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Chapter 14 Identification and Validation of Fractional Subdiffusion Dynamics Krzysztof Burnecki, Marcin Magdziarz and Aleksander Weron Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland

In this chapter we propose a systematic methodology on how to distinguish between three mechanisms leading to single molecule subdiffusion, namely fractional Brownian motion, fractional Lévy stable motion and Fractional Fokker–Planck equation. We illustrate step by step that the methods of sample mean-squared displacement and p-variation can be successfully applied for infinite and confined systems. This methodology is based on well-known and not so well-known statistical tools for identification and validation of the above three fractional dynamical systems.

1. 2.

3.

4. 5. 6. 7. 8.

Introduction . . . . . . . . . . . . . . . . . . . . . Three Models of Fractional Subdiffusion Dynamics 2.1. Fractional Brownian motion . . . . . . . . . 2.2. Fractional Lévy stable motion . . . . . . . . 2.3. Fractional Fokker–Planck equation . . . . . Selected Identification and Validation Tools . . . . 3.1. Kolmogorov–Smirnov statistic . . . . . . . . 3.2. FIRT estimator . . . . . . . . . . . . . . . . Sample MSD . . . . . . . . . . . . . . . . . . . . . Sample p-variation . . . . . . . . . . . . . . . . . . Statistical Validation . . . . . . . . . . . . . . . . . The Case of Confined Systems . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . .

. . . . . . . . . . . . .

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329 331 331 331 332 333 333 335 337 338 341 344 347

1. Introduction The issue of distinguishing between normal and anomalous diffusion, as such, concerns many fields of physics [1–13]. It is usually based on the analysis of the mean-squared displacement (MSD) of the diffusing particles. 329

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K. Burnecki, M. Magdziarz and A. Weron

In the case of classical diffusion, the second moment is linear in time, whereas anomalous diffusion processes exhibit distinct deviations from this fundamental property: x2 (t) ∼ ta , where for 0 < a < 1 is subdiffusive and for a > 1 is superdiffusive [12]. The origin of anomalous dynamics in a given system is often unknown. It is not always clear which model applies to a particular system [10,11, 14], information which is essential when diffusion-controlled processes are considered. Therefore, determining the appropriate model is an important and timely problem; see [10, 11, 15] for discussion on the origins of anomaly in the case of intracellular diffusion. The MSD can be obtained either by performing an average over an ensemble of particles, or by taking the temporal average over a single trajectory [16,17]. Recent advances in single molecule spectroscopy enabled single particle tracking experiments following individual particle trajectories [11, 15]. These require temporal moving averages. Therefore, determining the origins of subdiffusive transport is a vital and timely problem [10, 11, 15, 18, 19]. In Sec. 2, we describe shortly three different models of fractional subdiffusion dynamics. Section 3 contains description of basic identification and validation tools like Kolmogorov– Smirnov statistic and FIRT estimator. We show that they can be used for model identification of stationary or increment stationary data. In Sec. 4, we propose to replace the MSD with sample MSD since the sample (time average) MSD of the fractional Lévy α-stable motion (FLSM) behaves very differently from the corresponding ensemble average (second moment). Namely, while the ensemble average MSD diverges for α < 2, the sample MSD may exhibit either subdiffusion, normal diffusion or superdiffusion. Traditionally, α-stable processes like Lévy flights serve as examples of superdiffusive dynamics since for α < 2 their second moment is infinite and the MSD diverges. Nevertheless, H-self-similar Lévy stable processes can model either a subdiffusive, diffusive or superdiffusive dynamics in the sense of sample MSD. We also show that the character of the process is controlled by a sign of the memory parameter d = H − 1/α. In Sec. 5, we present in detail a new sample p-variation test (algorithm) which allows to distinguish between three models of subdiffusive dynamics on simulated data. We already identified [7–9] fractional subdiffusive dynamics on biological data describing the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells, but it may concern many other fields of contemporary experimental physics. Finally, in Secs. 6 and 7, we present the main results of this chapter. Namely, statistical validation of the proposed models in the case of freely moving particles and confined systems, respectively.

