Two Concepts of Probability

Statistical
Relative frequency in
repeated experiments

Inductive
Subjective
Based on incomplete
information, judgment
and logical reasoning
Bayesian
Line Diagram

From Kottegoda and Rosso, 1997 p3

Dot diagram

From Kottegoda and Rosso, 1997 p4

Histogram of minimum annual flow in
the Po river between 1918 and 1978
16

14

12
Number of occurrences

10

0
200 400 600 800 1000 1200
Minimum annual flow m3/s
 Minimum annual flow in the Po river between 1918 and 1978
Alternative histogram axis scaling
- Relative Frequency
- Density
0.25
Histogram 0.003
Relative frequency polygon

0.2 0.0025
Relative Frequency

0.002
0.15

Density
0.0015

0.1
0.001

0.05
0.0005

0 0
200 400 600 800 1000 1200
Minimum annual flow m3/s
 Po River, Minimum annual flow
cumulative relative frequency
(number of values ≤ n)/n (KR p 8)

0.9

0.8
Cumulative relative frequency

0.7

0.6

0.5
qs=sort(q)
0.4
n=length(q)
0.3 crf=(0:(n-1))/n
0.2 plot(qs,crf)
0.1

0
200 300 400 500 600 700 800 900 1000 1100
Minimum annual flow m3/s
 Po River, Minimum annual flow
Quantile plot (Q-plot)
1100
qs=sort(q)
1000 n=length(q)
crf=(0:(n-1))/n
900
plot(crf,qs)
Minimum annual flow m3/s

800
Interquartile range IQR

75% Quantile or quartile

700

600
Median
500
25% Quantile or quartile
400

300

200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cumulative relative frequency
 Quantile Definition

0.6pi
F(y)
p
0.2

qi
-3 -2 -1 0 1 2 3
x
y
A quantile qi is the random variable value
associated with a specific cumulative
probability pi
 Numerical Quantities

1 n
Mean x   xi
n i 1
n
1
Variance s 
2

n( 1) i 1
( x i  x )2

n
1
Std Deviation x 
n( 1) i1
( x i  x )2
n
xi  x
d
Mean absolute
deviation i 1 n
n

Skewness  (x i  x )3
g1  i 1
ns 3
Helsel and Hirsch page 21
 Time Series Box Plot
8

8
7

7
6

6
log(alafia)

Median
5

5
Box (Red
4

4
Lines) enclose
3

3
50% of the
values
1930 1940 1950 1960 1970 1980 1990 2000

Time
3 Box Plot
Outliers: beyond 1.5*IQR
2

Whiskers: 1.5*IQR or largest

value
1

Box: 25th %tile to 75th %tile

0

Line: Median (50th %tile) - not

the mean
-1

Note: The range shown by the box is

called the “Inter-Quartile Range” or
IQR.
-2

This is a robust measure of spread. It is

-3

insensitive to outliers since it is based

purely on the rank of the values.
 Seasonality of Flow Flow “Monthly Subseries Plot” - time series
for each month
1000 1500
Flow (cfs)
500
0

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Outliers Horizontal Line is the mean

Compare change in mean and
Flow median between Aug-Sep.
Note Skew in September
1000 1500

Box Plots
Flow (cfs)
500
0

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Scatter Plot - Flow v. Water Level

1500
Alafia Flow (cfs)

1000
500
0

20 25 30 35

MD-11 DP Water Level

 Multiple Scatterplots
0 5 10 15 0 5 10 15 20 25
Flow = f(Pumping)

1500
Causality?

1000
Flow.ALAFIA
Co-effect?

500
0
OR
15

Pumping = f(Flow)
10

Pcp.S259
5
0

35
30
WL.MB11DP Water Level = f(Pumping)

25
Logical relationship

20
25
20
15

Pump.MBTOTAL
10
5
0

0 500 1000 1500 20 25 30 35

 Scatterplot - between raw Q-Q plot - between sorted
x and y data x and y data

22
22

20
20

18
18

ys
y

16
16

14
14

12
12

12 14 16 18 20 12 14 16 18 20

x xs

Compares individual X Compares the

and Y values distributions of X and Y
Quantiles to compare to theoretical distribution
Rank the data
Theoretical distribution,
x1 e.g. Standard Normal
x2
x3
i pi
. prob( X  xi ) 

0.6
n 1
F(y)
.
.
0.2
xn
-3 -2 -1 0
qi1 2 3

y
qi is the distribution specific theoretical
quantile associated with ranked data value xi
 Quantile-Quantile Plots
Normal
QQ-plot for Q-Q
RawPlot
Flows Normal
QQ-plot for Q-Q Plot
Log-Transformed Flows
4000

ln(xi)

8
7
3000
Sample Quantiles

Sample Quantiles

6
xi
2000

5
1000

4
3
0

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

Theoretical Quantiles Theoretical Quantiles

qi qi

Used as a basis for finding transformation to

make the Raw flows Normally distributed.
 20 Quantile plots and Probability Plots

20
18

18
xs

xs
16

16
14

14
12
12

-2 -1 0 1 2 0.1 0.5 0.8 0.95

q p

Q-Q Plot Probability Plot

Theoretical quantile axis is relabeled
with corresponding probability values