Transcript Applied Hydrology Rainfall Analysis - RSLAB-NTU

Applied Hydrology
Hydrological Frequency Analysis
Prof. Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University
Outline
• Introductory thoughts
• General interpretation of hydrological
frequency analysis
• The general equation of frequency analysis
• Data series for frequency analysis
• Parameter estimation
• Techniques for GOF tests
• The Horner’s equation
• Further discussions
• More advanced topics
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Introductory thoughts
• Scientific perspective
– Joint distribution of storm duration and event-total
depth
• Engineering perspective
– Event-total depths of selected design durations
• The concept of design storms
– Duration
– Return period (not “recurrence interval”)
• Choice of return period resolution.
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• Bivariate frequency analysis
– What is the return period of a storm with observed
duration and event-total depth?
– Connection between bivariate frequency analysis
and conditional (duration-specific) frequency
analysis.
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General interpretation of
hydrological frequency analysis
• Hydrological frequency analysis is the work of
determining the magnitude of hydrological
variables that corresponds to a given
probability of exceedance. Frequency analysis
can be conducted for many hydrological
variables including floods, rainfalls, and
droughts.
• The work can be better perceived by treating
the interested variable as a random variable.
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• Let X represent the hydrological (random)
variable under investigation. A value xc
associating to some event is chosen such that if
X assumes a value exceeding xc the event is said
to occur. Every time when a random experiment
(or a trial) is conducted the event may or may
not occur.
• We are interested in the number of Bernoulli
trials in which the first success occur. This can
be described by the geometric distribution.
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Geometric distribution
• Geometric distribution represents the
probability of obtaining the first success in x
independent and identical Bernoulli trials.
f X ( x ; p )  (1  p )
x 1
p
x  1, 2 ,3, 
E[ X ]  1 / p
Var [ X ]  q / p
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Average number of trials to achieve the
first success.
Recurrence interval vs return period
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The general equation of frequency
analysis
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• It is apparent that calculation of x T involves
determining the type of distribution for X and
estimation of its mean and standard deviation.
The former can be done by GOF tests and the
latter is accomplished by parametric point
estimation.
1. Collecting required data.
2. Estimating the mean, standard deviation and
coefficient of skewness.
3. Determining appropriate distribution.
4. Calculating xT using the general eq.
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Data series for frequency analysis
• Complete duration series
– A complete duration series consists of all the
observed data.
• Partial duration series
– A partial duration series is a series of data which are
selected so that their magnitude is greater than a
predefined base value. If the base value is selected
so that the number of values in the series is equal
to the number of years of the record, the series is
called an “annual exceedance series”.
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• Extreme value series
– An extreme value series is a data series that
includes the largest or smallest values occurring in
each of the equally-long time intervals of the record.
If the time interval is taken as one year and the
largest values are used, then we have an “annual
maximum series”.
Annual exceedance series and
• Data independency
– Why is it important?
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annual maximum series are
different.
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Parameter estimation
• Method of moments
• Maximum likelihood method
• Depending on the distribution types, parameter
estimation may involve estimation of the mean,
standard deviation and/or coefficient of
skewness.
• Parameter estimation exemplified by the
gamma distribution.
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Gamma distribution parameter
estimation
• Gamma distribution is a special case of the Pearson type
III distribution (with zero location parameter).
• Gamma density
 x
f X ( x ; ,  ,  ) 
 
( )   
1
 


0
2
  

 1
e
(x / )
,
0  x  
2

  0

    
 0
 

2
where , , and  are the mean, standard deviation, and
coefficient of skewness of X (or Y), respectively, and 
and  are respectively the scale and shape parameters of
the gamma distribution.
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• MOM estimators
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• Maximum likelihood estimator
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Evaluating bias of different estimators of
coefficient of skewness
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Evaluating mean square error of different
estimators of coefficient of skewness
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Techniques for goodness-of-fit test
• A good reference for detailed discussion
about GOF test is:
Goodness-of-fit Techniques. Edited by R.B. D’Agostino and
M.A. Stephens, 1986.
– Probability plotting
– Chi-square test
– Kolmogorov-Smirnov Test
– Moment-ratios diagram method
• L-moments based GOF tests
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Probability plotting
• Fundamental concept
– Probability papers
– Empirical CDF vs theoretical CDF
• Example of misuse of probability plotting
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• Suppose the true underlying distribution
depends on a location parameter  and a scale
parameter  (they need not to be the mean
and standard deviation, respectively). The CDF
of such a distribution can be written as
 X 
F (X )  G
  G (Z )
 

