Anomalous spreading and misidentification of spatial random walk models

Applied Mathematical Modelling 40 (2016) 5283–5291

Contents lists available at ScienceDirect

Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Anomalous spreading and misidentification of spatial random walk modelsR Reiichiro Kawai∗ School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

a r t i c l e

i n f o

Article history: Received 2 July 2014 Revised 30 October 2015 Accepted 14 December 2015 Available online 30 December 2015 Keywords: Correlated random walk Fractional Brownian motion Long-range dependence Mean-squared displacement Turning angles Super-diffusion

a b s t r a c t Anomalous diffusive spreading behaviors are often modeled with continuous-time random walks or fractional Brownian motion, while spatial directional preference is the key attribute of correlated random walks. In this paper, we construct and analyze a spatial random walk model with particular attention to the synthesis of anomalous diffusive behaviors and spatial directional preference, as well as statistical inference under discrete sampling. Anisotropic anomalous diffusive behaviors of the proposed model emerge from the long memory among step lengths, standing in contrast to little tangling diffusive trajectories of correlated random walk models, designed directly through the relative turning angles. With the help of the proposed model, we demonstrate difficulty in model identification among spatial random walk models and suggest that the identification procedure be planned and carried out with a great deal of caution. © 2015 Elsevier Inc. All rights reserved.

1. Introduction A random walk is a mathematical model consisting of a succession of random steps and is quite often employed to simplify a continuous path. In front-end practice, one is technically unable to monitor trajectories in a fully continuous manner, so a continuous spatial path is observed at discrete time points and the resulting positions at certain observation times are modeled with a suitable random walk model. There are numerous statistical properties of continuous spatial trajectories that a random walk model is desired to capture in macroscopic and microscopic manners. Those include finite or infinite variance, step lengths with light- or heavy-tailed distribution, either persistence, anti-persistence or no correlation in direction and in step length, etc. A variety of random walk models have been introduced and studied in many different fields with a view towards such statistical properties. Those models include correlated random walks [1,2], Lévy walks and flights [3–5], continuous-time random walks [6–9], and many other variations. In addition, based upon those models, the issue of interpretation of the given data has been widely investigated in, for instance, [3,10–15]. Among a variety of desired statistical properties of spatial trajectories, the directional preference and anomalous diffusive spreading behaviors have long attracted a great deal of interest, for instance, [16,17]. To capture the directional preference, the class of correlated random walk models has often been chosen as the first candidate. Correlated random walks consist of two major components in their modeling; one is a series of steps with either fixed or random lengths, while the other is a series of randomly distributed and correlated turning angles, between successive step vectors. Correlated random walk R ∗

The author would like to thank two anonymous referees for valuable suggestions. Tel.: +61 2 9351 3376; fax: +61 2 9351 4534. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.apm.2015.12.028 S0307-904X(15)00840-9/© 2015 Elsevier Inc. All rights reserved.

5284

R. Kawai / Applied Mathematical Modelling 40 (2016) 5283–5291

models can naturally exhibit the desired spatial directional preference directly by assigning an appropriate probability distribution, usually peaked around the zero angle, to the relative turning angles. As we will show rigorously later, correlated random walk models are not capable of producing non-diffusive spreading. In the natural world, the mean-squared displacement of the observed spatial trajectories does not necessarily display linear growth in time. Instead, it is often sub-linear, for instance, in amorphous solids [9] and polymeric systems [18], and exhibits super-linear growth in turbulent media [19]. Anomalous diffusive behaviors are often modeled with fractional Brownian motion or continuous-time random walks mainly in one-dimensional setting. On the one hand, two-dimensional fractional Brownian motions studied in the literature (for instance, [4,20,21]) have spatially isotropic trajectories, that is, no directional preference can be exhibited. On the other hand, the class of continuous-time random walks have been investigated quite intensively for its capability of realizing a variety of diffusive modes through heavy-tailed distributions and correlations within and between waiting times and jump lengths. When making a statistical inference for trajectories based upon their discrete-time observations, the randomness of waiting times makes the law of increments quite involved in general. It is often difficult to go beyond checking whether the empirical mean-squared displacement evolves as expected relative to its theoretical behavior when the observation window increases. In this paper, we construct and analyze a spatial discrete-time random walk model that exhibits both anomalous diffusive behaviors and spatial directional preference in a relatively simple manner. The main feature of the proposed model is to make use of the memory in step length with the help of fractional dependence among increments to generate anomalies, rather than incorporating directional memory into the relative turning angles as in correlated random walks. The proposed model is kept within the realm of discrete-time random walks so as to ease a statistical inference under realistic discrete sampling of trajectories. The proposed model exhibits different modes of the mean-squared displacement around the average trajectory, either asymptotically diffusive or super-diffusive, depending on the characterizing parameters. The property of different spatial diffusive modes may be quite counterintuitive, unlike in one dimensional modeling. The proposed model also serves as a concrete example of quite likely misidentification among spatial random walk models. In a similar manner to the well known difficulty in distinguishing between correlated random walks and Lévy flights [11,13,15], we demonstrate that, especially in situations where statistics are scarce, the proposed model can be confused easily with uncorrelated and correlated random walks. 2. Correlated random walk As a direct counterpart of the proposed model, let us briefly review a class of correlated spatial random walk models. Consider a random walk in two dimension, whose location after n steps is defined by,









