FARIMA processes with application to biophysical data

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ournal of Statistical Mechanics:J Theory and Experiment

FARIMA processes with application tobiophysical data

Krzysztof Burnecki

Institute of Mathematics and Computer Science, Wroclaw University ofTechnology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, PolandE-mail: krzysztof.burnecki@pwr.wroc.pl

Received 3 April 2012Accepted 20 April 2012Published 16 May 2012

Online at stacks.iop.org/JSTAT/2012/P05015doi:10.1088/1742-5468/2012/05/P05015

Abstract. In this paper we show fractional autoregressive integrated movingaverage (FARIMA) time series with a negative memory parameter and stablenon-Gaussian noise model movements of mRNA molecules inside live E. coli cellsrecorded by means of a single particle tracing experiment. The phenomenon ofnegative memory is related to the so-called subdiffusion which is often observedin crowded media. We fit the FARIMA process by using a variant of Whittle’smethod introduced by Kokoszka and Taqqu (1996 Ann. Statist. 24 1880) for theFARIMA stable case with a positive memory parameter, which we extend to thenegative memory case. In order to show the goodness of fit we analyze residuals ofthe model. We check that they follow a non-Gaussian stable law and justify theirindependence. Finally, with the help of Monte Carlo simulations, we illustratethat the fitted FARIMA model reproduces statistical properties of the analyzedbiophysical data.

Keywords: stochastic particle dynamics (theory), dynamical processes (theory),fluctuations (theory), stochastic processes (theory)

c©2012 IOP Publishing Ltd and SISSA Medialab srl 1742-5468/12/P05015+18$33.00

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Contents

1. Introduction 2

2. FARIMA models 42.1. Parameter estimation for FARIMA processes . . . . . . . . . . . . . . . . . 6

3. Fitting FARIMA parameters to the data 93.1. Long memory of the data . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2. Estimation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4. Diagnostics of the residuals 13

5. Simulations of the model 14

6. Conclusions 15

Acknowledgments 17

References 17

1. Introduction

The phenomenon of long-range dependence (or long memory or the Joseph effect firstused by Mandelbrot and Wallis [1]) has a long history and has remained a topic of activeresearch in the study of economic and financial time series (see e.g. [2, 3], and referencestherein). It is widespread in the physical and natural sciences (see e.g. [4] and [5]) and hasbeen extensively documented in hydrology, meteorology and geophysics (see e.g. [6]–[9]).The long-range dependence has been discovered in DNA sequences with very importantimplications on chromatin-mediated regulation of nuclear functions [10]–[12]. Recently,it has also started to play an important role in the engineering sciences, especially inthe analysis and performance modeling of traffic measurements from modern high-speedcommunications networks (for a bibliographical survey of this area, see [13]).

The long memory can be empirically observed, e.g. by a slowly decayingautocovariance function (ACF) [5, 14]. The classic example of a long-range dependentprocess is the fractional autoregressive integrated moving average (FARIMA) model witha power-law ACF. It appears that the values of FARIMA with Gaussian noise, for thememory parameter d greater than 0, have such a slowly decaying ACF that it is notabsolutely summable. This behavior serves as a classical definition of the long-rangedependence [5]. When d < 0, the ACF still follows a power law, hence exhibiting moresignificant dependence than any other process with exponentially decaying ACF, suchas, e.g. an autoregressive moving average (ARMA) time series, but the rate of decay isslower than for the d-positive case making the ACF absolutely summable. This negativememory phenomenon can be described as follows: increases in the values of the timeseries are likely to be followed by decreases and, conversely, decreases are more likely tobe followed by increases (negative correlation). Such a time series is called short memoryor antipersistent. Antipersistence has been observed in financial time series for electricity

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price processes [15], in climatology [16] and is widely seen in nanoscale biophysics [17]–[20]. The latter case is related to so-called subdiffusion which describes processes whichdiffuse at a rate much slower than the Brownian motion (standard diffusion) with respectto the second moment (mean square displacement). The classic examples of subdiffusivecharacter are the fractional Brownian motion (FBM) and continuous-time random walk(CTRW) [21, 22]. FBM is a limit process of an appropriately normalized FARIMA processwith Gaussian noise [23]. However, the processes essentially differ as the latter allows forshort-term dependences modeled by the ARMA part.

The study of non-Gaussian FARIMA processes was initiated more recently, seee.g. [5, 3, 24, 25]. Ever since the pioneering work by Mandelbrot [26], stable processes haveenjoyed great popularity as flexible modeling tools in economics and natural sciences (seee.g. [27]–[30]). In the infinite variance case there is no standard definition of short and longmemory as the autocovariance function does not exist (is infinite). Therefore, definitionsused in the literature incorporate other measures of dependence, e.g. codifference, ordifferent functionals such as partial sums and maxima [3], [31]–[33]. Analogously to theGaussian case, we say that the FARIMA process with α-stable noise has a long memoryif d > 0 and short (negative) memory if d < 0. FARIMA with α-stable noise is relatedvia a limit theorem to a fractional stable motion (FSM) with self-similarity exponentH = d+1/α [23], which was recently proposed as a model for subdiffusive phenomena [35]and is a generalization of the FBM. The fractional stable motion has an infinite variance,hence it is sometimes wrongly regarded only as an example of superdiffusion, however, itwas shown in [35] that the basic statistic for assessing the diffusion type, namely the time-average (sample) mean-squared displacement, is well defined for that process. Moreover,its diffusive exponent can be either negative for H < 1/α, hence leading to a subdiffusion,or positive for H > 1/α, so exhibiting a superdiffusive behavior. As a consequence, thediffusion type is controlled by the sign of memory parameter d = H − 1/α.

