amath 575 talk
TRANSCRIPT
Reconstruction of Phase Space:
Embedding Dimensions
AMATH 575Kyle Mandli
June 2nd, 2005
Introduction
• Chaotic data is observed often in experiments in many fields
• Traditional techniques such as Fourier analysis are not effective when dealing with chaotic signals
• Need a similar systematic technique to linear signal analysis for non-linear dynamical systems
Outline
• Basic Steps in analysis of a time series
• Reconstruction of phase space
• Embedding Dimensions
• Conclusions
Analysis of Measured Signals• Separating signal from background noise or signal
contamination
• Determine the appropriate space to analyze the signal
• Classification of the signal
• Make predictions or models of the system
• A signal is some scalar quantity that we are sampling at regular time intervals which we will assume are constant
s(n) = s(t0 + n!s)
!s
Reconstructing Phase SpaceTwo steps to reconstructing phase space from a signal:
• Choosing a time delay
• Choosing a embedding dimension
Based on work by Mañé and Takens in 1981 and Sauer et al. in 1991.
Any smooth nonlinear change of variables will act as a coordinate basis for the dynamics which will be independent of the time lag T we choose.
The theory provides us with a sufficient embedding dimension for our attractor.
Reconstructing Phase SpaceTime-Lagged Coordinates
s(n + T ) = s(to + (n + T )!s)
We can then construct a vector in d dimensions that we can examine the dynamics of the signal directly on
y(n) = [s(n), s(n + T ), s(n + 2T ), ..., s(n + (d ! 1)T )]
New set of coordinates for phase space, different then physical coordinates but just as good.
Use time lagged coordinates in the signal to construct a new set of coordinates
Reconstructing Phase SpaceTime-Lagged Coordinates
• Embedding dimension independent of dynamics
• Two points close to each other should be property of the set, not of geometry
• Dynamics in too small of a dimension leads to orbits being close that should not
• When a proper dimension has been found, provides a phase space for the analysis for the dynamics
s(n) = x(n) + y(n)
x(n + 1) = 1 + y(n) ! 1.4x(n)2
y(n + 1) = 0.3x(n)
Example: Henon Map
Signal we receive:
Time delayed coordinates
S(n) = [s(n), s(n + 1)]
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y(t)2 + 1x(t) + c2y(t) ! c3y(t)3
dy
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c1 = 3/4 c2 = 1/2 c3 = 1/2
! = 1 ! = 14
Example: Goodwin Equations
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Example: Lorenz Attractorx(t) = !(y(t) ! x(t))
y(t) = !x(t)z(t) + rx(t) ! y(t)
z(t) = x(t)y(t) ! bz(t)
Signal we receive:
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r = 45.92
b = 4
s(n) = x(t0 + n!s)
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Choosing an Embedding Dimension
Reconstructing Phase Space
• Computational costs increases quickly with increasing
• If noise is present in our signal, the higher will be populated by this extra noise instead of the meaningful dynamics of the system
In theory, as long as we pick a high enough, we are fine but there are other considerations
d
d
We then want a systematic way to choose as small of a as possible but still have unfolded the dynamics of the system
d
d
• Many different ways to choose a function that is invariant under increase of dimensions
• Indirect method using the dynamics of the system to compute a geometrical property of the system
Choosing an Embedding Dimension: Invariants of the System
Reconstructing Phase Space
Idea: Any property of the system that is dependent on the distance between two points will stop changing when we reach a sufficient d
Choosing an Embedding Dimension: False Nearest Neighbors
Reconstructing Phase Space
Idea: Measure the distances between a point and its nearest neighbor, as this dimension increases, this distance should not change if the points are really nearest neighbors
Define the distance between a point and its nearest neighbor using a Euclidean distance
Rd(k)2 = [s(k)!sNN (k)]2+[s(k+T )!s
NN (k+T )]2+. . .+[s(k+T (d!1))!sNN (k+T (d!1))]2
and the change in distance by adding one more dimension is
Rd+1(k)2 = Rd(k)2 + [s(k + dT ) ! sNN (k + dT )]2
we can now look at the relative change in the distanceas a way to see if our points were not really close together but a projection form a higher phase space
Rd+1
Choosing an Embedding Dimension: False Nearest Neighbors
Reconstructing Phase Space
Using a threshold we can then write a criteria for false neighbors
RT
|s(k + Td) ! sNN (k + Td)|
Rd(k)> RT
Using this criterion we can then test our sequence of points and, as increases, find where the percentage of nearest neighbors goes to 0
d
In practice values of in the range work well for most situations.
RT 10 ! RT ! 50
Choosing an Embedding Dimension: False Nearest Neighbors
Reconstructing Phase Space
Apply the criterion to a data from a random-number generator. We find that embedding dimension is small.
Need another criterion taking into account the distances as measured with respect to the size of the attractor RA
If thenRd(k) ! RA Rd+1(k) ! 2Rd(k) =!
Rd+1(k)
RA
! 2
as another test for false nearest neighbors, a common way to estimate is by using the rms value of observationsRA
R2
A =1
N
N!
k=1
[s(k) ! s]2 s =1
N
N!
k=1
s(k)
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60
10
20
30
40
50
60
70
80
90
100Henon Map False Nearest Neighbors
Dimension
Perc
ent F
als
e N
eig
hbors
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60
10
20
30
40
50
60
70
80
90
100Lorenz Equations False Nearest Neighbors
Dimension
Perc
ent F
als
e N
eig
hbors
• One of our original problems was noise and how it is often indistinguishable from our signal in terms of traditional fourier techniques if our signal is chaotic
• Noise appears as high-dimensional chaos
• We can develop a requirement that if our required dimension goes beyond 20, our signal is too noisy and statistical techniques are the better choice for practical analysis
• We can also use the false nearest neighbors test to find the relative contamination of noise in the signal
Reconstructing Phase Space
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100Noisy Lorenz Data False Nearest Neighbors
Dimension
Perc
ent F
als
e N
eare
st N
eig
hbors
! = 1.0
! = 0.5
! = 0.1
! = 0.05
! = 0.01
Conclusions
• Finding an embedding dimension turns scalar time data into a multivariable system
• False Nearest Neighbors provides a robust way to determine necessary embedding dimensions
• Noise appears as high dimensional chaos and can be examined using the false nearest neighbors technique
References
• Abarbanel et al., 1993, “Analysis of Observed Chaotic Data,” Rev. Mod. Phys., Vol. 65, 4
• Sauer, T., A. Yorke, and M. Casdagli, 1991, Phys. Lett. A 160, 411
• Kennel, M. B., R. Brown, and H. D. I. Abarbanel, 1992, Phys. Rev. A 45, 3403