Numerical recipes for disaster: changing hazard and the stand-origin-map

The quantitative study of predictable temporal characteristics of disturbance phenomena is of tremendous consequence for both an effective understanding of biological process and for successful implementation of sustainable practices. This reliance is well demonstrated in both pure (e.g. biodiversity research) and applied (e.g. forestry sciences) research into patterns of fire-recurrence and its consequent interpretation and management implications. Much of fire-frequency prediction is dependent on the data manifest in a stand-origin-map. In this paper, we review the statistical methods for fire-frequency estimation from a stand-origin-map, where fire-recurrence is governed by an underlying probabilistic process that changes with time since last occurrence (i.e. non-constant hazard). We argue that both patterns of fire-recurrence that change with time are likely to be commonplace biological phenomena and that the conventionally prescribed methods for non-constant hazard fire-recurrence estimation from a stand-origin-map are erroneous. Even modified maximum-likelihood techniques that account for different degrees of data censoring are likely to be severely restrained in their capacity for rigorous estimation. We propose a novel formulation whereby the two-parameter Weibull distribution can be fitted retrospectively to the data in a stand-origin-map. This formulation is explored through an example from the Canadian Rockies.

[1]  Fu-Kwun Wang,et al.  Weibull maximum likelihood parameter estimates with censored data , 1997 .

[2]  William L. Baker,et al.  Effect of scale and spatial heterogeneity on fire-interval distributions , 1989 .

[3]  C. Hempel.  Philosophy of Natural Science , 1966 .

[4]  M. Finney.  The Missing Tail and Other Considerations for the Use of Fire History Models , 1995 .

[5]  E. Johnson.  Fire recurrence in the subarctic and its implications for vegetation composition , 1979 .

[6]  B. W. Wilgen,et al.  Fire and Plants , 1995, Population and Community Biology Series.

[7]  James S. Clark,et al.  FIRE AND CLIMATE CHANGE DURING THE LAST 750 YR IN NORTHWESTERN MINNESOTA , 1990 .

[8]  William J. Bond,et al.  Kill thy neighbour: an individulalistic argument for the evolution of flammability , 1995 .

[9]  S. Levin,et al.  The role of mosaic phenomena in natural communities. , 1977, Theoretical population biology.

[10]  D. Cox,et al.  A General Definition of Residuals , 1968 .

[11]  W. Reed,et al.  Estimation of Temporal Variations in Historical Fire Frequency from Time-Since-Fire Map Data , 1998, Forest Science.

[12]  David J. Strauss,et al.  Do One Percent of the Forest Fires Cause Ninety-Nine Percent of the Damage? , 1989, Forest Science.

[13]  A. M. Masters.  Changes in forest fire frequency in Kootenay National Park, Canadian Rockies , 1990 .

[14]  Wayne Nelson,et al.  Applied life data analysis , 1983 .

[15]  John H. K. Kao.  A Graphical Estimation of Mixed Weibull Parameters in Life-Testing of Electron Tubes , 1959 .

[16]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[17]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[18]  Michael E. Gilpin,et al.  Viable Populations for Conservation: Spatial structure and population vulnerability , 1987 .

[19]  D. Boychuk,et al.  A Multistage Stochastic Programming Model for Sustainable Forest-Level Timber Supply Under Risk of Fire , 1996, Forest Science.

[20]  J. Yarie,et al.  Forest fire cycles and life tables: a case study from interior Alaska , 1981 .

[21]  Peter F. Heil,et al.  Survival Distributions: Reliability Applications in the Biomedical Sciences , 1976 .

[22]  G. Tande.  Fire history and vegetation pattern of coniferous forests in Jasper National Park, Albert , 1979 .

[23]  W. Reed.  Estimating historical forest-fire frequencies from time-since-last-fire-sample data , 1997 .

[24]  E. Johnson,et al.  Fire Frequency Models, Methods and Interpretations* , 1994 .

[25]  W. Reed.  Estimating the Historic Probability of Stand-Replacement Fire Using the Age-Class Distribution of Undisturbed Forest , 1994, Forest Science.

[26]  S. Spurr.  The Forests of Itasca in the Nineteenth Century as Related to Fire , 1954 .

[27]  Edward A. Johnson,et al.  Historical vegetation change in the Kananaskis Valley, Canadian Rockies , 1987 .

[28]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[29]  W. Nelson.  Weibull Analysis of Reliability Data with Few or No Failures , 1985 .

[30]  P. Brown,et al.  Prescribed burning as a conservation management practice: A case history from the Cederberg mountains, Cape Province, South Africa , 1991 .

[31]  Ajith H. Perera,et al.  Modelling the effect of spatial scale and correlated fire disturbances on forest age distribution , 1997 .

[32]  C. E. Van Wagner,et al.  Age-class distribution and the forest fire cycle , 1978 .

[33]  M. L. Heinselman.  Fire in the Virgin Forests of the Boundary Waters Canoe Area, Minnesota , 1973, Quaternary Research.

[34]  C. E. Van Wagner,et al.  The theory and use of two fire history models , 1985 .

[35]  J. Fox,et al.  Bias in Estimating Forest Disturbance Rates and Tree Lifetimes , 1989 .

[36]  W. Romme,et al.  FIRE AND LANDSCAPE DIVERSITY IN SUBALPINE FORESTS OF YELLOWSTONE NATIONAL PARK , 1982 .

[37]  A. Cohen,et al.  Maximum Likelihood Estimation in the Weibull Distribution Based On Complete and On Censored Samples , 1965 .

[38]  G. Wood,et al.  Stand table modelling through the Weibull distribution and usage of skewness information , 1996 .

[39]  James S. Clark,et al.  Ecological disturbance as a renewal process: theory and application to fire history , 1989 .

[40]  Philip J. Riggan,et al.  Interaction of Fire and Community Development in Chaparral of Southern California , 1988 .

[41]  David M. Richardson,et al.  A Computer-Based System for Fire Management in the Mountains of the Cape Province, South-Africa , 1994 .

[42]  M. Soulé,et al.  Viable Populations for Conservation , 1987 .

[43]  E. Johnson,et al.  CLIMATICALLY INDUCED CHANGE IN FIRE FREQUENCY IN THE SOUTHERN CANADIAN ROCKIES , 1991 .

[44]  J. Bernardo,et al.  Psi (Digamma) Function , 1976 .

[45]  Daniel Polakow,et al.  Modelling fire-return interval T: stochasticity and censoring in the two-parameter Weibull model , 1999 .