Recap Fractal Functions

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Recap Fractal Functions
They are similar to themselves when transformed
by anisotropic dilations
Self Affine Functions
If f(x) is a self-affine function, then ?xo ? ?,
?H ? ? such that for any l gt 0,
H is called Hurst exponent

• Remarks
• If H lt 1, then f is not differentiable and the
  smaller the H, the more singular f.
• Thus, H indicates the global irregularity or
  roughness of f.
• DF H 2, DF is the fractal dimension of the
  graph of f

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Fractal functions can possess multi-affine
properties so that their roughness or the
irregularity can fluctuate from point to point.
Thus, the definition of Hurst regularity becomes
a local quantity.
The local Hurst exponent h(x) is also called
Hölder exponent of f at the point x. This is
primarily related with the strength of the
singularity of f at that point. At any given
point x0, the Hölder exponent is given by the
largest exponent such that there exist a
polynomial Pn(x-x0) of order n lt h(x0) and a
constant C gt 0, so that for any point x in the
neighborhood of x0, the following relation
holds h(x0) measures how irregular f is at
x0. The higher the exponent h(x0), the more
regular the function h.
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Multifractal Formalism
Halsey (1986) Phys. Rev. E
Statistical description of singular measures by
determining the singularity spectrum f(a)
f(a) characterizes the relative contribution of
each singularity of the measure Let Sa be he
subset of points x where the measure of an e-box
Bx(e) centered at x scales like m(Bx(e)) ea in
the limit e ? 0. Then, f(a) is the Hausdorff
dimension of Sa f(a) dimH(Sa)
Now, consider a sum taken over a partition of the
support of the singular measure into boxes of
size e
It can be proved that Zq(e) et(q) f(a)
minq(qa - t(q))
Generalized fractal dimension, Dq t(q)/(q-1)
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Structure Function (SF) Approach
One way to apply the concept of multifractals to
singular functions. Initially proposed in the
context of fully developed turbulence.
The intermittent character of turbulent velocity
signals could be investigated by studying the
moments of the PDF of the (longitudinal) velocity
increments,
over initial separation By Legendre
transform, one can estimate the Hausdorff
dimension D(h) of the subset of ? where velocity
increments behave as dvl lh
SF of order p
D(h) can be defined as the spectrum of Hölder
exponents for the signal in the same way as f(a)
is the singularity spectrum for singular
measures.
But this approach has limitations to fully
characterize the D(h)
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An alternative approach is based on WAVELET or
Wavelet Transform (WT)

• WT has been very efficient to detect
  singularities
• WT is well adapted to study fractal objects
• Since a wavelet can also be treated as an
  oscillating variant of a box, one can
  generalize the multifractal formalism by defining
  new partition functions in terms of wavelet
  coefficients
• Wavelet can also be chosen to be orthogonal to
  polynomial behavior up to some preferable
  order thus eliminating the regular (or trendlike)
  component
• The skeleton defined by the wavelet transform
  modulus maxima (WTMM) provides an adaptive
  space-scale partition of the fractal distribution
  from which D(h) can be obtained
• Wavelet can also be used to investigate
  correlations in both space and scales