X 
1
Z  G [ F ( X )] 

Z 



1

X
where Z is referred to as the standardized
variable and G(z) is the CDF of Z.
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• Also let Fn(X) represents the empirical
cumulative distribution function (ECDF) of X
based on a random sample of size n. A
probability plot is a plot of
1
z  G [ Fn ( x )] on x
where x represents the observed values of the
random variable X.
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Most of the plotting position methods are empirical. If n is the total number of
values to be plotted and m is the rank of a value in a list ordered by
descending magnitude, the exceedence probability of the mth largest value,
xm, is , for large n, shown in the following table.
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Misuse of probability plotting
Log Pearson Type III ?
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Misuse of probability plotting
48-hr rainfall depth
Log Pearson Type III ?
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Fitting a probability distribution to annual
maximum series (Non-parametric GOF tests)
• How do we fit a probability distribution to a
random sample?
– What type of distribution should be adopted?
– What are the parameter values for the distribution?
– How good is our fit?
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Chi-square GOF test
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Kolmogorov-Smirnov GOF test
• The chi-square test compares the empirical
histogram against the theoretical histogram.
• In contrast, the K-S test compares the empirical
cumulative distribution function (ECDF) against
the theoretical CDF.
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• In order to measure the difference between
Fn(X) and F(X), ECDF statistics based on the
vertical distances between Fn(X) and F(X) have
been proposed.
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Hypothesis test using Dn
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Values of
for the
Kolmogorov-Smirnov test
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IDF curve fitting using the Horner’s
equation
• The intensity-duration-frequency (IDF) relationship of
the design storm depths
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DDF curves
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IDF curves
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• Alternative IDF fitting
a (T )
I (T ) 
(Return-period specific)
( t  b (T ))
t
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c (T )
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Further discussions on frequency
analysis
• Partitioning of storm events
• Extracting annual maximum series
• Probabilistic interpretation of the design total
depth
• Joint distribution of duration and total depth
• Frequency analysis based on design-durationspecific total depth vs frequency analysis based on
joint distribution of duration and total depth of
real storms (need to consider certain storm type
that yields annual maximum rainfalls).
• Selection of the best-fit distribution
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Partitioning of storm events
• Consider event-total rainfall at a location.
– What is a storm event?
• Parameters related to partition of storm events
– Minimum inter-event-time
– A threshold value for rainfall depth
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Annual maximum series
• Data in an annual maximum series are
considered IID and therefore form a random
sample.
• For a given design duration tr, we continuously
move a window of size tr along the time axis
and select the maximum total values within the
window in each year.
• Determination of the annual maximum rainfall
is NOT based on the real storm duration;
instead, a design duration which is artificially
picked is used for this purpose.
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Random sample for estimation of
design storm depth
• The design storm depth of a specified duration
with return period T is the value of D(tr) with
the probability of exceedance equals  /T.
• Estimation of the design storm depth requires
collecting a random sample of size n, i.e., {x1,
x2, …, xn}.
• A random sample is a collection of
independently observed and identically
distributed (IID) data.
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Probabilistic interpretation of the
design storm depth
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• It should also be noted that since the total
depth in the depth-duration-frequency
relationship only represents the total amount of
rainfall of the design duration (not the real
storm duration), the probability distributions in
the preceding figure do not represent
distributions of total depth of real storm events.
• Or, more specifically, the preceding figure does
not represent the bivariate distribution of
duration and total depth of real storm events.
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• The usage of annual maximum series for rainfall
frequency analysis is more of an intelligent and
convenient engineering practice and the annual
maximum data do not provide much
information about the characteristics of the
duration and total depth of real storm events.
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Joint distribution of the total depth
and duration
• Total rainfall depth of a storm event varies with
its storm duration. [A bivariate distribution for
(D, tr).]
• For a given storm duration tr, the total depth
D(tr) is considered as a random variable and its
magnitudes corresponding to specific
exceedance probabilities are estimated.
[Conditional distribution]
• In general, E [ D ( tr1 )]  E [ D ( tr2 )] if tr1  tr2 .
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Selection of the best-fit distribution
• Methods of model selection based on loss of
information.
– Akaike information criterion (AIC)
– Schwarz's Bayesian information criterion (BIC)
– Hannan-Quinn (HQIC) information criterion
• Common practices of WRA-Taiwan
– SE and U
– SSE and SE
• Can the p-value be used for selection of the
best-fit distribution?
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Information-criteria-based
model selection
AIC 
p   ( )
BIC 
n
HQIC 
p log( n )  2  ( )
2n
p log(log( n ))   ( )
n
where  ( ) is the log-likelihood function for the
parameter  associated with the model, n is the
sample size, and p is the dimension of the
parametric space.
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WRA Practice


