A Time-Varying Model for Disturbance StormTime (Dst) Index Analysis
Presentation: Yang Li (Phd student)
Supervisor: Prof. Billings S.A. and Dr. Hua-Liang Wei
Department of Automatic Control and Systems Engineering,
the University of Sheffield
1. Background
1) Empirical modelling approach
----some physical insights or assumptions
2) Time-invariant system identification approach
----family of NARMAX
----advantage and disadvantage
3) Time-varying system identification approach
---- wavelet basis function approximation
(a) model coefficients approximated by wavelet basis function
(b) multi-resolution wavelet decompositions
2. Time-Varying ARX Model
1) TVARX process:
P
Q
y  t    ai  t y  t  i    bk  t u  t  k   e  t 
i 1
k 0
where
time-varying parameters :
model orders : P , Q
model residual : e  t 
ai  t  , bk  t 
(1)
Expanding ai  t  and bk  t  by wavelet basis function
L
a i  t    i ,m m  t  ,
m 1
L
b k  t     k ,m m  t 
m 1
where
expansion parameters : i ,m , k ,m
set of basis functions:  m  t  , m  1, 2, , L.
(2)
Substituting (2) into (1), yields,
P
Q
L
L
y  t   i ,m m  t y  t  i    k ,m m  t  u  t  k   e  t 
i 1 m 1
(3)
k 1 m 1
new variables can be defined:
ym t  i    m t  y t  i  ,
um t  k    m t  u t  k  .
(4)
Q
(5)
Substituting (4) into (3), yields,
P
L
L
y  t   i ,m ym  t  i    k ,mum  t  k   e  t  ,
i 1 m 1
k 1 m 1
model (5) in the form of matrix:
y  t    T  t   t   e  t 
(6)
where
T
regression vector:   t    ym t  1 , , ym t  P  , um t 1 , , um t  Q 
coefficient vector:   t   1,m , ,  P,m , 1,m , , Q,m 
T
The definition of time-dependent spectrum:
ˆ  t  e  j 2 f / f s
b
 k 0 k
Q
H  f ,t 
1   i 1 aˆi  t  e  j 2 f / f s
P
2
(7)
3. The Multi-Wavelet Basis Functions
B-spline function of m-th order:
x
m x
Nm  x  
Nm1  x  
Nm1  x  1 ,
m 1
m 1
with

1
N1  x   0,1  x   

0
if
x   0,1
otherwise
m2
. (Haar funciton)
(8)
(9)
fourth-order B-spline:
where
1 4  4
j
3
  x   N 4  x       1  x  j  ,
6 j 0  j 
xn  x n
for x  0 and xn
 0 for x  0.
(10)
coefficients expression ai  t  and bk  t  by B-spline function:
 N
ai  t     i,ql l q  t
lq
 N
bk  t     k ,qll q  t
l q
 N
   i,rl l r  t
lr
l r
where
1  q  r  s  4, t  1, 2,
 N
   j ,rll r  t
, N,
recursive coefficient:  i ,l and  k ,l .
 N
   i,sl l s  t
l s
 N
   j ,sll s  t
l s
(11)
4. Model Identification and Parameter Estimation
1) Identification and parameter estimation algorithm
normalized least mean square (NLMS),
2) model order determination:
Bayesian information criterion (BIC),
BIC  n  
N  n  ln  N   1
N n
MSE  n  ,
(12)
where
mean-squared-error (MSE):
1
MSE 
N
N
  y  t   yˆ  t 
t 1
2
,
(13)
5. Dst index Modelling and Analysis
1) Modelling data
Input VBs[mV/m]
15
10
5
0
0
200
400
600
800
1000
1200
0
200
400
600
Time [Hours]
800
1000
1200
Output Dst[nT]
100
0
-100
-200
-300
Fig,1 The input (the solar wind parameter VBs) and the output (the Dst index) measured, with the
sampling interval of 1-h. A total of 1176 observation, with a time resolution of -1-h, were
involved.
2). Result analysis
0
0
0
200
400
600
800
1000
2
-1
0
0.5
-2
0
200
400
600
800
1000
1
3
a (t)
0
0
-1
200
400
600
800
200
400
600
800
1000
0
200
400
600
800
1000
0
200
400
800
1000
0
-0.5
-1
0
0
1
2
b (t)
1
-1
a (t)
1
b (t)
1
a (t)
2
0.5
1000
b (t)
4
1
2
a (t)
2
0
0
-1
-0.5
0
200
400
600
Time [hours]
800
1000
600
Time [hours]
Fig. 2. The estimates of the six time-varying coefficients ai(t) (i=1,2,3,4) output estimated
parameters and bk(t) (k=0,1,2) input estimated parameters for the magnetosphere data.
50
0
Dst [nT]
-50
-100
-150
-200
Measurements
OSA (OHA)
-250
0
200
400
600
Time [hours]
800
1000
1200
Fig. 3, on the left hand side: A comparison of the recovered signal (one-step-ahead (1-h-ahead)
predictions) from the identified TVARX (4, 2) model and the original observations for the
magnetosphere data. Solid (blue) line indicates the observations and the dashed (red) line
indicates the signal recovered from the TVARX (4, 2) model.
On the right hand side: the 3-D topographical map of the time-dependent spectrum estimated
from the TVARX (4, 2) model for the magnetosphere system data.
Figure 4 The overlap of the transient power spectra calculated at different time instants from
the limited 1176 observation data for the problem described by (6).
6. Conclusions
1) modelling approach are locally defined.
---- more flexible and adaptable
2) time-dependent spectrum of transient properties.
---- the global dynamical behaviour
Thank you!