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Proof To compute the costs [MATH] for all [MATH] , Algorithm 2 constructs and store the costs with [MATH] increasing, with a temporary vector containing the required values of [MATH] at the step [MATH] , the step [MATH] updating the values [MATH] using [MATH] computations [MATH] By induction, at the end of the iteratio... |
Proposition 5 Algorithm 3 computes the cluster costs [MATH] for all [MATH] for a given [MATH] with a complexity in [MATH] time and in [MATH] memory space. |
Proof : Algorithm 3 is similar with Algorithm 2: to compute the costs [MATH] for all [MATH] for a given [MATH] , Algorithm 3 constructs and store the costs with [MATH] decreasing, with a temporary vector containing the required values of [MATH] at the step [MATH] , the step [MATH] updating the values [MATH] using [MATH... |
Dynamic Programming algorithm and complexity results in this section, the polynomial complexity of [MATH] [MATH] -Med2dPF is proven, distinguishing the case [MATH] from the general case with [MATH] |
6.1 The case [MATH] [MATH] -Med2dPF In the case [MATH] [MATH] [MATH] -Med2dPF can be reformulated with Proposition considering the following optimization problem instead of ( ): |
[EQUATION] Algorithm 4 solves this reformulated problem enumerating all the possibilities following [MATH] , with an efficient computation of costs [MATH] using Algorithm 2 and [MATH] with Algorithm 3. Both computations have a complexity in [MATH] time and [MATH] space. |
With a similar algorithm, Algorithm 4 explore all the local minima of [MATH] [MATH] -Med2dPF. Once the costs are computed in [MATH] time, the costs and [MATH] -medoids can be stored. Denoting [MATH] (resp [MATH] ) the [MATH] -medoids of cluster [MATH] (resp [MATH] |
[MATH] defines a local minimum if [MATH] and [MATH] , such remaining computations are in [MATH] for each possible local optimum, and thus in [MATH] to test the [MATH] possible local optimums. Computing all the local optimums (that are at most [MATH] ) has thus a complexity in [MATH] |
6.2 General cases [MATH] [MATH] -Med2dPF with [MATH] In the general case, Proposition allows to design a DP algorithm, enumerating the possibilities among the possibly global optimal solutions. |
Proposition 6 (Bellman equations) Defining [MATH] as the optimal cost of [MATH] [MATH] -Med2dPF among the points indexed in [MATH] for all [MATH] and [MATH] we have: |
[EQUATION] [EQUATION] Proof : ( 23 ) is trivial. We suppose [MATH] and prove ( 24 ). Let [MATH] Selecting for each [MATH] an optimal solution of [MATH] [MATH] -Med2dPF among points indexed in [MATH] and adding cluster [MATH] , it makes a feasible solution for [MATH] [MATH] -Med2dPF among points indexed in [MATH] |
with a cost [MATH] This last cost is greater than the optimal [MATH] [MATH] -Med2dPF cost, thus [MATH] [EQUATION] Let [MATH] indexes such that [MATH] |
defines an optimal solution of [MATH] [MATH] -Med2dPF, its cost is [MATH] Necessarily, [MATH] defines an optimal solution of [MATH] [MATH] -Med2dPF among points indexed in [MATH] . On the contrary, a strictly better solution for [MATH] would be constructed adding [MATH] We have thus: |
[MATH] Combined with ( 25 ), it proves : [MATH] [MATH] These relations allow to compute the optimal values of [MATH] by dynamic programming in the Algorithm 4. |
[MATH] is the optimal value of [MATH] [MATH] -Med2dPF, backtracking on the matrix [MATH] computes the optimal partitioning clusters. |
Actually, to compute the optimal value [MATH] and to recover the indexes of this optimal solution by backtracking, some elements of the matrix [MATH] are useless to compute. For [MATH] , there is no need to compute the values [MATH] For [MATH] , there is no need to compute the values [MATH] In the line [MATH] , the ele... |
To compute [MATH] , it requires the previous optimal computations of [MATH] with [MATH] It is possible (similarly with dupin2019planar dupin2020polynomial ), to compute the matrix [MATH] following the index [MATH] . In Algorithm 4, the matrix [MATH] is computed line by line following the index [MATH] increasing. Indeed... |
6.3 Complexity results Theorem 6.1 Let [MATH] a subset of [MATH] points of [MATH] , such that for all [MATH] [MATH] Clustering [MATH] with [MATH] [MATH] -Med2dPF is solvable to optimality in polynomial time with Algorithm 4. The complexity is [MATH] time and in [MATH] memory space , and [MATH] time and in [MATH] memory... |