  Zn,x cos (ξk ) := Vk := Zk . Zn,y sin (ξk ) n

n

k=1

k=1

(2.1)

Here, {Vk }k∈N indicate step vectors, {Zk }k∈N are nonnegative step lengths, while {ξk }k∈N are absolute turning angles. Throughout the paper, we assume that {Zk }k∈N and {ξk }k∈N are independent and that the step lengths {Zk }k∈N are independent and identically distributed (iid) nonnegative random variables. In order to introduce the correlation among steps to the framework of the simple spatial random walk model, the absolute turning angles {ξk }k∈N is formulated as a successive sum of relative turning angles {ηk }k∈N , which are iid random variables in (−π , +π ). Clearly, unless the relative turning angles are iid U (−π , +π ), the absolute turning angles {ξk }k∈N are correlated among themselves and the resulting random walk model is thus non-Markovian. With the relative turning angles {ηk }k∈N being iid U (−π , +π ), the correlated random walk reduces to a uncorrelated random walk and is then Markovian. Without assuming a particular distribution of the relative turning angles {ηk }k∈N , Kareiva and Shigesada [1] derived the asymptotic behavior of the mean-squared displacement,







 

2 2 E Zn,x + Zn,y ∼ n E Z12 +



E[cos(ξ1 )] (E[Z1 ] )2 , 1 − E[cos(ξ1 )]

(2.2)

where the asymptotic equivalence holds as n  +∞. This indicates that the correlated random walk spreads only in a diffusive manner. We can also show that for k1 > k2 ,

E[Vk1 , Vk2 ] −1 2 (E[Z1 ] )2 − k1 −k e 2 (ln a1 )  0, as k1 − k2  +∞,

≤ Var( Vk1 )Var( Vk2 ) Var(Z1 )

(2.3)

where a1 := E[cos(η1 )]2 + E[sin(η1 )]2 . (For completeness sake, we provide the derivation of (2.3) in Appendix A in a concise manner.) Importantly, the constant a1 is non-negative and strictly less than one, regardless of the distribution of the angle 2 η1 . For instance, if {ηk }k ≥ 1 are iid centered Gaussian N (0, λ2 ) for some λ > 0, then a1 = e−λ . Therefore, there may exist some directional dependence among steps, but still decaying exponentially fast. In the case of uncorrelated random walks, 2 + Z 2 ] = nE[Z 2 ], while the directional correlation (2.3) vanishes. the mean-squared displacement (2.2) is explicit E[Zn,x n,y 1 To quantify the degree of trajectory tangling, we estimate the average number of steps a trajectory requires for one  complete rotation, which we denote by T±2π indicating the expected first passage time of the series nk=1 ηk to the boundaries either +2π or −2π . Clearly, a large first passage time implies little tangling. As mentioned earlier, the setting of

R. Kawai / Applied Mathematical Modelling 40 (2016) 5283–5291

5285

Fig. 1. Typical trajectories with 50 0 0 steps of correlated random walks, with iid half-normal step lengths {Zk }k∈N and four different distributions for the relative turning angles {ηk }k∈N and the average rotation periods T±2π .

{ηk }k∈N ∼ U (−π , +π ) corresponds to a uncorrelated spatial random walk, while we set {ηk }k∈N ∼ N (0, λ2 ) for the corre-

lated spatial random walk model of this section. Although the Gaussian law is not entirely realistic for relative turning angles as the support of the law is unbounded, it does the job sufficiently for our purpose as most of the probability mass is concentrated on the circle (−π , +π ) if variance λ2 is very small. (In fact, high estimation accuracy is of minor importance for our purpose.) In Fig. 1, we present typical trajectories of the spatial random walk (2.1) where the step lengths {Zk }k∈N are iid with the



2 − z

half-normal density function (d/dz )P(Z1 ≤ z ) = 2/(π σ 2 )e 2σ 2 for z > 0 and some σ > 0, and four different distributions for the relative turning angles {ηk }k∈N , as indicated in the caption. Note that the half-normal distribution corresponds to the magnitude |Z| of a centered normal Z ∼ N (0, σ 2 ), thus it drops the sign of Z. As can be observed, the distribution of turning angles plays an essential role in forming spatial trajectories. The important point is, however, that as (2.2) indicates, the motion is still spatially diffusive. The average rotation periods T±2π indicate less tangling of correlated spatial random walk trajectories. 3. The model We are now ready to present the proposed model, where correlation among steps emerge not from the directional memory like in the correlated random walk of Section 2. We use the following framework, looking similar to (2.1), of spatial random walk:









  Zn,x cos (ζk ) := Vk := Wk , Zn,y sin (ζk ) n

n

k=1

k=1

(3.1)

where {Vk }k∈N indicate step vectors, {Wk }k∈N and {ζk }k∈N are independent as before. Here, {Wk }k∈N are centered fractional Brownian noise with Hurst index H ∈ (0, 1), that is, they are identically distributed centered (E[Wk ] = 0, k ∈ N) Gaussian, with autocovariance,