Since the short and long memory have been observed in many real-world phenomenathere is a need to build efficient estimators of the parameters of the FARIMA process, seee.g. [5, 34, 8, 36, 37]. In the literature different methods of assessing type of dependenceand estimating the memory parameter d have been developed [5, 3, 23]. It is importantto know what are the assumptions and limitations of various tools and what is the exactresult of different estimators [35, 38, 39]. For example, the classic rescaled range (R/S)method, in the general α-stable case, does not return the self-similarity parameter H ,which is true only in the Gaussian case, but the value d+ 1/2, where d = H − 1/α and αis the index of stability. Estimating the parameters of the FARIMA process seems a morecomplex task. In practice, the estimation is based either on the maximum likelihood orWhittle’s method [5, 40]. The former is widely used in the Gaussian case due to its well-known properties, but it has little application for FARIMA processes with stable noiseas density functions of stable non-Gaussian distributions have no simple form [31]. Thatis why Whittle’s method, which employs the idea of a periodogram, is often used in theinfinite variance case. However, in the literature, the consistency of Whittle’s estimatorhas been proved only for the long memory case, i.e. for d > 0 [41].

In this paper we show and justify statistically that a FARIMA time series with stableskewed innovations and a negative parameter d model mRNA molecule movements inE. coli cells. In order to estimate the parameters of the FARIMA process with negatived we use a variant of Whittle’s method which, we show, is well defined for d < 0 and

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skewed stable distributions. The fitted FARIMA model describes the data much betterthan the fractional stable motion, which was proposed for the same dataset in [35], as itincorporates short-term dependences clearly present in the data.

This paper is organized as follows: in section 2 we recall basic facts about a classicexample of long-range dependent processes, namely the FARIMA time series. We alsopresent known facts about the estimation of FARIMA parameters and concentrate onthe method introduced by Kokoszka and Taqqu [41] for both the stable case and positived. We extend their idea to the case of negative d. We justify the consistency of theestimator by means of Monte Carlo simulations. In section 3 we fit the FARIMA modelto experimental biological data. First, we check the power-law memory property byemploying three different methods of estimation of the memory parameter d: Lo’s modifiedR/S, sample mean-squared displacement and variance. Next, we estimate the order of themodel by investigating autocovariance and partial autocovariance functions. Finally, wefit parameters of the FARIMA(1, d, 1) process using the extension of the Kokoszka andTaqqu method. In section 4 we study residuals of the fitted process. We show they areskewed α-stable and test their randomness using different statistical tests. In section 5,with the help of Monte Carlo simulations, we illustrate that the identified model possessessimilar statistical properties to the observed biological data. In section 6 a summary ofthe results is presented.

2. FARIMA models

This section is a brief description of the FARIMA time series, introduced in [42] and [43].The FARIMA process {Xt}, denoted by FARIMA(p, d, q), is defined by:

Φp(B)Xt = Θq(B)(1 −B)−dZt, (1)

where Φp and Θq are polynomials of the order p and q respectively, B is the backwardoperator, and Zts are independent and identically distributed (iid) random variables witheither finite or infinite variance.

Polynomials Φp and Θq have classical forms known in time series theory, i.e. Φp(z) =1−φ1z−φ2z

2−· · ·−φpzp is the autoregressive polynomial and Θq(z) = 1+θ1z+θ2z

2+· · ·+θqz

q is the moving average polynomial. The backward operator B satisfies: BXt = Xt−1

and BjXt = Xt−j , j ∈ N. The operator (1 − B)−d is called the integration operator andthe number d is called the memory parameter. When d = 0, the integration operator(1 − B)−d reduces to the identity operator and the definition given by (1) reduces to:

Φp(B)Xt = Θq(B)Zt. (2)

Therefore, FARIMA(p, 0, q) is equivalent to the ARMA(p, q) process. In particular,when Φp = Θq ≡ 1, the sequence {Xt} is a pure noise process {Zt} and there is nodependence between observations. For negative integer values of the memory parameterd, i.e. d = −1,−2, . . ., the operator (1−B)−d has a finite extension and positive exponent−d. Therefore, it determines the full differencing of order −d and is often called the fulldifferencing operator.

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If polynomials Φp and Θq do not have common roots and Φp has no roots in the closedunit disk {z : |z| ≤ 1}, equation (1) can be rewritten in an equivalent form:

Xt︸︷︷︸

FARIMA(p,d,q)

=

full differencing of order−d︷ ︸︸ ︷

(1 − B)−d Θq(B)

Φp(B)Zt

︸ ︷︷ ︸

ARMA(p,q)

. (3)

The sequence {Zt} is often called the noise process (sequence) or innovations. In thispaper we assume that innovations Zt are iid and belong to the domain of attraction of anα-stable law with 0 < α < 2, i.e.

P (|Zt| > x) = x−αL(x), as x→ ∞, (4)

where L is a slowly varying function at infinity, and

P (Zt > x)

P (|Zt| > x)→ a,

P (Zt < −x)P (|Zt| > x)

→ b, as x → ∞, (5)

where a and b are nonnegative numbers such that a + b = 1. This corresponds to theinfinite variance case.