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2. Three Models of Fractional Subdiffusion Dynamics 2.1. Fractional Brownian motion In the literature, two popular stochastic models have been used to account for anomalous diffusion. The first one is the fractional Brownian motion (FBM) introduced by A. N. Kolmogorov in 1940 [20–22]. The second model of subdiffusion is the continuous time random walk (CTRW) and the corresponding fractional Fokker–Planck equation [12,23]. However, they do not exhaust all possible sources of anomalous diffusion. Another source could be random walks on fractal structures, fractional Langevin equations, generalized Langevin equations, percolation, etc. [12, 24]. FBM is a generalization of the classical Brownian motion (BM). Most of its statistical properties are characterized by the Hurst exponent 0 < H < 1. In particular, the MSD of FBM satisfies x2 (t) ∼ t2H , thus for H < 1/2 we obtain the subdiffusive dynamics, whereas for H > 1/2 the superdiffusive one. For further properties of FBM and its applications to physics see [25–29]. For any 0 < H < 1, FBM of index H (Hurst exponent) is the mean-zero Gaussian process BH (t) with the following integral representation [21, 26]: BH (t) =

∞ −∞

H− 1 H− 1 (t − u)+ 2 − (−u)+ 2 dB(u),

(1)

where B(t) is a standard Brownian motion and (x)+ = max(x, 0). FBM is H-self-similar, namely for every c > 0 we have BH (ct) = cH BH (t) in distribution, and has stationary increments. It is the only Gaussian process satisfying these properties. For H > 1/2, the increments of the process are positively correlated and exhibit long-range dependence (long memory, persistence), whereas for H < 1/2, the increments of the process are negatively correlated and exhibit short-range dependence (short memory, antipersistence) [26]. For the second moment of the FBM we have 2 (t) = σ 2 t2H , where σ > 0, which for H < 1/2 gives the subdiffusive BH dynamics and for H > 1/2 the superdiffusive one. 2.2. Fractional L´ evy stable motion FBM can be generalized to a fractional Lévy stable motion (FLSM) [25,26, 30–32]: Lα H (t)

∞

= −∞

{(t − u)d+ − (−u)d+ }dLα (u),

(2)

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where Lα (t) is a Lévy α-stable motion (LSM), 0 < α ≤ 2, 0 < H < 1, and d = H − 1/α. The process is α-stable (for α = 2 it becomes a FBM), H-selfsimilar and has stationary increments. Analogously to the FBM case, we say the increments of the process exhibit positive (long-range) dependence if d > 0 (H > 1/α), and negative dependence when d < 0 (H < 1/α) [31, 32]. This is due to the behavior of the integrand in (2). Therefore, as in the Gaussian case, the parameter d controls sign of dependence. We show in Sec. 4 that the time average MSD of FLSM behaves very differently from the corresponding ensemble average (second moment). This is a timely subject since single molecule experiments exhibit both anomalous kinetics and a large scatter of the time average MSD. While the ensemble average MSD diverges, the time average MSD may exhibit either subdiffusion, normal diffusion or superdiffusion. Thus in experiment what seems subdiffusive from a single trajectory analysis could in fact be superdiffusive in the ensemble sense. 2.3. Fractional Fokker–Planck equation Force-free subdiffusion in the framework of CTRW with heavy-tailed waiting times is convenintly described by the fractional Fokker–Planck equation (FFPE) [12, 33] 2 ∂w(x, t) 1−β 1 ∂ = 0 Dt w(x, t) ∂t 2 ∂x2

(3)

with the initial condition w(x, 0) = δ(x). The operator 0 Dt1−β , 0 < β < 1, is the fractional derivative of the Riemann–Liouville type. The tβ , which is characteristic for MSD corresponding to w(x, t) equals Γ(β+1) subdiffusive dynamics. In Eq. (3), w(x, t) denotes the PDF of some subdiffusive stochastic process Zβ (t). The process Zβ (t) can be explicitly represented in the following subordination form [34–36] Zβ (t) = B(Sβ (t)),

(4)

where B(t) is the standard Brownian motion and Sβ (t) is the so-called inverse β-stable subordinator defined as Sβ (t) = inf{τ > 0: Uβ (τ ) > t}.

