Basic concepts such as conditional probability distributions, conditional return periods, and joi... more Basic concepts such as conditional probability distributions, conditional return periods, and joint return periods are important to understand and to interpret multivariate hydrological events such as floods and storms. However, these concepts are not well documented in the open literature. This paper assembles and clarifies these concepts, and illustrates their practical utility. Relationships between joint return periods and univariate return periods are also derived. These concepts and relationships are demonstrated by applying a bivariate extreme value distribution to represent the joint distribution of flood peak and volume from an actual basin.
... The former is for detecting a monotonie trend and the latter for identifying a shift in mean ... more ... The former is for detecting a monotonie trend and the latter for identifying a shift in mean or median in a time series. Comparing to the parametric ?-test, nonparametric tests are more suitable fornon-normal data and ... Fanta et al. (2001) applied Fisher's test to assess the ...
Stochastic Environmental Research and Risk Assessment (SERRA), 2002
. The non-parametric Mann–Whitney (MW) statistic test has been popularly used to assess the sig... more . The non-parametric Mann–Whitney (MW) statistic test has been popularly used to assess the significance of a shift in median or mean of hydro-meteorological time series. It has been considered that the test is more suitable for non-normally distributed data and it may be not sensitive to the distribution type of sample data. However, no evidence has been provided to demonstrate
ABSTRACT This article proposes to use the computer-based bootstraptechnique to test the significa... more ABSTRACT This article proposes to use the computer-based bootstraptechnique to test the significance of the lag-k sampleserial correlation coefficient (rk) of a time series.The bootstrap approach is free of sample distribution types, freeof restrictive assumptions, and easy to understand and implement. Monte Carlo simulation indicates that the bootstrap test has thesame power as the conventional parametric test for assessing thesignificance of a lag-one serial correlation coefficient (r1). The usefulness of the bootstrap test is demonstrated by applying it to assess the significance of rk of the annual mean flow data of twenty pristine river basins in Ontario, Canada.
Stochastic Environmental Research and Risk Assessment, 2004
ABSTRACT Scaling properties of Canadian low flows, namely annual minimum mean 1-, 5- and 7-day fl... more ABSTRACT Scaling properties of Canadian low flows, namely annual minimum mean 1-, 5- and 7-day flows, are evaluated across Canada and in its sub-climatic regions. Across the entire country, the log relationship between the kth product moments (PMs, E[Qik]) of low flows and drainage area (Ai) can be represented by: ln(E[Qik])=ak+bkln(Ai)and bk=k, with = 0.86, 0.94 and 0.93 for annual minimum mean 1-, 5- and 7-day flows, respectively. The log linear relationships between the kth probability weighted moments (PWMs, ) and Ai are ln()=ck+Hln(Ai), in which H is constant and is independent of k. The values of H are 0.87, 0.97, and 0.96 for annual minimum mean 1-, 5- and 7-day flows, respectively, which are almost the same as the values. The coefficients of variation (Cv) are almost independent of drainage area. These results demonstrate that Canadian low flows generally exhibit simple scaling and drainage area alone describes most of the variability in the moments of the low flows. Low flows in each of the sub-climatic regions also obey a simple scaling law. The values of , H and Cv are different in each region, which may stem from physiographical and climatological differences among these regions. The finding lays a basis for applying the index flood method to conduct regional low flow frequency analysis as simple scaling is equivalent to the index flood method.
Stochastic Environmental Research and Risk Assessment - STOCH ENVIRON RES RISK ASSESS, 2004
There are two distinct bivariate extreme value distributions constructed from Gumbel marginals, n... more There are two distinct bivariate extreme value distributions constructed from Gumbel marginals, namely Gumbel mixed (GM) model and Gumbel logistic (GL) model. These two models have completely different structures and their dependence ranges are different. The product-moment correlation coefficient for the former is ??[0,2/3] and the latter is ??[0,1]. It is natural to ask which one is more appropriate for representing the joint probabilistic behavior of two correlated Gumbel-distributed variables. This study compares these two models by numerical experiments. The comparison is based on that: (i) if the two distribution models are identical, then the joint probability and the joint return period computed by the GM model should be the same as those by the GL model; and (ii) if a selected distribution is the true distribution from which sample data are drawn, then the probabilities computed by the theoretical model should provide a good fit to empirical ones. Comparison results indicat...
A univariate gamma distribution is one of the most commonly adopted statistical distributions in ... more A univariate gamma distribution is one of the most commonly adopted statistical distributions in hydrological frequency analysis. A bivariate gamma distribution constructed from specified gamma marginals may be useful for representing joint probabilistic properties of multivariate hydrological events such as floods and storms. This article presents a review of various bivariate gamma distribution models that are constructed from gamma marginals.
... Country Number of sites Catchment area (km ) Proportion to total area (%) Botswana Malawi Moz... more ... Country Number of sites Catchment area (km ) Proportion to total area (%) Botswana Malawi Mozambique Namibia S. Africa Swaziland Tanzania Zambia Zimbabwe Total 13 24 14 26 260 22 47 24 72 502 ... Page 6. 518 Berhanu Fanta et al. ...