What is essential in FARIMA series methodology is to allow for fractional values ofthe memory parameter d. Then the operator (1−B)−d is called the fractional integrationoperator and has an infinite binomial expansion:

(1 − B)−dZt =

∞∑

j=0

bj(d)Zt−j, (6)

where the bj(d)s are the coefficients in the expansion of the function f(z) = (1−z)−d, |z| <1, i.e.

bj(d) =Γ(j + d)

Γ(d)Γ(j + 1), j = 0, 1, . . . , (7)

where Γ is the gamma function.The series (6) is convergent and the FARIMA definition given by (1) is correct if and

only if

α(d− 1) < −1 ⇔ d < 1 − 1

α. (8)

In particular, in the well-known Gaussian case when α = 2 we obtain d < 1/2.Under assumption (8) and the above-mentioned conditions for polynomials Φp and Θq,the FARIMA(p, d, q) time series defined by (1) has a causal moving average form:

Xt︸︷︷︸

FARIMA(p,d,q)

=

d−fractional integration︷ ︸︸ ︷

(1 −B)−d Θq(B)

Φp(B)Zt

︸ ︷︷ ︸

ARMA(p,q)

=∞

∑

j=0

cj(d)Zt−j , (9)

for details see [44]. Therefore the FARIMA(p, d, q) time series can be obtained by d-fractional integration of the ARMA(p, q) series. In particular, the FARIMA(0, d, 0) seriesis just a d-fractionally integrated noise process. The d-fractional integration operation

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through the (1−B)−d operator builds a dependence between observations in the FARIMAsequence, even though they are distant in time.

For a Gaussian FARIMA series, i.e. when α = 2, the rate of decay of theautocovariance function is t2d−1 (see [43]). Therefore, for d > 0 its autocovariance functionsatisfies:

∞∑

t=0

|Cov(t)| = ∞. (10)

In the literature this phenomenon is called the long memory property, see e.g. [45]. Forprocesses with infinite variance there are no standard definitions of long-range dependence.For α-stable stationary processes with α < 2 the covariance may be replaced by thecodifference, which reduces to the covariance in the Gaussian case, for details, see [31].For stable FARIMA series the rate of decay of the codifference function is tα(d−1)+1 andtherefore the long memory property (10) holds, under this measure of dependence, ford > 1 − 2/α, for details see [44]. Other definitions of long-range dependence lead to asomehow more natural conclusion that stable FARIMA processes have long memory ifd > 0, see e.g. [32], which is analogous to the finite variance case.

In this paper we focus on the case when the memory parameter d is negative. Thenthe operator (1 − B)−d has a positive exponent −d. All coefficients bj(d)s are negative(when −1 < d < 0), except for b0(d) = 1, and converge to zero at a power rate, but slowerthan in the d-positive case.

If d < 1− 1/α and m is an integer number such that d0 := d+m ∈ [−1/α, 1− 1/α),then

Xt(d) = (1 − B)mXt(d0) a.s., (11)

where {Xt(d)} and {Xt(d0)} are FARIMA processes with memory parameters d and d0

respectively. Therefore, any FARIMA(p, d, q) time series can be obtained by a number offull differencings applied to FARIMA(p, d0, q) with d0 ∈ [−1/α, 1 − 1/α). According tothis fact, without any loss of generality, we may consider FARIMA models with memoryparameter only in the interval [−1/α, 1 − 1/α).

2.1. Parameter estimation for FARIMA processes

Let us recall now a method of estimation of FARIMA parameters, which was developedin [41]. This method was introduced for the FARIMA time series with the positive memoryparameter d ∈ (0, 1 − 1/α). It is well defined for FARIMA series with noise satisfyingconditions (4) and (5). Since d ∈ (0, 1 − 1/α), the method is only defined for α ∈ (1, 2].

The method is a variant of Whittle’s method [46]. In [46] Whittle’s method wasapplied to a finite variance ARMA time series. These results were extended in [47] toGaussian FARIMA processes with positive memory. For ARMA time series with noisein the domain of attraction of a non-Gaussian stable law, the estimation techniques weredeveloped in [40]. The method described in [41] is an extension of these results to FARIMAprocesses with stable noise and positive memory property, i.e. for d ∈ (0, 1 − 1/α) andα > 1.

Following [41], we define the (p+ q + 1)-dimensional vector

β0 := (φ1, φ2, . . . , φp, θ1, θ2, . . . , θq, d), (12)

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where φ1, φ2, . . . , φp and θ1, θ2, . . . , θq are the coefficients of the polynomials Φp and Θq

respectively. The vector β0 is from the parameter space E := {β : φp, θq �= 0,Φp(z)Θq(z) �=0 for |z| ≤ 1,Φp,Θq have no common roots, d ∈ (0, 1 − 1/α)}.

We denote the normalized periodogram by

In(λ) :=|∑n

t=1Xte−iλn|2

∑nt=1X

2(t), −π ≤ λ ≤ π, (13)

and let

γ(0) :=∞

∑

j=0

c2j(d), (14)

where the coefficients cj(d)s are defined in (9).We also introduce the power transfer function (here we make the dependence on β

explicit)

g(λ, β) :=

∣

∣

∣

∣

Θq(e−iλ, β)

Φp(e−iλ, β)(1 − e−iλ)d(β)

∣

∣

∣

∣

2

. (15)

Finally, the estimator βn of the true parameter vector β0 is defined as

βn := arg minβ∈E

∫ π

−π

In(λ)

g(λ, β)dλ. (16)

The main result presented in [41] is the following consistency condition. If β0 is thetrue parameter vector and βn is the estimator defined in (16), then

βnP→ β0 (17)

and∫ π

−π

In(λ)

g(λ, βn)dλ

P→ 2π

γ(0), (18)

where the coefficient γ(0) is defined in (14) andP→ denotes convergence in probability.

Now, we study an extension of the FARIMA estimation method to the d-negative case,for d ∈ (−1/2, 0). In view of assumption (8) it yields that α ∈ (2/3, 2]. We constructan estimator in this new case which is an extension of the one proposed in the positivememory case. Moreover, we justify the consistency result for the estimator using MonteCarlo simulations.

