The power spectral density (PSD) of a stochastic process , which is formally defined as (see, e.g., Ref.def )
provides important insights into the spectral content of . Here and in what follows, the symbol denotes averaging with respect to all possible realizations of the process, i. e., the expectation. Often the PSD as defined in Eq. (1) has the form , where is an amplitude and , (typically, one has mike ), is the exponent characteristic of the statistical properties of
. In experiments and numerical modeling, the PSD has been determined using a periodogram estimate for a wide variety of systems in physics, biophysics, geology etc. A by far non-exhaustive list of systems exhibitingspectra includes electrical signals in vacuum tubes, semiconductor devices, metal films and other condensed matter systems 1 ; 2 ; 3 , quantum dots 4 ; 5 ; 14 , nano-electrodes 6 and two-dimensional graphene layers with a widely tunable concentration of carriers 69 . The PSD has been analysed, as well, for the trajectories of tracers in artificial crowded fluids weiss , for active micro-rheology of colloidal suspensions 66 , Kardar-Parisi-Zhang interface fluctuations 77 , for sequences of earthquakes 7 , weather data 8 ; talk1 ; talk2 , biological evolution 9 , human cognition 10 , network traffic 11 and even for the loudness of music recordings (see, e.g., Refs. 777 ; 888 ).
On the theoretical side, the PSDs showing the dependences have been calculated analytically for diverse situations, including, e.g., the dynamics in chaotic Hamiltonian systems 999 , periodically-driven bistable systems han , fluctuations of the phase-separating interface ale , several diffusive and non-diffusive transport processes (see, e.g., Refs. 14 ; 15 ; 150 ; 16 ; 17 ), the running maximum of Brownian motion 29 , diffusion in presence of strong quenched disorder 30 ; 31 ; 32 ; 33 and the electric-field-driven ion transport through nanometer-scale membrane pores 333 . The PSD of Brownian motion in an optical trap has been scrutinized in Refs. flyv ; sch , which permitted the calibration of optical tweezers, making them a powerful tool for force spectroscopy, local viscometry, and other applications (see, e.g. Ref. chris ; lenerev ).
The systematic measurements of single particle trajectories started with Perrin more than a century ago perrin . Just a few years later Nordlund nordlund developed impressive experimental techniques, followed by a line of further refinements up to Kappler’s measurements kappler , to record sufficiently long trajectories to enable quantitative analysis on the basis of individual trajectories—without the need of prior ensemble averaging. Nowadays, with the advent of modern microscopy and supercomputing, scientists routinely measure long trajectories of submicron tracer particles or even single molecules weiss ; phystoday ; tak ; 18 ; 181 ; 19 ; lenerev .
In parallel to this experimental progress, there was a general shift of interest towards understanding statistical properties of individual realizations of stochastic processes. In particular, a conceptually important question often raised within this context is if one can reliably extract information about the ensemble-averaged properties of random processes from single-trajectory data phystoday . Considerable theoretical progress has been achieved, for instance, in finding the way to get the ensemble-averaged diffusion coefficient from a single Brownian trajectory, which task amounts to seeking properly defined functionals of the trajectory which possess an ergodic property (see, e. g., Refs. 18 ; 181 ; 19 ; phystoday ; 20 ; 201 ; 2011 ; 21 ; 210 ; 22 ; 23 ; 24 ; norre for a general overview).
where the bar denotes time averaging and the lag time acts as the size of the analysis window sliding over the time series . For finite observation time , the time averaged MSD varies randomly for different trajectories even under identical physical conditions. For both normal and anomalous diffusion this variation is mainly seen as an amplitude scatter of at given lag times, which remains remarkably constant (apart of relatively small local fluctuations) as function of pccp ; thiel ; jeon ; ghosh .
For the time averaged MSD the fluctuations are often quantified in terms of the so-called ergodicity breaking parameter , where the dimensionless variable for a given measures the deviations from the trajectory-average he ; pccp , with being the number of observed trajectories. Clearly, with being the coefficient of variation of the distribution .