2
(
X

X
)
 i i
SE  

N  p





1/ 2
N
SSE 

2
( X i  Xˆ i )
i 1
p: Number of distribution parameters
Weibull plotting position formula is used for calculation of cumulative
probability.
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


  X i  X i 






2
(
X

X
)
 i
i
SE  

N

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



1/ 2
U 
 X
2
i
N

1/ 2
2

N



  X


2
i
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N


1/ 2
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More advanced topics
• LMRD-based GOF tests
– Reference:
Liou, J.J., Wu, Y.C., Cheng, K.S., 2008. Establishing
acceptance regions for L-moments-based goodness-offit test by stochastic simulation. Journal of Hydrology,
Vol. 355, No.1-4, 49-62.
Wu, Y.C., Liou, J.J., Cheng, K.S., 2011. Establishing
acceptance regions for L-moments based goodness-offit tests for the Pearson type III distribution. Stochastic
Environmental Research and Risk Assessment, DOI
10.1007/s00477-011-0519-z.
• Regional frequency analysis
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GOF test using L-moment-ratios diagram
(LMRD)
• Concept of identifying appropriate distributions
using moment-ratio diagrams (MRD).
• Product-moment-ratio diagram (PMRD)
• L-moment-ratio diagram (LMRD)
– Two-parameter distributions
• Normal, Gumbel (EV-1), etc.
– Three-parameter distributions
• Log-normal, Pearson type III, GEV, etc.
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• Moment ratios are unique properties of
probability distributions and sample moment
ratios of ordinary skewness and kurtosis have
been used for selection of probability
distribution.
• The L-moments uniquely define the distribution
if the mean of the distribution exists, and the Lskewness and L-kurtosis are much less biased
than the ordinary skewness and kurtosis.
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• A two-parameter distribution with a location and a
scale parameter plots as a single point on the LMRD,
whereas a three-parameter distribution with location,
scale and shape parameters plots as a curve on the
LMRD, and distributions with more than one shape
parameter generally are associated with regions on the
diagram.
• However, theoretical points or curves of various
probability distributions on the LMRD cannot
accommodate for uncertainties induced by parameter
estimation using random samples.
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Ordinary (or product) moment-ratios
diagram (PMRD)
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The ordinary (or product) moment
ratios diagram
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Sample estimates of product moment
ratios
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(D'Agostino and Stephens, 1986)
90％
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• Even though joint distribution of the ordinary
sample skewness and sample kurtosis is
asymptotically normal, such asymptotic
property is a poor approximation in small and
moderately samples, particularly when the
underlying distribution is even moderately skew.
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Scattering of sample moment ratios of the
normal distribution
(100,000 random samples)
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L-moments and the L-moment ratios
diagram
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L-moment-ratio diagram of various
distributions
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Sample estimates of L-moment ratios
(probability weighted moment estimators)
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Sample estimates of L-moment ratios
(plotting-position estimators)
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~
that  r is
• Hosking and Wallis (1997) indicated
not
an unbiased estimator of  r , but its bias tends
to zero in large samples.
~
t