Cov(Wk1 , Wk2 ) =

σ2 2

(|k1 − k2 + 1|2H − 2|k1 − k2 |2H + |k1 − k2 − 1|2H ) =: σ 2CH (k1 − k2 ),

(3.2)

for k1 , k2 ∈ N, with σ > 0. Clearly, the function CH is even and defined on the set of all integers. Here, the parameter σ 2 indicates variance of a single step length due to Cov(W1 , W1 ) = Var(W1 ) = σ 2 . The fractional Brownian noise can be thought of as an equidistant discrete sample of a fractional Brownian motion [22]. For illustration purpose, we give the correlation matrices (CH (k1 − k2 ))1≤k1 ,k2 ≤3 of three adjacent increments, which are significantly distinct for different values of H;



1 0.414 0.269

H = 0.75

0.414 0.269 1 0.414 , 0.414 1

H = 0.50 1 0 0

0 1 0

0 0 , 1



1 −0.293 −0.048

H = 0.25

−0.293 −0.048 1 −0.293 , −0.293 1

where the numbers are rounded. When H = 0.5, the correlation matrix reduces to an identity, indicating that increments are independent of each other, obviously corresponding to the Brownian motion. However, if H > 0.5, increments are positively (and strongly) correlated and thus the trajectory tends to look persistent, while if H < 0.5, increments are negatively (but weakly) correlated and thus the trajectory tends to turn back anti-persistently. We keep the angles {ζk }k∈N in (3.1) iid and set them to take values only in the upper half circle (0, +π ]. (We may work on any half circle (θ , θ + π ] for any θ , which is nevertheless nothing more than a simple rotation of the coordinates.) Precise specification of their distribution on (0, +π ] turns out to be of minor importance for our purpose. In what follows, we call model (2.1) the fractional Brownian spatial random walk. Unless H = 0.5, the model is obviously non-Markovian.

5286

R. Kawai / Applied Mathematical Modelling 40 (2016) 5283–5291

Fig. 2. Absolute turning angles (left) and relative turning angles (middle) of typical first four steps. The absolute turning is taken always counterclockwise. Clockwise relative turning (only η4 above) should read a negative value. The right figure illustrates the unique decomposition into the step lengths {Wk }k∈N and the angles {ζk }k∈N .

There are two reasons for employing the notations {Wk }k∈N and {ζk }k∈N in formulation (3.1) on purpose, different from

{Zk }k∈N and {ξk }k∈N of the previous model (2.1);

(a) the random variables {Wk }k∈N here not only indicate the length but also contain the information of sign (+ or −) due to Gaussianity, unlike nonnegative iid step lengths {Zk }k∈N of the previous model (2.1), (b) the angles {ζk }k∈N take values only in (0, +π ], rather than the full circle (−π , +π ] for {ξk }k∈N in previous model (2.1) so that the sign of {Wk }k∈N can be preserved. On account of the restriction of the domain of the angles {ζk }k∈N to the half circle (0, +π ], all the elements {Wk }k∈N and {ζk }k∈N can be specified uniquely from any observations {(Zk,x , Zk,y )}k∈N through formulation (3.1). Note that even when observing a common set of {(Zk,x , Zk,y )}k∈N , the angles {ζk }k∈N specified through (3.1) are not necessarily identical to the absolute turning angles {ξk }k∈N in the sense of (2.1). In Fig. 2, we illustrate the relation among the absolute turning angles {ξk }k∈N , the relative turning angles {ηk }k∈N , and the model angles {ζk }k∈N . We now turn to the relevant statistical property of model (3.1). First, in the x- and y-directions, it is straightforward that the model has zero mean, that is,

E[Zn,x ] =

n 

E[Wk ]E[cos (ζk )] = 0,

E[Zn,y ] =

k=1

n 

E[Wk ]E[sin (ζk )] = 0.

k=1

Unlike the correlated random walk model, the mean-squared displacement here is not always linear in steps. In the x-direction, it holds that



Var(Zn,x ) = nσ Var(cos(ζ1 )) + n 2

2H

σ (E[cos(ζ1 )] ) ∼ 2

2

O(n2H ), if E[cos(ζ1 )] = 0 and H > 0.5, O ( n ), otherwise,

(3.3)

where the asymptotic equivalence holds as n ↑ +∞. Moreover, we have,



Var(Zn,y ) = nσ Var(sin(ζ1 )) + n 2

2H

σ (E[sin(ζ1 )] ) ∼ 2

2

O(n2H ), if H > 0.5, O ( n ), otherwise,

(3.4)

and the covariance,

Cov(Zn,x , Zn,y ) = nσ 2 E[cos (ζ1 ) sin (ζ1 )] + n2H

σ2 2

E[cos (ζ1 )]E[sin (ζ1 )],

(3.5)

