Our Multifractal World: Evidence from Human Decision Making and Activity Monitoring

Rachel HeathUniversity of Newcastle, Australiarachel.heath@newcastle.edu.au

In collaboration with Professors Andrew Heathcote and Greg Murray

SykTek®

Society for Mathematical Psychology Conference, July 2015

What I will discuss today

RTs from a four choice task (Kelly et al., 2001)

Activity recorded over 103 days (Bullock & Murray, 2014)

Models for predicting multifractal aspects of RT

and activity based on multiplicative cascade

processes

The data The models

A restricted gamma process drives a multiplicative cascade in a speeded CRT task.

A Gaussian multifractal predicts a manic episode for a Bipolar Disorder patient.

The Findings

Four Choice Task (Kelly, Heathcote, Heath & Longstaff, 2001)

On each trial, one of four visual stimuli were illuminated and the subject was required to depress the associated response button.

Experiment 1: 10 subjects, 3 sessions, each containing 2400 trials

Experiment 2: 12 subjects, similar task to Experiment 1Conditions: 1. Self-paced: Respond as quickly and as accurately as you can 2. Fast-paced Fast: ISI = mean RT in condition 1. Mean ISI = 449 ms 3. Fast-paced Slow: ISI = mean RT + 2SD as estimated from condition 1. Mean ISI = 578 ms

Dependent variable: The time series of RTs from stimulus onset to response execution (2400 per subject/session).• No distinction was made between correct and error responses.• The only trials not analyzed were those for which no response could be given.• No attempt was made to estimate residual time for stimulus transduction + motor response execution.

Each block of trials lasted no more than 4 minutes to minimize fatigue.

Activity Measurement (Bullock & Murray, 2014)

Activity counts were recorded every 2 mins using a wrist-worn Actiwatch.

Data were acquired from a young man diagnosed with bipolar disorder who wore the device for 103 days.

The patient suffered a severe manic episode and was hospitalized on day 104 at which time the Actiwatch was removed.

Data Analysis: Data were pooled over five successive time periods to produce a total count each 10 mins.

A log transform was applied and the resulting time series was differenced to remove linear trend.

Weekly transformed activity series (n = 1008) were analyzed to detect any precursor of the manic episode.

What is a fractal?• A fractal exhibits fluctuations that occur on only one time scale.• Self-similarity is a common characteristic of a fractal so that short-duration samples resemble

long-duration samples within the same time scale.• A fractal may result from a chaotic nonlinear process but this is not necessary.• A fractal has one characteristic index that represents its time-scale independence.

• A multifractal involves processes that occur on many different time scales, with a continuum of characteristic indices.

• For example, in language comprehension, processing of syllables, words, sentences and paragraphs occurs at increasingly longer time scales and in different regions of the brain (Lerner et al., 2011).

• The multifractal spectrum shows the relative strength of fractal activity occurring on these various time scales.

• Multifractal processes exhibit intermittency, with rapid fluctuations being interspersed with fluctuations occurring on much longer time scales.

What is a multifractal?

The Mathematics of Multifractal Processes

A stochastic process {X(t)} is a multifractal if it satisfies the q-th moment expectation equation:

is a continuous function of .

The expected increment for fractional Brownian motion, is obtained when q = 1, and, , where H is the Hurst index.

When , the expected increment of a stationary diffusion model is given by .

Time series data can be used to estimate the singularity scaling function, , from which can be obtained the q-order singularity strength function:

and the multifractal spectrum amplitude function:

The multifractal spectrum is obtained by plotting .

Multiplicative Cascade

p(x) Elementary processing pdf𝑇 0

𝑇 1

𝑇 2

𝑇 3

𝑇 4

𝑇 5

𝑇 6

𝑇 𝑘

Time scale

RT time seriesFast-paced fast

Subject 1

…….

Elementary Process PDF Multifractal spectrum

Poisson

Gamma

Restricted Gamma

Gaussian

E[p(x)] = value of when , i.e. the location of the peak of the multifractal spectrum

Var[p(x)] is determined by the width of the multifractal spectrum.

From: Calvert, Fisher and Mandelbrot (1997)

Evidence for Multifractal Scaling in Cognitive Tasks

• Kutnetsov and Wallot (2011): multifractal scaling in 22 of 30 time estimation sessions during which subjects provided continuation estimates of time intervals.• Ihlen and Vereijken (2010): multifractal scaling for simple and choice RT tasks but

not for a time interval estimation task (data from Wagenmakers et al., 2004). The width of the multifractal spectrum was related to the variance of a lognormal distribution of interaction multipliers.• Kelty-Stephen et al. (2013): evidence of multifractal scaling in rhythmic clapping

and visual tracking.