For Brownian motion, the fluctuations decrease with growing observation time such that 181 ; 2011 ; pccp , which permits to deduce the diffusion coefficient specific to an ensemble of trajectories from just a single, sufficiently long, trajectory. A similar decay of EB to zero for large is observed for fractional Brownian motion deng and scaled Brownian motion sbm . However, for anomalous diffusion with scale-free distributions of waiting times he and processes with systematic, spatially varying diffusion coefficient hdp the inequality persists even in the limit , which reveals ageing in the sense that the properties of the system, such as the effective diffusivity, are perpetually changing during the measurement and depend on phystoday ; pccp ; johannes . In these systems, the fluctuations as measured by do not decay to zero with increasing , and thus time averages of physical observables remain random quantities, albeit with well defined distributions (see, e.g. Refs. 2011 ; he ; pccp ). Many other facets of the time averaged MSD in Eq. (2) and of the parameter were exhaustively well studied to date for a variety of diffusive processes, providing a solid mathematical framework for the analysis of individual trajectories pccp ; lenerev ; igorrev ; hoefling ; deng ; maria ; bau .
Single-trajectory PSDs have been studied in several cases. In particular, the PSD of the loudness of musical recordings was discussed in Refs. 777 ; 888 and the spectra of temperature data were presented in Refs. talk1 ; talk2 . We note that, of course, for these two examples averaging over ”an ensemble” does not make any sense since the latter simply does not exist, likewise, e.g., the case of the financial markets data for which the single-trajectory MSD has been recently analyzed money . Further on, power spectra of individual time-series were examined for a two-state stochastic model describing blinking quantum dots 15 and also for single-particle tracking experiments with tracers in artificially crowded fluids weiss . It was shown that, surprisingly enough, the estimation of the exponent characterising the power spectrum of an ensemble from the single-trajectory PSD is rather robust. Next, in Refs. ameb ; ameb2 the power spectra of the velocities of independent motile amoeba (see also Refs. tak ; bod ) were analysed, revealing a robust high-frequency asymptotic form, persisting for all recorded individual trajectories.
Clearly, these numerical observations raise several challenging questions. Indeed, could it be possible that the exponent characterising the frequency-dependence of the standard PSD, defined as an ensemble-average property, can be observed already on a single-trajectory level? If true, does it hold for any stochastic transport process or just for some particular examples? Moreover, in which range of frequencies can such a behavior be observed? What are the distributions of the amplitudes entering the relation between a single-trajectory PSD and its ensemble-average counterpart and how broad are they? Evidently, numerical analyses may give a certain degree of understanding of some particular features, but a deep insight can be obtained only in conjunction with a full mathematical solution. Unlike the single-trajectory MSD, for which a deep knowledge has been already acquired via an exact analytical analysis, a similar analysis of the statistical properties of a single-trajectory PSD is lacking at present, although its ensemble-averaged counterpart is widely used as an important quantifier of different properties of random trajectories in diverse areas of engineering, physics and chemistry.
In this paper, going beyond the text-book definition (1), we concentrate on the question what information can be reliably obtained if one defines the PSD of a single, finite-time realization of
. We here focus on the paradigmatic process of Brownian motion (BM). This choice is two-fold: first, Brownian motion is ubiquitous in nature which renders this analysis particularly important. Second, it permits us to obtain an exact mathematical solution of the problem: we calculate exactly the moment-generating function and the full probability density function of the single-trajectory PSD and its moments of arbitrary order in the most general case of arbitrary frequency, arbitrary (finite) observation time and arbitrary number of the projections of a
-dimensional BM onto the coordinate axes. This furnishes yet another example of a time-averaged functional of BM, whose moment-generating function and full probability distribution can be calculated exactly (see, e. g., Ref.satya ).
Capitalizing on these results, we observe that for a sufficiently large (and frequency bounded away from zero) for any realization of the process a single-trajectory PSD is proportional to its first moment (1), and the latter embodies the full dependence on the frequency and on the diffusion coefficient specific to an ensemble of the trajectories. This means that the frequency-dependence can be deduced from a single trajectory. In other words, there is no need to perform averaging over an ensemble of trajectories—one long trajectory suffices. However, the proportionality factor, connecting a single-trajectory PSD and its ensemble-averaged counterpart—a numerical amplitude—is random and varies from realization to realization. Due to this fact, one cannot infer the value of the ensemble-averaged diffusion coefficient from the amplitude of a single-trajectory PSD. The distribution function of this amplitude is calculated exactly here and its effective width is quantified using standard criteria.