[MATH] [MATH] -Med2dPF is solvable in [MATH] time in general, the cases of a convex or concave 2d PF with [MATH] induces a time complexity in [MATH] |
Proof :The case [MATH] = is given in section 4.3 with Proposition . The case [MATH] is given using Algorithm 4 as described in section 6.1. In the following, we focus on the cases [MATH] and Algorithm 5. 24 ) uses only values [MATH] with [MATH] in Algorithm 5. Induction proves that |
[MATH] has its final value for all [MATH] at the end of the for loops from [MATH] to [MATH] [MATH] is thus at the end of these loops the optimal value of k- [MATH] -Med2dPF clustering among the [MATH] points of [MATH] The backtracking phase searches for the equalities in [MATH] |
to return the optimal clusters [MATH] Let us analyze the complexity. Sorting and indexing the elements of [MATH] following Lemma has a complexity in [MATH] The first line [MATH] is computed in [MATH] time using Algorithm 2 and Proposition Then, to compute the line [MATH] of the DP matrix [MATH] , the cost computations ... |
[MATH] Each line [MATH] is thus computed in [MATH] time, the bottleneck is the computations of cluster costs, and the total complexity to compute the DP matrix [MATH] is in [MATH] time. The backtracking phase requires [MATH] computations having a complexity in [MATH] time, re-computing the cluster costs, the complexity... |
Speeding-up the DP algorithms In this section, it is studied how to speed-up the DP algorithms in practice, without improving the theoretical complexity proven in section 6. On one hand, it is investigated how to remove useless computations in Algorithms 4 and 5. On the other hand, parallelization issues are discussed. |
7.1 Additional stopping criterion An additional stopping criterion can be designed to stop the enumeration of cases [MATH] to compute [MATH] |
Lemma 7 Let [MATH] and [MATH] . Let [MATH] an upper bound for [MATH] We suppose it exist [MATH] such that [MATH] Then, each optimal index [MATH] such that [MATH] fulfills necessarily [MATH] In other words, [MATH] |
Proof : We have [MATH] . Lemma implies that for all [MATH] [MATH] . Using [MATH] [MATH] , and thus [MATH] [MATH] Lemma allows to compute optimal value of each value of the DP matrix using less computations than the algorithms proposed in the last section. A strong interest of this property is that it avoids to compute ... |
7.2 Speeding-up the case [MATH] In the case of [MATH] [MATH] -Med2dPF, the stopping criterion can be processed also with index [MATH] increasing: |
Lemma 8 Let [MATH] an upper bound for [MATH] We suppose it exist [MATH] such that [MATH] We suppose it exist [MATH] such that [MATH] Then, [MATH] |
Proof : Lemma implies that [MATH] is a lower bound for the optimal separation index. Similarly, [MATH] is an upper bound for the optimal separation index, using that [MATH] is increasing with Lemma |
Incorporating the stopping criterion of Lemma in Algorithm 4 shall minimize the computations of [MATH] and [MATH] Costs computations of Algorithm 2 and 3 will be proceeded only when needed. Firstly, cost computations [MATH] (resp [MATH] ) are proceeded for heuristic [MATH] (resp [MATH] ), with an interruption of Algori... |
[MATH] [MATH] defines a feasible solution of [MATH] [MATH] -Med2dPF, and upper bound of the optimal cost. Then, the interrupted cost computations are continued till the stopping criterion is not reached, computing costs [MATH] |
and updating [MATH] to the best found value which may activate earlier the stopping criterion of Lemma 7.3 Improved algorithm in the general case |
Lemma can be used to stop earlier the optimal computations of each value in the DP matrix [MATH] For each [MATH] , the stopping criterion is used to stop the cost computations when all the [MATH] for [MATH] are proven optimal using Lemma . The advantage of such stopping criterion is to avoid the most time consuming cos... |
Lemma can also be used in a recursive approach using memoisation. In such approach, the cost computations and some values of the DP matrix are memoised, and Lemma allows to reduce the number of values to compute in the DP matrix. However, such memoisation can induce a memory space in [MATH] |
7.4 Parallelization issues A parallel implementation is a practical issue to speed up the DP algorithms. In Algorithms 2 and 3, the inner loop induces independent computations that can be processed in parallel, the implementation is straightforward in an environment like OpenMP. This makes the parallelization of the cr... |