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Continuous Wavelet Transform (CWT)
WT is a space-scale analysis which consists in
expanding signals in terms of wavelets which are
constructed from a single function, the analyzing
or mother wavelet y by means of translation and
dilation.
The WT of a real-valued function f is defined
as x0 is the space parameter a (gt0) is the
scale parameter
The analyzing wavelet is usually well localized
in both space and in frequency. Usually y is
required to be zero mean so that WT is
invertible. Y is required to be orthogonal to
lower order polynomials
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Mallat (1992) IEEE-PAMI, IEEE-Inf. Th.
If singularity of a function is of prime
interest, concentrate only on the WT skeleton
by its modulus maxima.
These maxima are defined, at each scale a, as the
local maxima of Tyf(x,a). Maxima lines ?
Connected curves of local maxima in the
space-scale plane. Lets ?(a0) is the set of all
the maxima lines that exist at the scale a0 and
whichcontain maxima at any scale a a0.
Important property of maxima lines ? There is
at least one maxima line pointing towards
each singularity
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Scanning Singularities by WT
Let assume that f has a local scaling Hölder
exponent h(x0) at the point x0 If the
singularity is not oscillating, it can be proved
that the local behavior of f is mirrored by the
WT which locally behaves like if ny gt h(x0),
where ny is the number of vanishing moments of y.
Thus, one can estimate h(x0) as the slope of
log-log plot of WTMM vs. scale a.
If one chooses ny lt h(x0), the WT still behaves
as a power-law but with a scaling exponent ny
Thus, around a given point x0, the faster the WT
decreases when the scale goes to zero, the more
regular f is around that point.
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WT of f(x) by using a first-order derivative of
Gaussian function g(1)
Horizontal slice of Tyf(b,a) at the scale a
a0
Maxima lines of WT in (b,a) plane as found by
the first-order (-) or second -order (- -)
derivative of Gaussian
Local measurement of scaling exponent along the
maxima lines when using g(1) (ny1) h(x0)0.6,
h(x1)ny1. The slope of the curve log2Tyf vs
log2a gives the scaling exponent.
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WTMM Method
One natural way to perform multifractal analysis
of fractal functions consists in generalizing
the classical multifractal formalism using
wavelets instead of boxes. Since wavelets is
considered to be a generalized oscillating
boxes, one can avoid the possible smooth
behavior which could potentially mask the
singularities. But how to define the covering
of the support of the singular part of the
function with the chosen set of wavelets of
different sizes?
One solution would be to define a partition
function in terms of WT coefficients
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1. By using WTMM method, one changes the
continuous sum over space into a discrete
sum over the local maxima
2. One can define the exponent t(q) from the
power-law behavior of partition function
3. The singularity spectrum can be determined
from the Legendre transform of the partition
function scaling exponent t(q)
where h ?t/?q
A linear t(q) curve indicates a homogenous
fractal function. A nonlinear t(q) curve
indicates a non-homogenous function exhibiting
multifractalproperties, i.e. Hölder exponent
h(x) depends on the spatial position x.
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Remarks

• t(2) is related to the spectral exponent b with
  b 2t(2)
• Commonly used wavelets Successive derivatives
  of the Gaussian function
• Structure Function (SF) and WTMM

SF
WTMM
Then, t(q) lq - 1
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Exmples - Monofractal Multifractal
Fractional Brownian Motion (fBM)
Signals Modeling physical phenomenon with
long-range dependence, 1/f noise It exhibits a
power spectral density S(f) 1/fb, where b 2H
1
A fBM BH(x), where H?0,1, is a Gaussian process
of zero mean and whose correlation function
The variance of such process is
The fBM yields classical BM when H 0.5
The increments of a fBM, dBH,l BH(xl) BH(x),
are stationary. because the correlation function
depends only on x-y and l.
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When H 0.5, correlation function is zero, i.e.,
increments of the classical Brownian motion are
independent. For H gt 0.5, the increments are
positively correlated (persistent random
walk) For H lt0.5, the increments are negatively
correlated (antipersistent random
walk). Additionally,
Thus, fBMs are self affine processes with Hurst
exponent H. Since the above equation holds for
any x and y, there is only one Hölder exponent
h(x) H ? fBMs are homogenous fractals
characterized by a singularity spectrum which
reduces to a single point D(h) 1 if h
H -? h ? H Then, t(q) qH - 1
? t(q) is a linear function of q with a slope of
H,
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Random W-Cascades
Variant of multiplicative cascade models (cascade
refers to a self similar process whose
properties are defined multiplicatively from
coarse to fine scales) Cascade models are
cornerstone of turbulence (intermittency
phenomenon in fullydeveloped turbulence when
energy is transferred from large eddies down to
small scales through a cascade process in which
transfer rate at a given scale is not spatially
homogenous)
If m and s2 are the mean and variance of lnW (W
is a multiplicative random variable with
log-normal distribution, then
The corresponding singularity spectrum will be
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fBM with H1/3
Log-normal W-cascades
Realizations
WT for different scales
WT skeleton definedby the maxima lines
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fBM with H1/3
Log-normal W-cascades
Nonlinear function For multi-fractals
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Biological Signal Analysis
System Features

• Macroscopic phenomena emerges out of microscopic
  interactions
• System often includes multiple components with a
  large number of degrees of freedom
• System may be driven by competing force
• System may be driven away from common equilibrium