First, we start with noticing that the denominator of the formula (13) for theperiodogram function In(·) does not depend on λ. Therefore, we can simplify the form (16)of the estimator βn by replacing In(·) with

In(λ) :=

∣

∣

∣

∣

∣

n∑

t=1

Xte−iλt

∣

∣

∣

∣

∣

2

, −π ≤ λ ≤ π. (19)

The next simplifying step may be done by noticing the symmetry of In(·) and powertransfer function g(·, β) with respect to λ. Hence, the simplified form for the estimator

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βn can be written as

βn := arg minβ∈E

∫ π

0

In(λ)

g(λ, β)dλ. (20)

The subject of our analysis is the estimator βn of the above form, but with revisedparameter space E, the same space as E but with d ∈ (−1/2, 0). The crucial differencebetween the classical d-positive case and the considered situation with d ∈ (−1/2, 0) is thefact that function g(·, ·) is continuous on space [0, π]×E whereas 1/g(·, ·) is discontinuous.More precisely, the function 1/g(·, β) diverges at λ = 0 for any β ∈ E.

First, we check whether the integral given in (20) is well defined when d ∈ (−1/2, 0).We obtain that

∫ π

0

In(λ)

g(λ, β)dλ = 2d(β)

∫ π

0

W (λ, β)(1− cosλ)d(β) dλ, (21)

where

W (λ, β) = In(λ)

∣

∣

∣

∣

Φ(e−iλ, β)

Θ(e−iλ, β)

∣

∣

∣

∣

2

. (22)

Let us recall that∫ π

0

(1 − cosλ)d dλ =2d√πΓ (d+ 1/2)

Γ(d+ 1)(23)

for d > −1/2. Since the function W (·, β) is a continuous function of the variable λ ∈ [0, π]a.s. we conclude the existence of the estimator. Let us also note that the integral in (23)is divergent for d ≤ −1/2.

Now, we may state the following consistency fact for the estimator βn. If β0 ∈ E isthe true parameter, then

βn := arg minβ∈E

∫ π

0

In(λ)

g(λ, β)dλ

P→ β0.

We justify the consistency of the FARIMA estimator βn by means of the Monte Carlomethod. To this end we perform simulations for sample trajectories of FARIMA(1, d, 1)with α-stable innovations. In such a case the parameter space E has three dimensions,i.e. β = (φ1, θ1, d) ∈ E, polynomials Φp(z) = 1 − φ1z, Θq(z) = 1 + θ1z, and the powertransfer function has the form

g(λ, β) =

∣

∣

∣

∣

1 + θ1e−iλ

1 − φ1e−iλ

∣

∣

∣

∣

21

(2 − 2 cosλ)d. (24)

Therefore, the estimator βn can be rewritten as

arg minβ∈E

∫ π

0

In(λ)

[

1 − 2φ1 cosλ+ φ21

1 + 2θ1 cosλ+ θ21

]

(2 − 2 cosλ)d dλ. (25)

We consider the following sets of parameters:

• (α, d) ∈ {1.8} × {−0.4,−0.2,−0.1} and β = 0.5.

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Figure 1. The box plots of estimators (d, φ1, θ1) for the FARIMA(1, d, 1) serieswith skewed α-stable innovations with α = 1.8 and β = 0.5 for three trajectorysubsets: (1) n = 1000, (2) n = 10000, and (3) n = 50000 of total lengthN = 50000. The black dotted lines correspond to the true values of parameters(d, φ1, θ1).

For the above three sets we choose the following polynomial coefficients:

• φ1 = −0.1 and θ1 = 0.2,

• φ1 = 0.6 and θ1 = −0.3,

• φ1 = −0.1 and θ1 = −0.3,

respectively.We simulated 100 trajectories of FARIMA(1, d, 1), using the algorithm presented

in [23], of length N = 50 000. Next, we calculated the values of FARIMA estimators (thememory parameter and polynomial coefficients) for increasing subsets of the trajectories,namely for the first n = 1000, n = 10 000, and n = 50 000 (i.e. the whole trajectory)points. The results are presented in the form of box plots in figure 1. We can see thatwith the increase of a subset’s length n, the estimators converge to the true values as theirvariance decreases.

3. Fitting FARIMA parameters to the data

We study here the data of Golding and Cox describing the motion of individualfluorescently labeled mRNA molecules inside live E. coli cells [48]. The data clearlyfollow the subdiffusive character and consist of 27 two-dimensional trajectories [21]. Weanalyze the y-coordinate of the longest trajectory of 1600 points. Since the trajectoryhas missing points, we fit the FARIMA model to the first 970 points of the trajectorywhich constitute the longest continuous part: {Yn : n = 1, 2, . . . , 970}. In this paperwe study the increments of the data. All further analysis is based on increments{Xn = Yn+1 − Yn : n = 1, 2, . . . , 969}. Both the data and its increments are presented infigure 2.

3.1. Long memory of the data

We first check the hypothesis of long memory in the analyzed time series. To this end weapply three methods of estimation of the memory parameter d, namely Lo’s modified

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Figure 2. A plot of the y-coordinate of particle position during the motion of atagged RNA molecule inside an E. coli cell (left panel) and its increments (rightpanel).

R/S statistic, variance method, and a method based on the notion of mean-squareddisplacement.