• r and r are respectively referred to as the
probability-weighted-moment estimator and
the plotting-position estimator of the Lmoment ratio  r .
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Establishing acceptance region for Lmoment ratios
• The standard normal and standard Gumbel
distributions (zero mean and unit standard deviation)
are used to exemplify the approach for construction of
acceptance regions for L-moment ratio diagram.
• L-moment-ratios ( 3 , 4 ) of the normal and Gumbel
distributions are respectively (0, 0.1226) and (0.1699,
0.1504).
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Stochastic simulation of the normal and
Gumbel distributions
• For either of the standard normal and standard
Gumbel distribution, a total of 100,000 random
samples were generated with respect to the specified
sample size20, 30, 40, 50, 60, 75, 100, 150, 250, 500,
and 1,000.
• For each of the 100,000 samples, sample L-skewness
and L-kurtosis were calculated using the probabilityweighted-moment estimator and the plotting-position
estimator.
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Scattering of sample L-moment ratios
Normal distribution
Normal distribution !
(100,000 random samples)
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Normal distribution ?
(100,000 random samples)
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95% acceptance region
99% acceptance region
Non-normal distribution !
(100,000 random samples)
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Scattering of sample L-moment ratios
Gumbel distribution
(100,000 random samples)
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(100,000 random samples)
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(100,000 random samples)
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• For both distribution types, the joint
distribution of sample L-skewness and Lkurtosis seem to resemble a bivariate normal
distribution for a larger sample size (n = 100).
• However, for sample size n = 20, the joint
distribution of sample L-skewness and Lkurtosis seems to differ from the bivariate
normal. Particularly for Gumbel distribution,
sample L-moments of both estimators are
positively skewed.
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• For smaller sample sizes (n = 20 and 50), the
distribution cloud of sample L-moment-ratios
estimated by the plotting-position method
appears to have its center located away from
( 3 , 4 ), an indication of biased estimation.
• However, for sample size n = 100, the bias is
almost unnoticeable, suggesting that the bias in
L-moment-ratio estimation using the plottingposition estimator is negligible for larger
sample sizes.
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• In contrast, the distribution cloud of the sample
L-moment-ratios estimated by the probabilityweighted-moment method appears to have its
center almost coincide with ( 3 ,  4 ).
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Bias of sample L-skewness and L-kurtosis Normal distribution
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Bias of sample L-skewness and L-kurtosis Gumbel distribution
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Mardia test for bivariate normality of sample Lskewness and L-kurtosis
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Mardia test for bivariate normality of
sample L-skewness and L-kurtosis
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Mardia test for bivariate normality of
sample L-skewness and L-kurtosis
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• It appears that the assumption of bivariate
normal distribution for sample L-skewness and
L-kurtosis of both distributions is valid for
moderate to large sample sizes. However, for
random samples of normal distribution with
sample size n  30 , the bivariate normal
assumption may not be adequate. Similarly, the
bivariate normal assumption for sample Lskewness and L-kurtosis of the Gumbel
distribution may not be adequate for sample
size n  60 .
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Establishing acceptance regions for LMRDbased GOF tests
• For moderate to large sample sizes, the sample Lskewness and L-kurtosis of both the normal and
Gumbel distributions have asymptotic bivariate normal
distributions.
• Using this property, the 100 (1   )% acceptance region of
a GOF test based on sample L-skewness and L-kurtosis
can be determined by the equiprobable density
contour of the bivariate normal distribution with its
encompassing area equivalent to 1   .
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• The probability density function of a
multivariate normal distribution is generally
expressed by
f (X ) 

1
2 
p
2
1
2

e
1
2
 X   T

1
X  
• The probability density function depends on
the random vector X only through the quadratic
form Q   X      X    which has a chi-square
distribution with p degrees of freedom.
T
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1
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• Therefore, probability density contours of a
multivariate normal distribution can be
expressed by
T
1
Q  X     X     c
for any constant c  0 .
• For a bivariate normal distribution (p=2) the
above equation represents an equiprobable
ellipse, and a set of equiprobable ellipses can
be constructed by assigning  22, to c for various
values of .
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• Consequently, the 100 (1   )% acceptance region
of a GOF test based on the sample L-skewness
and L-kurtosis is expressed by
 X   T   1  X      22,
 2 ,
2
is the upper quantile of the  2
where
distribution at significance level .
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124
• For bivariate normal random vector X  ( X X ) ,
T
1


 X     c can
X



the density contour of
also be expressed as
T
1
2
2
  X 1   1 2
2   X 1   1  X 2   2   X 2   2  


c
2 
2
2
1  
1
 1 2
2

1
• However, the expected values and covariance
matrix of sample L-skewness and L-kurtosis are
unknown and can only be estimated from
random samples generated by stochastic
simulation.
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• Thus, in construction of the equiprobable
ellipses, population parameters  ,  , and  must
be respectively replaced by their sample
estimates x , S , and r .
• The Hotelling’s T2 statistic
T
2
 X  x  S
T
1
X
 x
2
  X 1  x1 2
2 r  X 1  x1  X 2  x 2   X 2  x 2  