(5)

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Here, Uβ (τ ) is the β-stable subordinator [31,32] with the Laplace transform β given in the stretched exponential form E(e−uUβ (τ ) ) = e−τ u . Moreover, Sβ (t) is assumed to be independent of B(t). The Langevin-type process Zβ (t) reveals the detailed structure of trajectories corresponding to FFPE (3). Therefore, it allows one to study the statistical properties of the trajectories of subdiffusion in the framework of CTRW with heavy-tailed waiting times. 3. Selected Identification and Validation Tools We now collect a list of identification and validation tools for fractional subdiffusive dynamics used in this Chapter. In Table 1 they are presented along with the related characteristics of the data. Kolmogorov–Smirnov statistic and sample p-variation stand for examples of both identification and validation tools, whereas FIRT method and sample MSD are examples of identification tools. The first tool depicted in Table 1 assumes that the data are stationary. Stationary and nonstationary processes are very different in their properties, and they require different inference procedures. At this point, note that a simple and useful method to tell if a process is stationary in empirical studies is to plot the data. Loosely speaking, if a series does seem to have a trend, or a varying volatility, then very likely, it is not stationary. To make the process stationary it is sometimes enough to calculate its increments. 3.1. Kolmogorov–Smirnov statistic Once the distribution class is selected and the parameters are estimated using one of the available methods, the goodness-of-fit has to be tested. A standard approach consists of measuring the distance between the empirical and the fitted analytical distribution function. A group of statistics and tests based on this idea has been discussed [37].

Table 1.

List of simple identification and validation tools.

Tools Kolmogorov–Smirnov statistic FIRT method Sample mean-squared displacement Sample p-variation

Characteristic of the data Distribution type Self-similarity Diffusion type, long memory Diffusion type

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A statistics measuring the difference between the empirical Fn (x) and the fitted F (x) distribution functions, called an empirical distribution function (edf) statistic, is based on the vertical difference between the distributions. This distance is usually measured either by a supremum or a quadratic norm [38]. The most popular supremum statistic: D = sup |Fn (x) − F (x)|,

(6)

x

is known as the Kolmogorov or Kolmogorov–Smirnov (KS) statistic. For other classes of measures of discrepancy see, e.g. [37]. Suppose that a sample x1 , . . . , xn gives values zi = F (xi ), i = 1, . . . , n. It can be easily shown that, for values z and x related by z = F (x), the corresponding vertical differences in the edf diagrams for X and Z are equal. Consequently, edf statistics calculated from the empirical distribution function of the zi ’s compared with the uniform distribution will take the same values as if they were calculated from the empirical distribution function of the xi ’s, compared with F (x). This leads to the following formulas given in terms of the order statistics z(1) < z(2) < · · · < z(n) : i + − z(i) , D = max (7) 1≤i≤n n (i − 1) D− = max z(i) − , (8) 1≤i≤n n D = max(D+ , D− ),

(9)

Kolmogorov–Smirnov statistic can serve as a simple identification tool in the following way. We calculate the statistic for different choices of F (x), e.g., Gaussian and stable distributions. W choose such distribution for further analysis that has smaller value of the statistic. In Table 2 we can see calculated values of the KS statistic for a sample generated from a Lévy stable distribution with α = 1.85, σ = 1, β = µ = 0, for two possible choices of F (x), namely Gaussian and Lévy stable. The parameters of the distributions are maximum likelihood estimates [39]. Clearly, KS statistic for the stable distribution is much lower than the one calculated for the Gaussian case. In order to use Kolmogorov–Smirnov statistic for validation purposes, we have to construct a proper statistical test. The general test of fit is structured as follows. The null hypothesis is that a specific distribution is

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Table 2. Parameter estimates obtained via the maximum likelihood method and Kolmogorov–Smirnov test statistic for a sample generated from a L´ evy stable distribution with α = 1.85, σ = 1, β = µ = 0. The corresponding p-values based on 1000 simulated samples are given in parentheses. Distributions Parameters

Kolmogorov–Smirnov test

Gaussian

Stable

µ = 0.0123 σ = 2.4245 β = 0.0920

α = 1.8410 σ = 1.0048 µ = 0.0581

0.1239 ( 0 (H > 1/α), the character of the process changes to superdiffusive. What is even more amazing, it appears that Lévy α-stable processes for α < 2 can serve both as examples of subdiffusion and superdiffiusion. The subdiffusion pattern arises when the dependence is negative, so possible large positive jumps are quickly compensated by large negative jumps, and on average the process travels shorter distances than the light-tailed Brownian motion. 5. Sample p-variation Another method, which can be successfully applied to identify the type of subdiffusion in the experimental data, is the method of p-variation