First, we consider a Lo’s modified R/S statistic (MRS) Vq [2], which for a stationaryseries {Xn, n = 1, . . . , N} is defined by

Vq =max1≤k≤N X(k) − min1≤k≤N

∑ki=1X(k)

√

S2 + 2∑q

i=1 ωi(q)γi

, (26)

where X(k) =∑k

i=1(Xi−Xq), γi are sample autocovariances, Xq and Sq are sample meanand sample standard deviation of the qth subseries respectively, and weights ωi(q) aregiven by

ωi(q) = 1 − i

q + 1, q < N. (27)

This method is a version of the well-known R/S method [49]–[51].Plotting Vq with respect to q on a log–log scale and fitting the least squares line,

leads, for the finite variance case, to an estimate of H , namely the slope of the line equals1/2−H . One may check that the formula for the α-stable distribution can be generalizedto −d, where d = H−1/α. In figure 3 in the left panel we depict a plot of the Vq statisticagainst q for q = 1, 2, . . . , 200 on a double logarithmic scale with the fitted least squaresline. The slope is equal to −d = 0.17, hence d = −0.17.

In order to estimate the memory parameter d we also used a sample mean-squareddisplacement (sample MSD) [35]. Let {Yn, n = 1, . . . , N} be a sample of length N . Then,the sample MSD is defined by

MN (τ) =1

N − τ

N−τ∑

k=1

(Yk+τ − Yk)2. (28)

The sample MSD is a time-average MSD on a finite sample regarded as a functionof difference τ between observations. It is a random variable in contrast to the ensembleaverage which is deterministic and always infinite for the α-stable case with α < 2.

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Figure 3. Modified R/S Vq (left panel), sample MSD MN (τ) (middle panel) andvariance method S2(m) (right panel) statistics (blue crosses) with fitted lines (redsolid lines). The slopes of the lines equal 0.17 (hence d = −0.17), 0.66 (leadingto d = −0.17), and −1.19 (it yields d = −0.15), respectively.

If N becomes large and τ small, the sample comes from a FARIMA process withα-stable noise with α ≤ 2, then

MN (τ) ∼ Sτ 2d+1, (29)

where d = H − 1/α and S is a stable random variable which does not depend onτ [35]. Therefore, the memory parameter d characterizes the stochastic process in termsof the speed of transport, and for d �= 0 we obtain so-called anomalous dynamics, eithersuperdiffusive (for d > 0) or subdiffusive (for d < 0). In particular, for d = 0, so anARIMA, we obtain that MN (τ) behaves like τ exactly as for a Brownian motion.

As a by-product, the sample MSD can serve as a method of estimating d. The methodis well defined for the stable case and the estimator has a very small variance, which willbe discussed in another paper. In order to apply it, first, we calculated the sample MSDfor the trajectory values {Yn =

∑ni=1Xi:n = 1, 2, . . . , 970}:

MN (τ) =1

970 − τ

970−τ∑

k=1

(Yk+τ − Yk)2 . (30)

Next, applying (29), we fitted the linear regression line according to:

ln(MN (τ)) = ln(C) + (2d+ 1) ln(τ), τ = 1, 2, . . . , 10, (31)

where C is assumed to be a constant. Our analysis yields 2d + 1 = 0.66 and therefored = −0.17, see figure 3, the middle panel.

The last considered method of estimation of the parameter d is the variance method(VM) [38]. Consider the aggregated series

X(m)(k) = 1/mkm∑

i=(k−1)m+1

Xi, k = 1, 2, . . . , [N/m], (32)

obtained by dividing a given series of length N into blocks of length m, andaveraging the series over each block. We compute its sample variance: S2(m) =

(1/N/m)∑N/m

k=1 (X(m)(k) − X)2. The series X(m) − EX(m) scales like mH−1, thus if theseries is Gaussian, or at least has finite variance, the sample variance will be asymptotically

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Figure 4. Autocorrelation (left panel) and partial autocorrelation (right panel)functions of the analyzed data. Blue horizontal lines represent 95% confidenceintervals.

proportional to m2H−2 for large N/m and m. In the non-Gaussian stable case, the methodproduces a random variable which is proportional to md−1, where d = H − 1/α, thusproviding an estimate of d. The plot of the statistic on a log–log scale with a fitted lineis depicted in figure 3 in the right panel. The method yields d = −0.19.

3.2. Estimation results

Next, we examined the FARIMA model order by plotting the autocorrelation and partialautocorrelation functions (ACF and PACF) [52], see figure 4. We can observe that onlythe very first correlations fall outside the confidence intervals. This situation is typical ofthe negative memory case for AR and MA orders less than or equal to 1. In our analysiswe set p = q = 1.

In addition, we performed the Ljung–Box test which takes into account the magnitudeof correlation as a group. The Ljung–Box statistic is given by:

Q = N(N + 2)

(

ρ21

N − 1+

ρ22

N − 2+ · · ·+ ρ2

K

N −K

)

, (33)

where ρi is a squared autocorrelation at lag i and K is a number of autocorrelation lagsincluded in the statistic [53].

The Q statistic is χ2 distributed as N → ∞. This allows us to test the null hypothesisthat there is no serial correlation in the data. We calculated the p-values forK = 1, . . . , 30.All p-values were lower than 0.001 clearly leading to a conclusion that there is a serialcorrelation in the data, which confirms the results from the previous section.

εt =√

htηt, ηt ∼ N(0, 1), (34)

ht = a0 +

q∑

i=1

aiε2t−i +

p∑

j=1

bjht−j . (35)

Finally, we employed the estimation procedure for FARIMA(1, d, 1) presented insection 2.1 and we obtained that d = −0.16, φ1 = −0.02, and θ1 = 0.12. We can

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observe that the value of d is close to those estimated in section 3.1 which confirms thecorrectness of the analysis.

