Analyzing stochastic time series Tutorial Kingston University, Dept. Computing, Information Systems and Mathematics Analyzing stochastic time series Tutorial Malgorzata Kotulska Department of Biomedical Engineering & Instrumentation Wroclaw University of Technology, Poland

Outline Data motivated analysis - time series in the real life Probability and time series - stochastic vs dterministic Stationarity Correlations in time series Modelling linear time series with short-range correlations – ARIMA processes Time series with long correlations – Gaussian and non-Gaussian self-similar processes, fractional ARIMA

Time series – examples P. J. Brockwell, R. A. Davis, Introduction to Time Series and Forecasting, Springer, 1987

Ionic channels in cell membrane M. Kullman, M. Winterhalter, S. Bezrukov, Biophys. J.82 (2003) p.802

Nile river J. Beran, Statistics for long-memory processes, Chapman and Hall, 1994

Objectives of time series analysis Data description Data interpretation Data forecasting Control Modelling / Hypothesis testing Prediction

Time series

Time series – realization of a stochastic process {Xt} is a stochastic time series if each component takes a value according to a certain probability distribution function. A time series model specifies the joint distribution of the sequence of random variables.

White noise - example of a time series model

Gaussian white noise

Stochastic properties of the process STATIONARITY System does not change its properties in time Well-developed analytical methods of signal analysis and stochastic processes

WHEN A STOCHASTIC PROCESS IS STATIONARY? {Xt} is a strictly stationary time series if (X1,...,Xn)=d (X1+h,...,Xn+h), where n1, h – integer, =d means distribution equality Properties: The random variables are identically distributed. An idependent identically distributed (iid) sequence is strictly stationary.

Weak stationarity {Xt} is a weakly stationary time series if EXt =  and Var(Xt)=2 are independent of time t Cov(Xs, Xr) depends on (s-r) only, independent of t. Properties: E(Xt2) is time-invariant.

Quantitative method for stationarity Reverse Arrangement Test Weak stationarity: Testing if E(Xt2) is time-invariant

Quantile line method P{Xt  k(t)}=  PROPERTIES: A quantile of order , 0  1, is such a value k(t) that probability of the series taking value less than k(t) at time t equals . P{Xt  k(t)}=  PROPERTIES: Lines parallel to the time axis  stationarity Lines parallel to each other, not to the time axis  constant variance, a variable mean (or median) Lines not parallel to each other  a variable variance (or scale parameter)

Quantile lines of the raw time series Nonstationarity with a variable mean and variance

Methods for nonstationary time series Trend removal Segmentation of the series Specific analytical methods (e.g. ambiguity function, variograms for autocorrelation function)

Trend estimation Filters, e.g. moving average filter, FIR, IIR filters Polynomial (or other, e.g. log) estimation and removal Filters, e.g. moving average filter, FIR, IIR filters Differencing

Seasonal models Classical decomposition model seasonal component trend Stochastic process random noise Xt = mt + Yt + st

Backshift operator B

Detrended series P. J. Brockwell, R. A. Davis, Introduction to Time Series and Forecasting, Springer, 1987

Quantile lines of the differenced time series

(Sample) autocorrelation function

Range of correlations Independent data (e.g. WN) Short-range correlations Long-range correlations (correlated or anti-correlated structure)

ACF for Gaussian WN

Short-range correlations Markov processes (e.g. ionic channel conformational transition) ARMA (ARIMA) linear processes Nonlinear processes (e.g. GARCH process)

ARMA (ARIMA) models Time series is an ARMA(p,q) process if Xt is stationary and if for every t: Xt  1Xt-1 ...  pXt-p= Zt + 1Zt-1 +...+ pZt-p where Zt represents white noise with mean 0 and variance 2 Left side of the equation represents Autoregresive AR(p) part, and right side Moving Average MA(q) component. The polynomials (1- 1z-...- pzp) cannot have (1+ 1z+...+ pzq) common factors.