SAMPLE MULTIFRACTAL SPECTRAFROM EXPERIMENT 2 SELF-PACED CONDITION

All calculations used Multifractal Detrended Fluctuation Analysis (Ihlen, 2012)

Experiment 2: Self-Paced Task Subject 4Data

Surrogates

The significant difference occursfor processes operating on short time scales

Multifractality revealed by the multifractal spectrum

Width(Data) > Width(Surrogates)

Experiment 2: Self-Paced Task Subject 6

Data

Surrogates

Monofractality revealed by the narrow multifractal spectrum

Width(Data) < Width(Surrogates)

Experiment 2: Self-Paced Task Subject 3

Fractional Gaussian Process

Peak > 0.5

Width(Data) = Width(Surrogates)Data and Surrogates Coincide

Session 1 Session 2 Session 3 Self-Paced

Fast Slow

EXPERIMENT 1 EXPERIMENT 2

Rows represent subjects in each experiment

Possible Spectrum Type

Exp 1Width

Exp 2Width

MultifractalData > Surrogate 0.68 0.62

MonofractalData < Surrogate 0.50 0.54

Fractional GaussianData ~ Surrogate 0.57 0.54

F(2,18) = 0.43, p = 0.66 F(2,22) = 8.8, p = .0016

V = 74, p = .003

V = 71, p = .009No Practice Effect

Multifractal Spectrum Width less in speeded tasks

A multiplicative cascade driven by a Restricted Gamma pdf provides a close fit to the multifractal spectra for all subjects in the Fast-Paced Fast and Fast-Paced Slow conditions of Experiment 2, as well as for some subjects in the Self-Paced condition.

For other subjects in the Self-Paced condition a multiplicative cascade driven by a Gaussian pdf provides a better fit.

RT Standard Deviation RT Mean

Fast – Paced Fast

Fast – Paced Slow

Self – Paced

Elementary restricted gamma rate parameter, β, is linearly related to RT Mean and SD with negative slopes.

Linear tradeoff between mean RT and processing rate for data pooled over subjects.

Detecting the Onset of a Manic Episode Before It Happens …..

Multifractal spectra estimated from activity data were fit by a multiplicative cascade driven by a Gaussian process.

The continuous Shannon entropy of a lognormal pdf

was computed for each week using parameters and estimated from the multifractal spectra.

As declining health is associated with a decrease in entropy, evidence of the impending manic episode may have been available several weeks before hospitalization was required.

Conclusions

• Multifractality occurs for some subjects in some conditions in a choice RT experiment.• Estimates of the elementary process that drives a multiplicative cascade mental

process can be obtained from the multifractal spectrum.• Evidence for a restricted gamma distributed process suggests that current

choice decision models need to include multiplicative cascaded processes that can produce multifractal predictions.• Placing time constraints on RT in simple choice tasks decreases mean RT and

increases the gamma processing rate.• Multifractal spectra estimated from activity data could be fit by a Gaussian

process.• An entropy index estimated from the Gaussian process can be used to detect

the early stage of a manic episode in a person diagnosed with bipolar disorder.

AcknowledgmentsThe RT data were provided by Dr Alice Kelly who completed a PhD thesis at the University of Newcastle, Australia in 2001.The activity data were provided by Professor Greg Murray, Psychology, Swinburne University of Technology, Melbourne, Australia.

The volunteer subjects consented to partake in the RT task and activity monitoring following approval of the projects by the Research Ethics Committees at the University of Newcastle and Swinburne University of Technology, respectively.

References• Bullock, B., & Murray, G. (2014). Reduced amplitude of the 24 hour activity rhythm: A biomarker of vulnerability to

bipolar disorder? Clinical Psychological Science, 2, 86-96.

• Calvert, L., Fisher, A., & Mandelbrot, B. (1997). Large deviations and the distribution of price changes. Cowes Foundation Discussion Paper No. 1165. New Haven, CT: Yale University.

• Ihlen, E.A. (2012). Introduction to multifractal detrended fluctuation analysis in Matlab. Frontiers in Physiology–Fractal Physiology, 3, 1–18.

• Ihlen, E.A.F., & Vereijken, B. (2010). Interaction dominant dynamics in human cognition: beyond 1/fα fluctuations. Journal of Experimental Psychology: General, 139, 436–463.

• Kelly, A., Heathcote, A., Heath, R., & Longstaff, M. (2001). Response-time dynamics: Evidence for linear and low-dimensional nonlinear structure in human choice sequences. Quarterly Journal of Experimental Psychology, 54, 805–840.

• Kelty-Stephen, D.G., Palatinus, K., Saltzman, E., & Dixon, J.A. (2013). A tutorial on multifractality, cascades, and interactivity for empirical time series in ecological science. Ecological Psychology, 25, 1–62.

• Kuznetsov, N.A., & Wallot, S. (2011). Effects of accuracy feedback on fractal characteristics of time estimation. Frontiers in Integrative Neuroscience. doi: 10.3389/fnint.2011.00062

• Lerner, Y., Honey, C.J., Silbert, L.J., & Hasson, U. (2011). Topographic mapping of a hierarchy of temporal receptive windows using a narrated story. Journal of Neuroscience, 31, 2906–2915.

• Wagenmakers, E.-J., Farrell, S., & Ratcliff, R. (2004). Estimation and interpretation of 1/fα noise in human cognition. Psychonomic Bulletin & Review, 11, 579–615.