As a proof of concept, we revisit our predictions for a continuous-time BM—an idealized process with infinitesimal increments—resorting to a numerical analysis based on Monte Carlo simulations of discrete-time random walks, and also using experimental single-trajectory data for the diffusive motion of micron-sized polystyrene beads in a flow cell. We demonstrate that our theoretical prediction for the relation connecting a single-trajectory PSD and its ensemble-averaged counterpart is corroborated by numerical and experimental results. Additionally we show that the predicted distribution of the amplitude is consistent with numerical results, which means that the framework developed here is justified and allows for a meaningful analysis of experimental trajectories.
Pursuing this issue further, we address several general questions emerging in connection with a comparison of our analytical predictions against numerical simulations and experimental data. To this end, we first consider Wiener’s representation of BM in form of an infinite Fourier series with random coefficients, whose truncated version is often used in numerical simulations. We show that the distribution of the single-trajectory PSD obtained from Wiener’s representation in which just terms are kept, instead of an infinite number, has exactly the same form as the one obtained for the continuous-time BM when is within the interval . Outside of this interval, the probability density function of the truncated PSD converges to a different form.
We examine the case when a trajectory of a continuous-time BM is recorded at some discrete time moments, so that it is represented by a set of points. The single-trajectory PSD, i.e., the periodogram, becomes a periodic function of with the prime period equal to . We analyze several aspects of this discrete-time problem: We study how large should be taken at a fixed observation time so that we may recover the results obtained for the continuous-time BM. We analyze the limiting forms of the distribution of a single-trajectory periodogram and show, in particular, that for kept fixed and , the distribution converges to the form obtained for a continuous-time BM. On the contrary, when is left arbitrary so that it may assume any value within the prime period, that is, , the distribution of a single-trajectory periodogram converges to a different limiting form as . Our analysis demonstrates that when belongs to a certain interval within the prime period, a single-trajectory periodogram equals, up to a random numerical amplitude, the ensemble-averaged periodogram, and the latter embodies the full dependence on and . Therefore, similarly to the continuous-time case, the correct spectrum can be obtained already from a single trajectory. These new results are the foundation for the use of single trajectory PSD in the quantitative analysis of single or few recorded particle trajectories, complementing the widely used concept of the single trajectory MSD.
we first derive explicit expressions for the variance of a single-trajectory PSD and the corresponding coefficient of variation of the probability density function, and present exact results for the moment-generating function of a single-trajectory PSD, its full probability density function and moments of arbitrary order (Sec.III.1). Section III.2 is devoted to the relation connecting a single-trajectory PSD and its ensemble-averaged counterpart, while Sec. III.3 discusses fluctuations of the amplitude in this relation. Next, in Sec. IV we analyze the probability density function of a single-trajectory PSD obtained by truncating Wiener’s representation of a continuous-time BM. Section V presents an analogous analysis for the case when a continuous-time trajectory is recorded at discrete time moments. In Sec. VI we conclude with a brief recapitulation of our results and outline some perspectives for further research. Additional details are relegated to Appendix A, in which we present exact results for the distributions in the special case , and to Appendix B, where we discuss several cases in which the full distribution of a single-trajectory periodogram can be evaluated exactly in the discrete-time settings.
Ii Brownian motion: Definitions and notations
Let , denote a Brownian trajectory in a -dimensional continuum and with stand for the projection of on the axis . The projections are statistically independent of each other and (likewise the BM itself) each projection is a Gaussian process with zero mean and variance
where is the diffusion coefficient of . In a random walk sense, , where is the variance of the step length and is the mean time interval between consecutive steps. Hence, contains the dimension of the unprojected motion.
In text-book notations, the PSD of each of the projections is defined by Eq. (1), that is,
where (taking into account that is real-valued) the -dependent function is given explicitly by
being the covariance of . For BM, one then finds the standard result def
Hence, for BM the standard power spectral density , defined as an average over an ensemble of trajectories, is described by a power-law with characteristic exponent and an amplitude which is linearly proportional to the diffusion coefficient .