Conclusion and perspectives This paper examined properties of an extended version of the K-medoid and K-median problems in the special case of a discrete set of non-dominated points in a two dimensional Euclidian space. A characterization of global and local optima is proven with interval clustering. It is proven that ... |
[MATH] [MATH] -Med2dPF problems are solvable in [MATH] time when [MATH] the 2d PF is concave or convex. Practical speed-up are also proposed, in relation with discussions on parallelization issues. |
The complexity in [MATH] may be a bottleneck to deal with very large 2d PF, which open new perspectives. Heuristics may apply efficiently for such cases. Initialization strategies can use optimal solutions of p-centre or p-dispersion problems in a 2d PF or in 1d cases after projection. Having a NP-hard complexity prove... |
# Source: arxiv 1806.02251 # Title: Reconstruction of substrate's diffusion landscape by the wavelet analysis of single particle diffusion tracks # Sections: all # Downloaded: 2026-03-02T08:49:21.324968+00:00 |
Reconstruction of substrate’s diffusion landscape by the wavelet analysis of single particle diffusion tracks Department of Theoretical Physics, Kursk State University, Radishcheva st., 33, 305000 Kursk, Russia. Fax: +7-4712-51-04-69; Tel: +7-4712-51-04-69; E-mail: postnicov@gmail.com |
Institut für Physik and IRIS Adlershof, Humboldt Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany. Abstract We propose an approach to analysing single trajectories of a particle, which moves randomly on a landscape distinct parts of which result in sufficiently various diffusion coefficients. The method ba... |
Introduction The method of single particle tracing is the method of choice for investigating diffusive properties of complex systems such as complex materials, biological cells or their membranes. What is of the major interest in many cases are not the properties of the diffusion itself, but its application as a probe ... |
Recently, several approaches to extract local properties of diffusive motion from single trajectories and to attributing them to specific spatially localized areas were proposed. In particular, one can mention clustering of diffusion parameters determined based on whole trajectory processing from a set of trajectories ... |
segmentation of a single trajectory in parts corresponding to different diffusivities or into regions of active (superdiffusive) and passive (diffusive and subdiffusive) either by a sliding window analysis |
or by the wavelet thresholding , enclosing parts of a single trajectory by convex hulls, geometrical properties of which distinguish between regions with different mobility rates |
, etc. However, in cellular environments the diffusion is not only often anomalous, being either sub- or superdiffusion , but can also be paradoxical. In this last case the width of the displacements’ distribution grows linearly in time, but the distribution itself is strongly non-Gaussian. A theoretical example of ext... |
. In other, milder cases, the moments of the distribution (at least the second one) are present. This situation is termed as “normal, yet non-Gaussian” diffusion |
see also and references therein. Such normal, yet non-Gaussian diffusion may have different reasons, and appear either due to the heterogeneity of tracers, each possessing its own diffusion coefficient in an essentially homogeneous environment (”superstatistics”), due to environmental heterogeneity, or due to fluctuati... |
The situation assumed is as follows. A particle diffuses in an environment which might be homogeneous or not. The particle’s local motion corresponds to an unbiased diffusion with the diffusion coefficient which might be a function of the position and time. These changes are considered gradual in space or slow enough i... |
The answers to our initial questions are given by analysing the fluctuations of the diffusion coefficient. If these are small, the total inhomogeneity has to be ascribed to the tracers’ differences. If these are ruled out, the next step would be analysing the difference of the diffusion coefficients on self-crossings o... |
The method The input data of the SPT measurement are the time series of the instantaneous particle’s positions [MATH] sampled at the instants of time [MATH] The total displacement from the beginning of the observation up to time [MATH] corresponding to [MATH] time steps can therefore be reprersented as a displacement i... |
