WFM 5201: Data Management and Statistical Analysis

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WFM 5201 Data Management and Statistical Analysis
Lecture-8 Probabilistic Analysis

• Akm Saiful Islam

Institute of Water and Flood Management
(IWFM) Bangladesh University of Engineering and
Technology (BUET)
June, 2008
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Frequency Analysis

• Continuous Distributions
• Normal distribution
• Lognormal distribution
• Pearson Type III distribution
• Gumbels Extremal distribution
• Confidence Interval

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Log-Normal Distribution

• The lognormal distribution (sometimes spelled
  out as the logarithmic normal distribution) of a
  random variable is one for which the logarithm
  of follows a normal or Gaussian distribution.
  Denote , then Y has a normal or
  Gaussian distribution given by
• , (1)

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• Derived distribution Since ,
• the distribution of X can be found as
• 
  (2)
• Note that equation (1) gives the distribution of
  Y as a normal distribution with mean and
  variance . Equation (2) gives the
  distribution of X as the lognormal distribution
  with parameters
• and .

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• Estimation of parameters ( , ) of
  lognormal distribution
• Note , ,
  Chow (1954) Method
• (1)
• (2)
• (3)
• (4)The mean and variance of the lognormal
  distribution are
  
• (5) The coefficient of variation of the Xs is
• (6) The coefficient of skew of the Xs is
• (7) Thus the lognormal distribution is skewed to
  the right the skewness increasing with
  increasing values of .

and
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Example-1

• Use the lognormal distribution and calculate the
  expected relative frequency for the third class
  interval on the discharge data in the next table

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Frequency of the discharge of a River
Class Number Observed Relative Frequency
25,000 2 0.03
35,000 3 0.045
45,000 10 0.152
55,000 9 0.136
65,000 11 0.167
75,000 10 0.152
85,000 12 0.182
95,000 6 0.091
105,000 0 0.000
115,000 3 0.045
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Solution

• According to the lognormal distribution is

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• So from the standard normal table we get
• The expected relative frequency according to the
  lognormal distribution is 0.145

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Example-2

• Assume the data of previous table follow the
  lognormal distribution. Calculate the magnitude
  of the 100-year peak flood.

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Solution

• The 100-year peak flow corresponds to a prob(X gt
  x) of 0.01. X must be evaluated such that Px(x)
  0.99. This can accomplished by evaluating Z such
  that Pz(z)0.99 and then transforming to X. From
  the standard normal tables the value of Z
  corresponding to Pz(Z) of 0.99 is 2.326.
• The values of Sy and are given
• The 100-year peak flow according to the lognormal
  distribution is about 1,30,700 cfs.

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Extreme Value Distributions

• Many times interest exists in extreme events such
  as the maximum peak discharge of a stream or
  minimum daily flows.
• The probability distribution of a set of random
  variables is also a random variable.
• The probability distribution of this extreme
  value random variable will in general depend on
  the sample size and the parent distribution from
  which the sample was obtained.

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Extreme value type-I Gumbel distribution

• Extreme Value Type I distribution, Chow (1953)
  derived the expression
• To express T in terms of , the above
  equation can be written as

(3)
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Example-3 Gumble

• Determine the 5-year return period rainfall for
  Chicago using the frequency factor method and the
  annual maximum rainfall data given below. (Chow
  et al., 1988, p. 391)

Year Rainfall (inch) Year Rainfall (inch) Year Rainfall (inch)
1913 0.49 1926 0.68 1938 0.52
1914 0.66 1927 0.61 1939 0.64
1915 0.36 1928 0.88 1940 0.34
1916 0.58 1929 0.49 1941 0.7
1917 0.41 1930 0.33 1942 0.57
1918 0.47 1931 0.96 1943 0.92
1920 0.74 1932 0.94 1944 0.66
1921 0.53 1933 0.8 1945 0.65
1922 0.76 1934 0.62 1946 0.63
1923 0.57 1935 0.71 1947 0.6
1924 0.8 1936 1.11    
1925 0.66 1937 0.64    
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Solution

• The mean and standard deviation of annual maximum
  rainfalls at Chicago are 0.67 inch and 0.177
  inch, respectively. For , T5, equation (3) gives

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Log Pearson Type III

• For this distribution, the first step is to take
  the logarithms of the hydrologic data, . Usually
  logarithms to base 10 are used. The mean ,
  standard deviation , and coefficient of skewness,
  Cs are calculated for the logarithms of the data.
  The frequency factor depends on the return period
  and the coefficient of skewness .
• When , the frequency factor is equal to
  the standard normal variable z .
• When , is approximated by Kite (1977)
  as

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Example-4 Calculate the 5- and 50-year return
period annual maximum discharges of the Gaudalupe
River near Victoria, Texas, using the lognormal
and log-pearson Type III distributions. The data
in cfs from 1935 to 1978 are given below. (Chow
et al., 1988, p. 393)
Year 1930   1940   1950   1960   1970
0     55900   13300   23700   9190
1     58000   12300   55800   9740
2     56000   28400   10800   58500
3     7710   11600   4100   33100
4     12300   8560   5720   25200
5 38500   22000   4950   15000   30200
6 179000   17900   1730   9790   14100
7 17200   46000   25300   70000   54500
8 25400   6970   58300   44300   12700
9 4940   20600   10100   15200    
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Solution
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• It can be seen that the effect of including the
  small negative coefficient of skewness in the
  calculations is to alter slightly the estimated
  flow with that effect being more pronounced at
  years than at years. Another feature of the
  results is that the 50-year return period
  estimates are about three times as large as the
  5-year return period estimates for this example,
  the increase in the estimated flood discharges is
  less than proportional to the increase in return
  period.

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Confidence Interval
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