Modelling monthly rainfall time series using Markov Chains

05/02/2023 ELFEKI&ALAMRI ( ICWRAE 2010)

MODELING MONTHLY RAINFALL RECORDS IN ARID ZONES USING

MARKOV CHAINS: SAUDI ARABIA CASE STUDY

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


Outline Research Objectives Typical Rainfall Station Data Methodology and Model Development Results Conclusions Outlook


Research Objectives Modeling monthly rainfall records in arid

zones for future predictions of monthly rainfall.

Application on a case study in Saudi Arabia (Three Stations).


Stations Considered in the Study


Typical Rainfall Record (Khules Station)

(411) YEAR LY SUMMARY OF RAINS FOR THE PE RIOD OF 19 65-1998 (DATED:14// )

========== ========== ========== ========== ========== ======= ============= ======

STATION: 00262 J212 خليص / GEO_ AREA: 0020 محافظ 8000 خليـص ة H YDRO_AREA: السادسـة 6-------- ----------- --------- ------------ ---------- ---------- ---------- ---------- ------- ------------- ------ ---------- ---------- -----------

YEAR JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC TOTALS

-------- ----------- --------- ------------ ---------- ---------- ---------- ---------- ------- ------------- ------ ---------- ---------- -----------1965 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01966 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.2 16.2 0.4 23.81967 0.0 0.0 0.0 0.0 0.4 0.2 0.0 2.0 1.0 0.0 21.4 0.2 25.21968 0.4 0.2 0.4 72.2 2.0 1.0 0.0 0.0 0.6 0.0 32.6 25.6 135.01969 55.8 8.8 2.4 0.0 0.0 0.0 0.0 0.0 0.4 0.0 35.2 7.4 110.01970 56.8 0.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 30.6 1.8 90.31971 0.8 4.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.0 4.0 9.81972 18.2 0.0 0.0 0.0 0.0 0.0 0.2 0.2 0.0 15.6 11.4 7.4 53.01973 1.0 0.0 0.0 0.0 0.0 0.0 0.2 0.0 0.0 0.0 0.4 9.5 11.11974 77.7 0.0 2.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 80.51975 21.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.8 2.0 10.4 34.41976 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.6 2.0 10.4 13.01977 5.4 0.2 0.0 0.2 0.0 0.0 17.6 0.0 0.2 0.8 0.0 35.6 60.01978 14.0 15.0 0.2 0.0 0.2 0.0 4.0 0.0 0.0 0.0 0.0 0.4 33.81979 63.0 0.4 1.6 0.2 0.0 0.0 0.0 40.6 0.2 6.2 0.0 0.0 112.21980 0.0 0.0 1.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.4 0.0 44.61981 0.0 0.6 6.6 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.8 9.0 17.21982 3.4 0.2 0.0 0.0 1.6 0.0 0.0 0.0 0.0 0.2 0.0 0.0 5.41983 3.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.0 0.0 3.41984 0.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.6 0.0 7.2 0.0 14.21985 9.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 24.2 34.4 68.21986 0.0 0.2 0.0 3.2 0.0 0.0 0.0 0.0 0.0 3.6 0.0 0.0 7.01987 0.0 0.0 35.2 0.0 0.2 0.0 0.0 0.6 0.2 0.0 0.4 0.2 36.81988 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.8 0.0 60.2 63.01989 0.0 0.0 0.0 5.0 0.0 0.0 0.0 0.0 0.0 0.0 3.8 15.6 24.41990 0.6 0.0 0.2 9.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 9.81991 6.0 0.0 2.4 0.0 0.4 0.0 0.0 0.0 0.0 0.0 4.0 0.2 13.01992 39.6 0.0 0.0 0.0 0.0 0.0 0.0 11.0 0.0 0.0 0.0 0.0 50.61993 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01994 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4.4 2.4 1.2 1.4 9.41995 0.0 12.0 7.4 4.2 1.4 0.0 0.0 0.0 0.0 0.4 0.4 0.0 25.81996 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01997 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 20.2 0.8 15.8 36.81998 51.0 10.2 0.0 0.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 0.0 64.2


Visualization of a Monthly Rainfall Record (Khules Station)


Methodology and Model Development

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.