[EQUATION] The squared displacement in the walk is [EQUATION] Our random walk may be a correlated or not. If the random walk does not possess any memory at the timestep of the data aquisition procedure, the ensemble mean of each of the contributions to the second sum vanishes, and we get for the MSD |
[EQUATION] If we assume the steps to be i.i.d. random variables possessing the zero first moment [MATH] and a finite second moment [MATH] the sum will grow linearly in [MATH] and therefore in time, and the slope of the dependence will define the diffusion coefficient [MATH] where [MATH] is the dimension of space: for a... |
Now let us imagine that the diffusion coefficient and therefore the mean squared step lengths vary in time, but the change in this coefficient is small on the timescale of a single step, so that [MATH] is a slowly changing function of [MATH] or [MATH] . Then looking at [MATH] we still can read out [MATH] as the local s... |
The same approach can be used for determining local diffusion coefficients in the case when the diffusion coefficient is position-dependent (but varies only slightly on the scales of a single step). In this case the local slope of the [MATH] curve gives us the diffusion coefficient at a position at which the particle i... |
One of the advantages of the method based on slope estimation is the fact that the approach based on moving time averaging of the local squared displacements poses a task of judicious choice of the averaging window, which, for achieving satisfactory accuracy, has to be chosen adaptively, depending on the local diffusio... |
Since at short times diffusion always wins over the deterministic motion, the estimate for the diffusion coefficient would not be bad even for a particle moving in some external potential, provided the interval [MATH] between the data acquisition points is short enough. The only problem with estimating the diffusion co... |
Local linear fits by wavelet transforms In the case of a random walk with independent and stationary individual steps, the problem of diffusivity identification reduces to the problem of linear fit of the experimentally determined function [MATH] which might however show strong local fluctuations. To understand the ide... |
[MATH] we see that the distribution of this variable is a convolution of the distribution of [MATH] and a rectangular distribution of width [MATH] centered at [MATH] . The properly defined width of the convolution of two distributions (i.e of the distribution of the sum of two independent random variables) is larger th... |
In the standard least square fit (assuming the Gaussian statistics of errors, and an absence of the systematic error, i.e. that [MATH] ) the width of the distribution is characterised by the mean squared value of [MATH] , so that what has to be minimized is the sum of deviations squared. This definition makes problems ... |
An alternative method is based on looking at the empirical characteristic function of [MATH] [EQUATION] being an approximation to the Fourier transform of the probability density of [MATH] The narrower the distribution of [MATH] the broader is its spectrum [MATH] (or [MATH] ) and vice versa. The method is robust since ... |
For Gaussian distribution of errors, or for any other distribution of errors possessing the second moment [MATH] the form of this characteristic function close to its maximum attained at [MATH] will be [MATH] where [MATH] is the second moment of the distribution of the values of [MATH] , i.e. approximately [MATH] . The... |
) and looking for the maximum of the value of [MATH] as a function of [MATH] (taking the absolute value removes the influence of the term linear in [MATH] so that for the Gaussian case the method is equivalent to the least square fit). We note that passing into the complex domain, and using the characteristic function ... |
, when the least square fit fails. Thus, to perform the linear fit one considers the function [EQUATION] where [MATH] is an appropriate scaling factor discussed above, and finds the value of [MATH] corresponding to the maximum of this function. |
The value [MATH] will give the desired diffusion coefficient [EQUATION] The prefactor of [MATH] in Eq.( ) is chosen in such a way that for the case of linear [MATH] growing linearly in time [MATH] |
