Modeling Monthly Rainfall Records in Arid Zones Using Markov Chains: Saudi Arabia Case Study, the 4th International Conference on Water Resources and Arid Environments, December, 2010, pp.141-146.
Introduction to Machine Learning Unit-5 Notes for II-II Mechanical Engineering
Modeling Monthly Rainfall Records in Arid Zones Using Markov Chains: Saudi Arabia Case Study
1. 8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
Amro Elfeki and Nassir Al-Amri
Dept. of Hydrology and Water Resources Management ,Faculty of
Meteorology, Environment & Arid Land Agriculture, King Abdulaziz
University , P.O. Box 80208 Jeddah 21589 Saudi Arabia
2. Research Objectives
Typical Rainfall Station Data
Methodology and Model Development
Results
Conclusions
Outlook
8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
3. Modeling monthly rainfall records in arid
zones for future predictions of monthly
rainfall.
Application on a case study in Saudi Arabia
(Three Stations).
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7. Modeling the Sequence of Wet-Dry Month by Markov Chain.
Modeling the Amount of Rain in the Wet Month by Probability
Density Function (PDF).
We need:
Markov Chain Theory
Theory of PDFs of Random Variables.
Testing of Hypothesis for Fitting a PDF to the Data.
PDF Parameter Estimation by Method of Statistical Moments.
8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
8. 8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
SS S
i0 1 i+1i-1 N2
l k q
Pr( )
Pr( ) : ,
i i -1 i -2 i -3 0k l n pr
i i -1k l lk
, , S ,...,S S S SX X X X X
pS SX X
...
.....
....
....
..
1
21
11211
nnn
lk
n
pp
p
p
ppp
p
1,...,0
1
pp
n
k
lklk
( )
limN
N
klkp
1
1
...,
0, 1
n
k klk
l
n
k k
k
, k 1 , np
Marginal prob.
Transition prob.
10. Probability to jump from state l to state k
Assume stationarity: independent of time
Transition probability matrix has the form:
Pr( ) : ,i i -1k l lkpS SX X
10
p
q
1-q1-p
8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
11. 8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
00 01
10 11
Transition Probabilities
0 1
0 1
1 1
# of times the chain goes from state 0 to state 1
# of times the chain goes from state 0 to state 0 and state 1
# of times the chain
p p p p
p p q q
p
q
0
1
1 11 01
goes from state 1 to state 0
# of times the chain goes from state 1 to state 0 and state 1
Marginal Probabilities
Persistent Parameter
(1 ) 1
Mean Length of Persistent Sequ
q
p q
p
p q
p p q p q p
0 1
0 1
ence
1 1
,
1 1
L L
10
p
q
1-q1-p
14. Theory of PDFs of random variables:
- Log-Normal.
- Truncated Gaussian.
- Exponential.
- Gamma.
- Gumbel (Double Exponential).
PDF parameter estimation by method of statistical
moments:
Testing of hypothesis for fitting a PDF to the data.
8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
1
( )
k
r
rc rj j
j
m = f x x
15. 8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
Ho: the data follows the claimed distribution
H1: the data does not follow the claimed distribution
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ˆmax ( ) - ( )n n
x
D D F x F x
Formal question: Is the
length of largest difference
between the “empirical
distribution function and the
theoretical distribution
function” statistically
significant?
if the distribution is acceptedn
t
D
n
2 2
-1 -2
1
for a give , is computed from 2 (-1)i i t
i
t e
Dmax
17. 8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
ungrouped
mean 9.934090909
var 258.9335618
sd 16.09141267
skew 2.296349169
kurt 5.062391357
Median 2.4
Mode 0.2
Geomean 2.479539251
harmonic
mean 0.709048777
Quadratic
mean 18.85889992
average
deviation 11.1768595
range 77.5
relative
range 7.80141844
CV 1.619817336
mean(ln)= 0.908072757
sd (ln)= 1.843922054
Arithmatic mean 10.956
harmonic mean 0.5306
quadratic mean 19.184
variance 247.98
sd 15.747
skew 2.3485
kurt 8.2063
groupedAn Excel Sheet has been developed to
calculate the descriptive statistics and
perform hypothesis testing to fit a
distribution
19. 8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
An Excel Sheet has been
developed to
perform coding of the station
record,
calculate the transition
probability of the sequence,
and
perform simulations
of the sequence based on the
data and the parameters
estimated from the other
sheet.
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Data
Single Realization Simulation Animation of few Realizations
Log-Normal Distribution
Exponential
Distribution
Single Realization Simulation
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Exponential
Distribution
Data
Single Realization Simulation
25. 8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
Log-Normal Distribution
Data
Single Realization Simulation
Exponential
Distribution
26. K-S test shows that Log-normal and Exponential distributions are best
suited to the monthly data at 5% significant level.
Chi2 test rejects the probability distributions considered except at Khules
station where the Gamma distribution seems to fit the data, however, for
Amlog and Tabouk, the exponential distribution seems to fit the monthly
data visually.
The Markov chain analysis shows that (q > p): q(w→d) and p(d→w)
and therefore ( 1-p > 1-q): 1-p (d→d) and 1-q (w→w)
On average, 30% of the year is rainy and 70% of the year is dry.
Mean length of rainy months ~ 1.5 month.
Mean length of dry months ~ 4 month.
8/5/2016 ELFEKI&ALAMRI ( ICWRAE 2010)
27. Improving the model by:
1. Incorporating non-stationary transition probability
(Seasonality).
2. Providing uncertainty bounds in the predicted
rainfall records.
3. Introducing more pdfs.
4. Applying the developed model on many stations
in the Kingdom.
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