which behaves like O(n2H ) when E[cos(ζ1 )] = 0 and H > 0.5, for instance. (For completeness sake, we provide the derivation of (3.3)–(3.5) in Appendix B in a concise manner.) As the turning angles {ζk }k∈N take values in the upper half circle (0, +π ], its sine sin (ζ 1 ) is always positive, while the expectation E[cos(ζ1 )] may be zero, in particular, when its distribution is symmetric about the origin, such as {ζk }k∈N ∼ U (0, +π ). In light of (3.3) and (3.4), let us summarize the three distinct behaviors; (1) if H > 0.5 and E[cos(ζ1 )] = 0, then the motion is super-diffusive along the y-axis, while only diffusive along the x-axis, (2) if H > 0.5 and E[cos(ζ1 )] = 0, then the motion is super-diffusive in both the x- and y-directions in the long run, (3) if H ≤ 0.5, then the motion is spatially diffusive. (In particular, if H = 0.5, then all steps are mutually independent in both length and direction.) It is noteworthy that the short-range behavior (that is, small n) is determined through the competition between coefficients of two terms, for instance, Var(sin (ζ 1 )) and (E[sin(ζ1 )] )2 in (3.4). In the extreme case of Var(sin(ζ1 ))  (E[sin(ζ1 )] )2 , the second term (with n2H ) will dominate its short-range spreading behavior, which may look sub-diffusive, that is, O(n2H ) with 2H < 1.

R. Kawai / Applied Mathematical Modelling 40 (2016) 5283–5291

5287

Fig. 3. Typical trajectories with 50 0 0 steps (top) and mean-squared displacement (bottom) of the spatial fractional Brownian random walk with three different Hurst indices: H = 0.75, H = 0.5 and H = 0.25. The x- and y-axes of the top figures are displayed in the same scale. The average rotation periods T±2π are all Monte Carlo estimates.

We can show that the dimensionless directional correlation, similar to (2.3), is given by,



E[Vk1 , Vk2 ] Var( Vk1 )Var( Vk2 )

2(H−1 )

∼ a2 H (2H − 1 )|k1 − k2 |

,

|k1 − k2 | ↑ +∞,

(3.6)

with a positive constant a2 , depending only on the distribution of the turning angles {ζk }k∈N . Note that the right-hand side of (3.6) is positive if H > 0.5, while negative if H < 0.5. This polynomial decay of the directional correlation is much slower than the exponential decay (2.3) of the correlated random walk model. In fact, the dependence (3.6) is often said to be  long-ranged if H > 0.5, in the sense of limn↑+∞ nk=1 |CH (k − 1 )| = +∞. (This limit is however finite if H < 0.5, although the decay is still polynomial. The dependence is then said to be short-ranged.) In Fig. 3, we present typical trajectories with 50 0 0 steps and their mean-squared displacement of the fractional Brownian spatial random walk with three different Hurst indices: H = 0.75, 0.5 and 0.25. The mean-squared displacements are plotted against the number of steps separately in the x and y directions, along with the respective theoretical values Var(Zn, x ) and Var(Zn, y ) given in (3.3) and (3.4). (We did not plot the covariance (3.5) to avoid overload the figures with negative values.) Generation of a trajectory (3.1) consist of {Wk }k=1,...,N and {ζk }k=1,...,N . Generation of the latter sequence is straightforward. For {Wk }k=1,...,N , we need to generate N (= 50 0 0 ) standard normal random variables, construct the correlation matrix (3.2) in RN×N and its Cholesky decomposition. With the help of the Toeplitz structure of the correlation matrix (3.2), the construction and Cholesky decomposition of such a large matrix can be performed quickly on recent ordinary computers. With no surprise, the model with H = 0.5 and iid uniform absolute angles (no directional preference) reduces to an isotropic spatial random walk and looks just like an isotropic two-dimensional Brownian motion. Even with iid uniform absolute turning angles, the model with H = 0.5 may yield either diffusive or super-diffusive movements along different axes due to (3.3) and (3.4), depending on the characterizing parameters. This anomalous diffusive behavior is very difficult to foresee at the modeling stage as its origin is substantially different from the well known ones of super-diffusion, such as a short range observation [15] or (very) strong directional persistence [2] of correlated random walks, quick laminar motion of a Lévy flight (when every step is assumed to spend a common amount of time), or short waiting time in the continuous-time random walk model [23]. The trajectory from H = 0.75 (the top left two of Fig. 3) is super-diffusive and look strongly biased along one axis. Such trajectory patterns may look too peculiar in some fields of applications, while they may be rather natural as animal movements on forcibly directed areas in the search problem, for instance, deers and wolves moving along a flat ridge or a river valley. To the best of our knowledge, there are no spatial random walk models for such directional patterns without reflection operations. Even correlated random walks may display trajectories with strong directional bias (just like the left two of Fig. 3), while correlated random walk trajectories are only diffusive, no matter how they look, as result (2.2) rigorously asserts. In fact, both CRW and FBM trajectories may tend to look ballistic, respectively, by vanishing relative turning angles and, for instance, by taking the limit H → 1 with E[cos(ζ1 )] = 0. Again, the former is still diffusive, while the latter is super-diffusive due to (3.3) and (3.4). For further comparisons to the previous models, we have indicated in Fig. 3 the average rotation periods T±2π , which are all Monte Carlo estimates. Those times indicate that the proposed model tends to generate more tangling trajectories and is much closer to the uncorrelated spatial random walk in this respect, than to the correlated spatial random walk. This fact gives a relevant conclusion that as discussed earlier, little trajectory tangling (as in correlated random walks) is required neither for directional preference nor for super-diffusive spreading.

