Transcript Lab for Remote Sensing Hydrology and Spatial Modeling

Applied Hydrology
Frequency Analysis
Professor Ke-Sheng Cheng
Dept. of Bioenvironmental Systems Engineering
National Taiwan University
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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 frequency factor equation
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 It is apparent that
calculation of xT
involves determining
the type of distribution
for X and estimation of
its mean and standard
deviation. The former
can be done by GOF
test and the latter is
accomplished by
parametric point
estimation.
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1. Collecting required data.
2. Determining appropriate
distribution.
3. Estimating the mean and
standard deviation.
4. Calculating xT using the
general eq.
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Data series used 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”.
Data independency
Why is it important?
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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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Rainfall frequency analysis
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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Total depths of storm events
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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Probabilistic Interpretation of the
Design Storm Depth
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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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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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Fitting A Probability Distribution to
Annual Maximum Series
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 Dn , for the KolmogorovSmirnov test
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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 L-skewness 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) momentratios 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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95％
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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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~
Hosking and Wallis (1997) indicated that r
is not an unbiased estimator of r , but its
bias tends to zero in large samples.
 tr 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
L-moment 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 Lskewness and L-kurtosis were calculated using
the probability-weighted-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 Lmoment-ratios estimated by the plottingposition 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 plotting-position 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
probability-weighted-moment method
appears to have its center almost coincide
with ( 3 ,  4 ).
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Bias of sample L-skewness and Lkurtosis - Normal distribution
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Bias of sample L-skewness and Lkurtosis - Gumbel distribution
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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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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 L-skewness and Lkurtosis of the Gumbel distribution may
not be adequate for sample size n  60.
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Establishing acceptance regions for
LMRD-based 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

e
1
2

1
 X   T  1  X   
2
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 Lskewness and L-kurtosis is expressed by
X   
T
1  X      22,
2

where  is the upper quantile of the 2
distribution at significance level .
2
2,
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For bivariate normal random vector X
T
 ( X1 X 2 ) ,
the density contour of  X   T  1  X     c can
also be expressed as
2
2
1   X 1  1  2   X 1  1  X 2   2   X 2   2  


c
2 
2
2
1    1
 1 2
2

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   X  x  S 1  X  x 
2
T
2
2
1   X 1  x1  2r  X 1  x1  X 2  x2   X 2  x2  




2 
2
1 r 
s1
s1s2
s22

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The Hotelling’s T2 is distributed as a multiple
of an F-distribution, i.e.,
2
2
(
N
 1)
2
T ~
F( 2, N  2)
N ( N  2)
For large N,
2( N 2  1)
F2, N  2 ( )  2F2, N  2 ( )   22,
N ( N  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   X  xn  S
2
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 ˆ(n)  a  b  c
n
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Empirical relationships between the sample size and
parameters of the bivariate distribution of sample Lskewness and L-kurtosis
2
2









1
X

x
2
r
X

x
X

x
X

x
2
1
1
1
1
2
2
2
2
T 



2 
2
1 r 
s1
s1s2
s22

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Empirical relationships between the sample size and
parameters of the bivariate distribution of sample Lskewness and L-kurtosis
2
2









1
X

x
2
r
X

x
X

x
X

x
2
1
1
1
1
2
2
2
2
T 



2 
2
1 r 
s1
s1s2
s22

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Example
Suppose that a random sample of size n = 44
is available, and the plotting-position sample
L-skewness and L-kurtosis are calculated as
(~3 , ~4) = (0.214, 0.116). We want to test
whether the sample is originated from the
Gumbel distribution.
2
2


1
X 1  x1  2r  X 1  x1  X 2  x2   X 2  x2  
2
T 



2 
2
2
1 r 
s1
s1s2
s2

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From the regression models for plottingposition estimators, we find ˆ L CS , ˆ L CK ,ˆ L2CS ,
ˆ L2  CK , 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

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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
correspond to various
sample sizes (n = 20,
30, 40, 50, 60, 75, 100,
150, 250, 500, and
1,000).
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Acceptance ellipses
correspond 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
correspond to various
sample sizes (n = 20, 30,
40, 50, 60, 75, 100, 150,
250, 500, and 1,000).
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Acceptance ellipses
correspond 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 ̂ 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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End of this session.
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