Book contents
- Systems of Frequency Distributions for Water and Environmental Engineering
- Systems of Frequency Distributions for Water and Environmental Engineering
- Copyright page
- Dedication
- Contents
- Preface
- Acknowledgments
- 1 Introduction
- 2 Pearson System of Frequency Distributions
- 3 Burr System of Frequency Distributions
- 4 D’Addario System of Frequency Distributions
- 5 Dagum System of Frequency Distributions
- 6 Stoppa System of Frequency Distributions
- 7 Esteban System of Frequency Distributions
- 8 Singh System of Frequency Distributions
- 9 Systems of Frequency Distributions Using Bessel Functions and Cumulants
- 10 Frequency Distributions by Entropy Maximization
- 11 Transformations for Frequency Distributions
- 12 Genetic Theory of Frequency
- Appendix Datasets for Applications
- Index
- References
11 - Transformations for Frequency Distributions
Published online by Cambridge University Press: 06 November 2020
Book contents
- Systems of Frequency Distributions for Water and Environmental Engineering
- Systems of Frequency Distributions for Water and Environmental Engineering
- Copyright page
- Dedication
- Contents
- Preface
- Acknowledgments
- 1 Introduction
- 2 Pearson System of Frequency Distributions
- 3 Burr System of Frequency Distributions
- 4 D’Addario System of Frequency Distributions
- 5 Dagum System of Frequency Distributions
- 6 Stoppa System of Frequency Distributions
- 7 Esteban System of Frequency Distributions
- 8 Singh System of Frequency Distributions
- 9 Systems of Frequency Distributions Using Bessel Functions and Cumulants
- 10 Frequency Distributions by Entropy Maximization
- 11 Transformations for Frequency Distributions
- 12 Genetic Theory of Frequency
- Appendix Datasets for Applications
- Index
- References
Summary
A wide spectrum of frequency distributions, used in hydrologic, hydraulic, environmental, and water resources engineering, are derived using transformations of some basic frequency distributions. The basic distributions that have been used are normal, logistic, beta, Laplace, and other distributions, and the transformations used are logarithmic, power, and exponential. This chapter revisits the distributions obtained by transformation and transformations applied to basic distributions.
- Type
- Chapter
- Information
- Publisher: Cambridge University PressPrint publication year: 2020
References
Box, G. E. P., and Cox, G. P. (1964). An analysis of transformation. Journal of the Royal Statistical Society B26, pp. 211‒252.Google Scholar
Box, G. E. P., and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. New York: Addison Wesley Publishing Company, pp. 156‒160.Google Scholar
Chander, S., Spolia, S. K., and Kumar, A. (1978). Flood frequency analysis by power transformation. Journal of the Hydraulics Division, ASCE 104, no. HY1, pp. 1495‒1504.Google Scholar
D’Addario, R. (1936). Le transformate Euleriane. Annali dell’Instituto di Statistica, Vol. 8, p. 67. Bari, Italy: Universita di Bari.Google Scholar
Edgeworth, F. Y. (1898). On the representation of statistics by mathematical formulae. Journal of the Royal Society 1, pp. 670‒700.Google Scholar
Frechet, M. (1939). Sur les formules de repartition des renenus. Revue de l’Institut International de Statistique 7, pp. 32‒38.Google Scholar
Gupta, D. K., Asthana, B. N., and Bhargawa, A. N. (1989). Flood frequency analysis by two-step power transformation. IE(I) Journal-CV 70, pp. 127‒131.Google Scholar
Hirsch, R. M. (1979). Synthetic hydrology and water supply reliability. Water Resources Research 15, pp. 1603‒1615.CrossRefGoogle Scholar
Hirsch, R. M. (1982). A comparison of four streamflow record extension techniques. Water Resources Research 18, pp. 1081‒1088.CrossRefGoogle Scholar
Jain, D., and Singh, V. P. (1986). A comparison of transformation methods for flood frequency analysis. Water Resources Bulletin 22, no. 6, pp. 903‒912.Google Scholar
Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika 36, pp. 147‒176.Google Scholar
Johnson, N. L. (1954). Systems of frequency curves derived from the first law of Laplace. Trabajos de Estadistica 5, pp. 283‒291.Google Scholar
Johnson, N. L., and Kotz, S. (1970). Continuous Univariate Distributions (1). New York: John Wiley & Sons.Google Scholar
Johnson, N. L., and Tadikamalla, P. R. (1992). Translated families of distributions. In Handbook of the Logistic Distribution, edited by Balakrishnan, N., pp. 189‒208. New York: Marcel Dekker.Google Scholar
Kapteyn, J. C. (1903). Skew Frequency Curves in Biology and Statistics. Groningen: Noordhoff.Google Scholar
Kloek, T., and van Dijk, N. K. (1977). Efficient estimation of income distribution parameters. Journal of Econometrics 8, pp. 61‒74.Google Scholar
Mahmoud, M. R., and Abe El Ghafour, A. S. (2013). Shannon entropy for the generalized Feller-Pareto (GFP) family and order statistics of GFP subfamilies. Applied Mathematical Sciences 7, no. 65, pp. 3247‒3253.Google Scholar
Majumder, A., and Chakravarty, S. R. (1990). Distribution of personal income: Development of a new model and its application to U.S. data. Journal of Applied Economics 5, pp. 189‒196.Google Scholar
McDonald, J. B. (1984). Some generalized functions for the size distribution of income. Econometrics 52, no. 3, pp. 647‒663.Google Scholar
Moog, D. B., Whiting, P. J., and Thomas, R. B. (1999). Streamflow record extension using power transformation and application to sediment transport. Water Resources Research 35, no. 1, pp. 243‒254.Google Scholar
Rasheed, H. R., Ramamoorthy, M. V., and Aldabbagh, A. S. (1982). Modified SMEMAX transformation for frequency analysis. Water Resources Bulletin 18, no. 3, pp. 509‒512.Google Scholar
Rietz, H. L. (1922). Frequency distributions obtained by certain transformations of normally distributed variates. Annals of Mathematics 23, no. 4, pp. 292‒300.Google Scholar
Snyder, W. M., Mills, W. C., and Knisel, W. G. (1978). A unifying set of probability transforms for stochastic hydrology. Water Resources Bulletin 14, no. 1, pp. 83‒98.Google Scholar
Tadikamalla, P. R., and Johnson, N. L. (1982). Systems of frequency curves generated by transformations of logistic variables. Biometrika 69, pp. 461‒465.Google Scholar
Tahmasebi, S., and Behboodian, J. (2010). Shannon entropy for the Feller-Pareto (FP) family and order statistics of FP subfamilies. Applied Mathematical Sciences 4, no. 10, pp. 495‒504.Google Scholar
Tiao, G. C., and Lund, D. R. (1970). The use of OLUMY estimates in inference robustness studies of the location parameter of a class of symmetric distribution. Journal of the American Statistical Association 65, pp. 370‒386.CrossRefGoogle Scholar
van Uven, M. J. (1917). Logarithmic frequency distributions. Koninklijke Akademie van Wetenschappen te Amsterdam Proceedings 19, no. 4, pp. 670‒694.Google Scholar
Venugopal, K. (1980). Flood analysis by SMEMAX transformation ‒ a review. Journal of the Hydraulics Division, ASCE 106, no. HY2, pp. 338‒340.CrossRefGoogle Scholar
Wilson, B. G., Adams, B. J., and Karney, B. W. (1980). Bias in log-transformed frequency distributions. Journal of Hydrology 118, pp. 19‒37.CrossRefGoogle Scholar