Going beyond the text-book definition in Eq. (5), we now define the PSD of a single component :
and another property of interest here, the partial PSD of the trajectory
-parameterized random variables: the first momentof a single-trajectory PSD (8) is given by the standard result (6), while the first moment of the partial PSD (9), due to statistical independence of the components, is given by expression (6) multiplied by . Our goal is to evaluate exactly the full probability density function for .
Iii Brownian motion: Results
To get an idea of how representative of the actual behavior of a single-trajectory PSD the result (6) is, we first look at the variance of a single-component single-trajectory PSD,
Expression (10) permits us to determine the corresponding coefficient of variation
of the yet unknown distribution of a single-component single-trajectory PSD.
In Fig. 1 we depict , which is a function of the product exclusively. We observe that approaches for fixed and , and tends to the asymptotic value (thin horizontal line in Fig. 1) when at any fixed . Overall, for any value of
the coefficient of variation appears to be greater than unity, meaning that the standard deviationis greater than the average, , which signals that the parental distribution is effectively ”broad”, despite the fact that it evidently has well-defined moments of arbitrary order. As a consequence, the average described by Eq. (6) may indeed not be representative of the actual behavior of a single-trajectory PSD. This fully validates our quest for the distribution .
iii.1 Brownian motion: moment-generating-function, distribution and its moments.
Our first goal is to calculate the moment-generating function of in Eq. (9), defined formally as the following Laplace transform
To calculate exactly, it appears convenient to use Wiener’s representation of a given Brownian path in the form of a Fourier series with random coefficients
where are independent, identically-distributed random variables with the distribution
The corresponding single-trajectory partial PSD in Eq. (9) can be formally rewritten as
where the functions and are given explicitly by
we rewrite Eq. (12) in the factorized form
where the subscript in the averaging operator signifies that averaging over all paths of the component is replaced by an equivalent operation, the averaging over all possible values of . Performing this averaging, as well as the integrations over and , we get
We focus next on the infinite sums entering Eq. (III.1). They can be calculated exactly, and expressed via the first and the second moments of a single-trajectory PSD,
Consequently we realize that the moment-generating function in Eq. (III.1) can be cast into a more compact and physically meaningful form, which involves only the first moment and the variance (through the coefficient of variation) of the single-trajectory PSD,
This expression holds for any value of , and .
Performing next the inverse Laplace transform of the function defined by Eq. (21), we arrive at the following expression for the desired probability density function of the single-trajectory partial PSD defined in Eq. (9), which also (as the result in Eq. (21)) holds for arbitrary , arbitrary and arbitrary number of the projections of the trajectory onto the coordinate axes,
Here, is the modified Bessel function of the st kind and is the Gamma-function.
Before we proceed further, two remarks are in order. First, we note that the distribution (22) is the Bessel function distribution that has been used in mathematical statistics years ago as an example of a distribution with heavier than Gaussian tails (see, e.g., Refs. math2 ; math1 ). Second, as already mentioned, for the coefficient of variation is exactly equal to and hence, the coefficient in front of the term quadratic in in expression (21) vanishes. In this particular case the distribution (22) simplifies to the -distribution with degrees of freedom, which is presented in Appendix A.
Expression (22) permits us to straightforwardly calculate the moments of the partial single-trajectory PSD of arbitrary, not necessarily integer order ,
where is the Gauss hypergeometric function. Note that the moments of order for an arbitrary are all expressed through the first and the second (via ) moments of the single-component single-trajectory PSD only, since the parental Gaussian process is entirely defined by its first two moments.
where is the first moment of this random variable, i.e., a deterministic function of and which sets the scale of variation of , and is a dimensionless random amplitude, whose moments of order are defined by the expression in the right-hand-side of Eq. (23). The relation in Eq. (24) has important conceptual consequences on which we will elaborate below.
iii.2 Brownian motion: Single-trajectory PSD
One infers from Fig. 1 that upon an increase of the oscillatory terms in fade out, and saturates at the value . Let us define as the value of when the amplitude of the oscillatory terms in equals , where is a small fixed number. Given that the amplitude of the oscillatory terms is a monotonically decreasing function of , one has that for the variation coefficient . Next, the decay law of the amplitude of oscillations can be readily derived from Eqs. (6) and (10) to give that, in the leading in order, .
