Stochastic Hydrology, Civil Engineering - Notes - Civil Engineering (CE)

 Page 1


STOCHASTIC HYDROLOGY 
Lecture -23 
Course Instructor :  Prof. P. P. MUJUMDAR 
                                     Department of Civil Engg., IISc. 
 
INDIAN	
  INSTITUTE	
  OF	
  SCIENCE	
  
Page 2


STOCHASTIC HYDROLOGY 
Lecture -23 
Course Instructor :  Prof. P. P. MUJUMDAR 
                                     Department of Civil Engg., IISc. 
 
INDIAN	
  INSTITUTE	
  OF	
  SCIENCE	
  
2	
  
Summary	
  of	
  the	
  previous	
  lecture	
  
•? Markov chains 
–? Transition probabilities 
–? Transition probability matrix (TPM) 
–? Steady state Markov chains 
 
 
Page 3


STOCHASTIC HYDROLOGY 
Lecture -23 
Course Instructor :  Prof. P. P. MUJUMDAR 
                                     Department of Civil Engg., IISc. 
 
INDIAN	
  INSTITUTE	
  OF	
  SCIENCE	
  
2	
  
Summary	
  of	
  the	
  previous	
  lecture	
  
•? Markov chains 
–? Transition probabilities 
–? Transition probability matrix (TPM) 
–? Steady state Markov chains 
 
 
•? stochastic process with the property that value of 
process X
t 
 at time t depends on its value at time t-1 
and not on the sequence of other values 
•? At steady state,   
 
  
3	
  
Markov Chains 
1
t
tj t i ij
PX a X a P
-
????
===
????
( ) ( )
0 n
n
pp P = ×
[ ] [ ]
12 0 1
,,.....
tt t tt
PX X X X PX X
-- -
=
ppP = ×
Page 4


STOCHASTIC HYDROLOGY 
Lecture -23 
Course Instructor :  Prof. P. P. MUJUMDAR 
                                     Department of Civil Engg., IISc. 
 
INDIAN	
  INSTITUTE	
  OF	
  SCIENCE	
  
2	
  
Summary	
  of	
  the	
  previous	
  lecture	
  
•? Markov chains 
–? Transition probabilities 
–? Transition probability matrix (TPM) 
–? Steady state Markov chains 
 
 
•? stochastic process with the property that value of 
process X
t 
 at time t depends on its value at time t-1 
and not on the sequence of other values 
•? At steady state,   
 
  
3	
  
Markov Chains 
1
t
tj t i ij
PX a X a P
-
????
===
????
( ) ( )
0 n
n
pp P = ×
[ ] [ ]
12 0 1
,,.....
tt t tt
PX X X X PX X
-- -
=
ppP = ×
Example – 1 
4	
  
Consider the TPM for a 2-state first order homogeneous 
Markov chain as  
 
 
 
State 1 is a non-rainy day and state 2 is a rainy day 
Obtain the  
1.? probability that day 1 is a non-rainy day given that day 0 is 
a rainy day  
2.? probability that day 2 is a rainy day given that day 0 is a 
non-rainy day  
3.? probability that day 100 is a rainy day given that day 0 is a 
non-rainy day 
0.7 0.3
0.4 0.6
TPM
????
=
????
????
Page 5


STOCHASTIC HYDROLOGY 
Lecture -23 
Course Instructor :  Prof. P. P. MUJUMDAR 
                                     Department of Civil Engg., IISc. 
 
INDIAN	
  INSTITUTE	
  OF	
  SCIENCE	
  
2	
  
Summary	
  of	
  the	
  previous	
  lecture	
  
•? Markov chains 
–? Transition probabilities 
–? Transition probability matrix (TPM) 
–? Steady state Markov chains 
 
 
•? stochastic process with the property that value of 
process X
t 
 at time t depends on its value at time t-1 
and not on the sequence of other values 
•? At steady state,   
 
  
3	
  
Markov Chains 
1
t
tj t i ij
PX a X a P
-
????
===
????
( ) ( )
0 n
n
pp P = ×
[ ] [ ]
12 0 1
,,.....
tt t tt
PX X X X PX X
-- -
=
ppP = ×
Example – 1 
4	
  
Consider the TPM for a 2-state first order homogeneous 
Markov chain as  
 
 
 
State 1 is a non-rainy day and state 2 is a rainy day 
Obtain the  
1.? probability that day 1 is a non-rainy day given that day 0 is 
a rainy day  
2.? probability that day 2 is a rainy day given that day 0 is a 
non-rainy day  
3.? probability that day 100 is a rainy day given that day 0 is a 
non-rainy day 
0.7 0.3
0.4 0.6
TPM
????
=
????
????
Example – 1 (contd.) 
5	
  
1.? probability that day 1 is a non-rainy day given that day 
0 is a rainy day  
The probability is 0.4 
 
2.? probability that day 2 is a rainy day given that day 0 is 
a non-rainy day 
p
(1)
, in this case is [0.7 0.3] because it is given that 
day 0 is a non-rainy day.  
0.7 0.3
0.4 0.6
TPM
????
=
????
????
No rain
 
rain
 
No rain
 
rain
 
( ) ( )
21
pp P = ×