Recurrence Quantification Analysis (RQA):Tutorial & application to eye-movement data

Deborah J. Aks -- Rutgers University

SCTPLS Workshop 7/25/13 Portland State University

Oscillations can be a sign of system coupling, info transfer & memory

Synchronization of oscillatory activity of separable systems (e.g., distant neural ensembles) can serve as mechanism for information transfer,*& memory.

*Fries P (2001). "A mechanism for cognitive dynamics: neuronalcommunication through neuronal coherence". TICS 9: 474-480.

Application to tracking objects with & without interruptions:

•Do pattern of eye-movements reveal recurring dynamic?•Can this serve as a memory to sustain effective tracking over time and when tracking undergoes intermittent interruptions.

Aks, D. J. (2008). Studying temporal and spatial patterns in perceptual behavior: Implications for dynamical structure. In S. Guastello, M. Koopmans, & D. Pincus (Eds.), Chaos and Complexity in Psychology: The Theory of Nonlinear Dynamical Systems (Chapter 5). New York: Cambridge University Press.

Dynamical Tools• Descriptive & Correlational Statistics • Scatter & Delay plots• Probability Distributions (PDFs)• Relative Dispersion (SD/M)• Power spectra (FFT) • Autocorrelation (Tau)• Rescaled range (R/S)

• Recurrence Quantification Analysis (RQA)

Scaling

Rate of Decay

Detect & quantify recurring patterns + graphical representation of generating

system

“When you solve a nonlinear equation …[using tools of complexity theory] ... the result is not [just] a formula but a visual shape, .. an ‘attractor.’ Capra Fritjof, Juarrero Alicia..(2007)

Reframing Complexity: Perspectives from the North and South.

Rossler Attractor

Equations can describe a path in space

Dynamic system will tend toward certain states as time goes on = Attractor

Poincaré’s recurrence theorem (1890)Certain systems will, after a sufficient time, return

to a state very close to its initial state.

Only 1 (scalar) time series!

Takens’ embedding theorem (1981)Topological representation of original (multidimensional) behavior can be derived from time series of a single observable variable!

F. Takens (1981). "Detecting strange attractors in turbulence". In D. A. Rand and L.-S. Young. Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898. Springer-Verlag. pp. 366–381.

Poincaré, Henri. (1890). “On the Three-body Problem and the Equations of Dynamics” (“Sur le Probleme des trios corps ci les equations de dynamique”), Acta mathematica, 13: 1-270, in The Kinetic Theory of Gases (pgs. 368-81),

RQA & RP

* Shalizi (2009) notes that Packard, Crutchfield, Farmer and Shaw (1980) first published what is widely known as Taken’s theorem, but credit David Ruelle as originator of the solution to use time-delays to reconstruct the SS of a system.

• Eckmann, J.P., Kamphorst, S. O., Ruelle, D. (1987). Recurrence Plots of Dynamical Systems, Europhysics Letters, 5, 973–977.*

• Webber, C.L., Jr., & Zbilut, J. P. (1994). Dynamical assessment of physiological systems and states using recurrence plot strategies. Journal of Applied Physiology, 76, 965-973.

• Zbilut, J. P. Webber, C.L., (1992) Embeddings and delays as derived from quantification of recurrence plots, Physics Letters A, 171(3–4), 199–203.

• Marwan, N. , Romano, M. C. Thiel, M. Kurths, J. (2007). Recurrence Plots for the Analysis of Complex Systems, Physics Reports, 438(5-6), 237-329.

What is the path & topology of state-space trajectory?

• Time delayed reconstruction captures system states at each observation of system’s behavior (over time.)

State Space (SS) Recurrence Plot (RP)

State-space analysis:

– Delayed coordinate embedding = Time Delay Reconstruction; (TDR) – Dynamic State Space portrait = Phase Space

–Visualize system trajectories to infer properties of system that generated the time series.

–Topological representation preserves temporal & spatial dependence in the time series.

–Detect repeating patterns & behavioral transitions --> RP

•

Increasing Delay

Rotating around (Rossler) attractor

State-space analysis:•Delayed coordinate embedding (Time Delay Reconstruction) State Space portrait

• Topological representation preserves temporal & spatial dependence in the time series.

• Detect repeating patterns & behavioral transitions.

• Learn about a system’s attractor(s):

Main Steps of RQA: Time Delay Reconstruction (TDR)

based on Webber and Zbilut’s (2005)

Parameters: Embedding Dimension (M); Delay (L); Window Range (W); Norm (N); Distance Matrix (DM), Radius(r), and Line (Ln).

