Download Matalas (1967) Multisite Normal Generation Model for Multivariate Stochastic Processes and more Study notes Mathematical Statistics in PDF only on Docsity!
- Matalas (1967) has given a multisite normal
generation model that preserves the mean,
variance, lag one serial correlation, lag one cross-
correlation and lag zero cross-correlation.
where
X
t
and X
t+
are p x 1 vectors representing standardized
data corresponding to p sites at time steps t and t+
resp.
Multivariate Stochastic models
3
t 1 t t 1
X AX B ε
= +
Ref.: Matalas, N.C. (1967) Mathematical assessment of synthetic hydrology, Water
Resources Research 3(4):937-
Assumption is that the model is
multivariate normal.
ε
t+
is N(0,1); p x 1 vector with ε
t+
independent of X
t
A and B are coefficient matrices of size p x p. B is
assumed to be lower triangular matrix
Multivariate Stochastic models
4
0 t t
M E X X ⎡ ⎤ =
⎣ ⎦
( )
, ,
0
1
n
j t j i t i
t i j
Q Q
Q Q
m i j
n s s
=
∑
M
0
is the cross-correlation matrix (size pxp)
of lag zero
i.e., m
( i , j ) represents lag one cross correlation
between the data at sites i and j.
Therefore M
is the cross-correlation matrix of lag
one.
Multivariate Stochastic models
6
( )
, 1 ,
1
2
n
j t j i t i
t i j
Q Q
Q Q
m i j
n s s
−
=
∑
Q is the original random variable before
standardization e.g., stream flow
Considering the model,
Post multiplying with X
t
on both sides and taking the
expectation,.
Multivariate Stochastic models
7
t 1 t t 1
X AX B ε
= +
t 1 t t t t 1 t
E X X AE X X BE ε X
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = +
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
M AM 0
A M M
= +
=
ε
t+
and X
t
are independent
Multivariate Stochastic models
9
1 t t 1
M E X X
t t
t t
t t
M E X X
E X X
E X X
1 t t 1
M E X X
or
Multivariate Stochastic models
10
Taking expectation on both sides,
{ }
t t t t t
t t t t
X AX B
X A B
ε ε ε
ε ε ε
= +
= +
0
t t t t t t
t t t t
E X E X A B
E X A E B
IB
B
ε ε ε ε
ε ε ε
⎡ ⎤ ⎡ ⎤ = +
⎣ ⎦ ⎣ ⎦
⎡ ⎤ ⎡ ⎤ = +
⎣ ⎦ ⎣ ⎦
= +
=
Since ε
t+
has
unit variance
- B does not have a unique solution.
- One method is to assume B to be a lower triangular
matrix.
Multivariate Stochastic models
12
( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
'
b b b b p
b b b b p
BB
b p b p b p p b p p
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
c c c c p
c c c c p
C
c p c p c p p
- The diagonal elements of the B matrix are obtained
as,
Multivariate Stochastic models
13
( ) ( )
( ) ( ) ( ) { }
1,1 1,
2, 2 2, 2 2,
b c
b c b
=
= −
( ) ( ) ( ) ( ) ( ) { }
1
2 2 2 2
b k k , = c k k , − b k k , − 1 − b k k , − 2 − ... − b k ,
These elements
are obtained one
by one, using also
the expressions for
the k
th
row
elements given in
the next slide
The annual flow in MCM at two sites P and Q is given
below. Generate the first two values of data from
these two sites.
15
Example – 1
Year 1 2 3 4 5 6 7 8 9 10
Annual flow at
site P (MCM)
Annual flow at
site Q (MCM)
Year 11 12 13 14 15 16 17 18 19
Annual flow at
site P (MCM)
Annual flow at
site Q (MCM)
M
matrix is cross correlation matrix of lag zero
16
Example – 1 (Contd.)
Site P Q
Mean 5333 5462
Std.dev. 1125.1 823.
( ) ( )
( ) ( )
0 0
0 0
P P P Q
Q P Q Q
r r
M
r r
⎡ ⎤
=
⎢ ⎥
⎣ ⎦
P
Q
P Q
18
Example – 1 (Contd.)
( )
( ) ( )
( )
, , 1
1
,
1
1
n
P i P Q i Q
i
P Q
P Q
x x x x
r
n s s
=
− −
=
−
∑
0.302 0.
0.02 0.
M
⎡ ⎤
=
⎢ ⎥
−
⎣ ⎦
A M M
=
2.73 2.
2.17 2.
M
− ⎡ ⎤
=
⎢ ⎥
−
⎣ ⎦
19
Example – 1 (Contd.)
A M M
=
0.302 0.164 2.73 2.
0.02 0.118 2.17 2.
− ⎡ ⎤ ⎡ ⎤
=
⎢ ⎥ ⎢ ⎥
− −
⎣ ⎦ ⎣ ⎦
0.47 0.
0.31 0.
A
− ⎡ ⎤
=
⎢ ⎥
−
⎣ ⎦
21
Example – 1 (Contd.)
( ) ( ) ( )
b 1,1 = c 1,1 = 0.89 = 0.
( ) ( ) ( )
( )
1,1 0.89, 1, 2 2,1 0.76,
2, 2 0.
c c c
c
= = =
=
( )
( )
( )
2,
2,1 0.
1,1 0.
c
b
b
= = =
( ) ( ) ( ) { }
{ }
2, 2 2, 2 2,
0.95 0.81 0.
b = c − b
= − =
( )
( )
( )
,
,
1,
c k
b k
b
=
22
Example – 1 (Contd.)
( ) ( ) ( )
b 1,1 = 0.94, b 2,1 = 0.81, b 2, 2 =0.
0.94 0
0.81 0.
B
⎡ ⎤
=
⎢ ⎥
⎣ ⎦
t 1 t t 1
X AX B ε
= +
0.47 0.21 0.94 0
0.31 0.37 0.81 0.
P t P t P t
Q t Q t Q t
x x
x x
ε
ε
⎡ ⎤ − ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
= +
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
−
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
