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

Contents Background Methodology Experiment of The

Precipitation Data Series Conclusion Remarks

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.

the linear modeling in short-range climatic prediction operation :

AR models MA models ARMA models ARIMA models.

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.

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

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

*

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

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

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

-

+ *

*y4

4 *

y2 y3

t e

y1

The binary tree in EM of a individual Pi

yetyyyy 3345 4

n

y

initialization

mutation

select

crossover

evaluate fitness

stop condition ？

end

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

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

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~

Sketch map of comparison between simulation

(dashed line) with actual figures (solid line) for X~

simulation of XXX ' ~

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

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 ~

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.

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.