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[7–9]. The idea of p-variation generalizes the well-established notions of total variation and quadratic variation, which have found applications in various branches of mathematics, physics and engineering, including: optimal control, numerical analysis of differential equations, and calculus of variations [46]. For a stochastic process X(t) observed on time interval [0, T ], the corresponding p-variation is defined as V (p) (t) = lim Vn(p) (t), n→∞

p > 0.

(15)

(p)

Here, Vn (t) is the sum of powers of increments of the process X(t) Vn(p) (t)

=

n 2 −1

j=0

X

p (j + 1)T jT ∧t −X ∧ t n n 2 2

(16)

(p)

with a∧b = min{a, b}. Vn (t) is called the sample p-variation. We underline (p) that Vn (t) is very easy to calculate numerically. For large enough n, the sample p-variation is a good approximation of V (p) (t). It appears that for each subdiffusion model (FBM, FLSM and FFPE) considered here, the p-variation displays completely different behavior. This interesting fact plays a crucial role in identifying the proper model of subdiffusion. The p-variation of the FBM BH (t) satisfies [47]  1  +∞ if p < ,    H    1 (p) V (t) = tE (|BH (1)|1/H ) if p = , (17)  H     1  0 if p > . H The expected value in the above formula equals E(|BH (1)|1/H ) = 1/2H σ1/H 1 √2 Γ( 2H + 12 ). Here, σ > 0 is the scale parameter of BH (t). Observe π that p = 1/H is the critical value in the above formula for FBM. Since the trajectories of the FLSM in the subdiffusive regime d < 0 (H < 1/α) are nowhere bounded, the p-variation corresponding to Lα H (t) is equal to infinity V (p) (t) = ∞ for any p > 0.

(18)

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The p-variation of the Langevin-type process Zβ (t) = B(Sβ (t)) corresponding to FFPE (3) satisfies [23]   +∞ (p) V (t) = Sβ (t)   0

if p < 2, if p = 2,

(19)

if p > 2.

Note that p = 2 is the critical value for the p-variation of Zβ (t). The above formulas (17)–(19) confirm that the p-variation of the discussed subdiffusion models differs considerably. This suggests the following p-variation test for distinguishing between FBM, FLSM and FFPE dynamics [7–9]: •

•

•

If the underlying model is FBM BH (t), then by (17) the sample p(p) variation Vn (t), as a function of n, should behave in the following (p) way: (a) for p < 1/H, Vn (t) should increase with increasing n; (b) for (p) p = 1/H, Vn (t) should behave as the linear function tE(|BH (1)|1/H ); (p) (c) for p > 1/H, Vn (t) should decrease with increasing n (see Fig. 3, (a) and (b)). If the underlying model is Zβ (t) corresponding to FFPE (3), then by (p) (19) the sample p-variation Vn (t), as a function of n, should behave in (p) the following way: (a) for p < 2, Vn (t) should increase with increasing (p) n; (b) for p = 2, Vn (t) should stabilize around Sβ (t); (c) for p > 2, (p) Vn (t) should decrease with increasing n (see Fig. 3, (c) and (d)). If the underlying model is FLSM in the subdiffusive regime d < 0, then (p) by (18) the sample p-variation Vn (t) should increase with increasing n for any parameter p > 0 (see Fig. 3, (e) and (f)).

In Fig. 3 we illustrate the behavior of sample p-variation for all three (p) subdiffusion models. The obvious differences in the behavior of Vn (t) corresponding to FBM, FLSM, and FFPE allow to identify the underlying subdiffusion mechanism. In practice, for a given time series {Xi , i = 1, 2, . . . , 2N } of experimental (p) observations, one needs to calculate the sample p-variation Vn (t) for various parameters p and n. Next, one compares the obtained results with the theoretical ones corresponding to FBM, FLSM, and FFPE, and chooses the best fitting model.