5288

R. Kawai / Applied Mathematical Modelling 40 (2016) 5283–5291

Fig. 4. Typical trajectories of four random walk models consisting of 50 0 0 steps, with the respective histogram of the 50 0 0 relative turning angles {ηk }k∈N .

4. Misidentification of spatial random models With the aid of the proposed model, we next demonstrate that very different types of spatial random walks may be confused with each other easily. In Fig. 4, we present typical trajectories consisting of 50 0 0 steps of four different types of random walk models; (a) the uncorrelated spatial random walk (2.1) with {ηk }k∈N ∼ U (−π , +π ), (b) the correlated spatial random walk (2.1) with {ηk }k∈N ∼ N (0, 0.22 ), (c) the fractional Brownian spatial random walk (3.1) with H = 0.25 and E[cos(ζ1 )] = 0, and (d) the fractional Brownian spatial random walk (3.1) with H = 0.75 and E[cos(ζ1 )] = 0. (We will use the abbreviation “CRW” and “FBW” for the correlated spatial random walk and the fractional Brownian spatial random walk, respectively. In the case of no correlation in CRW or H = 0.5 in FBW, we use “RW” for further short.) Then, we decompose those four random walk trajectories into the step lengths {Zk }k∈N and the relative turning angles {ηk }k∈N , as in Fig. 2, as if we were observing trajectories under equidistant discrete sampling. The information given in Fig. 4 are a typical set used for statistical analysis for spatial random walk models. The data on the proposed model however cast doubts on the conventional interpretation procedure. First, we compare FBWs with CRW. With a large number of data points, like 50 0 0 steps in Fig. 3, their trajectories do not look alike. Nonetheless, such a large number of observations may not be available all the time. Without a sufficiently large sample, the estimation quality for directional correlation is most likely poor, and thus it will be difficult to tell whether the decay is exponential (2.3) or polynomial (3.6). Also, when the given sample is not large enough to ensure the asymptotic equivalence in (3.3) and (3.4), the mean-squared displacement may remain in the diffusion regime even when H > 0.5. This is consistent with the results of [15] that correct identification can only be possible when the time series is long enough. If the observer fails to check the histogram of relative turning angles, the FBW with H > 0.5 could be mistaken for some CRW. Even the distinction between FBWs and RW is not very straightforward. Depending on available sample size, they could really resemble each other. In addition, trajectories may not only look alike but also exhibit similar spatial diffusive properties, due to (3.3) and (3.4). If the observer fails to notice some (anti-)persistence at small time lags along the y-axis, the FBWs may be misidentified undoubtedly as an RW. Also, regardless of the parameter H, relative turning angles look iid uniform in the naïve histograms. To make a convincing statistical inference, it is necessary to go beyond the histogram of relative turning angles, that is, one needs to look at the time series of relative angles. On the one hand, for the sample path with H = 0.25 in Fig. 4, we find clear term structure;









−0.0132 − 0.8054 cos(ηk−1 ) + 0.7084 kc + 0.4820 kc−1 cos(ηk ) = , sin(ηk ) 0.0271 + 0.0431 sin(ηk−1 ) + 0.6882 ks

respectively, ARMA(1,1) and AR(1), where { kc }k∈N and { ks }k∈N are independent sequences of iid standard normal random variables, while for H = 0.75, we find









0.3441 + 0.1766 cos(ηk−1 ) + 0.1208 cos(ηk−2 ) + 0.6667 kc cos(ηk ) = , sin(ηk ) −0.0327 − 0.1753 sin(ηk−1 ) + 0.6313 ks

respectively, AR(2) and AR(1). On the other hand, as a matter of course, our estimates for RW indicate ARMA(0,0), as