Signal Properties

• Nonstationarity
• Nonlinearity
• Irregular fluctuations may reflect scaling of
  fractal characteristics

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HRV Time Series
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WTMM HRV Time Series
Ivanov et al. (1999) Nature
Chosen wavelet Third derivative of
Gaussian Scale a 1.15i, I 0, , 41 q -5,
-4, , 0, , 5 (Larger positive q reflects the
scaling of large fluctuations and strong
singularities Larger negative q reflects the
scaling of small fluctuations and weak
singularities Data 6 hr records of data from
two groups of subjects Healthy, CHF (heart
failure)
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Two typical subjects
Nonlinear t(q) vs. q curve for healthy ?
Multifractality Linear t(q) vs. q curve for
CHF ? Monofractality
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Group Averaged Data
Singularity Spectrum
Clear loss of multifractality for heart failure
subjects Local Hurst exponents in the range
0.07lthlt0.17 are associated with fractal
dimensions close to one ? The subset
characterized by these local exponents
are
statistically dominant For heart failure,
statistically dominant exponents collapse, h ?
0.22
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Multifractality Nonlinearity
Two types of surrogates Sequence shuffled (same
mean, variance) Fourier phase
shuffled (same power spectrum)
Possible relationship between nonlinear features
and the multifractality of healthy cardiac
dynamics
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Discriminating Ability
(hmax -hmin)
std
t(q3)
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Local Hurst Exponents
Healthy Subjects
Heart Failure Subjects
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Multifractality Shuffling

• Multifractality due to a broad probability
  density function for the values of time
  series (multifractality cannot be removed by
  shuffling)
• Mutifractality due to a different long-range
  correlations of the small and large scale
  fluctuations (multifractality will be removed by
  shuffling)If both types of multifractality are
  present, the shuffled sequence will show
  weakermultifractality than the original sequence.

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Multifractal DFA
Assumptions xk, k1,2,, N is of compact
support The support of any sequence is defined
as the set of the indices k with nonzero values
of xk. The support is compact if xk0 for only
small number of cases.

• Determine the profile (Subtraction of mean is
  not always necessary)
• Divide the profile Y(i) into Ns ?int(N/s)
  nonoverlapping segments of equal length s.
• (As N may not be an integer multiple of s, some
  data points can be wasted. So,
• Repeat the same procedure form the opposite end
  of the sequence).
• So, 2Ns segments are obtained overall.

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3. Determine the local trend in each of the 2Ns
segments by simple least-square fit.
Determine the variance for each segment v,
v 1, 2, , Ns yv(i) is the fitting
polynomial in segment v. 4.
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Synchronization -

• Nonlinear Chaotic Systems

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What is Synchronization?
Syn ( Together) Chronos (Time) Some
agreement of correlation in time of different
process There is no general universally accepted
definition of synchronization. Common
definition an appearance of some relations
between functionals of two processes due to
some form of interaction.
System 2
?
System 1
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Some Common Examples

• Coupled pendulum clocks
• Synchronized flashing of fireflies
• Brain oscillations (dominantly during epileptic
  attack)
• Circadian rhythm
• Synchronized Clapping in Concert

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Synchronization in Chaotic Systems
Paradox Chaotic system is sensitive on initial
conditions. Two trajectories emerging from two
different closeby initial conditions will
diverge from each other at an exponential rate.
Preliminary Definition Synchronization of
chaos can be considered as a process where two
(or many) chaotic systems (either equivalent or
nonequivalent) adjust a given property of
theirmotion to a common behavior, due to weak
coupling or forcing which may rangefrom complete
agreement of trajectories to statistical locking
of phases.
Boccaletti et al. (2002) Phys. Rep.
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Types of Synchronization

• Complete Synchronization Perfect locking of
  the trajectories of two chaotic systems
  (unidirectionally coupled identical chaotic
  systems with conditional Lyapunov exponents are
  all negative)
• Generalized Synchronization A functional
  interdependence between two different chaotic
  systems
• Phase Synchronization Locking of phases but
  amplitudes may remain unrelated
• Lag Synchronization Two outputs lock their
  phases and amplitudes but with sometime lag
• Intermittent Lag Synchronization Same as
  earlier but with some intermitten burstsof
  nonsynchronous period (in this region, local
  Lyapunov exponent becomes positive)
• Intermittent Phase Synchronization
• Almost Synchronization Asymptotic boundedness
  of the difference between a subset of the
  variables of one system and the corresponding
  subset of variable of the other system