Theory of Markov ChainSS 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

pp

ppp

p 1,...,0

1

ppn

klklk

( )limN

Nklk p

1

1

...,

0, 1

n

k klkl

n

k kk

, k 1 , n p

Marginal prob.

Transition prob.


Modeling the Sequence of Wet-Dry Months by a Markov Chain

10

p

q

1-q1-p


2-State Transition Probability

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 lk pS SX X

10

p

q

1-q1-p


Estimation of Markov Chain Model Parameters

00 01

10 11

Transition Probabilities 0 10 11 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 pp 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

qp q

pp q

p p q p q p

0 1

0 1

ence1 1,

1 1L L

10

p

q

1-q1-p


Coding Sequence of Wet and Dry


Monthly Transition Probabilities and Stationary Distributions

Station Transition Probabilities Stationary Distribution

Persistent Parameter (Lag-1 Autocorrelation)

Mean Length of Persistent

Sequence (month)

Khules Dry Wet 0.32

Dry 0.78 0.22

0.68 3.1Wet

0.46 0.54 0.32 1.5

Amolg Dry Wet 0.25

Dry 0.82 0.18

0.76 4.2Wet

0.57 0.43 0.24 1.3

Tabouk Dry Wet 0.2

Dry 0.74 0.26

0.68 3.1Wet

0.54 0.46 0.33 1.5


Modeling the Amount of Rain in the Wet Month

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.1

( )k

rrc rj j

j

m = f x x


Fitting PDF by Χ2 TestHo: the data follows the claimed distribution

H1: the data does not follow the claimed distribution


Fitting CDF by Kolmogorov-Smirnov Test

ˆ max ( ) - ( )n nxD 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 acceptednt

Dn

2 2-1 -2

1

for a give , is computed from 2 (-1) i i t

i

t e

Dmax


Spreadsheet Model for Statistical Analysis (Khules Statistics)

ungroupedmean 9.934090909var 258.9335618sd 16.09141267skew 2.296349169kurt 5.062391357Median 2.4Mode 0.2Geomean 2.479539251harmonic mean 0.709048777Quadratic mean 18.85889992

average deviation 11.1768595range 77.5relative range 7.80141844CV 1.619817336

mean(ln)= 0.908072757sd (ln)= 1.843922054

Arithmatic mean 10.956harmonic mean 0.5306quadratic mean 19.184variance 247.98sd 15.747skew 2.3485kurt 8.2063

groupedAn Excel Sheet has been developed tocalculate the descriptive statistics and perform hypothesis testing to fit a distribution


Statistical Measures and Hypothesis Testing

Station Arith. Mean

SD (mm)

CV Geo. Mean Skew Kurt. χ2 K-S Test

(α=0.05) (mm) (mm) (α=0.05)

Khules 9.9 16 1.6 2.48 2.3 5 Gamma Log-normal

Amlog 14.1 17.6 1.2 7.8 2 4.5 ------- Exponential

Log-normal

Tabouk 7.5 9.75 1.3 3.7 2.5 7.1 ------- Exponential Log-normal


Spreadsheet Model for Markov Chain Analysis and Predictions

An Excel Sheet has been developed to

perform coding of the station record,

calculate the transition probability of the sequence,

and

perform simulationsof the sequence based on the data and the parameters estimated from the other sheet.


Results of Khules Station

Log-normal Gamma


Results of Amlog Station

ExponentialLog-normal


Results of Tabouk Station

ExponentialLog-normal


Amlog Station PredictionsData

Single Realization Simulation Animation of few Realizations

Log-Normal Distribution

Exponential Distribution

Single Realization Simulation


Khules Station Predictions


Data



Tabouk Station Predictions

Log-Normal Distribution

Data




Conclusions 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.


Outlook 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.

Date post:	22-Mar-2017
Category:	Engineering
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Modelling monthly rainfall time series using Markov Chains

Engineering