0.0135 + 0.6878 kc cos(ηk ) = , sin(ηk ) 0.0295 + 0.7252 ks

R. Kawai / Applied Mathematical Modelling 40 (2016) 5283–5291

5289

that is, no dependence on the past. We note that analyzing the time series of the raw angles {ηk }k∈N (rather than their trigonometric transforms) yields very misleading results, for instance, because two angles ηk = ±(π − 10−10 ) are taken as very far apart numbers, although they represent very similar angles in reality. 5. Contrast with existing random walk models We have constructed and analyzed the fractional Brownian spatial random walk model with particular attention to the synthesis of anomalous diffusive behaviors and spatial directional preference. Here, we brief contrast and connect the proposed model with other existing random walks of seemingly similar features, from both probabilistic and statistical point of view. Two-dimensional versions of the fractional Brownian motion have been studied by several authors (for instance, [4,20,21]), as a natural generalization of the univariate fractional Brownian motion. In principle, those boil down to stochastic processes with iid components, each of which is a univariate fractional Brownian motion with a common Hurst parameter. Due to the Gaussianity, the resulting multivariate trajectories are necessarily isotropic in law and thus exhibit no directional preference. Hence, once trajectories are projected onto two perpendicular axes, one may directly apply a variety of existing statistical methodologies for the univariate fractional Brownian motion ([10,24,25] and references therein). In fact, it is rather easy to realize directional preference by bundling two independent fractional Brownian motions with different Hurst indices along two axes. One can then observe different diffusive modes in different directions, whereas its physical motivation and interpretation do not seem as clear as of the proposed model. From a statistical standpoint, due to the lack isotropy, the joint distribution of discrete observations of its multivariate trajectory seems no longer tractable. The class of continuous-time random walks has attracted an overwhelming deal of attention [6–9,26–29] for its capability of realizing three diffusive modes, as well as its close relation to the fractional Fokker–Planck equation and, in special cases, to the fractional Brownian motion. In principle, continuous-time random walks behave super-diffusive in the long run when each jump tends to be large, whereas sub-diffusion when each jump tends to wait extremely long after its previous jump. In the literature, there have been numerous further studies on the distributions of jump sizes and waiting times and the correlations within (for instance, [28,29]) and/or between them (for instance, [30–32]), as well as the corresponding fractional Fokker–Planck equation. A primary difference from the proposed model is that, with some exceptions, the primary interest of continuous-time random walks lies in the one-dimensional diffusive modes, not in the multidimensional directional preference. When making a statistical inference for trajectories based upon their discrete-time observations (X0 , Xt1 , . . . ), their joint distribution is generally intractable due to the randomness and correlation in waiting times and jump sizes. One exception is the three-dimensional random walk model with correlated jump lengths and constant inter-jump time [29], which falls directly into the framework of discrete sampling. It has the mean-squared displacement of order t3 in the long run (in our context, H = 3/2), whereas its multivariate trajectories are kept isotropic. The usual practice is thus to check whether the empirical mean-squared displacement evolves as expected relative to its theoretical behavior when the observation window increases. It seems out of reach as yet to go beyond that realm, such as the rigorous maximum likelihood analysis, the local asymptotic normality, and the hypothesis testing for anomalous diffusion indices, all of which are certainly important topics to address. 6. Concluding remarks In this paper, we have developed the spatial random walk model that spreads in diffusive and super-diffusive manners along either one of or both axes. Turning angles of the proposed model are simply isotropic, quite far from the concept of the relative turning angles which is the key building block of the correlated random walk model. We have demonstrated that by directing steps essentially independently and isotropically, the univariate fractional Gaussian dependence may induce various modes of spatial spreading behaviors, depending on the characterizing parameters. As opposed to the key feature of correlated random walks, it is demonstrated with the proposed model that a tangling trajectory can still display strong directional preference and super-diffusive spreading. Analysis of the proposed model demonstrates that the absence of prior knowledge may pose great challenges in identification of the correct random walk model from spatial observations. Unlike in one dimension, spatial random walk models may be mistaken for other types so easily that the identification procedure should be planned and carried out in a very careful manner. We have demonstrated that the average rotation period and time series analysis of relative angles play an important role for a convincing statistical inference. In fact, our finding does not only urge a careful treatment in the identification, but also suggests not to restrict attention to correlated random walks and Lévy flights only in the misidentification issue. This finding suggests to expand the candidate pool of random walk models, not only correlated random walks, continuous-time random walks and Lévy flights, in the misidentification issue. In addition, the data sampling frequency and the total observation window are known to act as very important factors of model misidentification. In particular, the availability of large amount of high resolution data, thanks to recent developments in the methodology and equipment for field and laboratory experiments, poses a considerable challenge for data analysis. At different sampling frequencies, the results from the analysis of the random walk data could be so different that standard statistical tools may fail to distinguish between the two qualitatively different spatial movement patterns. As such, there remain a variety of open issues in interpretation of spatial trajectories.

5290

R. Kawai / Applied Mathematical Modelling 40 (2016) 5283–5291

A. Derivations of formula for correlated spatial random walk Formula (2.3) for the correlated spatial random walk model in Section 2 can be derived by,



E[Vk1 , Vk2 ] Var( Vk1 )Var( Vk2 )

=

( E[Z1 ] )2 E[cos(ξk1 − ξk2 )] Var(Z1 )

=

k1 −k2 ( E[Z1 ] )2 (E[cos(η1 )]2 +E[sin(η1 )]2 ) 2 cos((k1 −k2 )atan 2(E[sin(η1 )], E[cos(η1 )] )), Var(Z1 )

where we have used Var( Vk ) = Var(Z1 ), k ∈ N,

E[Vk1 , Vk2 ] = E[Zk1 Zk2 ]E[cos(ξk1 ) cos(ξk2 ) + sin(ξk1 ) sin(ξk2 )] = (E[Z1 ] )2 E[cos(ξk1 − ξk2 )], and the Euler formula. Finally, by the Cauchy–Schwarz inequality, we have,

E[cos(η1 )]2 + E[sin(η1 )]2 < E[cos2 (η1 )] + E[sin (η1 )] = 1, 2

where the equality never holds here. B. Derivations of formulas for fractional brownian spatial random walk We here derive formulas (3.3)–(3.6) for the fractional Brownian spatial random walk in Section 3. Along the x-axis, it holds that,