Then, for , up to terms proportional to , which can be made arbitrarily small, we see that the moments of the random amplitude in Eq. (24) are given by
meaning that in this limit becomes just a real number that is independent of and . This implies, in turn, that with any necessary accuracy, prescribed by the choice of , and for any realization of a trajectory ,
where the symbol signifies that the omitted terms, (stemming out of the oscillatory terms in , and hence, in ), have an amplitude smaller than .
Therefore, we arrive at the following conclusion, which is the main conceptual result of our analysis: for continuous-time BM and , the frequency-dependence of the PSD of any single trajectory is the same as of the ensemble-averaged PSD with any desired accuracy, set by . Consequently, in response to the title question of our work, we conclude that for BM the -dependence of the power spectrum can be deduced already from a single, sufficiently long trajectory, without any necessity to perform an additional averaging over an ensemble of such trajectories. In Sec. V (see Fig. 7) below we show that this prediction made for BM—a somewhat idealized stochastic process with infinitesimal increments—holds indeed for discrete-time random walks and single-trajectory experiments, in which a BM trajectory is recorded at discrete instants of time and for a finite observation time.
Lastly, we note that since is linearly proportional to the diffusion coefficient and the latter appears to be multiplied by a random numerical amplitude, one cannot infer the ensemble-averaged diffusion coefficient from a single-trajectory PSD. The error in estimating from a single trajectory is given precisely by the deviation of this amplitude from its averaged value. Below we discuss the fluctuations of this numerical amplitude.
iii.3 Fluctuations of the amplitude
In Fig. 2 we depict this distribution for and (curves from left to right). We observe that the very shape of the distribution depends on the number of components which are taken into account in order to evaluate the partial PSD. For (that is, for BM in one dimensional systems, or in two or three-dimensions but when only one of the components is being tracked), the distribution is a monotonically decreasing function with the maximal value at . In contrast, for and , is a bell-shaped function with a left power-law tail and an exponential right tail.
In Fig. 2 we also present a comparison of our analytical predictions in Eq. (27) against the results of numerical simulations. We use two methods to produce numerically the distributions of defined in Eq. (24) for the range of frequencies where is almost constant (see Secs. IV and V for more details). The first method hinges on Wiener’s representation of a BM in Eq. (13), which is truncated at the upper summation limit at (see Sec. IV below for more details). Crosses () in Fig. 2 depict the corresponding results obtained via averaging over trajectories generated using such a representation of BM. Further, we use a discrete-time representation of a BM—lattice random walks with unit spacing and stepping events at each tick of the clock—which are produced by Monte Carlos simulations. We set the observation time and generate trajectories with steps to get a single-trajectory periodogram. A thorough discussion of the domain of frequencies in which the periodogram yields the behavior specific to a continuous-time BM are presented below in Sec. V. Pluses () depict the results obtained numerically for random walks averaged over realizations of the process. Overall, we observe an excellent agreement between our analytical predictions, derived for BM—a continuous-time process with infinitesimal increments, and the numerical results. This signifies that the framework developed here is completely justified and can be used for a meaningful interpretation of stochastic trajectories obtained in single-trajectory experiments.
Moreover, the moments of the distribution in Eq. (27) are given by Eq. (25). We find that the average value is , as it should be, and the variance so that the distribution broadens with increasing . The coefficient of variation of the -dependent distribution in Eq. (27) is equal to meaning that fluctuations become progressively less important for increasing . We also remark that the most probable values for the cases and are and , respectively, and are well below their average values.
Lastly, we quantify the effective broadness of the distribution in Eq. (27), focusing on its heterogeneity index
where and are the amplitudes drawn from two different independent realizations of . Such a diagnostic tool has been proposed in Refs. 90 ; 91 ; 92 in order to quantify fluctuations in the first passage phenomena in bounded domains, compare also with the discussion in Ref. aljaz ; aljaz1 . In our context, shows the likeliness of the event that two values of the amplitudes and will be equal to each other. With the distribution (27) of the random variable the distribution of the heterogeneity index can be calculated via its integral representation 90 ; 91
Performing the integrals in relation (III.3), we find the following exact expressions for the distribution of the heterogeneity index: For we have
where is the complete elliptic integral of the second kind; for the distribution has the form
while for it follows that
Note that the third-order derivative with respect to in relation (III.3) can be taken explicitly producing, however, a rather cumbersome expression in terms of the complete elliptic integrals. For the sake of compactness, we nonetheless prefer the current notation.
