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WFM 5201: Data Management and Statistical Analysis

Akm Saiful Islam. WFM 5201: Data Management and Statistical Analysis. Lecture-8: Probabilistic Analysis. Institute of Water and Flood Management (IWFM) Bangladesh University of Engineering and Technology (BUET). June, 2008. Frequency Analysis. Continuous Distributions Normal distribution

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WFM 5201: Data Management and Statistical Analysis

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  1. Akm Saiful Islam WFM 5201: Data Management and Statistical Analysis Lecture-8: Probabilistic Analysis Institute of Water and Flood Management (IWFM) Bangladesh University of Engineering and Technology (BUET) June, 2008

  2. Frequency Analysis • Continuous Distributions • Normal distribution • Lognormal distribution • Pearson Type III distribution • Gumbel’s Extremal distribution • Confidence Interval

  3. 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)

  4. 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 .

  5. 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

  6. 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

  7. Frequency of the discharge of a River

  8. Solution • According to the lognormal distribution is

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

  10. Example-2: • Assume the data of previous table follow the lognormal distribution. Calculate the magnitude of the 100-year peak flood.

  11. 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.

  12. 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.

  13. 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)

  14. 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)

  15. 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

  16. 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

  17. 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)

  18. Solution

  19. 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.

  20. Confidence Interval

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