⎡

Var(Zn,x ) = E⎣

n 

2 ⎤

Wk cos (ζk )



n 

⎦=E

k=1

=

n 



 

E Wk2 E (cos (ζk ) )

2



+2

k=1

= σ2

(Wk cos (ζn ) )2 + 2

k=1

n k 1 −1 

Wk1 Wk2 cos







ζk1 cos ζk2



k1 =2 k2 =1

n k 1 −1 



 

E Wk1 Wk2 E cos



    ζk1 E cos ζk2

k1 =2 k2 =1 n 



n k 1 −1   2 2 E (cos (ζk ) ) + 2σ 2 (E[cos (ζ1 )] ) CH (k1 − k2 )

k=1

k1 =2 k2 =1

= nσ Var(cos (ζ1 ) ) + n

σ (E[cos (ζ1 )] ) ⎧ 2 2 H 2 ⎨n σ (E[cos (ζ1 )]) , if E[cos (ζ1 )] = 0 and H > 1/2, ∼ nσ 2 E (cos (ζ1 ) )2 , if H = 1/2, ⎩ 2 nσ Var(cos (ζ1 ) ), otherwise, 2

2H

2

2

 where the asymptotic equivalence holds as n ↑ +∞ and we have used the formula nk ,k =1 CH (k1 − k2 ) = n2H . The variance 1 2 in the y-direction and the covariance between two axes can be derived in a similar manner as,

⎧ ⎨n2H σ 2(E[sin (ζ1 )])2 , if H > 1/2, 2 2 2H 2 Var(Zn,y ) = nσ Var(sin (ζ1 ) ) + n σ (E[sin (ζ1 )] ) ∼ nσ 2 E (sin (ζ1 ) )2 , if H = 1/2, ⎩ 2 nσ Var(sin (ζ1 ) ), if H < 1/2.

and

Cov(Zn,x , Zn,y ) = E[Zn,x Zn,y ] =

n  n 



E Wk1 Wk2 cos



   ζk1 sin ζk2

k1 =1 k2 =1

=

n  n 



 

E Wk1 Wk2 E cos



   ζk1 sin ζk2

k1 =1 k2 =1

= E[cos (ζ1 ) sin (ζ1 )]

n 





E Wk2 +

k=1

= nσ 2 E[cos (ζ1 ) sin (ζ1 )] + n

n  



 

E Wk1 Wk2 E cos



    ζk1 E sin ζk2

k1 =k2 k2 =1 2 2H σ

2

E[cos (ζ1 )]E[sin (ζ1 )].

Corresponding to the variance Var(Zn, x ), the covariance Cov(Zn, x , Zn, y ) behaves like,



O(n2H ), if E[cos (ζ1 )] = 0 and H > 1/2, O(n ), if H = 1/2 and if either E[cos (ζ1 ) sin (ζ1 )] = 0 or E[cos (ζ1 )] = 0, 0, if E[cos (ζ1 ) sin (ζ1 )] = E[cos (ζ1 )] = 0.

R. Kawai / Applied Mathematical Modelling 40 (2016) 5283–5291

5291

Finally, it holds that,



E[Vk1 , Vk2 ] Var( Vk1 )Var( Vk2 )

=

σ 2CH (k1 − k2 ) E[cos(ζk1 − ζk2 )] = a2CH (k1 − k2 ), σ2

where we have used Var( Vk ) = Var(W1 ) = σ 2 , k ∈ N,

E[Vk1 , Vk2 ] = E[Wk1 Wk2 ]E[cos(ζk1 ) cos(ζk2 ) + sin(ζk1 ) sin(ζk2 )] = a2 σ 2CH (k1 − k2 ). Here, a2 = (E[cos(ζ1 )] )2 + (E[sin(ζ1 )] )2 > 0, which is independent of k1 and k2 as ζk1 and ζk2 are iid. Finally, with the