4. Diagnostics of the residuals

We now study residuals of the fitted FARIMA model. To validate the goodness of fit of theFARIMA model we need to investigate the stationarity and independence. We calculatedresiduals as rescaled one-step prediction errors. To this end we used the linear predictorfor FARIMA with the stable noise model which was derived in [54]. First, we studiedthe autocorrelation and partial autocorrelation functions with 95% confidence intervalsfor the residuals. It appeared that roughly all correlations fall inside the intervals, hencesuggesting stationarity and independence of the data.

Next, we applied the Ljung–Box test to residuals. Similarly as for the original data,we calculated p-values for K = 1, . . . , 30. The lowest p-value was 0.5425, clearly indicatingthat there is no serial correlation in the data. To confirm the hypothesis, we have alsoperformed other tests of randomness, namely turning points, difference-sign, and rank [52].The obtained values were 0.13, 1, and 0.79, respectively. From these results we can inferthat there is no serial correlation in the residuals.

Finally, to confirm independence of the residuals we repeated the calculations fromthe previous section of three different memory estimators. The estimated values for MRS,MSD, and variance methods were −0.08, 0.01, and −0.01. They are close to zero whichjustifies independence.

We now consider in detail two possible probability laws underlying the residuals (and,consequently, the data): Gaussian and α-stable. To check if residuals come from apopulation with a specific distribution, we performed various statistical tests based onthe empirical distribution function.

We used two classes of measures of distance between the empirical FN(x) andtheoretical F (x) cumulative distribution functions, namely the Kolmogorov–Smirnov andCramer–von Mises, see e.g. [55, 56]. The former statistic is given by

D = supx

|FN(x) − F (x)|. (36)

This statistic can be written as a maximum of two nonnegative supremum statistics:

D+ = supx{FN(x) − F (x)} and D− = sup

x{F (x) − FN(x)}. (37)

Belonging to the same class is the Kuiper statistic given by

V = D+ +D−. (38)

The second class forms the Cramer–von Mises family

CM = N

∫ ∞

−∞(FN (x) − F (x))2ψ(x) dF (x). (39)

ψ(x) is a special weight function, for ψ(x) = 1 we obtain a W 2 Cramer–von Mises statisticand for ψ(x) = [F (x)(1 − F (x))]−1 we arrive at an A2 Anderson and Darling (AD) one.

The maximum likelihood method was used to estimate parameters of the Gaussiandistribution. We obtained that μG = 0.0003 and σG = 0.0133. To estimate the parameters

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Table 1. Values of the test statistics and corresponding p-values (in parentheses,based on 1000 simulated samples) under the assumption of the Gaussian, stable,and NIG laws.D V W 2 A2

Gaussian

0.0452 0.0830 0.5062 3.6180(<0.001) (<0.001) (<0.001) (<0.001)

Stable

0.0159 0.0244 0.0320 0.2927(0.7) (0.93) (0.65) (0.42)

NIG

0.0228 0.0401 0.0755 0.5092(0.2) (0.13) (0.17) (0.12)

of the stable distribution, we employed a regression-type estimator, which is regarded asboth accurate and fast [57, 58]. The estimated parameters were: α = 1.8328, σ = 0.0081,β = 0.4645, and μ = 0.0008. We note that the positive skewness parameter βS suggeststhe data to be right-skewed which is observed for the data, see figure 2. Finally, weconsidered the normal-inverse Gaussian (NIG) distribution which is a popular distributionin modeling semi-heavy phenomena [59, 57, 60]. The maximum likelihood estimates were:αNIG = 96.0410, βNIG = −0.5141, δNIG = 0.0165, and μNIG = 0.0004.

To calculate p-values, we used the procedure based on the Monte Carlo simulationsdescribed in [56]. The results are presented in table 1. For the Gaussian distribution, thecalculated p-values for all considered tests were lower than 0.001, showing that a Gaussiandistribution hypothesis should be definitely rejected for the data. In contrast to this, p-values for the stable distribution were very high. This clearly indicates stability of thedata. The values for the NIG distribution were much lower than for the stable case.

5. Simulations of the model

In this section we check whether the fitted FARIMA (1,−0.16, 1) model reproduces wellthe values of different estimators calculated for the biological data in section 3. To this endwe simulated 1000 trajectories of the fitted FARIMA of the same length as the biologicaldata. First, we present a sample simulated trajectory and its partial sum process, seefigure 5. We can see they resemble the analyzed data, see figure 2. We also calculatedsample ACF and PACF for the simulated trajectory and depict the results in figure 6. Wecan see the plots are similar to the plots of samples ACF and PACF for the biological datapresented in figure 4. Next, we calculated MRS, MSD, and VM estimators and presentedthe results in the form of box plots, see figure 7. We compared the output with the valuesthat were estimated from the data in section 3 (in figure 7 they are marked with black solidlines). We can notice that in all three cases the data estimates fall within the interquartilerange which justifies the fact that the model possesses similar statistical properties to theempirical data. Finally, we checked whether the estimates of FARIMA model parameters

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Figure 5. A plot of the simulated FARIMA(1,−0.16, 1) trajectory (left panel)and the corresponding partial sum process (right panel).

Figure 6. Autocorrelation (left panel) and partial autocorrelation (right panel)functions of a sample simulated trajectory. Blue horizontal lines represent 95%confidence intervals.

for the simulated model are similar to those obtained for the original data. In figure 8we can see box plots of estimated values of d (memory parameter), φ1 (AR part), and θ1(MA). The black solid line represents estimates obtained for the original data. We cansee that they fall inside the interquartile range, hence confirming the goodness of fit ofthe FARIMA model.