2 
2
2
1 r 
s1
s1 s 2
s2

1
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• The Hotelling’s T2 is distributed as a multiple of
an F-distribution, i.e.,
2 ( N  1)
2
T
2
~
N ( N  2)
F( 2 , N  2 )
• For large N,
2( N
2
 1)
N ( N  2)
F 2 , N  2 ( )  2 F 2 , N  2 ( )   2 , 
2
Therefore, the distribution of the Hotelling’s T2
can be well approximated by the chi-square
distribution with degree of freedom 2.
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• Thus, if the sample L-moments of a random
sample of size n falls outside of the
corresponding ellipse, i.e.
T
2
  X  xn  S
T
1
n
X
 xn   
2
2 ,
the null hypothesis that the random sample is
originated from a normal or Gumbel
distribution is rejected.
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Scattering of sample L-moment ratios
Normal distribution
(100,000 random samples)
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Normal distribution ?
(100,000 random samples)
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Variation of 95% acceptance regions with
respect to sample size n
95% acceptance region
n=50
n=20
n=100
Non-normal distribution !
What if n=36?
(100,000 random samples)
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Empirical relationships between parameters of
acceptance regions and sample size
• Since the 95% acceptance regions of the
proposed GOF tests are dependent on the
sample size n, it is therefore worthy to
investigate the feasibility of establishing
empirical relationships between the 95%
acceptance region and the sample size. Such
empirical relationships can be established using
the following regression model
a
b
ˆ
 (n)   2  c
n n
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Empirical relationships between the sample size and
parameters of the bivariate distribution of sample Lskewness and L-kurtosis
T
2
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  X 1  x1 2 2 r  X 1  x1  X 2  x 2   X 2  x 2 2 




2 
2
2
1 r 
s1
s1 s 2
s2

1
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Empirical relationships between the sample size and
parameters of the bivariate distribution of sample Lskewness and L-kurtosis
T
2
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  X 1  x1 2 2 r  X 1  x1  X 2  x 2   X 2  x 2 2 




2 
2
2
1 r 
s1
s1 s 2
s2

1
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Example
• Suppose that a random sample of size n = 44 is
available, and the plotting-position sample Lskewness and L-kurtosis are calculated as (~ ,~4 )
= (0.214, 0.116). We want to test whether the
sample is originated from the Gumbel
distribution.
3
T
2
2
2

 X 1  x1  2 r  X 1  x1  X 2  x 2   X 2  x 2  
1




2 
2
2
1 r 
s1
s1 s 2
s2

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• From the regression models for plottingposition estimators, we find ˆ , ˆ , ˆ ,
ˆ
, and r to be respectively 0.1784, 0.1369,
0.005119, 0.002924, and 0.6039. The
Hotelling’s T2 is then calculated as 0.9908.
• The value of T2 is much smaller than the
threshold value
L  CS
L  CK
2
L  CS
2
L  CK

