1. . Multifractal dynamics of activity data in Bipolar Disorder: Techniques for predicting episode risk Rachel Heath School of Psychology University of Newcastle, Australia Email: rachel.heath@newcastle.edu.aurachel.heath@newcastle.edu.au Greg Murray Brain and Psychological Sciences Research Institute Swinburne University of Technology Australia SykTek ®

2. Our menu for today ……… The importance of predicting and preventing episodes in bipolar disorder What is a multifractal? A homogeneous stochastic multiplicative cascade model Computing Hurst functions and the multifractal spectrum Checking for nonlinearity Estimating the model parameters Shannon entropy for a continuous function Detecting the early signs of a manic episode using activity data.

3. The importance of predicting and preventing episodes in bipolar disorder Bipolar disorder is a serious mental illness affecting about 2% of people worldwide. Those diagnosed with the disorder suffer extremes of mood, depression and mania, sometimes requiring hospitalization when the episodes lead to severe psychomotor retardation, suicide risk and psychosis. Hospitalization is distressing for the patient and family, and very costly for the community. Treatment involves the judicious use of mood stabilizers such as lithium and lamotrigine, as well as antipsychotics such as quetiapine. Undesirable side-effects are common. It is thought that each episode leads to greater mental dysfunction, especially when rapid cycling of episodes occurs as the disorder progresses.

4. Our Strategy to Ameliorate Suffering in Bipolar Disorder Use a wrist-worn monitor to keep track of activity and sleep. Analyze the activity data so that the complexity of the data is recognized. Use nonlinear methods to summarize the complexity within the data sets. Use a change detection procedure to detect early signs of an episode. Warn the patient and the medical team that intervention is needed when early signs of an episode are detected. Make use of advanced technology, such as mobile devices and cloud computing to maximize the benefit of such a system.

5. What is a multifractal? We assume that human activity operates on multiple time scales and is intrinsically nonlinear. A fractal is an object constructed using nonlinear processes that occur on only one time scale. A multifractal is an object constructed using nonlinear processes that occur on more than one time scale. Multifractals occur throughout nature, e.g. atmospheric turbulence, waterfalls, physiological processes such as heart-rate, etc. Static multifractals that occur on multiple spatial scales include mountain ranges, forests, etc. Our computational challenge is to represent an arbitrary multifractal in terms of a homogeneous stochastic process that operates on a tree structure, check for nonlinearity and fit an appropriate model to data for prediction purposes.

6. A homogeneous stochastic multiplicative cascade model

7. Data Sets for Weeks 9 and 14

10. Computing Hurst functions and the multifractal spectrum

11. We will generalize this method to produce MultiFractal Detrended Fluctuation Analysis (Kandelhardt, 2002)

13. Checking for nonlinearity For each time series compute 30 surrogate series that have the same linear properties as the original series (mean, variance, autocorrelation function). Use surrogate series as linear controls for the Hurst function estimates. If the Hurst function estimated from the activity series exceeds these confidence bounds, reject the Linearity hypothesis. For comparison purposes, a Gaussian control with the same mean and variance as the activity data was also analyzed.

14. Activity series Surrogate series Gaussian series Monofractal DFA is linear Activity series is antipersistent

15. Estimating the model parameters In the multiplicative cascade model assume p(x) is Gaussian with mean μ and variance σ 2. The multifractal spectrum is given by (Calvert, Fisher & Mandelbrot, 1997): This means that the parameter estimates can be read off the empirical multifractal spectrum but for good measure we use Global Optimization by Differential Evolution methods (DEoptim package in R) to estimate parameters (Ardia et al., 2011).

16. Activity data Surrogate Gaussian Multifractal Spectrum

17. The width of the multifractal spectrum was less in week 14 than week 9.

18. Shannon entropy for a continuous pdf which yields The parameters μ and variance σ 2 are estimated from the multifractal spectrum generated by a homogeneous stochastic multiplicative cascade that best fits the data.

19. Detecting the early signs of a manic episode using activity data Activity data were collected for 14 weeks with only a few missing observations from week 6 from a young man diagnosed with bipolar disorder. At the end of week 14 he was admitted to hospital with a severe manic episode at which point activity recording ceased. We pooled activity data over each 10 minute period and transformed the data by taking logarithms then first differences to remove trend. The multifractal spectra and entropy were estimated for each week except week 6. In the entropy graph we interpolated the data to provide an estimate of entropy for week 6. In many medical applications, a reduction in entropy signals ill health. Change detection employed Guan’s (2004) semiparametric changepoint model as implemented by the sac R package. Epidemic mode was used to estimate more than one change point. This is a special form of the conventional CUSUM method.

20. Euthymia Early sign of episode A warning of an increasing episode risk would be sent to the patient and treating team at the end of week 12

21. Summary Activity data is complex and nonlinear. Activity data can be fit by the output of a homogeneous stochastic multiplicative cascade driven by a Gaussian elementary random variable. An entropy index derived from this multifractal model is sensitive to the precursors of a serious manic episode in a young person diagnosed with bipolar disorder. This technology can be implemented in mobile devices with use of the cloud to inform patients and their medical team of the risk of a future serious mood episode that might lead to hospitalization. By so doing, the suffering of patients and their carers might be reduced, and the high cost of hospitalization minimized by emphasizing preventative mental health.

22. References Ardia, D., Boudt, K., Carl, P., Mullen, K.M., & Peterson, B.G. (2011). Differential Evolution with DEoptim. An Application to Non-Convex Portfolio Optimization. The R Journal, 3, 27-34. URL http://journal.r- project.org/2011-1/. 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. Guan, Z. (2004). A semiparametric changepoint model. Biometrika, 91, 849-862. Heath, R.A., & Murray, G. (2016, in press). Multifractal dynamics of activity data in Bipolar Disorder: Towards automated early warning of manic relapse. Fractal Geometry and Nonlinear Analysis in Biology and Medicine. Ihlen, E.A. (2012). Introduction to multifractal detrended fluctuation analysis in Matlab. Frontiers in Physiology–Fractal Physiology, 3, 1–18. Kantelhardt, J.W., Zschiegner, S.A., Koscielny-Bunde, E., Bunde, A., Havlin, S., Stanley, H.E. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica A, 316, 87-114.