If [MATH] is position-dependent, the approach ( ) needs to be restricted to the analysis of parts of random walk trajectories localized within relatively small spatial regions where the diffusivity can be considered as approximately constant. This translates into the analysis of the parts of trajectories bounded to som... |
[EQUATION] (here [MATH] stands for [MATH] and integral notation is used instead of the sum). This approach is similar to the method applied in |
for the analysis of temporal evolution of non-stationary relaxation processes. The variable [MATH] parametrizes the position of the part of the trajectory (in time) where the diffusion coefficient is estimated. The variable [MATH] is called “scale” and plays a twofold role. First, it determines the duration of the corr... |
[MATH] , where [MATH] is the scale corresponding to the maximum of the absolute value of the wavelet transform, [MATH] at a fixed [MATH] Due to the properties of complex exponential, ( ) can be rewritten as |
[EQUATION] Note that the factor moved outside of the integral does not influence [MATH] and, therefore, can be further omitted while one considers the absolute value only. The integrand consists of the term coinciding with ( ) multiplied by the bell-shaped Gaussian sliding filter, which assures time localization |
for the regions, where the residual [MATH] tends to a constant. The Morlet wavelet provides the best possible simultaneous localization in time and frequency domains and an exact correspondence between the maximum of its absolute value and the Fourier frequency of a harmonic signal, which in our case corresponds to the... |
In principle, the Gaussian function can be replaced by another well-localized filter of self-similar shape. To get better time localization mentioned above the filters with finite support could be preferred. Thus, an appropriate replacement of ( ) in the case of time-localized trajectory analysis will be |
[EQUATION] A good candidate for the window function [MATH] is a Tukey filter of length [MATH] with [MATH] [EQUATION] It comprises a flat top, within of which ( ) reduces to ( ), and rapidly decaying side parts smoothly connected to the central one that assures locality of the filter and absence of significant spectral ... |
, the length of the Tukey filter considered within a general multiscale theory of wavelets, should be coordinated with the characteristic periods corresponding to the values [MATH] at which [MATH] is sampled, so that |
[MATH] rounded to a next integer. Here [MATH] counts a number of periods [MATH] stacked on the length of the filter. In further examples we use [MATH] and [MATH] , which provides a uniform averaging over two trial periods of oscillations with half-period-long smooth transition to zero value. The resulting local diffusi... |
Before going into details of the procedures as applied to real data, let us discuss a small simulation of a random walk in a patchy environment, where the diffusion in each patch is normal but is characterized by a different diffusion coefficient. |
Numerical example To discuss the method let us consider a simple example of the discrete lattice random walk determined as [EQUATION] |
where [MATH] is the step displacement. This displacement is either 0 or [MATH] in one of the directions, and its value depends on the waiting time [MATH] on a site (taken to be a whole number). If the walker arrives at a site at [MATH] -th time step, it stays there for the next [MATH] subsequent time steps, and the nex... |
As an example we consider a random walk generated according to Eqs. ( )–( ) on a plane subdivided into quadrants with the waiting times [MATH] counter-clockwise starting from the right lower quadrant, see Fig. (a). In this plot the coloured background indicates regions with different constant diffusion coefficients and... |
Fig. (b) demonstrates the behaviour of [MATH] , the sum of the step lengths squared, which will be substituted into Eq. ( ), resulting in abdolute values of the wavelet transform as color-coded in Fig. (d). The ordinate [MATH] |
is scaled from the parameter [MATH] in the same way as Eq.( ) defines it for the case of wavelet maximum [MATH] The corresponding estimates for [MATH] correspond to a stripe of the brightest colour in the plot of the absolute value of the transform. |
One can see that the relatively long first part of trajectory corresponding to the random walk on the quadrant with [MATH] results in the constant slope subinterval of around 300 steps in Fig. (b), which is mapped into the horizontal bright stripe the left side of in Fig. (d) of the same duration. The position of the m... |
