Full text of DOI:10.1007/BF02876208

Vol. 44 No. 6
SCIENCE IN CHINA (Series D)
June 2001
A new inverse method and application to ocean data
WEI Enbo (魏恩泊)1, TIAN Jiwei (田纪伟)2 & GU Guoqing (顾国庆)1
1. College of Power Engineering, University of Shanghai Science and Technology, Shanghai 200093, China;
2. Physical Oceanography Laboratory, Ocean University of Qingdao, Qingdao 266003, China
Correspondence should be addressed to Gu Guoqing (email:guoguoqk@onlin.sh.cn)
Received September 10, 1999
Abstract A new method is proposed to inverse normalization data of hidden variables in a dy-
namical system by embedding a time series in multidimensional spaces and applying a normaliza-
tion analysis to the conditional probability density of points in the reconstructed phase spaces. The
method is robust in the application to Lorenz system and 4-dimensional Rössler system by testing
quantitatively and qualitatively the correlation coefficient between inverse data and original data in
time domain and in frequency domain, respectively. By applying the method to analyzing the South
China Sea data, the normalization data of wind speed is extracted from the sea surface tempera-
ture time series.
Keywords: reconstructed phase space, conditional probability density, normalization.
In recent years, as the embedding space theorem developed, the inverse method of hidden
variables has become an important problem to study. For example, Breeden and Hübler[1] have
developed a method for reconstructing equivalent equation of motion for a system with one or
several hidden variables. Gouesbet and Sceller[2—5] have designed a reconstruction technique of
global vector field to recover dynamical system from a scalar time series. These information in-
verse methods are all on the basis of recovering original equations in reconstructed phase spaces.
In 1995, Ortega[6] devised a useful procedure which explored the approximate attractor invariant
measure in reconstructed phase spaces to detect the dominant frequencies of hidden variables by
use of embedding theorem. However, the procedure cannot extract the normalization data and
power spectra of hidden variables from a chaotic time series. Considering a time series coming
from a nonlinear dynamical system, it is possible to inverse the normalization data and power
spectra of hidden variables by embedding the data in phase spaces because the nonlinear interac-
tion information of other variables in the system is hidden in the data.
In this paper, we will discuss the inverse problem in accordance with the diffeomorphism
properties between physical spaces and reconstructed phase spaces and then give a new method
used to extract the normalization data and power spectra of hidden variables from the chosen time
series by means of normalization analysis and spectral analysis over the conditional probability
density of points that the trajectory encounters as it evolves in the phase space.
1
Inverse method
At present, there are few useful methods for extracting the data of hidden variables although

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a number of methods to evaluate attractor invariants from numerical scalar time series, such as
generalized dimensions, Lyapunov exponent and generalized entropies, are based on embedding
theorem. In fact, Takens’[7] embedding theorem assures that the topology of the attractor in physi-
cal space can be preserved by time delay maps only if the embedding dimension is large enough to
fully unfold the attractor structure, so the topology of original system can be recovered in embed-
ding space. Moreover, Eckman and Ruelle’s[8] ergodic theory of strange attractor shows that
among many possible invariant distributions only the natural density of points in phase space is
stable. Therefore, according to these theorems, we introduce the conditional probability density of
points in reconstructed phase space to inverse the hidden variable data by using only one time se-
ries.
Consider a time series x(i) (i = 1, 2, …, N). Using time delay or differences coordinates[9], we
embed the data in m-dimension phase space S = {v(i)},
v(i) = (x(i), x(i +τ),…, x(i + (m −1)τ)) (or v(i) = (x(i), x (i),…, x(m−1) (i)) )
′
= (v1(i),v2 (i),…,vm (i)) ,
where τ and m are delay time and embedding dimension, respectively. Let σk be the standard de-
viation of the values of the component vk, and let tk stand for the statement |vk(i)−vk(j)|≤μkσk,
where k≤m and μk is a constant. If we perform a partition of m-dimension phase space, we denote
each partition element by a label, i, and the conditional probability density of the ith element of
the partition is
ni (t1 (t2 ,t3 ,ꢀ,tm ))
pi (t1 /(t2 ,t3 ,ꢀ,tm )) =
,
(1)
l
ni (t1 /(t2 ,t3 ,ꢀ,tm ))
∑
i=1
where the useful length l = N − (m − 1)τ, ni(t1/(t2, t3, …, tm)) represents the number of points v1(j)
within |v1(i) − v1(j)|≤μ1σ1 when the points vk(j) of the partition element are within |vk(i) − vk(j)|≤
μkσk (k = 2, 3, …, m). As each partition element i evolves with time, the nature density visits dif-
ferent regions with different densities under the condition (t2, t3, …, tm) along the axis v1 in the
reconstructed phase space, and each specific recurring motion of the system is probed[6,8]. In fact,
for an attractor in an m-dimension phase space, we can extract the information of hidden variables
by means of slicing the attractor with m − 1 dimensional hypersheets[9] and the probability density
sliced at each point obeys δ distribution, where if x = a, δ(x − a) =∞, otherwise δ(x − a) = 0. The
attractor can be sufficiently shown in the reconstructed phase space if parameters μk(k = 2, 3, …,
m), which are multiples of variation scales at the center of partition element along the directions of
embedding coordinates, are properly chosen with the intrinsic properties of the maps from the
physical space to the reconstructed space. Thus the density time series with suitable μk can ap-
proximately describe the periodic variation of hidden variables in the system and the normaliza-
tion data of hidden variables can be extracted by applying the normalization analysis to the density

