Sin ttulo de diapositiva

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European Geophysical Society XXV General Assembly
EXTRACTING DYNAMICS FROM EMPIRICAL DATA AN
EVOLUTIONARY COMPUTATION APPROACH
A. Orfila (1,2), A. Álvarez (1,3), ,J.
Tintoré(1,2), E. Hernández-García(1,2)
(1)Instituto Mediterráneo de Estudios
Avanzados(CSIC-UIB), Palma de Mallorca,
Spain (2)Dept. de Física. Universitat de les
Illes Balears, Palma de Mallorca,
Spain (3)SACLANT Undersea Research Centre, La
Spezia, Italy
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INDEX
1. Characterising the nature of the signal
1.1. Singular Spectrum Analysis
1.2. Monte Carlo SSA
2. Forecasting futere events
2.1. Evolutionary Computation
3. General Results and Conclusions
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1. INTRODUCTION
Apparent randomness in oceanic/atmospheric time
series may be due to the non-linear but
deterministic behaviour of ocean
dynamics Irregularity of ENSO events may be not
completely random but related to low order
chaotic process (Vallis,1986, Jin et al.,1994,
Tziperman et al. 1994)

If true
Some aspects of the oceanic variability are
predictable
Here, we develop a new method to further
establish, from observations, the chaotic nature
of ENSO and predict future events
G. K. Vallis. El Niñoa chaotic dynamical
system?. Science, 232, 243-245, 1986 Jin, F. F.,
Neelin, J. D., Ghil, M. El Niño on the Devils
Staircase Annual subharmonic steps to chaos.
Science, 264, 70, 1994 Tziperman, E., Stone, L.,
M. Cane, M. Jarosh H. El Niño chaos
Overlapping of resonances .... Science, 264,
7274 (1994).
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2. OBJECTIVE

CHARACTERIZE and FORECAST the ocean variability
using an evolutionary genetic algorithm
Application to El Niño phenomena
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3. CHARACTERISING ENSO DYNAMICS
Given a time series from a physical phenomena

What can we learn from it?
We can characterize the chaotic nature of the
time series by discriminating the environmental
noise from the deterministic signal
Time series from 1949 to 1998 built up with 5
months running average of spatially averaged sea
surface temperature (SST) over an area in the
Tropical Pacific Ocean located between
4ºS-4ºN,150ºW-90ºW.
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3. CHARACTERISING ENSO DYNAMICS
A) Singular Spectrum Analysis (SSA)
Reconstruction of the Time Series with only the
relevant eigenvectors
To discriminate signal and noise in the Time
Series, we employ

• Method

• Embed the time series in a vector space of
  dimension M (14 in our case)
• Compute the covariance matrix C
• Diagonalize C
• Projections of the eigenvectors are the
  reconstructed components of the time series
• Eigenvalues give the variance of the
  reconstructed components

Which eigenvectors should be chosen?

• To obtain the new time series we need to
  establish the relevant eigenvectors.

For this purpose we use MCSSA
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3. CHARACTERISING ENSO DYNAMICS
B) Monte Carlo Singular Spectrum Analysis (MCSSA)
To determine which eigenvalues could represent
a deterministic process, the surrogate data
method is employed
Environmental noise contaminate the signal an
assumption must be done to discriminate the noise

• Method (Allen,1996)

• Null Hypothesis The signal is generated by an
  AR(1) (Theiler et al,1992Mardia et al, 1979)
  process (red noise)
• Surrogate time series generated by Monte Carlo
  methods (Vatuard et al, 1992)
• Comparison with the original time series
• Discriminating Statistics

The clean time series is generated using the
relevant eigenvectors within the limits given by
the red noise
Theiler J., Eubank, S., Longtin, A., Galdrikian,
B. Farmer, J. D. Testing for nonlinearity in
time series the method of surrogate data.
Physica D, 58, 7794 (1992). Mardia, K. V., Kent,
J. T. Bybby, J. M., Multivariate analysis
Academic Press, 496 pp. (1979). Vatuard, R.,
Yiou, P. Ghil, M. Singular Spectrum Analysis A
toolkit for short, noisy and chaotic series.
Physica D, 58, 95126 (1992).
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3. CHARACTERISING ENSO DYNAMICS

• The SSA gives the eigenvalues (blue points)
• The MCSSA gives the 95 confidence level with an
  AR(1) noise (red line)

• The new clean time series is obtained using only
  the first four Principal Components, the most
  relevant obtained by the the SSA

• The new clean time series explains 60 of the
  variance of the original data.