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x 104 (1/H)

Vn

2

15000 Vn(2)(t)

(t)

10000

1.5 1

5000

0.5 0 0

5000

10000

t 15000

0 0

(a) 4000

(1/H)

Vn

10000

t 15000

10000

t 15000

5000

(b) 1500 V (2)(t) n

(t)

3000

1000

2000 500

1000 0 0

5000

10000

t 15000

0 0

5000

(c)

(d)

4

x 10 15

(1/H)

Vn

(2)

10000 Vn (t)

(t)

10 5000 5 0 0

5000

10000

t 15000

(e) (1/H) Vn (t)

0 0

5000

10000

t 15000

(f) (2) Vn (t)

Fig. 3. Behavior of and corresponding to FBM ((a) and (b)), FFPE ((c) and (d)) and FLSM ((e) and (f)). The parameters are: H = β/2 = 0.35, α = 1.5. Notation used: x (n = 11), +(n=12), (n=13), o (n = 14).

6. Statistical Validation The p-variation test described in the previous section, specifically the comparison of empirical data with theoretical predictions and the classification into one of the three underlying scenarios (FBM, FLSM, FFPE), was done in a non-quantitative (visual) manner. Now, we give the p-variation method more formal and quantitative shape. To verify the agreement between experimental data and theory, we introduce a proper statistical test (a similar test was introduced in a recent paper [8]), which

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allows to classify the experimental data into one scenario or another using quantitative confidence level. First, for a given empirical data, we verify statistically if it originates from the FBM. To be more precise, our null hypothesis is that the data comes from BH (t) for given H. To verify such null hypothesis, we examine the value of the following statistic (1/H)

Dm,n =

|Vˆm

(1/H)

(T ) − Vˆn (1/H) Vˆn (T )

(T )|

.

(1/H) (1/H) Here, Vˆm (T ) and Vˆn (T ), m < n, are the values of sample 1/Hvariation calculated for the empirical data. Note that by (17), under the null hypothesis, the difference Dm,n should be small for large enough m and n. Therefore, the null hypothesis is rejected at confidence level α ˆ (usually α ˆ= 0.05 or α ˆ = 0.01), if

Dm,n > Dαˆ . The constant Dαˆ is found from (1/H) (1/H) |Vm (T ) − Vn (T )| P ˆ, ≤ Dαˆ = 1 − α (1/H) Vn (T ) (1/H)

(1/H)

where Vm (T ) and Vn (T ) are the values of sample 1/H-variation corresponding to FBM, i.e. n 1/H

2 −1 (1/H) BH (j + 1)T ∧ T − BH jT ∧ T (T ) = , Vn n n 2 2 j=0 (1/H)

and similarly Vm (T ). Note that the constant Dαˆ can be easily found via Monte Carlo techniques. The methods of simulating BH (t) can be found [48]. We applied the above statistical test to the simulated trajectories of BH (t), Zβ (t) and Lα H (t). The obtained p-values are shown in Fig. 4(a). Next, let us consider the null hypothesis that the data comes from FFPE. In such case the following statistic should be investigated Em,n =

(2) (2) |Vˆm (T ) − Vˆn (T )| . (2) Vˆn (T )

(2) (2) Here, Vˆm (T ) and Vˆn (T ), m < n, are the values of sample 2-variation corresponding to the empirical trajectory. Now, by (19), under the null

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1

1

0.8

0.8

p-value

p-value


0.6 0.4

343

0.6 0.4 0.2

0.2 0

0 FBM

FFPE

FLSM

FBM

FFPE

(a)

FLSM

(b)

Fig. 4. (a) Results of the introduced statistical test with the null hypothesis that the data comes from FBM. Shown are three boxplots, each with the p-values calculated for 100 simulated trajectories of FBM, FFPE and FLSM, respectively. The test accepted almost all the trajectories of FMB, whereas the median of p-values obtained for FFPE and FLSM is lower than 0.05. Recall that if the p-value is smaller than the confidence level α ˆ = 0.05, the null hypothesis should be rejected. (b) Results of the introduced test for the null hypothesis that the data comes from FFPE model Zβ (t). In this case almost all the simulated trajectories of Zβ (t) were accepted, while the median of p-values obtained for FBM and FLSM is lower than 0.05. The length of trajectories is 212 , number of trajectories equals 100 for each model, and the parameters are: H = β/2 = 0.4, α = 1.5, m = 9 and n = 12.

