[5] Effect of binning and flipping

The following relates to [2]-[4]:

Martin has pointed out that the natural binning of the raw data and the different flipping operation of Gonzalez et al. could be partly responsible for the marked differences between their and our results. In particular, the double-peak structure might be reduced if our raw data are artificially binned and the same flipping rule (most occupied cell at right hand side) is used.

When applied to the experimental microbead data, the results are considerably closer to the single-peaked, triangular distributions of Gonzalez:

[4] Finite Size PDF of independent points in 1D

The following relates to posts [2] and [3]:

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Simplifications:

In order to indentify the minimum conditions of a double-peaked universal PDF, the situation can be simplified in several aspects:

A) 1D space:

By restricting the trajectories to one spatial dimension x, rotations are not required any longer.

B) Sets of N independent points:

Trajectories are normally generated by subsequently adding random increments (steps) to the respective last position of a walker. Here, in order to avoid any correlations between successive points, the N positions in each point set are drawn independently from a fixed probability density.

C) Random points equally distributed in [0,1]:

Let this probability density be constant in the intervall [0,1] and zero outside.

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Statistical quantities of point sets:

For each point set {x_i}, the center of mass is computed,

\overline{x} = \left\langle x_i \right\rangle_i = \frac{1}{N}\sum_{i=1}^N x_i \;,

as well as the standard deviation

\sigma = \sqrt{\left\langle (x_i-\overline{x})^2\right\rangle_i} \;.

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Transformations on point sets:

C-Operation (Center):

x_i \rightarrow (x_i-\overline{x}) \;\;\forall i

S-Operation (Scale):

x_i \rightarrow \overline{x}+(x_i-\overline{x})/\sigma \;\;\forall i

Note that C and S commute with each other.

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Effects of transformations:

The effects of C and S on the resulting averaged distributions P(x) are discussed in the following:

* No C, no S:

Direct averaging, without any C or S, yields the expected box-shaped distribution in [0,1], centered around 1/2.

* Only C:

Applying only the C-operation yields PDFs centered around zero. For N=1 one obtains a delta-function, for N=2 a triangular function, etc. For very large N, the box is recovered:


* Only S:

Scaling alone produces already a double-peak structure, centered around 1/2. However, it is a finite size effect that quickly disappears for long trajectories:

* C and S:

The combined effect of C and S yields a double-peak centered around zero:



If instead of the box-shaped distribution, a Gaussian distribution is used for the random points, the results are qualitatively the same. However, the double peak is much weaker pronounced and visible only for very small trajectory lengths.

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Summary:

Summing up, the double-peak is not a very remarkable feature of 1D trajectories.

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Origin of double-peak:

Take the extreme case of N=2: After a C-operation the two points lie symmetrically left and right from x=0. An additional S-operation scales the distance between the two points to the norm value, and the average distribution P(x) consequently consists of two delta-functions.

On the other hand, when N becomes very large, the histogram of each individual trajectory will already reflect the ensemble average very closely. The CS-operations therefore do not change the shape of this distribution qualitatively and one expects for P(x) to recover the fundamental distribution (in our case: box-shape).

Then, for intermediate lengths N, one expects a gradual interpolation between the double-delta-peak and a box-distribution.

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[3] Universal PDF of 2D Brownian and Persistent Trajectories

I am grateful to Martin Reichelsdorfer, who has programmed and applied the CRFSA-method (center, rotate, flip, scale and average), as described in post [2], to various synthetic and experimental trajectories in 2D.

In contrast to the original method of Gonzalez et al., his raw data trajectories consisted of real points in 2D. Cells were only introduced in the final averaging step. Therefore, the flipping operation had to be redefined: It was done in such a way that the last point of the trajectory was always located right of the first point.

He first tested the method for a standard Brownian Random Walk. For a given time period dt, the PDF of a Random Walk is known to be a Gaussian, centered at the origin, with a variance proportional to dt. This is also what one naively expects to see as the ensemble averaged spatial distribution. However, after CRFSA one obtaines a double-peak structure:


Second, Martin applied CRFSA to measured trajectories of micro-beads bound to living MEVO cells. On long time scales, those trajectories show superdiffusive behaviour with fractional powerlaw exponents in the mean squared displeacement (MSD), corresponding to directional persistence in the bead motion. This persistence is a qualitative difference to the Brownian Random Walk and one might expect new features for the ensemble averaged spatial distribution function (SDF). However, the results for the Persistent Walks are not much different from before.


No significant changes are observed when the ensemble of trajectories is divided into groups of similar powerlaw exponent.

These results lead to a couple of questions:

• Why is the CRFSA-SDF of a Random Walk not a Gaussian ?
  
• What exactly is the origin of the double-peak ?
• What makes the trajectories of mobile phone users so qualitatively different from Brownian or peristent walks ?
  

[2] Universal Spatial Distribution of Mobile Agents

In a recent paper "Understanding individual human mobility patterns" published by Marta C. Gonzalez et al. in Nature, the authors tracked the coordinates of mobile phone users over an extended period.

In this study, space was divided into cells (the range of a single mobile phone antenna) and the actual raw data consisted of the sequence of cells in which each user was found. In most of the cells, the same user was found several times. In particular, there was a cell of maximum occupancy for each user.

For each individual, the spatial distribution pattern (formally equivalant to a mass distribution) was artificially centered around the origin. Next, it was rotated around its center of mass, so that the main axis of the inertia tensor became alligned along the x-axis. Next, the pattern was flipped, such that the cell of maximum occupancy was located at the right side. It was then scaled to a fixed standard deviation in both x- and y-directions. The centered, rotated, flipped and scaled distributions were finally averaged over all individual trajectories (CRFSA method) . The result was a "universal" and surprisingly non-trivial distribution pattern:



It seems worthwhile to apply a similar procedure to the trajectories of microbeads attached to the living cytoskeleton. I shall name this project "Universal PDF".