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4. FORECAST ENSO EVENTS


Building a dynamical model directly from data
involves two steps
a) STATE SPACE RECONSTRUCTION b)
EVOLUTIONARY COMPUTATION
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4. FORECAST ENSO EVENTS
a) State space reconstruction


1. The goal of state space reconstruction is to
use the immediate past behaviour of the time
series to reconstruct the current state of the
variables T
2. We assume that the reconstructed time
series T(ti) is a projection of the dynamical
system (the tropical ocean)
where s is a d-dimensional state that defines
the dynamical system F and h is a non-linear
function
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4. FORECAST ENSO EVENTS
3. The past behaviour of the time series
contains information about the present state.
For a deterministic system there is a function
P verifying (Takens theorem) (Takens,1991)
T(t1)P(T(t- ?),T(t-2?)..............T(t-M?))
where ? is the time delay , one year in our case
(Abardanel, 1993)
How do we get the mapping function P?
Takens, F. Detecting strange attractors in
turbulence. In Dynamical systems and turbulence
SpringerVerlag, Berlin (1991). Abarbanel, H. D.
I., Brown, R., Sidorowich, J. J. Tsimring, L.
Sh. The analysis of observed chaotic data in
physical systems. Rev. Modern Phys., 65,
13311392 (1993).
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4. FORECAST ENSO EVENTS
b) Evolutionary computation

A population of potential solutions of the
problem are subjected to an evolutionary process
based on Darwinian theories of natural selection
involving

• Mutation
• Reproduction
• Survival of the strongest individuals
  characterised by those which best fits the
  environmental conditions, i.e. the data
  (Kozac,1994 Spiro,1997 Goldberg ,1989)

K. R. Kozac, Genetic programming MIT Press, 745
pp. (1994). Szpiro, G. G. Forecasting chaotic
time series with genetic algorithms. Phys. Rev.
E, 55, 25572568 (1997). Goldberg, D. E.,
Genetic algorithms in search, optimization and
machine learning AddisonWesley Longman, Inc.,
412 pp. (1989).
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4. FORECAST ENSO EVENTS
P1 (x1,x2,x3)(ax1x2-c) P2(x1,x2,x3)bx1x3/x2
Initial random population


1. Reproduction P1P2 (Crossing)
P3(ax1-x2) P4bx1x3/(x2-c)
P1(ax1-x2-c) P4b/x1x3/(x2-c)
2. Mutation
P1,P3 best fit survive P3,P4 die
3. Survival of the strongest individuals
(Selection)
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4. FORECAST ENSO EVENTS
Using only 34 years (1950-1983) of the filtered
SST (training set) Maximum numbers of symbols
allowed for each tentative equation20 Each
generation consists of a population of 120
equations After 10000 generations, we obtain the
following mapping equation


(Eq.1)
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4. FORECAST ENSO EVENTS

• Estimation of the forecast skill of Eq (1) in red
  in the fig. is based on the retroactive real-time
  forecast method (Tangnag, 1997) where we trained
  the evolutionary program with the record of
  1950-1983 and made the forecasts during 1984-1993

The red points are the forecast obtained by the
Eq.1 and the blue line are the original data

• The percentage of the testing set's total
  variance explained by Eq.(1) is 98.6. of the
  cleaned time-series. This part of ENSO signal is
  deterministic

Tangang, F. T., Hsieh, W. W., Tang, B.,
Forecastingthe equatorial Pacific sea surface
temperatures by neural network models.Climate
Dyn., 13, 135-147 (1997).
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4. FORECAST ENSO EVENTS

• In order to discriminate if the agreement between
  data and predictions in the testing set comes
  from artificial dependencies in the data
  introduced by the filtering procedure or from an
  intrinsic dynamical behaviour captured by the
  evolutionary algorithm, Eq. (1) is validated
  against a subset of five years of unfiltered
  records (1994-1998)

The blue line are the original data and the red
points are the validation obtained by Eq.(1) for
a subset of unfiltered records (1994-1998) with
an accuracy of 96 and a two years forecast
(1999-2000)
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4. CONCLUSIONS

• We have shown that the ENSO signal is partly
  deterministic
• This deterministic part of the interannual SST
  variability can be predicted by Eq.1 with two
  years in advance
• Specifically, the forecast of Eq.1 for the next
  two years is the existence of a cooling event (La
  Niña) during 1999 and an El Niño event in the
  year 2001

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5. SUMMARY

• We discriminate the environmental noise from the
  deterministic signal with the MCSSA method
• We build a new time series of the dynamical
  system with the first four EOFs
• Using an evolutionary algorithm we find the
  function that best maps the system
• The percentage of the total variance explained
  by the model Eq.(1) is 96 in the dataset from
  1994 to 1998.
• From these results we can infer that the
  symbolic regression technique has been able to
  capture a deterministic behaviour intrinsic to
  the interannual SST variability of the tropical
  Pacific ocean

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