Basic concepts such as conditional probability distributions, conditional return periods, and joi... more Basic concepts such as conditional probability distributions, conditional return periods, and joint return periods are important to understand and to interpret multivariate hydrological events such as floods and storms. However, these concepts are not well documented in the open literature. This paper assembles and clarifies these concepts, and illustrates their practical utility. Relationships between joint return periods and univariate return periods are also derived. These concepts and relationships are demonstrated by applying a bivariate extreme value distribution to represent the joint distribution of flood peak and volume from an actual basin.
... The former is for detecting a monotonie trend and the latter for identifying a shift in mean ... more ... The former is for detecting a monotonie trend and the latter for identifying a shift in mean or median in a time series. Comparing to the parametric ?-test, nonparametric tests are more suitable fornon-normal data and ... Fanta et al. (2001) applied Fisher's test to assess the ...
Stochastic Environmental Research and Risk Assessment (SERRA), 2002
. The non-parametric Mann–Whitney (MW) statistic test has been popularly used to assess the sig... more . The non-parametric Mann–Whitney (MW) statistic test has been popularly used to assess the significance of a shift in median or mean of hydro-meteorological time series. It has been considered that the test is more suitable for non-normally distributed data and it may be not sensitive to the distribution type of sample data. However, no evidence has been provided to demonstrate
ABSTRACT This article proposes to use the computer-based bootstraptechnique to test the significa... more ABSTRACT This article proposes to use the computer-based bootstraptechnique to test the significance of the lag-k sampleserial correlation coefficient (rk) of a time series.The bootstrap approach is free of sample distribution types, freeof restrictive assumptions, and easy to understand and implement. Monte Carlo simulation indicates that the bootstrap test has thesame power as the conventional parametric test for assessing thesignificance of a lag-one serial correlation coefficient (r1). The usefulness of the bootstrap test is demonstrated by applying it to assess the significance of rk of the annual mean flow data of twenty pristine river basins in Ontario, Canada.
Stochastic Environmental Research and Risk Assessment, 2004
ABSTRACT Scaling properties of Canadian low flows, namely annual minimum mean 1-, 5- and 7-day fl... more ABSTRACT Scaling properties of Canadian low flows, namely annual minimum mean 1-, 5- and 7-day flows, are evaluated across Canada and in its sub-climatic regions. Across the entire country, the log relationship between the kth product moments (PMs, E[Qik]) of low flows and drainage area (Ai) can be represented by: ln(E[Qik])=ak+bkln(Ai)and bk=k, with = 0.86, 0.94 and 0.93 for annual minimum mean 1-, 5- and 7-day flows, respectively. The log linear relationships between the kth probability weighted moments (PWMs, ) and Ai are ln()=ck+Hln(Ai), in which H is constant and is independent of k. The values of H are 0.87, 0.97, and 0.96 for annual minimum mean 1-, 5- and 7-day flows, respectively, which are almost the same as the values. The coefficients of variation (Cv) are almost independent of drainage area. These results demonstrate that Canadian low flows generally exhibit simple scaling and drainage area alone describes most of the variability in the moments of the low flows. Low flows in each of the sub-climatic regions also obey a simple scaling law. The values of , H and Cv are different in each region, which may stem from physiographical and climatological differences among these regions. The finding lays a basis for applying the index flood method to conduct regional low flow frequency analysis as simple scaling is equivalent to the index flood method.
Stochastic Environmental Research and Risk Assessment - STOCH ENVIRON RES RISK ASSESS, 2004
There are two distinct bivariate extreme value distributions constructed from Gumbel marginals, n... more There are two distinct bivariate extreme value distributions constructed from Gumbel marginals, namely Gumbel mixed (GM) model and Gumbel logistic (GL) model. These two models have completely different structures and their dependence ranges are different. The product-moment correlation coefficient for the former is ??[0,2/3] and the latter is ??[0,1]. It is natural to ask which one is more appropriate for representing the joint probabilistic behavior of two correlated Gumbel-distributed variables. This study compares these two models by numerical experiments. The comparison is based on that: (i) if the two distribution models are identical, then the joint probability and the joint return period computed by the GM model should be the same as those by the GL model; and (ii) if a selected distribution is the true distribution from which sample data are drawn, then the probabilities computed by the theoretical model should provide a good fit to empirical ones. Comparison results indicat...
A univariate gamma distribution is one of the most commonly adopted statistical distributions in ... more A univariate gamma distribution is one of the most commonly adopted statistical distributions in hydrological frequency analysis. A bivariate gamma distribution constructed from specified gamma marginals may be useful for representing joint probabilistic properties of multivariate hydrological events such as floods and storms. This article presents a review of various bivariate gamma distribution models that are constructed from gamma marginals.
... Country Number of sites Catchment area (km ) Proportion to total area (%) Botswana Malawi Moz... more ... Country Number of sites Catchment area (km ) Proportion to total area (%) Botswana Malawi Mozambique Namibia S. Africa Swaziland Tanzania Zambia Zimbabwe Total 13 24 14 26 260 22 47 24 72 502 ... Page 6. 518 Berhanu Fanta et al. ...
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