6. Conclusions

In this paper we show that the FARIMA(1,−0.16, 1) process with skewed 1.85-stablenoise can model the movement of mRNA molecules inside live E. coli cells. The modeldescribes both short-term (via the ARMA part) and negative power-law (via the fractionaldifferencing parameter d) dependences, heavy tails, and skewness clearly present inthe data. To the best of the author’s knowledge, it is the first approach to fitting aFARIMA model with non-Gaussian stable innovations to biological data. The model isan essential improvement over FLM which was studied in [35] as it incorporates short-termdependences which are observed in the data and influence the values of different estimators.

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Figure 7. Box plots of the estimated memory parameter d obtained viamodified R/S (left panel), sample mean-squared displacement (middle panel),and variance (right panel) methods for 1000 simulated trajectories of the fittedFARIMA(1,−0.16, 1) model. Black solid lines represent estimated values of d forthe empirical data.

Figure 8. Box plots of the estimated memory parameter d (left panel), first-order autoregressive (middle panel), and moving average coefficients (right panel)obtained via the FARIMA estimation procedure for 1000 simulated trajectoriesof the fitted FARIMA(1,−0.16, 1) model. Black solid lines represent estimatedvalues of d, φ1, and θ1 for the empirical data.

To this end, first, we checked the property of negative power-law memory by differentmethods of estimation d, namely MRS, sample MSD, and variance. They, together withthe plot of ACF and PACF, clearly indicate that the data possess a negative memoryproperty. In order to find parameters of the FARIMA model we proposed an estimationprocedure for FARIMA processes with the negative memory parameter d and skewed α-stable noise. The procedure is an extension of the method introduced by Kokoszka andTaqqu [41] for positive d and symmetric distributions. We showed that the procedure iswell defined and the obtained estimator is consistent by means of Monte Carlo simulations.

Next, we studied the residuals of the process. We analyzed three possible underlyingdistributions, namely Gaussian, NIG, and stable. The latter comes as a unanimous winner,which was justified by various goodness of fit statistical tests. We also investigated the

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independence of residuals. The results clearly indicate that the fitted FARIMA model iswell fitted.

Finally, we checked whether the model reproduces the statistical properties of thestudied data. To this end we simulated 1000 trajectories of the same length as the analyzeddata and estimated the memory and all FARIMA parameters via the same methods asfor the mRNA trajectory. We presented the results in the form of box plots. We showedthat the values of the estimators obtained for the biological data fall within correspondinginterquartile ranges constructed for the simulated processes, which confirms the goodnessof fit.

Observe also that the shape of a cell and crowded fluid characteristic of the cytoplasminfluence the dynamics of the labeled mRNA molecules. In particular, we believe thatthe parameters of the fitted FARIMA models can provide some insight into the physicalreasons for subdiffusive motion of the molecule. Namely, the parameter d is influenced bythe shape of the cell. Simulations show that, for example, if the cell has size smaller withrespect to some direction, then the molecule has less space there, hence it ‘bounces’ offthe cell walls more frequently, which makes the memory parameter more negative in thisdirection, which is equivalent to a stronger subdiffusive behavior [60].

We note that the analysis was performed for the longest trajectory recorded duringthe experiment but similar conclusions can be drawn for other trajectories. We do notpresent them due to the short lengths of these samples which makes the estimation resultsless definite. We have also observed a similar statistical picture for the data describingthe epidermal growth factor receptor labeled with quantum dots in the plasma membraneof live cells presented in [61]. We hope that the methodology proposed in this paper willbe useful in determining the appropriate stochastic model behind biological data.

Acknowledgments

The author would like to thank Ido Golding and Arnauld Serge for providing the empiricaldata. This research has been partially supported by the Polish Ministry of Science andHigher Education grant No. N N201 417639.

References

[1] Mandelbrot B B and Wallis J R, 1968 Water Resour. Res. 4 909[2] Lou A W, 1991 Econometrica 59 1279[3] Doukhan P, Oppenheim G and Taqqu M S, 2003 Theory and Applications of Long-range Dependence

(Boston, MA: Birkhauser)[4] Mandelbrot B B, 1982 The Fractal Geometry of Nature (San Francisco: Freeman)[5] Beran J, 1994 Statistics for Long-Memory Processes (New York: Chapman and Hall)[6] Mandelbrot B B and Van Ness J W, 1968 SIAM Rev. 10 422[7] Bertacca M, Berizzi F and Mese E, 2002 IEEE Trans. Geosci. Remote Sens. 43 2484[8] Stanislavsky A, Burnecki K, Magdziarz M, Weron A and Weron K, 2009 Astrophys. J. 693 1877[9] Horvatic D, Stanley H E and Podobnik B, 2011 Europhys. Lett. 94 18007

[10] Peng C K, Buldyrev S V, Goldberger A L, Havlin S, Sciortino F, Simons M and Stanley H E, 1992 Nature356 168

[11] Arneodo A, Bacry E, Graves P V and Muzy J F, 1995 Phys. Rev. Lett. 74 3293[12] Arneodo A, Vaillant C, Audit B, Argoul F, Daubenton-Carafa Y and Thermes C, 2011 Phys. Rep. 498 45[13] Willinger W, Taqqu M S and Erramilli A, A bibliographical guide to self-similar traffic and performance

modeling for modern high-speed networks, 1996 Stochastic Networks: Theory and Applicationsed F Kelly et al (Oxford: Clarendon) pp 339–66