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2
2 , 0 . 05
 5 . 99
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• The null hypothesis that the random sample is
originated from the Gumbel distribution is not
rejected.
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95% acceptance regions of L-moments-based
GOF test for the normal distribution
Acceptance ellipses
corresponding to
various sample sizes
(n = 20, 30, 40, 50, 60,
75, 100, 150, 250, 500,
and 1,000).
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Acceptance ellipses
corresponding to
various sample sizes
(n = 20, 30, 40, 50, 60,
75, 100, 150, 250, 500,
and 1,000).
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95% acceptance regions of L-moments-based
GOF test for the Gumbel distribution
Acceptance ellipses
corresponding to
various sample sizes (n
= 20, 30, 40, 50, 60, 75,
100, 150, 250, 500, and
1,000).
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Acceptance ellipses
corresponding to
various sample sizes (n
= 20, 30, 40, 50, 60, 75,
100, 150, 250, 500, and
1,000).
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Validity check of the LMRD acceptance
regions
• The sample-size-dependent confidence
intervals established using empirical
relationships described in the last section are
further checked for their validity. This is done
by stochastically generating 10,000 random
samples for both the standard normal and
Gumbel distributions, with sample size20, 30,
40, 50, 60, 75, 100, 150, 250, 500, and 1,000.
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• For validity of the sample-size-dependent 95%
acceptance regions, the rejection rate ˆ should
be very close to the level of significance
(  0.05) or the acceptance rate be very close
to 0.95.
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Acceptance rate of the validity check for sample-size-dependent
95% acceptance regions of sample L-skewness and L-kurtosis pairs.
Based on 10,000 random samples
for any given sample size n.
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Regional frequency analysis
• Fundamental concept of regional frequency
analysis
• The index-flood approach
• The frequency-factor approach
• Validation of RFA using simulated data
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Fundamental concept of regional frequency
analysis
• Hydrological frequency analysis is generally
conducted for sites with rainfall or flow
measurements.
• For areas with short record length or without
rainfall or flow measurements, hydrological
frequency analysis needs to be conducted using
data from sites of similar hydrological
characteristics.
• Data observed at different sites within a
“homogeneous region” can be combined and used
in regional frequency analysis.
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• Catastrophic storm rainfalls (or extraordinary
rainfalls) often are considered as outliers.
Whether or not such rainfalls should be
included in site-specific frequency analysis is
disputable.
– 24-hr annual max. rainfalls (Morakot) of 2009
•
•
•
•
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甲仙
泰武
大湖山
阿禮
1077 mm
1747 mm
1329 mm
1237 mm
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Morakot rainfalls were not included in frequency analysis.
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The Index-Flood Approach for RFA –
Concept
• Proposed by Dalrymple (1960) for flood
frequency analysis.
• Let Q be the hydrological variable of interest,
for example annual maximum rainfall of a
specific duration or annual maximum flow.
Suppose that observed data of Q are available
at N different sites and ni represents the sample
size for the i-th (i = 1, 2, …, N) site. Also, let Qi(F)
be the quantile function of Q at site-i.
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Observed data: Q ij , j  1, 2 , , n i ; i  1, 2 , , N
Quantile function
P Q i  Q i ( F )   F , 0  F  1
• Assume the quantile function of hydrological variables
at different sites can be expressed by
Q i ( F )   i q ( F ), i  1,  , N .
where i is the index flood (Dalrymple, 1960) and q(F),
known as the regional growth curve, is an adjusted
dimensionless quantile function common to every site.
• The index flood i is often taken to be the mean of Q
at site-i.
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• The regional growth curve q(F) is considered as
the quantile function of a common distribution
Qij/i .
• It is usually assumed that the distribution type
for the rescaled data Qij/i (i.e. the regional
frequency distribution q ( F ;  1 ,   p ) ) is known.
Thus, it is necessary to estimate parameters of
this common distribution using observed data
available at different sites.
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The Index-Flood Approach for RFA –
Estimations
• Parameter estimation
ˆ i  Q i 
ˆk 
N

