the evolutionary modeling and short-range climatic prediction for meteorological element time series...
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The Evolutionary Modeling and Short-range Climatic Prediction for Meteorological Element Time Series
K. Q. Yu, Y. H. Zhou, J. A. Yang Institute of Heavy Rain, China Meteorological Administration,
Wuhan 430074
Z.KangComputation Center, Wuhan University, Wuhan 430072
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Contents Background Methodology Experiment of The
Precipitation Data Series Conclusion Remarks
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Background
the meteorological element series is the set of solution by integrating a
perfect climatic numerical model with certain initial, boundary etc. conditions;
is also the concentrated expression of all climatic factors (including itself) nonlinear interaction in the model.
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the linear modeling in short-range climatic prediction operation :
AR models MA models ARMA models ARIMA models.
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Asking a question !
can we find out another nonlinear modeling method except climatic numerical model to simulate the change with time of meteorological element ?
The evolutionary modeling (EM) has interested many researchers in various fields.
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MethodologyA series of observed meteorological data X(t) (x can be
temperature, or pressure, or rainfall etc.) at the past m steps can be written as
(1) If we can find out a higher-order ordinary differential
equation which contains some arithmetic operations, logic operations and elementary functions ( + ―, , × ∕, , sin , cos , exp , ln) involving variable x* and time t ,
(2)
1
*1
2
*2**
*
,,,),(,n
nn
dt
xd
dt
xd
dt
dxtxtf
dt
xd
Tmi tx, tx ,tx ,txt )()()()()( 1210 ,X
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with the initial conditions
(3)
minimizing the norm of n-dimensional column vector
(4)
FXX f ,min *
1
0
2* )()(m
iii txtxXX*
n , , , ,j dt
txd
dt
txd
,txtx
j
j
j
j
)1321()()(
)()(
00
*
00
*
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Eq. [2] is called NODE model of the meteorological data series
(2)
so the values of x in future can be predicted by integrating the Eq.2 based on the “past” and “now” of it.
1
*1
2
*2**
*
,,,),(,n
nn
dt
xd
dt
xd
dt
dxtxtf
dt
xd
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NODE evolutionary modelingAs the initial value problem of NODE with the
form of Eq. [2] can be converted into a system
of ordinary differential equations (SODE)
(7)
yyyt fyy
yy
yy
yy
nnn
nn
,,,'
'
211
'
1
23
'
12
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with the initial conditions
(8)
Once a HODE is converted into a SODE, the key part of the model (or model structure) is the
last (nth) equation (10)
n n , , , ,j dt
txdty
,txty
j
j
j ),1321()(
)(
)()(
)1(
0
)1(
0
001
nnn yyyt fyy ,,, 21
'
1
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-
+ *
*y4
4 *
y2 y3
t e
y1
The binary tree in EM of a individual Pi
yetyyyy 3345 4
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n
y
initialization
mutation
select
crossover
evaluate fitness
stop condition ?
end
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How to use EM for Prediction of meteorological
element series X (which is precipitation in flood season
May---Sep. at Wuhan)
observed data series annual variation series that is
necessary for prediction
The first-order NODE is enough !
XY1
'12 YY
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Experiment of the precipitation data series X
data series X is split into two parts : main series includes macro climatic time-scale period waves, is approximated by EM
: perturbation series includes micro climatic time-scale period waves superimposed on the macro one, is approximated by Natural
Fractals
Model reflects the effect of multi-time scale exterior forced factors in climate system
'X XX ~
X~
'X
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The expressions of simulation for in EM model are complex trigonometric functions that cause the irregular waves
Expressions of NODE from ten runs of EM
(avg=10,dep=3,gen=20,bre=100,order=1) 1 dx/dt=(t-8.602427)*cos(x)* x 2 dx/dt=(x-t+391.443512)/sin(x)
3 dx/dt=x/sin(x)+(t+0.266459)/sin(x) 4 dx/dt=cos(-113.667633*x)*(4278803.5/x)
5 dx/dt=6327.78479*cos(841403.5625/t) 6 dx/dt=x/sin(x*17.448830) 7 dx/dt=(exp(t)-0.453206-x)/sin(x) 8 dx/dt=(x+12.219144)/sin(x^3.0) 9 dx/dt=(6.818380* x+x)*sin(x) 10 dx/dt=ln(|x|)-(x+92.175331)/sin(x)
X~
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Sketch map of comparison between simulation
(dashed line) with actual figures (solid line) for X~
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simulation of XXX ' ~
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evaluating the forecast capability of
EM
Data series for EM 2002 2003 2004 2005 Prediction 631 846 1058 811
Rwh55(1947-2001) Deviation % -8% +14% -11% Prediction 794 945 733
Rwh55(1948-2002) Deviation % +7% -20% Prediction 939 783
Rwh55(1949-2003) Deviation % -21%
Observed data 692 739 1184
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Conclusion remarks
(1) For the meteorological data series that is of chaos character, it is an efficient way to split the series into two parts
and to simulate them by evolutionary
modeling and natural fractals respectively, the model reflects multi-time scale effects of forced factors in climate system.
'X XX ~
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Conclusion remarks
(2) The NODE in EM can approximate more efficiently the nonlinear character of meteorological element data series but linear models can’t do so, and it is also of benefit to thinking over short-range climatic analysis and prediction.
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Conclusion remarks
(3) So far as the EM algorithms technique the EM for single element time series can be extended to the EM for several element time series because the prediction is a problem of the first-order derivative equation of meteorological element x with respect to time t , each one in the system of ordinary differential equations Eq. [7] can denote respectively an xi of a complex climatic system. The EM algorithms are still same only with the values of those xi to be known at initial time. This is the job we are going to do.
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