Deecing nonlinear processes in eperimenal daa: Applicaions in Psychology and Medicine Richard A. Heah Division of Psychology, Universiy of Sunderland, UK richard.heah@sunderland.ac.uk
Menu For Today Time series in psychology Sequenial RT N = 4800, Kelly e al, QJEP Mood ime series N = 1035 Deecing nonlinear deerminism in a ime series Surrogae series Time asymmery Noise reducion Recurrence Plos Nonlinear predicion Lyapunov specrum Useful Sofware Resources Norhumbria2002 2
Time Series in Psychology Phase Plo and Specral Represenaions Lag 1 Phase Plo for Sequenial RT daa Subjec 3 650 Power Specrum for RT daa Eperimenal and Surrogae Series 550 450 30 350 250 Power 20 150 150 250 350 450 550 650 RTn 10 0.0 0.5 1.0 1.5 Frequency Hz Norhumbria2002 3
Use of Surrogae Series Norhumbria2002 4
Use of Surrogae Daa Ses o Disinguish Correlaed Noise wih Chaos Generae Surrogae Daa By Gaussifying he daa se Compuing Power and Phase Specra Randomise Phase Specral Componens Reconsruc he Time Series by Inverse FourierTransform Compare Dynamic Indices for Noise Reduced and Surrogae Time series Deerminism possible if here is a significan difference beween hese indices Norhumbria2002 5
Deecing Deerminism Using Hypohesis Tesing Null Hypohesis: Time series generaed by linearly filered Gaussian process Research Hypohesis: Deerminism presen Compue Deerminism Inde, e.g. Time asymmery inde, D2 Compue 95% CI for Inde from Surrogae samples or use Resampling Techniques Rejec Null Hypohesis if eperimenal inde lies ouside he CI Norhumbria2002 6
Time Asymmery Inde r e v N n 1 N n 1 X n X n X n X n 2 3 3 2 Xi, i=1,n is a ime series = ime delay Norhumbria2002 7
Time Asymmery Tes for Nonlineariy RT Sequences Using a 4 Choice Task 0.2 0.1 14.8 Ep z = 6.67 0.0-4-3-2-1 0 1 2 3 4 5 Surrogae Time Asymmery Inde The eperimenal series is significanly differen from he surrogae series => nonlineariy Norhumbria2002 8
Noise Reducion Procedures Norhumbria2002 9
Local Singular Value Decomposiion Embed daa in an E-dimensional space Compue he covariance mari using neares neighbour poins in phase space Compue singular values and corresponding eigenfuncions Projec he original daa ino he space spanned by he firs few eigenfuncions Generaes increase in signal-noise raio Norhumbria2002 10
RT series before op and afer boom noise reducion 700 600 RT ms 500 400 300 200 500 1000 Time secs 1500 2000 700 600 RT ms 500 400 300 200 500 1000 1500 2000 Time secs Norhumbria2002 11
Noise Reducion RT Sequence Phase Plo: Original Daa 400 350 300-10 0 10 20 RT' RT Sequence Phase Plo Noise Reduced Daa 400 350 300-20 -10 0 10 20 30 40 RT' Norhumbria2002 12
Recurrence Plos Norhumbria2002 13
Visualizing a Nonlinear Time Series Given ime series X1, X2, X3, X4, X5, X6, Selec a ime lag based on, say, he firs zero of he auocorrelaion funcion, e.g. L = 2. Selec dimensionaliy of embedding space a leas wice ha of he unknown aracor, e.g. E = 3 The firs poin is {X1, X3, X5}, second poin is {X2, X4, X6}, ec Topological properies equivalen o original ime series Norhumbria2002 14
Compuing a Recurrence Plo Plo successive embedded poin indices for RT ime series The colour of he poin is proporional o he row-column iner-poin disance Indices Ep Surr % Recurrence 0.03 0.04 % Deerminism 41.8 0.91 Enropy 2.19 0.65 Norhumbria2002 15
Recurrence Plo RT Eperimenal Series %recurr=0.03 %deer = 41.8 enropy=2.19 Norhumbria2002 16
Recurrence Plo RT Surrogae Series %recurr=0.04 %deer=0.91 enropy=0.65 Norhumbria2002 17
Nonlinear Predicion Tes An auocorrelaed ime series generaes a monoonic increasing predicion error over ime However, a chaoic ime series generaes a similar rend wih consisenly lower predicion error. Verificaion of chaos requires a leas one posiive Lyapunov eponen and a negaive Lyapunov eponen sum. Norhumbria2002 18