i 1
1
ni
ni
Q
ij
j 1
(i)
n iˆk
N
n
i
, k  1,  , p .
i 1
qˆ ( F )  q ( F ; ˆk , k  1,  , p )
• Regional frequency analysis
Qˆ i ( F )  ˆ i  qˆ ( F )  ˆ i  q ( F ; ˆk , k  1,  , p )
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The Index-Flood Approach for RFA –
Implicit Assumptions
•
•
•
•
•
Observations at any given site are identically
distributed.
Observations at any given site are serially
independent.
Observations at different sites are
independent.
The distributions of the rescaled variable at
different sites are identical.
The distribution type of the rescaled variable
is correctly specified.
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• The assumption that distributions of the
rescaled variable at different sites are identical
implicitly imply the existence of a
homogeneous region.
• A homogeneous region is considered as an area
within which rescaled variables at different sites
have approximately the same probability
distributions.
• The homogeneous region need not to be
geographically continuous.
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• Implicit in the definition of a homogeneous
region, is the condition that all sites can be
described by one common probability
distribution after the site data are rescaled by
their at-site mean. Thus, all sites within a
homogeneous region have a common regional
growth curve.
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General procedures of regional frequency
analysis
1. Data screening
– Correctness check
– Data should be stationary over time.
2. Identifying homogeneous regions
– A set of characteristic variables should be chosen
and used for delineation of homogeneous regions.
– Characteristic variables may include geographic
and hydrological variables.
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3. Choice of an appropriate regional frequency
distribution
– GOF test using rescaled samples from different
sites within the same homogeneous region.
– The chosen distribution not only should fit the
data well but also yield quantile estimates that are
robust to physically plausible deviations of the
true frequency distribution from the chosen
frequency distribution.
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4. Parameter estimation of the regional
frequency distribution
– Estimating parameters of the site-specific
frequency distribution
– Estimating parameters of the regional frequency
distribution using weighted average.
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Situations for application of RFA
• General application to individual sites with
observed data
– Regionalization is valuable. Even though a region
may be moderately heterogeneous, regional
frequency analysis will still yield much more
accurate quantile estimates than at-site analysis.
• Application to one site of special interest.
– Special care should be taken (by choosing
appropriate characteristic variables) to make the site
typical of the region to which it is assigned.
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• Application to one or more ungauged sites
(PUB program – http://iahs.info/ ).
– An ungauged site can be assigned to a
homogeneous region based on its characteristic
variables. The regional growth curve at an
ungauged site is then estimated using the
characteristic variables.
– The index flood (or index quantity, if the variable of
interest is not flood flow) can be considered as a
function of characteristic variables and to calibrate
the function by using data from the gauged sites.
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Data screening using a measure of
discordance Di
• Assuming that there are N
sites in a region and we
want to identify those sites
that are grossly discordant
with the group as a whole.
Hosking and Wallis (1997)
proposed a measure of
discordance in terms of Lmoments (t, t3, and t4) of
the sites’ data.
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• It can be shown that Di satisfies the algebraic
bound D i  ( N  1) / 3 . Thus, the value of Di can exceed
3 only in regions having 11 or more sites.
• The criterion for discordance should be an increasing
function of the number of sites in the region since
regions with more sites are more likely to contain sites
with large values of Di. Hosking and Wallis (1997)
recommend that any site with Di >3 be regarded as
discordant, as such sites have L-moments ratios that
are markedly different from the average for the other
sites in the region.
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Defining homogeneous sub-regions
• Homogeneous sub-regions (grouping of sites/gages)
can be determined based on the similarity of the
physical and/or meteorological characteristics of the
sites. This can be done by performing cluster analysis.
• L-moment statistics can then used to estimate the
variability and skewness of the pooled regional data
and to test for heterogeneity as a basis for accepting
or rejecting the proposed sub-region formulation.
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• Candidates for physical features included such
measures as: site elevation; elevation averaged
over some grid size; localized topographic slope;
macro topographic slope averaged over some
grid size; distance from the coast or source of
moisture; distance to sheltering mountains or
ridgelines; and latitude or longitude.
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• Candidate climatological characteristics
included such measures as: mean annual
precipitation; precipitation during a given
season; seasonality of extreme storms; and
seasonal temperature/dewpoint indices.
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Example
• A review of the topographic and climatological
characteristics in the Oregon study area showed
only two measures, mean annual precipitation
(MAP) and latitude were needed for grouping
of sites/gages into homogeneous sub-regions
within a given climatic region. Homogeneous
sub-regions were therefore formed with
gages/sites within small ranges of MAP and
latitude.
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• The output from the cluster analysis need not, and
usually should not, be final. Subjective adjustments
can often be found to improve the physical coherence
of the regions. Several kinds of adjustment of regions
may be useful:
–
–
–
–
–
–
–
move a site or a few sites from one region to another;
delete a site or a few sites from the data set;
subdivide the region;
break up the region by reassigning its sites to other regions;
merge the region with another or others;
merge two or more regions and redefine groups; and
obtain more data and redefine groups.
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Test of regional homogeneity
• Once a set of physically plausible regions has
been defined, it is desirable to assess whether
the regions are meaningful. This involves
testing whether a proposed region may be
accepted as being homogeneous and whether
two or more homogeneous regions are
sufficiently similar that they should be
combined into a single region.
• The hypothesis of homogeneity is that the atsite frequency distributions are the same except
for a site-specific scale factor.
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Rationale of test of regional homogeneity
• Comparing the between-site dispersion of the
sample L-moment ratios for the group of sites
under consideration and the expected
dispersion of a homogeneous region.
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Test of regional homogeneity
• A heterogeneity measure proposed by Hosking
and Wallis (1997).
• Suppose that the proposed region has N sites,
with site i having record length ni and sample L(i)
(i)
(i)
t
,
t
,
and
t
.
moment ratios
3
4
R
R
R
• Let t , t 3 , and t 4 represent the regional average
L-CV, L-skewness, and L-kurtosis, weighted
proportionally to the sites’ record length; for
example
N
t
R


i 1
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nit
(i )
n
i
i 1
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• Calculate the weighted standard deviation of
the at-site sample L-CVs,