Using Fig. (d) we determined the [MATH] , the position of the maximum of the absolute value of our wavelet transform (which in the following will simply be called the “wavelet maximum” for the sake of brevity) for each time step and represented the corresponding value of diffusivity on the walker’s trajectory by a colo... |
To demonstrate that the algorithm described above actually reconstructs the spatial patches of the diffusion coefficient and allows for distinguishing them from possible time-dependent diffusivities, we consider another simple simulation, when the waiting times governing [MATH] in Eqs. ( )–( ) are assumed to be random ... |
[MATH] and, therefore the averaged instant displacement squared determines the effective averaged diffusivity as [MATH] with [MATH] |
Fig. illustrates such process for [MATH] . From Fig. (d) one readily infers that the maxima of the absolute value of the wavelet transform are located in a stripe around [MATH] corresponding to [MATH] that is close to the average slope of the line in Fig. (b). However, this stripe shows bursts, which reflect jumps in d... |
An additional representation of the trajectory color-coded for the local value of the diffusion coefficient, see Fig. (c) unveils the effect which now is evident from the fact that the overlapping parts of the trajectory are coded with different colors: in contrast to the case of spatially-dependent diffusion considere... |
Application to real data As a practical example of the proposed algorithm application to real data of single molecule walk on a biological membrane, we analyse one of the data sets measured experimentally and presented in the work |
(the data provided by courtesy of C. Manzo), see (a), where actual walker locations are marked by black dots connected by lines for visibility. The plot [MATH] shown in Fig. (b) is quite far from a straight line. However, there are sufficiently long intervals (almost a half of the track in total), where it can be appro... |
The result of application of Eq. ( ) with ( ) with the function [MATH] depicted in (b) is shown in Fig. (c) that allows for the direct comparison of the behaviors of [MATH] and of the wavelet’s maximum. Here warmer colours correspond to larger values of the wavelet maximum. |
The dashed black straight line is the global average diffusion coefficient determined by the time-averaged mean-square displacement |
[EQUATION] with [MATH] [MATH] ) equal to [MATH] that agrees with the result given in The local maxima of the wavelet transform are close to the [MATH] -based value of diffusivity, fluctuating around this line but also show bursts typical for rapid changes. |
Fig. (a) also presents the trajectory of this random walk color-coded for the local diffusivity obtained via the our wavelet maxima (maxima with magnitudes less than 40% of the largest value of the wavelet maxima are excluded from the analysis since they correspond to finite size effects. The resulting spatial picture ... |
Caveats and safeguarding procedures As it has been mentioned above, the proposed procedure will result in an accurate representation of the local diffusion coefficients if the underlying random displacements are indeed diffusive i.e. not correlated on the time scale of the step of data acquisition procedure. As a safeg... |
[MATH] would not change, but the changes are visible for correlated one. Such a procedure can be given a slightly different flavor, as we will discuss below. |
We consider the localized smoothing of the random walk data based on the local multiscale averaging, each iteration of which is computed in the discrete case |
[EQUATION] for each component of the vector of elementary steps [MATH] (indices [MATH] are omitted in Eq. ( 10 )), which correspond to [MATH] |
Thus, let us analyse now a set of the coarse-grained cumulative displacements squared [EQUATION] where the factor [MATH] is introduced to compensate scaling in Eq. ( 10 ), and time steps with respect to initial ones are defined as |
[MATH] , i.e. via power two of the elementary time steps. In such representation, the function [MATH] is invariant on the average for the true Brownian motion with uncorrelated steps and the wavelet-based algorithm gives the same value of the local diffusion coefficients that follows from the multiscale construction of... |
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