The probability density functions in Eqs. (III.3), (31) and (III.3) are depicted in Fig. 3. We observe that for the event in which two values of deduced from two different independent trajectories are equal to each other, (that is, when and ), is the most unlikely, since it corresponds to the minimum of the distribution in Eq. (III.3). Therefore, for the case the most probable outcome is that two values of obtained for two different realizations of BM will be very different from each other. For and , corresponds to the most probable event but still the distributions in Eqs. (31) and (III.3) appear to be rather broad so that a pronounced realization-to-realization variation of the amplitude is expected.
Iv Truncated Wiener’s representation
In simulations of a continuous-time BM one often uses Wiener’s representation (13), truncating it at the upper summation limit at some integer ,
The sample paths of such a partial sum process are known to converge to the sample paths of the BM at the rate . Focusing on a single-component single-trajectory PSD in Eq. (8) (generalization to the -dimensional case and a partial PSD in Eq. (9) is straightforward) we have the following estimate for ,
Below we examine how accurately reproduces . We first consider the first moment of the truncated series and its variance, which are given by
In Fig. 4 we present a comparison of the averaged single-component single-trajectory PSD and of its counterpart obtained from the truncated Wiener’s series (34), as well as of the corresponding coefficients of variation of the two probability density functions. We observe a perfect agreement between the results obtained from the complete series and the truncated ones. Moreover, we see that and exhibit a uniform convergence to and , respectively, for . This implies that keeping just terms in the truncated Wiener’s series permits us to describe reliably well the behavior of the pertinent properties over more than two decades of variation of . Extending this interval up to three decades will require keeping terms, for four decades terms, and so on.
We finally focus on the moment-generating function of , and find
We note that has exactly the same form as the moment-generating function (21) (with ) evaluated for a complete Wiener’s series and hence, has the distribution of exactly the same form as the one in Eq. (22) with the only difference that the first moment and the variance have to be replaced by their counterparts obtained via truncation of the Wiener’s representation at some level . Given that for these properties are identical (see Fig. 4), it means that in this interval of variation of equation (37) coincides with (21), and the probability density function of coincides with result (22). Outside of this interval, i.e., for , the coefficient of variation of the truncated PSD jumps from to meaning that the distribution of becomes the -distribution (see Appendix A), which is evidently a spurious behavior.
V Discrete sets of data
In experiments or in Monte Carlo simulations of BM, the particle position is recorded at some discrete time moments such that one stores a given trajectory as a finite set of data. Here we analyze how the features unveiled in the previous Sections will change, if instead of continuous-time BM we rather use a picture based on a discrete-time random walk.
Suppose that the time interval is divided into equally-sized subintervals , such that the particle position is recorded at time moments , . We use the convention that at the particle starts at the origin and focus on the behavior of a single-component PSD—the extension of our analysis over the general case of components is straightforward but results in rather cumbersome expressions.
As a first step, we convert the integrals in Eq. (8) into the corresponding sums to get a periodogram
where we now denote a single-trajectory PSD as to emphasize that it is a different mathematical object as compared to the PSD in Eq. (8). Since we now have only a finite amount of points instead of a continuum, in Eq. (38) will be a periodic function of so that it may only approximate the behavior of the PSD for continuous-time BM in some range of frequencies at a given observation time. Further on, we use the scaling property of BM to rewrite the latter expression as
where is now a trajectory of a standard lattice random walk with unit spacing and stepping at each clock tick , . Next, we write
where are independent increments, and . Then, expression (39) becomes
v.1 Discrete-time case: Mean and variance of a single-trajectory periodogram
At this point, it may be expedient to first look at the ensemble-averaged single-trajectory periodogram in Eq. (41) and at its variance, and to compare them against their continuous-time counterparts. Averaging the expression in Eq. (41) and its squared value over all possible realizations of the increments , we have
To illustrate the behavior of