amath 575 talk

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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dx

dt= ! sin "t ! c1

y(t)2 ! 1

y(t)2 + 1x(t) + c2y(t) ! c3y(t)3

dy

dt= x(t)

c1 = 3/4 c2 = 1/2 c3 = 1/2

! = 1 ! = 14

Example: Goodwin Equations

Signal we receive: s(n) = x(t0 + n!s)

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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:

! = 16

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


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


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



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



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


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

amath 575 talk

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