hypothesis the distance Em,n should be small for large enough m and n. Consequently, the null hypothesis is rejected at confidence level α ˆ , if Em,n > Eαˆ , where the constant Eαˆ is derived from (2) (2) |Vm (T ) − Vn (T )| P ˆ. ≤ Eαˆ = 1 − α (2) Vn (T ) (2)

(2)

Here Vm (T ) and Vn (T ) are the values of sample 2-variation corresponding to Zβ (t), i.e. Vn(2) (T )

=

n 2 −1

j=0 (2)

Zβ

(j + 1)T ∧T 2n

− Zβ

2 jT ∧ T , n 2

and similarly Vm (T ). Similarly as before, the constant Eαˆ should be determined via Monte Carlo techniques. The method of simulating Zβ (t) can be found [35, 49]. The above statistical test was applied to the simulated trajectories of BH (t), Zβ (t) and Lα H (t). The results are presented in Fig. 4(b).

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The above tests allow one to validate statistically that the data originates from FBM (or from FFPE). However, the same methodology cannot be applied in a straightforward manner to the case of FLSM. The reason is that for the last process, there is no critical value of p, for which the corresponding sample p-variation would stabilize. Thus, a different approach to this problem is necessary. Some steps in this direction are in progress. 7. The Case of Confined Systems The results and tests introduced in previous sections were done for the case of freely moving particles. The considered system had no boundaries and no external force was present. Such setting is convenient for a number of physical and biological systems, however, there are some cases for which confinement influences considerably the dynamics of particles. Therefore, it is of great interest to extend the statistical methods of identifying and validating the type of subdiffusion also to the case of confined systems. FBM in confinement is governed by the fractional Langevin equation of the form dYH (t) = F (YH (t))dt + dBH (t),

(20)

with F (x) being the external force. Similarly, confined subdiffusion in the framework of CTRW with heavytailed waiting times is described by the fractional Fokker–Planck equation [12, 33] ∂2 ∂F (x) ∂w(x, t) = 0 Dt1−β − + K 2 w(x, t). (21) ∂t ∂x ∂x The force F (x), which is assumed to be continuous, is related to the binding potential v(x) through F (x) = −v (x). The constant K denotes the anomalous diffusion coefficient. Equation (21) describes the evolution in time of the PDF w(x, t) of some subdiffusion process Wβ (t). This Langevintype process has the following representation [34–36] Wβ (t) = X(Sβ (t)),

(22)

where X(t) is the solution of the Itˆ o stochastic differential equation dX(τ ) = F (X(τ ))dτ + (2K)1/2 dB(τ )

(23)

driven by the standard Brownian motion B(τ ) and Sβ (t) is the inverse stable subordinator (5).

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In this section we restrict ourselves only to the above two models of subdiffusion (YH (t) and Wβ (t)). The case of FLSM in confinement requires further extensive studies. Now, our goal is to extend the previously introduced methods of identification based on p-variation to the case of confined systems described by (20) and (22). The crucial, and somewhat surprising fact is that the presence of external force does not modify the properties of p-variation. One can show [8] that the p-variation corresponding to YH (t) yields  1   +∞ if 0 < p < ,   H    1 (p) V (t) = tE(|BH (1)|1/H ) if p = ,  H     1  0 if p > . H

(24)

Analogously, one proves [8] that V (p) (t) corresponding to Wβ (t) is given by  if 0 < p < 2,  +∞ (p) V (t) = 2KSα (t) if p = 2,   0 if p > 2.

(25)

Moreover, formulas (24) and (25) are also valid for the case of finite system with reflecting boundaries. Comparing the above results with (17) and (19), we observe that indeed the confinement has no influence on the behavior of p-variation (however, (p) the shape of the potential may influence the speed of convergence of Vn (t) (p) to V (t)). Therefore, visual identification of the type of subdiffusion, based on the differences in the behavior of p-variation corresponding to FBM and FFPE models, follows exactly the same line as in Sec. 5. In additional, validation of the model via statistical hypothesis testing for the confined systems can be performed in the analogous manner as in the force-free seetting. Below, for completeness, we present the details of the testing procedures. Assume that the null hypothesis is that the data comes from YH (t). To verify such hypothesis one examines the statistic (1/H)

Dm,n =

|Vˆm

(1/H)

(T ) − Vˆn (1/H) Vˆn (T )

(T )|

.