[14] Mantegna R N and Stanley H E, 1999 An Introduction to Econophysics (Cambridge: Cambridge UniversityPress)

doi:10.1088/1742-5468/2012/05/P05015 17

J.Stat.M

ech.(2012)

P05015

FARIMA processes with application to biophysical data

[15] Weron R, 2006 Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach (Chichester:Wiley)

[16] Carvalho L M V, Tsonis A A, Jones C, Rocha H R and Polito P S, 2007 Nonlinear Process. Geophys.14 723

[17] Kou S C and Xie X S, 2004 Phys. Rev. Lett. 93 180603[18] Kotulska M, 2007 Biophys. J. 92 2412[19] Kou S C, 2008 Ann. Appl. Stat. 2 501[20] Jeon J H, Tejedor V, Burov S, Barkai E, Selhuber-Unkel C, Berg-Sørensen K, Oddershede L and Metzler R,

2011 Phys. Rev. Lett. 106 048103[21] Magdziarz M, Weron A, Burnecki K and Klafter J, 2009 Phys. Rev. Lett. 103 180602[22] Burnecki K, Magdziarz M and Weron A, Identification and validation of fractional subdiffusion dynamics,

2011 Fractional Dynamics ed J Klafter, S C Lim and R Metzler (Singapore: World Scientific) pp 331–52[23] Stoev S and Taqqu M S, 2004 Fractals 12 95[24] Eberlein E and Taqqu M S, 1986 Dependence in Probability and Statistics (Boston, MA: Birkhauser)[25] Embrechts P and Maejima M, 2002 Selfsimilar Processes (Princeton, NJ: Princeton University Press)[26] Mandelbrot B, 1963 J. Bus. 36[27] McCulloch J H, Financial applications of stable distributions, 1996 Handbook of Statistics in Statistical

Methods in Finance ed G S Maddala and C R Rao (Amsterdam: North-Holland) pp 393–425[28] Cont R and Tankov P, 2004 Financial Modelling with Jump Processes (Boca Raton, FL: Chapman and

Hall/CRC Press)[29] Metzler R and Klafter J, 2000 Phys. Rep. 339 77[30] Sabatini A M, 2000 IEEE Trans. Biomed. Eng. 47 1219[31] Samorodnitsky G and Taqqu M S, 1994 Stable Non-Gaussian Random Processes (New York: Chapman and

Hall)[32] Heyde C C and Yang Y, 1997 J. Appl. Probab. 34 939[33] Perpete N and Schmitt F G, 2011 J. Stat. Mech. P12013[34] Burnecki K, Klafter J, Magdziarz M and Weron A, 2008 Physica A 387 1077[35] Burnecki K and Weron A, 2010 Phys. Rev. E 82 021130[36] Peng C K, Buldyrev S V, Havlin S, Simons M, Stanley H E and Goldberger A L, 1994 Phys. Rev. E

49 1685[37] Audit B, Bacry E, Muzy J F and Arneodo A, 2002 IEEE Trans. Inf. Theory 48 2938[38] Taqqu M S and Teverovsky V, On estimating the intensity of long-range dependence in finite and infinite

variance time series, 1998 A Practical Guide to Heavy Tails ed R Adler et al (Boston, MA: Birkhauser)pp 177–217

[39] Giraitis L, Kokoszka P and Leipus R, 2001 J. Appl. Probab. 38 1033[40] Mikosch T, Gadrich T, Kluppelberg C and Adler R J, 1995 Ann. Stat. 23 305[41] Kokoszka P S and Taqqu M S, 1996 Ann. Statist. 24 1880[42] Granger C W J and Joyeux R, 1980 J. Time Ser. Anal. 1 15[43] Hosking J R M, 1981 Biometrika 68 165[44] Kokoszka P S and Taqqu M S, 1995 Stoch. Process. Appl. 60 19[45] Samorodnitsky G, 2006 Found. Trends Stoch. Syst. 1 163[46] Hannan E J, 1973 J. Appl. Probab. 10 130[47] Fox R and Taqqu M S, 1986 Ann. Stat. 14 517[48] Golding I and Cox E C, 2006 Phys. Rev. Lett. 96 098102[49] Hurst H E, 1951 Trans. Am. Soc. Civ. Eng. 116 770[50] Teverovsky V, Taqqu M S and Willinger W, 1999 J. Stat. Plann. Inference 80 211[51] Weron R and Przybylowicz B, 2000 Physica A 283 462[52] Brockwell P and Davis R, 2002 Introduction to Time Series and Forecasting (New York: Springer)[53] Ljung G M and Box G E P, 1978 Biometrika 65 297[54] Kokoszka P S, 1996 Probab. Math. Stat. 16 65[55] D’Agostino R B and Stephens M A, 1986 Goodness-of-fit Techniques (New York: Dekker)[56] Burnecki K, Janczura J and Weron R, Building loss models, 2011 Statistical Tools for Finance and

Insurance 2nd edn, ed P Cizek et al (Berlin: Springer) pp 293–328[57] Weron R, Computationally intensive value at risk calculations, 2004 Handbook of Computational Statistics

(Berlin: Springer) pp 911–50[58] Koutrouvelis I A, 1980 J. Am. Stat. Assoc. 75 918[59] Barndorff-Nielsen O E, 1995 Normal//Inverse Gaussian Processes and the Modelling of Stock Returns

Research Report 300 Department of Theoretical Statistics, University of Aarhus[60] Burnecki K, Muszkieta M, Sikora G and Weron A, 2012 Europhys. Lett. 98 10004[61] Serge A, Bertaux N, Rigneault H and Marquet D, 2008 Nature Methods 5 687

doi:10.1088/1742-5468/2012/05/P05015 18