Nonlinear Predicion Tes Predicion Tes for RT Daa Subjec 1 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 d = 1, m = 4, r = 5 Ep Noise Reduced Surrogae 0.2 0 1 2 3 4 5 6 7 8 9 10 Predicion Lag Evidence for sensiive dependence on iniial condiions Norhumbria2002 19
Lyapunov Eponens Lyapunov eponen: Rae a which nearby iniial rajecory values diverge Lyapunov eponens of a sable sysem are negaive, while a chaoic sysem has a leas one posiive larges Lyapunov eponen and several oher eponens, heir oal sum being negaive. The more posiive he Larges Lyapunov Eponen he more chaoic he sysem, i.e. he more unpredicable he sysem. Norhumbria2002 20
Norhumbria2002 21 Lyapunov Specrum. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 2 2 2 2 1 2 1 1 2 1 1 1 m m i m m m m i i i i i m i m i 1 J 1 1 1 J i i Compue eigenvalues and eigenvecors of J
Confirming Chaos Lyapunov eponens for noise-reduced RT daa l1l4 esimaed using a feed-forward Neural Nework wih 4 inpu unis, 6 hidden unis and 1 oupu uni Gencay & Decher, 1992. Compued using he NETLE sofware SERIES l1 l2 l3 l4 SUM EXP 0.16-0.03-0.44-0.91-1.22 SURR -0.35-0.71-1.23-1.45-3.74 +ve Lyapunov eponen AND -ve Lyapunov eponen sum for eperimenal series ==> chaos Norhumbria2002 22
Mood Time-Series for Depressed Clien daa provided by Prof A. Young, Universiy of Newcasle Parial Auocorrelaion Funcion for Derended Raings Parial Auocorrelaion 1.0 0.8 0.6 0.4 0.2 0.0-0.2-0.4-0.6-0.8-1.0 R 0.43 R 1 0.16 R 2 5 15 25 35 45 55 65 75 Lag PAC T Lag PAC T Lag PAC T Lag PAC T Lag PAC T 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.52 0.17 0.02 0.03 0.04 0.07 0.02-0.03 0.02 0.01 0.04-0.03 0.01 0.02 0.02 16.87 5.54 0.71 1.11 1.41 2.20 0.71-0.97 0.76 0.40 1.13-0.82 0.22 0.65 0.60 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.00-0.02-0.02 0.02 0.03 0.01-0.03-0.00 0.01 0.05 0.06 0.01 0.01 0.02 0.00 0.12-0.68-0.60 0.72 1.03 0.19-0.88-0.01 0.21 1.64 2.02 0.37 0.37 0.50 0.15 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45-0.02-0.03 0.02-0.03-0.03-0.03 0.02-0.01 0.03-0.04 0.06-0.04-0.03 0.03-0.04-0.76-1.01 0.59-1.05-0.90-0.99 0.80-0.38 0.97-1.22 1.81-1.20-1.00 0.96-1.21 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0.06-0.02 0.01 0.01-0.05-0.04-0.05 0.01 0.06 0.02-0.03 0.03-0.00-0.04 0.05 2.08-0.63 0.35 0.33-1.69-1.45-1.57 0.28 2.09 0.71-0.81 0.98-0.09-1.13 1.70 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75-0.02-0.02 0.03 0.00-0.01-0.05-0.07-0.08 0.01-0.02-0.03 0.01-0.02-0.02 0.01-0.66-0.58 0.94 0.06-0.21-1.52-2.25-2.53 0.32-0.51-0.96 0.16-0.77-0.77 0.19 Norhumbria2002 23
1/f Specrum for Derended Mood Raings logp = -5.22670-0.754824 logf S = 0.315656 R-Sq = 84.4 % R-Sqadj = 84.0 % -1-2 logp -3-4 -5-5 -4-3 -2-1 logf log P 5.22 0.75log F, R 2 = 0.84, a highly significan resul Norhumbria2002 24
Evidence for Nonlineariy Timerev Inde for Derended Raing Daa ML Esimaes - 95% CI 99 ML Esimaes 95 90 Mean SDev -0.022125 0.236781 Percen 80 70 60 50 40 30 20 Raing Daa Goodness of Fi AD* 0.924 10 5 1-0.5 0.0 0.5 Timerev Approimae Enropy, a nonlinear inde of informaion conen, was compued using 2 runs and a filer level of 2.0, yielding a value of 0.41. This value suggess some deerminisic srucure in he daa, i being closer o 1.0 for pure whie noise. Norhumbria2002 25
Useful WWW Resources TISEAN 2.0: hp://www.mpipks-dresden.mpg.de/~isean/tisean_2.0/inde.hml SANTIS/DATAPLORE: hp://www.daan.de/daaplore/ CHAOS DATA ANALYZER: hp://spro.physics.wisc.edu/cda.hm VISUAL RECURRENCE ANALYSIS: hp://pweb.necom.com/~eugenek/download.hml NETLE SOFTWARE PACKAGE: hp://www.bsu.edu/econ/liu/download/inde.hml TOOLS FOR DYNAMICS: hp://www.zweb.com/apnonlin/ Norhumbria2002 26
Useful Book References Heah, R.A. 2000. Nonlinear dynamics: Techniques and applicaions in psychology. Mahwah, NJ: Erlbaum Associaes. Kanz, H., & Schreiber, T. 1997. Nonlinear ime series analysis. Cambridge, UK: Cambridge Universiy Press. Kaplan, D., & Glass, L. 1995. Undersanding nonlinear dynamics. New York: Springer Verlag. Norhumbria2002 27