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(1/H) (1/H) Recall that Vˆm (T ) and Vˆn (T ), m < n, are the values of sample 1/H-variation calculated for the empirical data. By (24), under the null hypothesis the difference Dm,n should be small for large enough m and n. Therefore, the null hypothesis is rejected at confidence level α ˆ if

Dm,n > Dαˆ . The constant Dαˆ is found from (1/H) (1/H) |Vm (T ) − Vn (T )| P ˆ, ≤ Dαˆ = 1 − α (1/H) Vn (T ) (1/H)

(1/H)

with Vm (T ) and Vn (T ) being the values of sample 1/H-variation corresponding to the process YH (t), i.e. Vn(1/H) (T )

=

n 2 −1

j=0

YH

(j + 1)T ∧T 2n

− YH

1/H jT ∧ T , n 2

(1/H)

and similarly Vm (T ). The constant Dαˆ can be found by the Monte Carlo method. If the null hypothesis is that the data comes from Wβ (t), one should investigate the statistic Em,n =

(2) (2) |Vˆm (T ) − Vˆn (T )| . (2) Vˆn (T )

(2) (2) Once again, Vˆm (T ) and Vˆn (T ), m < n, are the values of sample 2variation corresponding to the empirical trajectory. Now, by (25), under the null hypothesis the distance Em,n should be small for large enough m and n. Thus, the null hypothesis is rejected at confidence level α, ˆ if

Em,n > Eαˆ , where the constant Eαˆ is derived from (2) (2) |Vm (T ) − Vn (T )| ˆ. ≤ Eαˆ = 1 − α P (2) Vn (T ) (2)

(2)

Here, Vm (T ) and Vn (T ) are the values of sample 2-variation corresponding to Wβ (t), i.e. Vn(2) (T )

=

n 2 −1

j=0

Wβ

(j + 1)T ∧T 2n

− Wβ

2 jT , ∧ T 2n

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(2)

and similarly Vm (T ). The constant Eαˆ can be determined via Monte Carlo techniques. The method of simulating Wβ (t) can be found [35, 49]. The results of the above tests for the simulated trajectories of YH (t) and Wβ (t) are analogous to the ones presented in Sec. 6. The advantage of p-variation test is its universality – we do not need to know any information about the confinement of the system. Moreover, to perform the test only one sufficiently long trajectory is needed. 8. Conclusions We have described here the dynamics of the sample MSD and sample p-variation for CTRW process represented by the FFPE and for general Lévy stable processes, in particular, for a FBM and a FLSM. As a consequence, we constructed a new test which allows to identify the dynamics underlying the data and distinguish between three types of subdiffusive dynamics: FFPE, FBM and FLSM. This was done employing various statistical tests (Kolmogorov–Smirnov and FIRT) and introduced in this chapter sample MSD, and sample p-variation tests. We already showed [7–9] that some of the bacterial cytoplasm data [11] can be modeled by a FLSM or FBM with d < 0. We have also observed a similar effect for the data describing the epidermal growth factor receptor labeled with quantum dots in the plasma membrane of live cells [5]. We hope that the general statistical methodology proposed in this chapter will be useful in identification and validation of the appropriate fractional stochastic model behind the data. Acknowledgment The research of M.M. has been partially supported by the European Union within the European Social Funds. References X. S. Xie and H. P. Lu, J. Bio. Chem. 274, 15967 (1999). S. Weiss, Nature Struct. Biol. 7, 724 (2000). W. Moerner, J. Phys. Chem. B 106, 910 (2002). S. C. Kou, V. S. Xie and J. S. Lin, J. Roy. Stat. Soc. C 54, 469 (2005). A. Serge, N. Bertaux, H. Rigneault and D. Marquet, Nature Meth. 5, 687 (2008). 6. I. Bronstein, Y. Israel, E. Kepten, S. Mai, Y. Shav-Tal, E. Barkai and Y. Garini, Phys. Rev. Lett. 103, 018102 (2009).

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