Examples The range of MA component estimated by ACF (the lag number within Bartlett’s limits ), the range of AR component by PACF Confidence band is

Exponential decay of ACF MA(1) sample ACF AR(1)

Stationary processes with long memory Qualitative features Relatively long periods with high or low level of observation values In short periods there seems to be cycles and local trends. Looking at long series – no particular cycles or persisting trends Overall the series looks stationary

Stationary processes with long memory Quantitative features The variance of the sample mean decays to zero at a slower rate. Instead of there is The sample autocorrelation function decays to zero in a power-law manner instead of exponentially Similarly, the periodogram (frequency analysis) shows a power-law

Classical processes with long correlations Fractional ARIMA processes (fARIMA) Self similar processes

fARIMA ARMA (p,q): ARIMA (p,d,q): fractional ARIMA (p,d,q):

Self-similar process A process X={X(t)}t  0 is called self-similar if for some H > 0 Self similarity in distribution sense, not for the sample path. Every change in time scale a>0 corresponds to the space scale aH. The bigger H the more dramatic change. Self similarity can be observed also on the quantile lines. (Parallel to t H). H =1 – /2 – self-similarity index, (HR +)

Frequency-domain analysis Periodogram Periodogram - estimated PSD by a Fourier transform of a sample autocorrelation function. Periodogram of a long memory time series depends on frequency according to power-law relationship (straight line on log-log plot). It means that if one doubles the frequency - PSD diminishes by the same fraction regardless of the frequency

Basic features of self similar process APPEARANCE. If an amplitude of a self-similar process is rescaled by r H, X (rt) looks like X (t), statistically indistinguishable. VARIANCE of the signal changes as Var (X(t))  t 2H CORRELATION - correlated or anticorrelated structuring H=0.5 no memory H>0.5 long memory H<0.5 antipersistent long correlations – „short memory” PERIODOGRAM - power-law dependance on frequency Na podst. Eke.

Nile Example J. Beran, Statistics for long-memory processes, Chapman and Hall, 1994

Methods R/S analysis by Hurst DFA – Detrended Fluctuation Analysis Exponent-based (correlogram, periodogram of the residuals) : H =1 – /2 other (for appropriate PDFs, e.g. Orey index)

Hurst exponent – the algorithm A series with N elements is divided into shorter series – n elements each

Hurst exponent

Classical self-similar processes

Fractional Brownian motions (fBm) Gaussian noise A series n (n=1,...,N) of uncorrelated and random variables Each n - Gaussian distribution N(0,) Brownian motion A sum of Gaussian white-noise sequence yn(Bm)= n(Bm)=  n1/2 Definition based on standard devation of Bm. Bm – the classic example of nonstationary time series. No memory. Fractional Brownian motions (fBm) n(fBm)   nH

Fractional Brownian motions (fBm) X={X(t)}t  0 is a nonstationary Gaussian process with mean zero and an autocovariance function: Fractional Gaussian noise (fGn) A stationary process of increments in fBm (differences between values separated by some step)

Gaussian or non-gaussian process

Fractional Lévy Stable Motion In the fractional Lévy stable motion (FLSM) the distribution is Lévy‑stable. =0.5 Stable distribution (solid), the attraction domain of stable distribution: Burr and Pareto distributions (broken & dotted)

Scaling properties of PDF -stable distributions have scaling properties – a sum of independent and identically distributed random variables maintains the same shape of the distribution. Similarly as the Gaussian distribution, also a stable distribution (CLT). Only a few -stable distributions have direct formulas for their probability density function. Usually only the characteristic function is given. The distinctive properties of -stable distributions are their long tails, infinite variance and, in some cases, infinite mean value.

fractional Levy-stable motion fLSM process is a self‑similar non-stationary process which can be represented as Z(u) is a symmetric Lévy -stable motion, and  is the stability index of stable distribution. The increment process of FLSM is stationary and it is called a fractional stable noise (FSN).

Memory of a self-similar process d = H  1/ For a Gaussian process =2 For d > 0 the memory is long – a long-range persistent process. Otherwise (d < 0) – a long-range antipersistent process („short memory”). The time series looks very rough

Summary Time series can be deterministic or stochastic. Visual distinction not always possible. Stochastic time series may tested analytically by statistical methods and an appropriate model attributed if the series is stationary. Otherwise a pre-processing needed. Random data in time series may be correlated. Correlations are called memory. Independent data (e.g. WN) do not have a memory. Each element assumes a value according to an independent probability density function.

Summary (2) In short-range memory the time series is correlated with a few elements back; the autocorrelation function shows an exponential decay. Typical models of time series with short memory are linear ARMA models - linear combination of previous elements and white noise components. The range of AR component estimated by ACF, the range of MA component by PACF. In long-range memory the series is correlated with vary far away elements. The decay of the autocorrelation function is slower than exponential. ACF decays according to power-law. Long-range time series can be typically modelled by self-similar processes (fBm, FLSM) or fARIMA linear models

Recommended text books P. J. Brockwell, R. A. Davis, Introduction to Time Series and Forecasting, Springer, 1987 J. Beran, Statistics for long-memory processes, Chapman and Hall, 1994 G.E.P.Box, G.M. Jenkins, Time series analysis: forecasting and control, Holden Day, 1970