wfm 5201: data management and statistical analysis

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)


Frequency Analysis

Continuous Distributions Normal distribution Lognormal distribution Pearson Type III distribution Gumbel’s Extremal distribution

Confidence Interval


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


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


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


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


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


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


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


Example-2:

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


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(


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.


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)


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


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


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


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


Solution


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.


Confidence Interval

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