These control matrix distances, rescaling, RP density, and other thresholds to detect system dynamics.

•Select some variable x as an index of a behavior to track how it changes over time = Behavioral Time Series

• Replace each observation in Xt with a time lagged (L) version of itself: Yi = {xi, xi-L, x j-2L … xi-(m-1)L} Yi = {y1, y2, y3, … y(n-(m-1) L}.

• Calculate Distance Matrix (DM) = distances between all pairs of vectors in reconstructed space. Max, Min or the ave Euclidean distance

• Expand 1-D time series into multi-D space, where dynamic of generating mechanism is predicted to reside. = Embedding dimension (M)

•To recover dominant dynamics driving behavior. estimate w/:•average mutual information (AMI) L •false nearest neighbors (FNN) m

TDR shows:•state’s changing trajectory over time in the SS representation, •whether a sequence of behavior recurs with itself and its neighbors, & •how many neighbors share information.

Tracking an oscillating object

Subjects tracked a red dot, moving horizontally back and forth on a computer screen.

Visible objectFixations cluster around

end points

Object tracking: No interruptions

Endpoints as attractors may correspond to 2 dimensions of system. Add more dimensions w/ mechanisms driving eye-movements (e.g., gaze-object coupling; memory for object’s path).

Tracking an oscillating object

Occluded object

RQA Parameters

1. Delay (L)*2. Embedding Dimension (M)*3. Window Range (W); Norm (N)4. Radius (r)*5. Line (Ln)6. Distance Matrix (DM)

• Rescaled DM (RDM)

*difficult to estimate w/ human data

Embedding Dimension (M)–Dimension that dynamic is projected onto n-dimensions of reconstructed phase space–Estimate via False Nearest Neighbor (FNN)

• stationary and low-noise systems: plateau. • non-stationary, high noise systems

(dimension tends to be inflated) • Since human behavior is non-stationary & very noisy, stable FNN is

difficult to find

Guidelines: W = Weber method vs. S=Shelhamer method

W M>D. (Usually M is 10 or 20 and no higher in biological systems. when set to high get artifactual patterns of recurrence.

S M < 2D+1

Brute force approach RMS as M or as L changes.

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!

Window Range (W)

• Distinguish long- vs. short-range patterns.

• W= Pend - Pinit + 1when M>1 for N points: M-1> Pend

therefore: Pend < N-M+1

• Small W for small scale recurrences • Large W for long scale recurrences

• Global vs. Local Recurrence: • Additional parameters

• Norm= geometric size & shape of neighborhood surrounding each reference point.

• Rescaling- Distance Matrix (DM)– compare data at different scales. Max DM most common; Mean DM smoothes data

• Radius (r)=Small r -- examine local recurrence (Lacey patterns); Large r -- global recurrence

– threshold distance: (3 SD’s of DM)

• Line (Ln)= threshold for classifying as recurring feature. If length of recurrence is shorter than L, feature is rejected.

%DET & Line (Ln)

• Dynamical stability & Divergence

•%DET = % of recurring points forming diagonal lines & periodicity of behavior

• Consecutive recurring points in longest diagonal line (MAXLn).

• Inversely related to system complexity, & how much the systems trajectories diverge over time (i.e., the Lyapunov exponent).

• Small values of MAXLn = more divergence in the SS trajectories & greater complexity (ENT). Behavior that is less predictable over time.

• ENT loss of redundant info in recurrence structure= measure of predictability.

RQARP

1. Characterize recurring patterns w/ ..– time indices of each point on the SS – represent these as coordinates (i,j) in RP

2. Reference point selected on SS attractor, w/ circular radius r.

• If points on attractor falls within r, then a dot is placed at coordinates (i,j) on RP.

• Repeat for all points.

• An NxN plot is formed with N as # of points classified as part of the same attractor.

1. RP contains info on systems dynamics: e.g.,

1. deviations from random behavior,

2. periodicities, transitions,

3. local, global and other forms of recurrences.

http://www.recurrence-plot.tk/

N. Marwan, M. C. Romano, M. Thiel, J. Kurths: Recurrence Plots for the Analysis of Complex Systems, Physics Reports, 438(5-6), 237-329, 2007.

• Tutorial Data

Sine wave Periodic Limit Cycle

Sine wave + noise Quasi-Periodic Limit Cycle

Human Eye-Movements

Software

Visual Recurrence Analysis (VRA; v.4.6) by E. Kononov (2005), and RQA_X (v0.4) by A. Keller (2008).

