A study of complexity in GammaRay Burst using the Diffusion Entropy approach

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A study of complexity in Gamma-Ray Burst using
the Diffusion Entropy approach

• Nicola Omodei
• Jacopo Bellazzini,
• Simone Montangero

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Introduction

• Diffusion Entropy was introduced in Scafetta et
  al. (2001)
• Successfully applied in complex systems dynamics
• Fully developed turbulence in a fluid flow
  (Frisch 1996)
• Self organized critically (Bak et al. 1987)
• Solar Flares (Grigolini et al. 2002).

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DE the method

• xi Time series, i 0, N

In the standard language of the diffusive process
we can think to each xi as the length of a jump
of a walker moving in one dimensional space.
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The Diffusive Trajectories

• We create a set of many different diffusion
  trajectories by means of a moving window of size
  ?t with 1 lt ?t lt N, which is the same as a
  boxcar-smoothing forward in time from point t to
  any point t ?t.
• In practice, we generate N- ?t 1 trajectories
  yj, as

Each yj(?t) can be thought of as the final
position of the walker after ?t time steps
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The Poisson Noise
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The Shannon Entropy

• The normalized histogram that contains the final
  positions at time ?t (yi(?t)) is the probability
  distribution p(y, ?t) of the final positions y of
  the walker at time ?t.
• The Shannon entropy is defined as

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The Diffusion Entropy

• From the central limit theorem, if the time
  series has finite variance and is uncorrelated,
  the probability p(y,?t) satisfies the scaling
  condition

For normal (Brownian) diffusion ? ? N 0.5,
and F Gaussian function (Feller 1971).
If the scaling condition is satisfied
?
Independent of the value of ? and of the shape of
F !
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The normal diffusion
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The scaling index ?
We plot S(?t) in Log-Linear scale

• ? 0.5 gt This is what we call NOISE !

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The Diffusion Entropy Approach
A large class of diffusive processes are
characterized by the scaling condition, with ? ?
0.5 and F not gaussian. Such process are called
anomalous. This process have typically long
time correlations, and/or are not stationary. In
other words DE smoothes the signal by summing a
series of data contained in the window ?t and
characterizes how the information loss increases
with increasing window size.

• Uncorrelated time series (Noise) ? ?
  0.5
• - Stationarity
• Correlated Time Series (SIGNAL) ? ? ? 0.5
• - Non-Stationarity
• - Memory-Effects
• - Other

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Synthetic signals

• What we mean by complexity is essentially the
  same as anomalous diffusion it is an estimate of
  how far an anomalous diffusive process is from
  completely uncorrelated noise.
• DE measure the balance between uncorrelated noise
  and correlated signal
• No background estimation (the background has to
  be stationary)

The upper limit is ? 1 ballistic motion,
pure deterministic process
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Gamma Ray Burst

• Gamma Ray burst are non-stationary time series.
  The signal lasts only a fraction of the recorded
  time history. The complexity in GRBs arises from
  high time variability, the non- periodicity, and
  the typical non-sationarity of the signal.

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The anomalous diffusion
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Three Analysis

• Diffusion Entropy of a single burst in the four
  BATSE Energy Channels
• Diffusion Entropy for the whole BASTE catalog
• Diffusion Entropy index distribution
• The Power-Complexity relation
• Diffusion Entropy as a function of time

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DE of a single burst in four Energy Channels

• We have shown that
• The ? index describes univocally the behavior of
  a GRB in terms of complexity.
• Higher is the non-stationarity of the GRB,
  higher is complexity in the signal higher is
  the ? index.

GRB910429
GRB910601
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Diffusion Entropy and the BATSE catalog
The DE is correlated both with the intensity of
the signal (fluence) and with the duration of the
GRB (T90). We consider 2 x T90 portion of the
light curve (for Long bursts only) -gt We keep
only the correlation with the intensity !
)
)
(
)
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The Power-Complexity relation
The behavior of the delta indexes in the first
three channels is approximately the same, while
in the fourth channel is weaker. GRB are highly
correlated signals. For intense bursts ? 1
(upper bound). Also weak bursts are highly
correlated (? 0.8) if a portion of the light
curve is considered (proportional to the T90)
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The DE as a function of the Time
T90
T90
Re-flaring?
Noise value
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Synthetic GRB
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Simulated Power Complexity Relation
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Conclusion and future work
Diffusion entropy has been successfully applied
to Gamma-Ray Bursts, obtaining a new description
of the GRB phenomena in terms of their scaling
index ?. Diffusion Enropy has a big advantage
compared to other tools (CCF, ACF) DE does not
need a background identification (if the
background is stationary). It provides a measure
of the balance between background and
signal. Since that the Diffusion Entropy is
sensible to non-stationary signals and it detects
memory effects or correlations in temporal
series, we think that it can be applied for
searching anomalous diffusive processes (GRB or
something else) hidden in GLAST data. Data
Challenge data gt The background is not
stationary due to the orbital motion of the
satellite, but it is easy to de trend!)
(astro-ph/0309025)
AA 414, 11771184 (2004)