WFM 5201: Data Management and Statistical Analysis

1
WFM 5201 Data Management and Statistical Analysis
Lecture-11 Frequency 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

• Probability Position Formula and Probability Plot
• Analytical Frequency Analysis
• Normal and Log-normal distribution
• Gumbels Extreme Value distributions Type I
• Log Pearsons Type III distribution

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Introduction to Frequency Analysis

• The magnitude of an extreme event is inversely
  related to its frequency of occurrence, very
  severe events occurring less frequently than more
  moderate events.
• The objective of frequency analysis is to relate
  the magnitude of extreme events to their
  frequency of occurrence through the use of
  probability distributions.
• Frequency analysis is defined as the
  investigation of population sample data to
  estimate recurrence or probabilities of
  magnitudes.
• It is one of the earliest and most frequent uses
  of statistics in hydrology and natural sciences.
• Early applications of frequency analysis were
  largely in the area of flood flow estimation.
• Today nearly every phase of hydrology and natural
  sciences is subjected to frequency analyses.

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Methods

• Two methods of frequency analysis are described
  one is a straightforward plotting technique to
  obtain the cumulative distribution and the other
  uses the frequency factors.
• The cumulative distribution function provides a
  rapid means of determining the probability of an
  event equal to or less than some specified
  quantity. The inverse is used to obtain the
  recurrence intervals.
• The analytical frequency analysis is a simplified
  technique based on frequency factors depending on
  the distributional assumption that is made and of
  the mean, variance and for some distributions the
  coefficient of skew of the data.

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Plotting Position Formula

• The frequency of an even can be obtained by use
  of plotting position formulas. Where,
• P the probability of occurrence
• n the number of values
• m the rank of descending values with largest
  equal to 1
• T 1-P the mean number of exceedances
• ac parameters depending on n

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Plotting Position relationship
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Parameters
n 10 20 30 40 50
a 0.448 0.443 0.442 0.441 0.440
n 60 70 80 90 100
a 0.440 0.440 0.440 0.439 0.439
a is generally recommended as 0.4 . For normal
distribution a 3/8 For Gumbels (EV1)
distribution a 0.4
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Exercise-1

• Using the 23 years of annual precipitation depths
  for a station given in the table below,
  estimation the exceedance frequency and
  recurrence intervals of the highest ten values
  using Weibull equation

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Here, n 23
Year Rain depth (in) Rank, m P (m/(n1)) Tr (year)
1981 24 1 0.042 24
1986 23 3
1988 23 3 0.125 8
1978 21 4 0.167 6
1993 20 5 0.208 4.8
1999 19 8
1998 19 8
1980 19 8 0.333 3
1990 18 10
1983 18 10 0.417 2.4
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Probability plot

• A probability plot is a plot of a magnitude
  versus a probability.
• Determining the probability to assign a data
  point is commonly referred to as determining the
  plotting position.

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• Plotting position may be expressed as a
  probability from 0 to 1 or a percent from 0 to
  100. Which method is being used should be clear
  from the context. In some discussions of
  probability plotting, especially in hydrologic
  literature, the probability scale is used to
  denote prob or . One
  can always transform the probability scale
  to or even
  
• if desired.

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Gumbels (1958) Criteria

• The plotting position must be such that all
  observations can be plotted.
• The plotting position should lie between the
  observed frequencies m-1/n of m/n and n where is
  the rank of the observation beginning with m1
  for the largest (smallest) value and n is the
  number of years of record (if applicable) or the
  number of observations.
• The return period of a value equal to or larger
  than the largest observation and the return
  period of a value equal to or smaller than the
  smallest observation should converge toward n .
• The observations should be equally spaced on the
  frequency scale.
• The plotting position should have an initiative
  meaning, be analytically simple, and be easy to
  use.

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Steps for probability plot

• Rank the data from the largest (smallest) to the
  smallest (largest) value. If two or more
  observations have the same value, several
  procedures can be used for assigning a plotting
  position.
• Calculate the plotting position of each data
  point from relationship Table presented in
  earlier slide.
• Select the type of probability paper to be used.
• Plot the observations on the probability paper.
• A theoretical normal line is drawn. For normal
  distribution, the line should pass through the
  mean plus one standard deviation at 84.1 and the
  mean minus one standard deviation at 15.9.

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Analytical Frequency Analysis

• Chow has proposed
• where, K is the frequency factor
• s is the standard deviation and x bar is the
  mean value.

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Methods of Analytical Frequency Analysis

• Normal distribution
• Log-normal distribution
• Gumbels Extreme Value distributions Type I
• Log Pearsons Type III distribution

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Normal Distribution

• The probability that X is less than or equal to x
  when X can be evaluated from
• The parameters (mean) and
  (variance) are denoted as location and scale
  parameters, respectively. The normal distribution
  is a bell-shaped, continuous and symmetrical
  distribution (the coefficient of skew is zero).

(4.9)
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Standard normal distribution

• The probability that X is less than or equal to x
  when X is can be evaluated from
• (4.9)

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• The equation (4.9) cannot be evaluated
  analytically so that approximate methods of
  integration arc required. If a tabulation of the
  integral was made, a separate table would be
  required for each value of and . By using
  the liner transformation , the random
  variable Z will be N(0,1). The random variable Z
  is said to be standardized (has and )
  and N(0,1) is said to be the standard normal
  distribution. The standard normal distribution is
  given by
• (4.10)
• and the cumulative standard normal is given
  by
• (4.11)

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Figure 4.2.1.3 Standard normal distribution
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• Figure 4.2.1.3 shows the standard normal
  distribution which along with the transformation
  
• contains all of the
  information shown in Figures 4.1 and 4.2.
• Both and are widely
  tabulated.
• Most tables utilize the symmetry of the normal
  distribution so that only positive values of Z
  are shown.
• Tables of may show or
  
• prob
• Care must be exercised when using normal
  probability tables to see what values are
  tabulated.

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• Exercise-2 Assume the following data follows a
  normal distribution. Find the rain depth that
  would have a recurrence interval of 100 years.
• Year Annual Rainfall (in)
• 2000 43
• 1999 44
• 1998 38
• 1997 31
• 1996 47
• .. ..

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• Solution
• Mean 41.5, St. Dev 6.7 in (given)
• x Mean Std.Dev z
• x 41.5 z(6.7)
• P(z) 1/T 1/100 0.01
• F(z) 1.0 P(z) 0.99
• From Interpolation using Tables E.4
• Z 2.33
• X 41.5 (2.33 x 6.7) 57.11 in

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Table Area under standardized normal
distribution
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Linear Interpolation from Z-table

• For, p0.99 , find z ?
• From table, z1 2.32 , p1 0.9898 and z2 2.33
  p20.9901