible today. Nevertheless, caution
with regard to many stochastic models should be observed.
It is common to find articles describing such models which
state that a number of simplifying assumptions were neces-
sary in order to arrive at a model for which computation
was feasible, and which then go on to add that even with
these assumptions the computational limits of available
facilities were nearly exceeded, a combination which raises
the possibility that excessive simplification may have been
introduced. In these circumstances, less ambitious treat-
ment of the data might prove more satisfactory. Despite
these comments, however, it is clear that the environmental
field presents many problems to which the techniques of
stochastic modelling can be usefully applied.ADDITIONAL CONSIDERATIONSThe methods outlined in the previous sections represent a
brief introduction to the statistics used in environmental
studies. It appears that the importance of some of these
statistical methods, particularly analysis of variance, multi-
variate procedures and the use of stochastic modelling will
increase. The impact of computer techniques has been great
on statistical computations in environmental fields. Large
amounts of data may be collected and processed by com-
puter methods.ACKNOWLEDGMENTSThe author is greatly indebted to D.E.B. Fowlkes for his
many valuable suggestions and comments regarding this
paper and to Dr. J. M. Chambers for his critical reading of
sections of the paper.REFERENCESAitchison, J. and J. A. Brown The Lognormal Distribution, Cambridge
Univ. Press, 1957.
Anderson, T. W., The Statistical Analysis of Time Series, John Wiley and
Sons, Inc., New York, 1971.Bailey, N. T. J., The Elements of Stochastic Processes: With Applications to
the Natural Sciences, John Wiley and Sons, Inc., New York, 1964.
Bartlett, M. S., An Introduction to Stochastic Processes, 2nd Ed.,
Cambridge Univ. Press, 1966.
Bartlett, M. S., Stochastic Population Models in Ecology and Epidemi-
ology, Methuen and Co., Ltd, London, John Wiley and Sons, Inc.,
New York, 1960.
Bernier, J., On the design and evaluation of cloud seeding experiments
performed by Electricite de France, in Proceedings of the Fifth Berke-
ley Symposium on Mathematical Statistics and Probability, 5, Lecam,
L. M. and J. Neyman, Eds., University of California Press, Berkeley,
1967, p. 35.
Castillo, E., Extreme Value Theory in Engineering, Academic Press,
London, 1988.
Chernall, R. G., The impact of computers on archaeological theory, Com-
puters and the Humanities, 3, 1968. p. 15.
Cochran, W. G. and G. M. Cox, Experimental Designs, 2nd Ed., John Wiley
and Sons, Inc., New York, 1957.
Coles, S. G., An Introduction to Statistical Modelling of Extremes, Springer,
2001.
Computational Laboratory Staff, Tables on the Cumulative Binomial Prob-
ability Distribution, Harvard University Press, Cambridge, MA., 1955.
Cooley, William W. and Paul R. Lohnes, Multivariate Data Analysis, John
Wiley and Sons, Inc., New York, 1971.
Cox, B., J. Hunt, P. Mason, H. Wheater and P. Wolf, Eds., Flood Risk in a
Changing Climate, Phil. Trans. of the Royal Society, 2002.
Cox, D. R., and D. V. Hinkley, A note on the efficiency of least squares
estimates, J. R. Statis. Soc., B30, 284–289.
Cramer, H., The Elements of Probability Theory, John Wiley and Sons, Inc.,
New York, 1955.
Cramer, H., Mathematical Methods of Statistics, Princeton University Press,
Princeton, NJ, 1946.
Daniel, C. D. and F. S. Wood, Fitting Equations to Data, John Wiley and
Sons, Inc., New York, 1971.
Decker, Wayne L. and Paul T. Schickedanz, The evaluation of rainfall
records from a five year cloud seeding experiment in Missouri, in
Proceedings of the Fifth Berkeley Symposium on Mathematical Statis-
tics and Probability, 5, Lecam, L. M. and J. Neyman, Eds., Univ. of
California Press, Berkeley, 1967, p. 55.
Rapport Delta Commissie, Beschouwingen over Stormvloeden en Getijbe-
weging, III 1 S Bigdragen Rijkwaterstaat, The Hague, 1960.
Dixon, W. J., Ed., Biomedical Computer Programs, Univ. of California
Publications in Automatic Computation No. 2, Univ. of California
Press, Berkeley, 1967.
Fair, G. M. and J. C. Geyer, Water Supply and Wastewater Disposal, John
Wiley and Sons, Inc., New York, 1954.
Feller, W., An Introduction to Probability Theory and Its Applications, 3rd
Ed., 1, John Wiley and Sons, Inc., New York, 1968.
Fisher, N. I., Statistical Analysis for Circular Data, Cambridge University
Press, 1973.
Fisher, R. A., Statistical Methods for Research Workers, 13th Ed., Hafner,
New York, 1958.
Fisher, R. A. and L. H.C. Tippett, Limiting forms of the frequency distribu-
tion of the largest or smallest member of a sample, Proc. Camb. Phil.
Soc., 24, 180–190. 1928.
Fisher, R. A. and F. Yates, Statistical Tables for Biological, Agricultural and
Medical Research, 6th Ed., Hafner, New York, 1964.
Fowlkes, E. B., Some operators for ANOVA calculations, Technometrics,
11, 1969, p. 511.
Freund, J. E., Mathematical Statistics, Prentice-Hall, Inc., Englewood
Cliffs, NJ, 1962.
General Electric Company, defense Systems Department, Tables of the
Individual and Cumulative Terms of Poisson Distribution, Van Nos-
trand, Princeton, NJ, 1962.
Gumbel, E. J., Statistics of Extremes, Columbia University Press,
New York, 1958.
Gumbel, E. J., Statistical Theory of Extreme Values and Some Practical
Applications, Applied Mathematics Series No. 3, National Bureau of
Standards, US Government Printing Office, Washington, DC, 1954.
Harman, H. H., Modern Factor Analysis, 2nd Ed., University of Chicago
Press, Chicago, 1967.C019_004_r03.indd 1135C019_004_r03.indd 1135 11/18/2005 1:30:58 PM11/18/2005 1:30:58 PM