CVEEN 4410: Engineering Hydrology General Goal: Use frequency analysis of historical data to forecast hydrologic events Specific goals for this presentation: - plot data sample that fit a normal distribution using “probability paper” - plot a “linearized” representation of the CDF corresponding to the sample moments (mean and standard deviation) - judge fit of the data set to the normal distribution, using the plot Graphical Frequency Analysis: Normal Distribution Objectives “Recipe” Plotting Positions Probability Paper Plotting data with the Weibull formula and checking fit
Transcript
CVEEN 4410: Engineering Hydrology
General Goal: Use frequency analysis of historical data to forecast
hydrologic events
Specific goals for this presentation:
- plot data sample that fit a normal distribution using “probability paper”
- plot a “linearized” representation of the CDF corresponding to the
sample moments (mean and standard deviation)
- judge fit of the data set to the normal distribution, using the plot
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
3
Distribution-based (Analytical) Approach Computation of the magnitudes of extreme events
using this approach requires that the probability distribution function be invertible;
that is, given a value for T or [F(xT) = 1-1/T)], the corresponding value of xT can be determined.
Recall the 3 main probability distributions used in practice:
• Normal Distribution
• Log-Normal Distribution
• Log-Pearson Type III Distribution
Each distribution can be used to predict design floods and other
hydrologic events.
2
1. Assume the RV has normal distribution: create a series
that consists of the individual RV (e.g. annual maximum
flood series)
2. Compute the sample mean, ẍ, and standard deviation, s
3. Define the frequency curve plotting points: Point 1 = ẍ - s @ non-exceedance probability = 0.1587 Point 2 = ẍ + s @ non-exceedance probability = 0.8413
4. Note that 0.8413 and 0.1587 represent the probabilities that the an
observation is 1 SD away (either side, or z =1 and z =-1) from the mean of
a standard normal distribution. In very general terms, the frequency curve
(of item #3 above) provides a linear representation (line) of the CDF for the
population represented by the sample mean and sample SD of your
measured data set. The closer the “fit” of the RV values to the line, the
better the data are represented by the normal distribution. A poor fit to the
line usually means a different probability distribution applies (for example,
maybe a log-normal, if skewness is evident in the histogram).
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
6. If fit good, then use plot to make forecasts!
“Recipe” for Graphical Frequency Analysis: Normal Distribution
5. Check fit of data to the frequency curve: plot the data
using an appropriate plotting position.
2
1. Assume the RV has normal distribution: create a series
that consists of the individual RV (e.g. annual maximum
flood series)
2. Compute the sample mean, ẍ, and standard deviation, s
3. Define the frequency curve plotting points: Point 1 = ẍ - s @ non-exceedance probability = 0.1587 Point 2 = ẍ + s @ non-exceedance probability = 0.8413
4. Note that 0.8413 and 0.1587 represent the probabilities that the an
observation is 1 SD away (either side, or z =1 and z =-1) from the mean of
a standard normal distribution. In very general terms, the frequency curve
(of item #3 above) provides a linear representation (line) of the CDF for the
population represented by the sample mean and sample SD of your
measured data set. The closer the “fit” of the RV values to the line, the
better the data are represented by the normal distribution. A poor fit to the
line usually means a different probability distribution applies (for example,
maybe a log-normal, if skewness is evident in the histogram).
6. If fit good, then use plot to make forecasts!
“Recipe” for Graphical Frequency Analysis: Normal Distribution
5. Check fit of data to the frequency curve: plot
the data using an appropriate plotting position.
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
4
But first: Plotting Positions
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
5
But first: Plotting Positions
For a detailed primer (that exceeds detail of most textbooks), see: http://www.weibull.com/LifeDataWeb/probability_plotting.htm
Note that this primer is NOT assigned reading, but rather is optional.
According to Weibull (2006):
“The method of probability plotting takes the cdf of the distribution
and attempts to linearize it by employing a specially constructed
paper”
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
6
In this course, we will use only three
common plotting position formulas:
Weibull:
Hazen:
Cunnane:
[i is the rank and n is the number of data points]
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
CVEEN 4410 Hydrology - Class 19 7
Common plotting position formulas:
Weibull:
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
8
Bulletin 17B:
[a and b are constants specific to each
probability distribution]
Another common choice is Bulletin 17B, which has
tailored (unique) parameters for each distribution.
For sake of brevity, we’ll stick to Weibull, Hazen and
Cunnane, which are purported to be applicable for all
probability distributions.
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
March 28, 2011 CVEEN 4410 Hydrology - Class 19 9
Special probability paper is used for plotting data
for analysis. With the use of these plotting
positions, this special paper assists with
“linearizing” the CDF, such that you can compare
your data to the assumed probability distribution.
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
March 28, 2011 CVEEN 4410 Hydrology - Class 19 9
Recall step 5 in our “recipe” earlier:
5. Check fit of data to the frequency curve: plot the
data using an appropriate plotting position.
So, how exactly do you plot your data using “plotting
positions” on this probability paper?
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
March 28, 2011 CVEEN 4410 Hydrology - Class 19 9
So, how exactly do you plot your data using “plotting positions” on this
probability paper? Let’s consider the Weibull formula.
(1) Sort your random variable data from lowest to highest value; the
rank value is referred to as i, whereas the total number of data is n.
(2) For each data point, calculate the Weibull plot position value, using
.
(3) For data ranked from low to high, Pi is the non-exceedance
probability (x-axis) value, i.e., the lower x-axis; if you choose to rank
from high to low (as your text instructs), then Pi is the exceedance
probability, i.e., the upper x-axis.
(4) Scale your ordinate axis (y-axis) corresponding to the range of your
data.
(5) Plot each random variable value (y-axis) versus its Pi value (x-axis).
(6) Recall that for data that fit a normal distribution, you may check the
fit of data to that distribution by plotting the distribution line:
Point 1 = ẍ - s @ non-exceedance probability = 0.1587
Point 2 = ẍ + s @ non-exceedance probability = 0.8413
Graphical
Frequency
Analysis:
Normal
Distribution
Objectives
“Recipe”
Plotting
Positions
Probability
Paper
Plotting data with the Weibull
formula and checking fit
9
10,000
20,000
30,000
40,000
1 2 3 4 5 6 7 8 9 * * * * *
* *
* * * * * *
* * * * *
*
* * *
*
*
x
x
7,500
5,000
2,500
3. Define the frequency curve plotting points: Point 1 = ẍ - s @ non-exceedance probability = 0.1587 Point 2 = ẍ + s @ non-exceedance probability = 0.8413
