Download Frequency Analysis

Frequency Analysis
Learning Objectives (Bedient et al, Chapter 3)
• Quantitatively describe inherently random quantities
using the methods of probability and statistics
• Plot random data using a relative frequency histogram,
cumulative frequency histogram and probability plot
• Use the method of moments to fit a probability
distribution
• Use frequency analysis to calculate flood flows of a
given return period
• Calculate the probability and risk associated with
hydrologic events
A random variable X is a variable whose
outcomes (values) are governed by the laws of
chance.
Discrete Values
Probability Mass Function
0.35
0.3
PX(x)
0.25
PX(x)=Pr(X=x)
0.2
0.15
0.1
0.05
0
1
2
3
4
5
6
x
7
8
9 10
Probability Concepts
For Xi outcomes of mutually exclusive collectively exhaustive
events
0 ≤ P(Xi) ≤ 1
𝑁
𝑖=1 𝑃 𝑋𝑖 = 1
P(X1 X2) = P(X1)+P(X2) for X1, X2 mutually exclusive
P(X1 Y1) = P(X1) P(Y1) for X1, Y1 independent
More generally
P(X1  Y1) = P(X1)+ P(Y1) - P(X1 Y1)
(Note error in text. Equation 3-6 should read as above)
Conditional Probability
P(X1|Y1) = P(X1 Y1) /P(Y1)
Blacksmith Fork River example
•
•
•
•
BlacksmithForkPeak.txt
What is the probability distribution to describe this data
What are the parameters of the distribution
What is the 50 year and 100 year flood
Numerical Quantities
Mean
Variance
Std Deviation
1 n
x   xi
n i 1
1 n
2
s 
( xi  x) 2

n  1 i 1
x
3-37
3-38
1 n
( xi  x) 2

n  1 i 1
n
Skewness
Cs 
n
(n  1)( n  2)
 (x
i 1
i
 x) 3
s3
3-40
Continuous Variable
0.20
0.30
Probability density function
0.10
x1
0.00
Pr[ x1  X  x 2 ]   f ( x )dx
f(x)
x2
0
2
4
6
x
8
10
12
Cumulative distribution function
x
FX ( x )  Pr[ X  x ]   f ( t )dt
0.4
0.0
dF
f (x) 
dx
F(x)
0.8

0
2
4
6
x
8
10
12
Figure 3-8
Hydrology and Floodplain Analysis, Fourth Edition
By Philip B. Bedient, Wayne C. Huber, and Baxter E. Vieux
Copyright ©2008 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Moments of Random Variables
Moments of Random Variables
Population
Sample

Mean

1 N
X   Xi
N i 1
 xf ( x )dx


Expectation
 xf ( x )dx
E( X ) 


Expectation operator
E(g( X)) 
 g(x )f (x )dx

1 N
Ê( X ) 
Xi

N i 1
1 N
Ê( g( X ))   g( X i )
N i 1

2
(
x


)
f ( x )dx

2
 
Variance

 E([ X  E( X )] 2 )

Skewness
1
3
N
1
2
S 
(
X

X
)
 i
N ( 1) i 1
2

 ( x  )
 
3
f ( x )dx
 E([ X  E( X )] 3 ) /  3
ˆ 
1 N
(X i  X) 3

N i 1
S3
N/(N-1)(N-2)
Sample bias
correction
Problem 3-3
Normal Distribution
2

1
1 x  
f ( x) 
exp  
 
 2
 2    
3-57
Carl Friedrich Gauß, immortalized
Log Normal Distribution
X is log normally distributed if Y=ln(X) is normally distributed
dy 1
Y  ln( X ) 

dx x
Derived distribution
dy
1
1

Y  log10 (X)  
dx x ln(10) 2.302 x
f X ( x )  f Y ( y)
2

1
1  ln( x )   y  

fX (x) 
exp 
 
y
x y 2
 2 
 

1
f X (x) 
2.302 x y
dy
dx
Base e logs
 1  log ( x )    2 
10
y

  Base 10 logs

exp 
 
y
 2 
2
 

Probability Plots
0.4
0.0
F(x)
F(x)
0.8
Cumulative Distribution Function
0
2
4
6
xx
8
10
Switch Axes
12
21 01 8
•
•
x
6
•
4
2
•
0
0.0
0.4
F(x)
F(x)
0.8
Stretch X axis so that the target distribution
plots as a straight line
Achieve this by plotting on the X-axis quantiles
corresponding to specific F values and labeling
them with the F value
The quantile function is the inverse of the
cumulative distribution function
q = Q(F) = F-1(F), for a given F, evaluate q.
[If possible label x-axis with the F values, but
feel free to in Excel just leave quantiles on the
x-axis]