Download Log-Normal, Log Pearson`s Type III, Gumbel EV1Distribution

WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
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
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 , Y  ln X then Y has a
normal or Gaussian distribution given by:
f ( y) 
1
2
2
y
e
1  y y
 
2   y




2
,
  y  
(1)
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Derived distribution: Since Y  ln X , dy  1
dx x
the distribution of X can be found as:
dy
(2)
1
1
1
f ( x)  f ( y ) 

e
 
e

1  y y
 
2   y
dx

2 y2




2
1  y y
 
2   y
x




2
2x 2 y2
Note that equation (1) gives the
distribution of Y as a normal distribution
with mean  y and variance  y2 . Equation
(2) gives the distribution of X as the
lognormal distribution with parameters
 y and  y2 .
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Estimation of parameters (  y ,  y2 ) of lognormal
distribution:
y
 Note: Y  ln X , y   i , S   y  ny Chow (1954) Method:
n
n 1
 (1) C v  S x / X
1
X
 (2) Y  2 ln C  1
 (3) S 2  ln( C 2  1)
y
v
2
y
2
i
2
2
2
v

(4)The mean and variance of the lognormal distribution
are:
E( X )  exp(  y   y2 / 2) and
Var ( X )   e  1
2
x



 y2
Cv  e
 y2
1
(5) The coefficient of variation of the Xs is:
(6) The coefficient of skew of the Xs is:   3C v  C v3
(7) Thus the lognormal distribution is skewed to the right;
the skewness increasing with increasing values of C v .
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
35,000
45,000
55,000
2
3
10
9
0.03
0.045
0.152
0.136
65,000
75,000
85,000
11
10
12
0.167
0.152
0.182
95,000
105,000
115,000
6
0
3
0.091
0.000
0.045
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Solution

According to the lognormal distribution is
CV  S x / x  21,000 / 67,500  0.311
y  1 ln[ x 2 /(Cv2  1)]  1 ln[67,500 2 /(0.311 2  1)]  11.0737
2
2
s y  ln(Cv2  1)  0.311 2  1  0.30395
z  (ln x  y) / s y  (ln 45,000  11.0737) / 0.30395  1.182
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam

So from the standard normal table we get
p z ( z )  0.198
px ( x)  Pz ( z ) /( x  S y )  0.198 / 45,000(0.30395)
px ( x)  1.4476 105
f 45,000  10,000 (1.4476  10 5 )  0.145

The expected relative frequency according to the
lognormal distribution is 0.145
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.
y  sy z  y

The values of Sy and y are given
y  0.30395 (2.326 )  11.0737  11.781
x  exp( y)  130 ,700 cfs

The 100-year peak flow according to the lognormal
distribution is about 1,30,700 cfs.
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
  T  
6
KT  
 
0.5772  ln ln 
 
T

1
 
 

(3)
To express T in terms of K , the above
equation can be written as
T
T 
1

 
K T

1  exp  exp    

6
 

 

 
 

  0.5772
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)
Rainfall
Rainfall
Rainfall
Year
1913
1914
1915
1916
1917
1918
1920
1921
1922
1923
1924
1925
(inch)
0.49
0.66
0.36
0.58
0.41
0.47
0.74
0.53
0.76
0.57
0.8
0.66
Year
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
(inch)
0.68
0.61
0.88
0.49
0.33
0.96
0.94
0.8
0.62
0.71
1.11
0.64
Year
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
(inch)
0.52
0.64
0.34
0.7
0.57
0.92
0.66
0.65
0.63
0.6
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
  T  
6
KT  
 
0.5772  ln ln 
 
  T  1  
  5  
6
KT  
   0.719
0.5772  ln ln 
 
  5  1  
xT  x  KT s
xT = 0.649 + (0.719)(0. 177) = 0.78 in
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 , C s  0 the frequency factor is equal to the
standard normal variable z .
When ,C s  0 is approximated by Kite (1977) as
1 3
1 5
2
2
3
4
K T  z  ( z  1)k  ( z  6 z )k ( z  1)k  zk  k
3
3
2
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
0
1
2
3
4
5
38500
17900
6
0
7
17200
8
25400
9
4940
1940
55900
58000
56000
7710
12300
22000
1950
13300
12300
28400
11600
8560
4950
1960
23700
55800
10800
4100
5720
15000
1970
9190
9740
58500
33100
25200
30200
17900
46000
6970
20600
1730
25300
58300
10100
9790
70000
44300
15200
14100
54500
12700
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