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Types of Synchronization

• Complete Synchronization Perfect locking of
  the trajectories of two chaotic systems
  (unidirectionally coupled identical chaotic
  systems with conditional Lyapunov exponents are
  all negative)
• Generalized Synchronization A functional
  interdependence between two different chaotic
  systems
• Phase Synchronization Locking of phases but
  amplitudes may remain unrelated
• Lag Synchronization Two outputs lock their
  phases and amplitudes but with sometime lag
• Intermittent Lag Synchronization Same as
  earlier but with some intermitten burstsof
  nonsynchronous period (in this region, local
  Lyapunov exponent becomes positive)
• Intermittent Phase Synchronization
• Almost Synchronization Asymptotic boundedness
  of the difference between a subset of the
  variables of one system and the corresponding
  subset of variable of the other system

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Phase Synchronization
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Phase Synchronization Classically considered as
an adjustment of frequencies of periodic
oscillators due to some form of weak interaction
For periodic self-sustained oscillator,

• is coupling coefficient
• g1,2 are 2p periodic

For coupled oscillators,
Equation for generalized phase difference,
Phase locking condition,
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Coupled chaotic Rossler Equations
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Noisy Oscillators
Relative phase equation,
Small noise Only slight perturbation to
relative phase dynamics Particle is kicked by
noise to make a jump into next equilibrium
position ? phase slip of 2p Stationary segments
of relative phases but with interval of
?2p phase jumps Distributions of jn,m mod 2p will
be unimodal Strong noise, Irregular and
frequent phase slips Region of stable relative
phase shrinks and performs a biased random
walk (unbiased only at the center) Synchronizatio
n transition is smeared and Synchronization
appears as weakly
Case of synchrony
Outside synchrony
Thus, detection of synchronization is only
possible in statistical sense !!
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Chaotic Oscillators
A is the chaotically varying amplitude
For chaotic oscillator
The evolution of the phase for chaotic oscillator
is qualitatively same as the dynamics of phase
of noisy periodic oscillator. ? The phase
dynamics are generally diffusive
(Diffusion constant determines the phase
coherence of oscillations
and is inversely proportional to the width of
spectral peak) The phase performs a random
walk
In short, coupled chaotic oscillators can be
treated in a same way as coupled noisy periodic
oscillators. So, the synchronization region may
appear as weak!
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Coupled Rossler Equations with Dynamic Noise
Identical Oscillators, ?? 0
e 0
e 0.04
Detuned Oscillators, ?? 0.015
e 0
e 0.04
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Phase Synchrony Index (r)
Tass (1998) PRL

• Consider two signals x(t) and y(t). Find their
  phases fx and fy by Hilbert method xhilb
  x(t) () (1/pt). xanal x i.xhilb a(t)
  exp (i fx(t))
• Find relative phase j fx- fy (define on real
  axis)
• Find the distribution function of j mod 2p.
• Find the entropy of this distribution, H
  -Sk1,, M pkln pk and maximal entropy Hmax
  ln M,
• where M is the number of partitions.
• 5. Phase synchrony index, r (Hmax H)/Hmax

• measures uniformity of distribution of
• phase differences

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Mormann et al. (2000) Physica D
Mean Phase Coherence (R)
1. Obtain instantaneous phases by means of
Hilbert transform (confine the phase to
the interval 0, 2p) 2. Average phasor of
instantaneous phase difference

• and R are confined between 0 (no synchrony)
  and 1 (perfect synchrony)
• They are only sensitive to phase information

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Robustness of R
Coupled Chaotic Equation
SNR 0
SNR 1
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Wavelet Based Phase Synchrony
Phases are calculated by convolving each signal
with a complex wavelet function
where w0 is the center frequency and s
determines the rate of decay (frequency span)
The convolved signal
Quantify the relative phase
Define a phase synchronization index
The higher the gW, the higher the degree of phase
coupling.