(i)
R 2
V    ni (t  t )
 i 1
N
N

i 1

ni 

1 2
• Fit a four-parameter kappa distribution to the
regional average L-moment ratios
R
R
R
1, t , t 3 , and t 4 .
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• Simulate a large number Nsim of realizations of a region
with N sites, each having this kappa distribution as its
frequency distribution.
• The simulated regions are homogeneous and have no
cross-correlation or serial correlation; sites have the
same record lengths as their real-world counterparts.
• For each simulated region, calculate V.
• From the simulations determine mean and standard
deviation of the Nsim values of V. Call these  V and  V .
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• Calculate the heterogeneity measure
H 
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V  V
V
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Choosing a distribution for frequency
analysis
• For regional frequency analysis, a single probability
distribution is applied to all sites within a
homogeneous region. Thus, it is necessary to choose a
best-fit distribution from a set of candidate
distributions.
• Assume that the region is acceptably close to
homogeneous. The L-moment ratios of the sites in a
homogeneous region are well summarized by the
regional average and the scatter of the individual sites’
L-moment ratios about the regional average
represents no more than sampling variability.
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• The goodness-of-fit can be judged by how well
the L-skewness and L-kurtosis of the fitted
distribution match the regional average Lskewness and L-kurtosis of the observed data.
• Assume for convenience that the candidate
distribution is generalized extreme-value (GEV),
which has three parameters, and the sample Lskewness and L-kurtosis are exactly unbiased.
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• The GEV distribution fitted by the method of Lmoments has L-skewness equal to the regional
average L-skewness.
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• Note: When fitting a three-parameter candidate
distribution to at-sites L-moment ratios, we
only need to estimate the L-skewness of the
distribution by using the method of L-moments
(L-skewness equal to the regional average Lskewness). There is no need for estimation of
the L-kurtosis since the L-kurtosis is completely
dependent on the L-skewness.
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• We thus judge the quality of fit by the
GEV
difference between the L-kurtosis  4 of the
fitted GEV distribution and the regional average
R
L-kurtosis t 4 .
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( t ,
R
3
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DIST
4
)
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• Small values of ZGEV indicate that the GEV distribution can be
considered as the true underlying distribution for the region.
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Calculation of 4
• Theoretically, separate set of simulations must
be made for each candidate distribution in
order to obtain the appropriate 4 values. In
practice, we can obtain a 4 value by using the
same simulated realizations of a kappa
distribution for a homogeneous region used in
test of regional homogeneity.
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Bias correction for L-kurtosis
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Goodness-of-fit test
• Given a set of candidate three-parameter
distributions (Pearson type III, GEV, lognormal,
generalized Pareto, etc.). We first need to fit
each distribution to the regional average Lmoment ratios 1, t R , and t 3R .
• Denote by 4DIST the L-kurtosis of the fitted
distribution, where DIST represents a candidate
distribution.
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• Fit a kappa distribution to the regional average Lmoment ratios1, t R , t 3R and t 4R.
• Simulate a large number, Nsim, of realizations of a
region with N sites, each having this kappa distribution
as its frequency distribution. The simulated
realizations are homogeneous and have no crosscorrelation or serial correlation; sites have the same
record lengths as their real-world counterparts. The
fitting of a kappa distribution and simulation of at-site
realizations of the kappa distribution can use the same
simulations as those used for test of regional
homogeneity.
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• For the m-th simulated realization, regional
[m ]
[m ]
t
t
average L-skewness 3 and L-kurtosis 4 can be
calculated.
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The kappa distribution
• The kappa distribution has four parameters: 
(location),  (scale), k and h.
f ( x)  
1
1  k ( x   ) /  
1 k 1

F ( x )  1  h 1  k ( x   ) /  
F ( x ) 
1 h

1 k 1 h
1     (1  g 1 ) / k
2   ( g1  g 2 ) / k
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 3  (  g 1  3 g 2  2 g 3 ) /( g 1  g 2 )
 4  (  g 1  6 g 2  10 g 3  5 g 4 ) /( g 1  g 2 )
r  (1  k )  ( r / h )

 h 1  k  (1  k  r / h ) , h  0
gr  
r  (1  k )  (  k  r / h )

, h  0.
1 k
 (  h )  (1  r / h )
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