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time series. The normalization formula is defined as follows:
pi − p
yi =
,
(2)
l
1
2
(
pi − p
)
∑
l
i=1
l
1
where p =
pi and pi = pi (t1 /(t2 ,t3,…,tm )).
∑
l
i=1
2
Applications
We apply the above method to the chaotic Lorenz system,
ꢁ
ꢁ
ꢁ
x = −σ x +σ y, y = −y + rx − xz, z = −bz + xy,
with the parameters σ = 10.0, r = 28.0, b = 8/3, and 4-dimensional hyperchaotic Rössler sys-
tem[10]
,
ꢁ
ꢁ
ꢁ
ꢁ
x = −y − z, y = x + 0.25y + w,z = 3 + xz, w = −0.5z + 0.05w .
Two systems are integrated by means of four-order Runge-Kutta algorithm with a fixed step
(Lorenz system with a fixed step of 0.04 and 4-dimensional Rössler system with a fixed step of
0.001). We store these components of the systems for further analysis (the sampling time interval
equals the fixed integration step and the useful length of time series is 16384).
Because the x component (or y) hides the frequencies information of the z component (see fig.
[9]
ꢁ
ꢁꢁ
ꢁ
1), we embed the x component in the 3-dimensional phase space (x(i), x(i), x(i)) , where x and
ꢁꢁ
x are obtained by using center difference formula to extract the z component data. We choose
μ1 =1.5, μ2 = μ3 = 1.0 in order to show the different scales of axes in the Lorenz attractor in the
Fig. 1. The power spectra of x component (a), z component (b) and the inverse power spectra (c) of z component in Lorenz
system.

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reconstructed phase space[11] and perform a normalization analysis over the conditional probability
density time series by means of the procedure outlined in section 1. The inverse normalization
values are shown in fig. 2 (dashed line). The correlation coefficient between the inverse normali-
zation time series and original normalization time series (see fig. 2, real line ) of the z component
is about 0.83. The inverse errors are caused by the finite length of the data and the different am-
plitudes of vibration in the attractor, which restrict large amplitudes to be extracted exactly. In or-
der to diagnose the inverse data in frequencies domain qualitatively, we detect the power spectra
of inverse data (see fig. 1(c)). We can further show that the power spectra of the z component is
recovered by the x component, since the two power spectra, fig. 1(b) and fig. 1(c), are basically
the same and the correlation coefficient between the two power spectra is about 0.99.
Fig. 2. The normalization data of z component (real line) and the inverse time series (dashed line) of z component in Lorenz
system.
In the next application of the method to 4-dimensional Rössler system, we choose the time
series of the x component to inverse the normalization data of the w component because of the

