Visual Recurrence Analysis Deborah J' Aks Rutgers University

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Visual Recurrence AnalysisDeborah J. Aks --
Rutgers University

• Graphically detect..
• hidden patterns and changes in data
• similar patterns across time series
• Topological representation of original
  (multidimensional) behavior can be derived from
  time series of a single observable variable! (F.
  Takens, 1981).

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VRA/ VRQ Parameters

• Embedding Dimension (M)
• Delay (L)
• Window Range (W)
• Norm
• Rescaling Distance Matrix (DM)
• Radius
• Line
• Very challenging to estimate

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• Embedding Dimension (M)
• dimension that dynamic is projected onto
• n-dimensions of reconstructed phase space
• estimated via False Nearest Neighbor (FNN)
• stationary and low-noise systems plateau.
• non-stationary, high noise systems (dimension
  tends to be inflated) W MgtD. (Usually M is 10 or
  20 and no higher in biological systems. when set
  to high get artifactual patterns of
  recurrence.S Mlt2D1 Brute force approach RMS
  as Min or as L changes.

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Window Range (W)

• Pend - Pinit 1when Mgt1 for N points M-1gt
  Pendtherefore Pend lt N-M1short W focus on
  small scale recurrences vs. long W focus is on
  long scale
• distinguish long- vs. short-range patterns.

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Delay (L)

• L depends if time series is..
• Linear L 1st minimum in linear autocorrelation
• Non-linear L (average) mutual info (MI
  VRA)discontinuous signals (maps) L 1 to
  include all points.
• Physiological and psychological signals often
  have no optimal

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• Norm geometric size shape of neighborhood
  surrounding each reference point.
• Rescaling- Distance Matrix (DM)-- compare across
  data of diff scales. max DM most common mean DM
  smoothes data
• Radius (R) absolute distance Thresh distance
  3 SDs small R -- examine local recurrence
  (Lacey patterns) Lrg R global recurrence
• Line (L) 2 filter function too extract
  important features of time series.
• if length of recurrence feature is shorter than
  line parameter, feature is rejected.

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Sine Wave (M2 L23 n1K)
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Sine Wave Noise (M7 L6 n1K)
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Sine Wave (M2 L1 n1K)
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White Noise (M9 L1 n?)
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Brown Noise (M10 L1 n?)
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Pink Noise (M10 L1 n2K)
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Sine Wave Noise (M9 L25 n1K)
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Signatures of dynamical structure

• consecutive aligned points parallel to main
  diagonal recurrent path they are part of a
  larger attractor thus their paths follow each
  other.
• near points are recurrent points coupled for
  (at least) momentary points in time. if x(i) is
  similar to x(j) or if ?x(i)-x(j)? lt threshold
  distance (r)
• lines parallel to the main diagonalDeterministic
  or correlated structure e.g., Slow trajectory
• isolated points two near recurring points.
• Long lines indicate periodic signals short lines
  (can) indicate chaotic signals
• Stochastic signals produce no diagonals (unless
  radius is set too high)
• Lines orthogonal to main diagonal indicate a
  reversal in trajectory or nearby paths moving in
  opposite directions with time.
• white bands transient data e.g., sudden shift
  of an attractor in state space.
• 8a) Parallel diagonal lines that are uniformally
  spaced and equal in length to the main diagonal
  indicate clear periodicity in the system. 8b)
  Decrease density from main diagonal drift or
  non-stationarity
• 9) horizontal vertical lines path recurrent
  for several points with a single point such as in
  a looping around the point laminarity or
  transience 11) fuzziness additive noise
• very short parallel lines rapid divergence of
  trajectories after coming closely together at
  intermittent points in time.
• Lacey patterns local (short-range) recurrent
  patterns

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Search Displays
Complexity ---gt
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Search Displays w/ scan paths
Complexity
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Edge Scan (M7 L6 n772)
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xtra
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B-P Noise ?1.5 (M10 L1 n?)
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Brown Noise 2 (M10 L1 n?)