RQA (Recurrence Quantification Analysis) from C. Webber. RQA 8.1: http://homepages.luc.edu/~cwebber

Cross Recurrence Plots- Matlab Toolbox (CRP) by N. Marwan CRP 4.5: http://www.agnld.uni-potsdam.de/~marwan/toolbox.php.

Increasing Delay (L)

Sine wave

Info bits (I) at each lag (L) estimated from MI

FNN min = 2 = M

1st diff across x-positions

1 2 3 4

RQA measures

Recurrence Rate (RR), density of recurrence points in a recurrence plot. = correlation sum.

Determinism (DET), % recurrence points forming diagonal lines.

Diagonal lines = epochs of similar (time evolution of) states of the system. DL length = prediction time

Laminarity (LAM), % recurrence points forming vertical lines. ~ intermittency. ~ laminar states in the system.

Trapping time (TT), mean length of vertical lines, time trapped in one state or slowly changing state

Windowed calculation of these measures time dependent analysis.

Recurrence Rate (RR),

Determinism (DET),

Laminarity (LAM),

DET/RR

LAM/RR

ENT

1K sinusoid time series

Information bits (I) at each lag (L) estimated from Mutual Information (MI) calculation

M=2 (FNN min) Full cycle of the sine wave at L= 16 (~ min MI)

State Spaces

Recurrence Plots

Signatures of dynamical structure

– Lines parallel to main diagonal = •correlated structure•recurrent path•part of a larger attractor•paths follow each other.

– = spacing, & = length to main diagonal indicate clear periodicity in the system.

– Long lines = periodic sustained behaviorShort lines = transient (or chaotic) signals

L= 16

Signatures of dynamical structure

L=1• Near points are recurrent & coupled

for (at least) momentary points in time:• if x(i) is similar to x(j) or if x(i)-x(j) <

threshold distance (r)

• Lines orthogonal to main diagonal

indicate a reversal in trajectory or nearby paths moving in opposite directions with time.

White Noise

Brown Noise

(M=10; L=1)

Stochastic signals -->

8. No diagonals (unless radius is set too high)

• Bands = transient data e.g., sudden shift of an attractor in state space.

• Horizontal & vertical lines = path recurrent for several points with a single point = laminarity or transience

• Lacey patterns – local (short-range) recurrent patterns

• Decrease density from main diagonal = drift or non-stationarity

More signatures of dynamical structure

Eye Movements

Sine Wave + White Noise(M=2; n=1K)

L=1 L= 16

•Fuzziness = additive noise

Isolated points = near & recurring points.

•Very short parallel lines = rapid divergence of trajectories after coming closely together at intermittent points in time.

Eye-movement dynamics

• Highly coordinated w/ attention, overt behaviors & environmental events

• Oscillatory behaviors & memory

• ‘Critical’ propertiese.g., flexibility, coordination, efficiency & intermittency

(x,y position over time)

-------------------------------------------------------

• Saccades & Fixations:• Sequences (xi, xi+1.. xn) & (yi, yi+1.. yn) • Differences (xn – xn+1 ) & (yn – yn+1)• Distance (x2 + y2)1/2

• Duration (msec, sec…)• Direction Arctan (y/x)

Map trajectory of eye scan-paths:

Tracking a single oscillating object

• Track a red dot, moving horizontally back and forth on a computer screen.

Tracking a single oscillating objectEye positions sampled at fixed rate

1.6Hz

0.3Hz

Apparent Motion: Tracking two alternating dots on left vs. right of display

Real Motion Tracking: Track single-dot oscillating horizontally across display

Horiz eye-mvmts1st diff

AMI FNN

Real Motion

•

M=2; L=1 M=2; L=13 M=3; L=13

Xdiff OE8– FNN--1HZ samples

~ Saccade amplitude with eyes moving horizontally

from left to right

M

1

2

3

4

5

6

7

1st diff across x-positions

(L= 50)L= 2

AMI x-diff (direction+ distance)

Distance

Abs val1st diff across x-positions

1st diff across x-positions

Distance + Direction

FNN

Xdiff_abs val distance

Abs val1st diff across x-positions

ami opt L=2.1 FNN m=6

Det=0.0900 Lam=0.1644 rr=0.0228 m= 6.0000 L= 2.0000 e=1.0000

L =1 to 4 (similar Laminar RPs)L = 5, 7 diag struc starts to appear

x-diffdistdist x-diffx-diff

Tracking, Oscillations & Limit cycles

System coupling – Gaze-Object

Can recurring patterns reveal memory used in tracking?

Stay tuned..