help of the Taylor expansion, up to the quadratic term, (1 + x )b ≈ 1 + bx + b(b − 1 )x2 /2, for x in a neighborhood of the origin, we get the asymptotic equivalence CH (k1 − k2 ) ∼ H (2H − 1 )|k1 − k2 |2(H−1 ) , as |k1 − k2 | ↑ +∞. References [1] P.M. Kareiva, N. Shigesada, Analyzing insect movement as a correlated random walk, Oecologia 56 (2) (1983) 234–238. [2] N. Kumar, U. Harbola, K. Lindenberg, Memory-induced anomalous dynamics: emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model, Phys. Rev. E 82 (2010) 021101. [3] B.Q. Ai, Z.G. Shao, W.R. Zhong, Rectified Brownian transport in corrugated channels: fractional Brownian motion and Lévy flights, J. Chem. Phys. 137 (17) (2012) 174101. [4] A.M. Reynolds, Scale-free animal movement patterns: Lévy walks outperform fractional Brownian motions and fractional Lévy motions in random search scenarios, J. Phys. A: Math. Theor. 42 (43) (2009) 434006. [5] M.F. Shlesinger, G.M. Zaslavsky, U. Frisch (Eds.), Lévy Flights and Related Topics in Physics, Springer, Berlin, 1995. [6] F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Fractional calculus and continuous-time finance II: the waiting-time distribution, Phys. A: Stat. Mech. Appl. 287 (3–4) (20 0 0) 468–481. [7] J. Nelson, Continuous-time random-walk model of electron transport in nanocrystalline TiO2 electrodes, Phys. Rev. B 59 (1999) 15374–15380. [8] H. Scher, G. Margolin, R. Metzler, J. Klafter, B. Berkowitz, The dynamical foundation of fractal stream chemistry: the origin of extremely long retention times, Geophys. Res. Lett. 29 (5) (2002) 1061. [9] H. Scher, E.W. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B 12 (6) (1975) 2455–2477. [10] R. Kawai, S. Petrovskii, Multiscale properties of random walk models of animal movement: lessons from statistical inference, Proc. R. Soc. A 468 (2141) (2012) 1428–1451. [11] R. Kawai, Sampling rate of spatial stochastic processes with independent components in modeling random search paths, Phys. Rev. E 85 (2) (2012) 021907. [12] R. Kawai, Continuous-time modeling of random searches: statistical properties and inference, J. Phys. A: Math. Theor. 45 (23) (2012) 235004. [13] M.J. Plank, E.A. Codling, Sampling rate and misidentification of Lévy and non-Lévy movement paths, Ecology 90 (12) (2009) 3546–3553. [14] G. Rosser, A.G. Fletcher, P.K. Maini, R.E. Baker, The effect of sampling rate on observed statistics in a correlated random walk, J. R. Soc. Interface 10 (2013) 20130273. [15] G.M. Viswanathan, E.P. Raposo, F. Bartumeus, J. Catalan, M.G.E. da Luz, Necessary criterion for distinguishing true superdiffusion from correlated random walk processes, Phys. Rev. E 72 (2005) 011111. [16] R. Barkana, On correlated random walks and 21-cm fluctuations during cosmic reionization, Mon. Not. R. Astron. Soc. 376 (4) (2007) 1784–1792. [17] E. Koscielny-Bunde, A. Bunde, S. Havlin, H.E. Roman, Y. Goldreich, H.J. Schellenhuber, Indication of a universal persistence law governing atmospheric variability, Phys. Rev. Lett. 81 (3) (1998) 729. [18] S.C. Weber, J.A. Theriot, A.J. Spakowitz, Subdiffusive motion of a polymer composed of subdiffusive monomers, Phys. Rev. E 82 (2010) 011913. [19] T.H. Solomon, E.R. Weeks, H.L. Swinney, Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow, Phys. Rev. Lett. 71 (1993) 3975–3978. [20] H. Qian, G.M. Raymond, J.B. Bassingthwaighte, On two-dimensional fractional Brownian motion and fractional Brownian random field, J. Phys. A: Math. Theor. 31 (28) (1998) L527. [21] D. Saupe, Algorithm for random fractals, in: H.-O. Peitgen, D. Saupe (Eds.), The Science of Fractional Images, Springer New York, 1988, pp. 71–136. [22] B. Mandelbrot, J.W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (4) (1968) 422–437. [23] G. Zumofen, J. Klafter, Laminar-localized-phase coexistence in dynamical systems, Phys. Rev. E 51 (3) (1995) 1818–1821. [24] R. Kawai, Fisher information for fractional Brownian motion under high-frequency sampling, Commun. Stat. – Theory Methods 42 (9) (2013) 1628– 1636. [25] R. Kawai, On singularity of Fisher information matrix for stochastic processes under high frequency sampling, in: A. Cangiani, et al. (Eds.), Numerical Mathematics and Advanced Applications 2011, Springer-Verlag, 2013, pp. 841–849. [26] A.V. Chechkin, M. Hofmann, I.M. Sokolov, Continuous-time random walk with correlated waiting times, Phys. Rev. E 80 (3) (2009) 031112. [27] M.M. Meerschaert, E. Erkan Nane, Y. Xiao, Correlated continuous time random walks, Stat. Prob. Lett. 79 (9) (2009) 1194–1202. [28] J.H.P. Schulz, A.V. Chechkin, R. Metzler, Correlated continuous time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics, J. Phys. A: Math. Theor. 46 (47) (2013) 475001. [29] V. Tejedor, R. Metzler, Anomalous diffusion in correlated continuous time random walks, J. Phys. A: Math. Theor. 43 (8) (2010) 082002. [30] J. Liu, J.-D. Bao, Continuous time random walk with jump length correlated with waiting time, Phys. A: Stat. Mech. Appl. 392 (4) (2013) 612–617. [31] L. Shi, Z. Yu, Z. Mao, A. Xiao, H. Huang, Space-time fractional diffusion equations and asymptotic behaviors of a coupled continuous time random walk model, Phys. A: Stat. Mech. Appl. 392 (23) (2013) 5801–5807. [32] V.Y. Zaburdaev, Random walk model with waiting times depending on the preceding jump length, J. Stat. Phys. 123 (4) (2006) 871–881.