wfm 5201: data management and statistical analysis
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WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
WFM 5201: Data Management and Statistical Analysis
Akm Saiful Islam
Lecture-8: Probabilistic Analysis
June, 2008
Institute of Water and Flood Management (IWFM)Bangladesh University of Engineering and Technology (BUET)
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Frequency Analysis
Continuous Distributions Normal distribution Lognormal distribution Pearson Type III distribution Gumbel’s Extremal distribution
Confidence Interval
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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)
2
2
1
22
1)(
y
yy
y
eyf
y
XY ln
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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 .
22
2
1
22
2
1
2 2
11
2
1)()(
y
y
y
y y
y
y
y
exx
edx
dyyfxf
y 2y
y 2y
XY lnxdx
dy 1
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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 .
XSC xv /
1ln
2
12
2
vC
XY
)1ln( 22 vy CS
XY lnn
yy
i1
22
2
n
ynyS
i
y
)2/exp()( 2yyXE 1)(
22 yeXVar x
12
yeCv
33 vv CC
vC
y 2y
and
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Solution
According to the lognormal distribution is
311.0500,67/000,21/ xSC xV
0737.11)]1311.0/(500,67ln[21)]1/(ln[2
1 2222 vCxy
30395.01311.0)1ln( 22 vy Cs
182.130395.0/)0737.11000,45(ln/)(ln ysyxz
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
So from the standard normal table we get
The expected relative frequency according to the lognormal distribution is 0.145
5104476.1)(
)30395.0(000,45/198.0)/()()(
xp
SxzPxp
x
yzx
198.0)( zpz
145.0)104476.1(000,10 5000,45 f
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Example-2:
Assume the data of previous table follow the lognormal distribution. Calculate the magnitude of the 100-year peak flood.
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Solution: The 100-year peak flow corresponds to a prob(X > 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.
y
yzsy y
781.110737.11)326.2(30395.0 y
cfsyx 700,130)exp(
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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.
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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
1lnln5772.0
6
T
TKT
6expexp1
1
TKT
TK
5772.0
(3)
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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)
YearRainfall (inch) Year
Rainfall (inch) Year
Rainfall (inch)
1913 0.49 1926 0.68 1938 0.521914 0.66 1927 0.61 1939 0.641915 0.36 1928 0.88 1940 0.341916 0.58 1929 0.49 1941 0.71917 0.41 1930 0.33 1942 0.571918 0.47 1931 0.96 1943 0.921920 0.74 1932 0.94 1944 0.661921 0.53 1933 0.8 1945 0.651922 0.76 1934 0.62 1946 0.631923 0.57 1935 0.71 1947 0.61924 0.8 1936 1.11 1925 0.66 1937 0.64
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Solution
The mean and standard deviation of annual maximum rainfalls at Chicago are 0.67 inch and 0.177 inch, respectively. For , T=5, equation (3) gives
1lnln5772.0
6
T
TKT
0.71915
5lnln5772.0
6
TK
in 0.78= 177)(0.719)(0.+0.649=T
TT
x
sKxx
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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
5432232
3
1)1()6(
3
1)1( kzkkzkzzkzzKT
0sC
0sC
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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 91901 58000 12300 55800 9740
2 56000 28400 10800 585003 7710 11600 4100 33100
4 12300 8560 5720 252005 38500 22000 4950 15000 30200
6179000 17900 1730 9790 14100
7 17200 46000 25300 70000 545008 25400 6970 58300 44300 12700
9 4940 20600 10100 15200
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Solution
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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.
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Confidence Interval
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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