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information of the w component hidden in the x component (see fig. 3(a), (b)). We embed the time
series x in 4-dimensional phase space (x(i), x(i +τ), x(i + 2τ), x(i + 3τ)) using time delay method,
where τ = 80Δt and Δt is the sampling time interval. The inverse normalization data of the w
component are obtained by using formulae (1) and (2) with μ1 = 1.5, μ2 = 1.2, μ3 = 1.0, μ4 = 1.2.
The inverse data are of opposite phase to the original data since the density time series probes the
periodic vibration of hidden variable instead of the initial phase. Thus we correct the inverse data
with eiπ (fig. 4, dashed line). In fig. 4, the correlation coefficient between the two normalization
time series is about 0.8. Because of the finite length of the time series x, the inverse errors occur
on the large amplitude, which corresponds to the large period in the w component. The method is
also very successful in extracting the power spectrum of the w component from the x component
since the correlation coefficient between the two spectra is about 0.98 (see fig. 3(c)).
Fig. 3. The power spectra of x component (a), w component (b) and the inverse power spectra (c) of w component in Rössler
system.
Fig. 4. The normalization data of w component (real line) and the inverse normalization data (dashed line) of w component in
Rössler system.

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The last application is to assimilation data of sea surface temperature (SST) and wind speed
(WS) at (114.37 ꢂ E, 21.9ꢂ N) of the South China Sea from 1979 to 1995, where WS data are sam-
pled at a height of 10 m over the sea surface (the useful length of the data is 1220, the sampling
time interval is 5 days). We can observe that the periodic information of WS is hidden in SST
from their power spectra in fig. 5. We obtain the embedding dimension of the SST, which is about
5.3, using the correlation integration method[12,13]. If the data are not contaminated by noise, the
structure of air-sea interaction system can be recovered as long as the embedding dimension is
larger than 5.3. However, because the noise exists in actual data and leads the embedding dimen-
sion to increase and the valid embedding dimension of SST is no less than 5. In the actual applica-
tion, when we embed the SST data in 5-dimensional and 6-dimensional phase spaces to analyze
the SST data, we obtain the same inverse results by using the method. Now, for example, we ana-
lyze the results of SST in 5-dimensional reconstructed phase space (x(i), x(i+ τ ), …, x(i+4τ)),
where, τ = 75days. The inverse normalization data of WS are calculated by the formulae in section
1 with μ1 = 1.4, μ2 = 1.0, μ3 = μ4 = μ5 = 1.7 (see fig. 6). In fig. 6, the correlation coefficient be-
tween the inverse normalization data and original normalization data of WS is about 0.47, because
the original data of WS are random in high frequencies domain. Comparing the two power spectra,
we can clearly obtain the power spectrum of WS from the SST because the correlation coefficient
is up to 0.96. Applying the procedure to the SST data of other stations in the South China Sea, we
can reach the same conclusions. From all the above, we can conclude that the two variables, SST
and WS, have a specific nonlinear relationship in the air-sea interaction system, and the informa-
tion of WS hidden in SST can be extracted by using the method.
Fig. 5. The power spectra of SST (a), WS (b) and the inverse WS(c) in the South China Sea.

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Fig. 6. The normalization data of WS (a) and the inverse WS (b) in the South China Sea.
3
Conclusions
For a chaotic time series, we have proposed a novel method to recover the normalization data
and the power spectra of hidden variables in the data through normalization analysis and spectral
analysis over the conditional probability density of points that the trajectory of the chaotic system
encounters as it evolves. Our method is tested by applying it to two different chaotic systems, and
the numerical results show that it is possible to recover the normalization data of the hidden vari-
able from a chaotic time series and the power spectrum in particular. In further application, this
method provides not only the useful inverse idea of hidden variables but also credible evidence for
how to choose and detect a physical variable for unknown systems. Three parameters, μk, τ and m
of the method, are chosen for the actual application to a chaotic time series. The parameter μk
describes the variation scales of the embedding coordinates in an attractor, which can be chosen
within large range[11]. The delay time τ and embedding dimension m, which are two important
parameters in the reconstructed phase space[14,15], should be carefully calculated for acquiring ac-
curate information of a dynamical system. In the application to the South China Sea data, we ob-
tain the normalization data and power spectrum of wind speed from the sea surface temperature,
demonstrating that the two variables are necessary for studying the air-sea interaction.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No.
49476254).
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