AnDi-J › Trajectory Analysis

This tool allows to perform single-trajectory analysis of the input experiments. Every tab has a side Experiments panel where each experiment can be toggled on/off and, where relevant, restricted to the 2D, X, or Y component.

tMSD Visualization

This tab shows the time-averaged MSD curves for each experiment, allowing for an initial visual inspection of the diffusive properties of the loaded experiments.

• Per-experiment selectors for Avg (the average tMSD), All (every individual trajectory, faint), and eMSD (ensemble MSD, dashed).
• The lag axis is controlled by Min lag, Max lag (in frames) and Log-spaced. The number of lag points is chosen automatically (every integer lag when linear; capped, log-spaced sampling when log-spaced is selected). When the window opens, the tMSD is computed with min lag 1, max lag a quarter of the longest trajectory length and linear lags, the values shown in the fields.
• Draw line lets you place a reference line and read its slope live; in log–log mode the slope is labelled α, since it corresponds to the anomalous exponent.
• eMSD vs tMSD as an ergodicity test: comparing the eMSD curve to the Avg tMSD shows whether the process is ergodic (the two coincide) or non-ergodic (they differ).

Diffusion coefficient

Estimates the diffusion coefficient D per trajectory from a linear fit of the tMSD vs lag-time (mirroring andi_datasets’ get_diff_coeff), shown as a histogram. Dashed lines show the mean over the experiment.

You can configure the time lags at will, depending on the conditions of your experiment. Three formats are accepted:

1. [min, max] — a fixed integer range. When the second value is an integer ≥ 1 and larger than the first, the fit uses every lag from min to max, the same for all trajectories. Example: [1, 20] uses lags 1, 2, …, 20.

2. [min, fraction] — a per-trajectory (adaptive) range. When the second value is between 0 and 1, it is read as a fraction of each trajectory’s length: the fit uses lags from min up to floor(fraction × length) (with a small lower bound of 4), computed independently for every trajectory, so short trajectories automatically get a shorter maximum lag. Example (and default): [1, 0.1] fits from lag 1 up to 10 % of each trajectory’s length.

3. [t₁, t₂, t₃, …] — an explicit list. Any comma-separated list of positive integers is used verbatim as the set of lags for every trajectory. Example: [1, 2, 4, 8].

For guidance on what to choose, we recommend:

• Kepten, E., Weron, A., Sikora, G., Burnecki, K., & Garini, Y. (2015). Guidelines for the fitting of anomalous diffusion mean square displacement graphs from single particle tracking experiments. PLOS ONE, 10(2), e0117722.

You can analyse the 2D / X / Y components independently, which makes it possible, for instance, to detect anisotropic diffusion in your experiment. The values shown when the window opens use the default time lags [1,2]. The histogram toolbar has Normalize (%) and Log D axis; the latter shows D on a logarithmic axis with log-spaced bins (values D ≤ 0 are left out).

Anomalous exponent

Estimates the anomalous exponent α with two interchangeable methods, selected from the Method drop-down:

TMSD — log–log fit of the tMSD curve (click to expand)

The anomalous exponent is obtained from a log–log fit of the time-averaged MSD (tMSD) versus the time lag Δ: α is the slope of log(tMSD) vs log(Δ). The Time lags field controls which lags enter that fit, and follows the same format as for D above.

How to choose. The key trade-off is statistics versus bias. At a lag Δ, the tMSD of a trajectory of length L is averaged over only L − Δ time origins, so large lags are noisy and can distort the slope; conversely you need enough points to define a slope reliably.

• Keep the maximum lag small relative to the trajectory length (roughly 10–25 %). The adaptive form [1, 0.1] (the default) is the safest general choice because it scales automatically with each trajectory — recommended when your dataset mixes different lengths.
• Use a fixed [min, max] only when all trajectories are of similar length and you want identical lags across them (e.g. for reproducibility).
• Start at lag 1 for clean/simulated data. For experimental data affected by localization noise or motion blur at the shortest lag, starting at 2 (e.g. [2, 0.1]) can give a cleaner slope.
• Make sure the range spans at least a few lags (two is the minimum for a slope, but more gives a more stable fit).
• Because the fit is in log–log space, the frame interval does not affect α (it only rescales the lag axis); the interval matters for D, not for the exponent.

ML — deep-learning prediction (click to expand)

Uses a neural network to predict the exponent, typically a much more accurate technique than the tMSD approach (as shown in the 1st AnDi Challenge). Two options:

1. Default — uses a pre-trained model that outperforms every method of the 1st AnDi Challenge on 2D trajectories (see ML models). This model internally also predicts the diffusion model, so running it automatically populates the Diffusion Model tab (see below).
2. Load — allows you to load a custom α predictor. See ML models for details.

Once the exponent is calculated, a histogram of the exponent per trajectory is shown. The table below it reports the exponent computed from an ensemble MSD fit, the most robust method if the dataset is homogeneous. If ML is used to predict α, a new column appears with the average over those predictions.

D vs α

A scatter of one point per trajectory in the D–α plane, or a density of the same points. Both quantities are taken from the last computation in their respective tabs; change the parameters or methods there and this tab updates too.

The dropdown at the start of the toolbar chooses the view:

• Scatter — one point per trajectory, with the Select tool for population tagging (below).
• Density — a 2-D histogram of the same points, overlaid per experiment and tagged population in their colours, as in the Trajectory Segmentation D vs α tab, with α bins, D bins and Log count scale; the legend gives the number of trajectories of each. Selection and tagging are not available in this view; tagged populations appear as their own colour.

Features:

• Marginal KDEs — the plots along the top and right edges show a kernel density estimate of the population for α and D, respectively.
• Population tagging — in Select mode, circle (lasso) a cluster of points (see blue area in the figure below), type a name in Population tag: and click Tag population. The enclosed trajectories are recoloured to a shade of their parent colour and added as a new dataset to the Turning Angles and Diffusion Model tabs. A trajectory can belong to several populations; where they overlap, the latest population is drawn on top. A management table below the plot lists each population (tag, source datasets, number of trajectories, Show, colour — click the colour to change it in every tab). Untick Show to hide a population from this plot only: its points take their parent’s colour again. Delete selected deletes the highlighted populations (click, Ctrl- or Shift-click a row) from every tab, after a confirmation. Click Save… in the Save column to export the population’s trajectories as a csv in the input format (TRACK_ID, POSITION_X, POSITION_Y, POSITION_T, FRAME), plus an EXPERIMENT column naming the experiment each trajectory comes from. The file can be loaded again in the Data Manager, which ignores the extra column; with a calibration, positions are written in μm and POSITION_T in seconds, so the file loads back calibrated. TRACK_ID keeps the trajectory’s id when the population comes from one experiment and is renumbered from 0 when it mixes several, so ids stay unique; commas in experiment names are written as semicolons.
• Plot manipulation — Log D axis toggle, rubber-band Zoom, Pan, Reset view, in both views.

Diffusion Model

Classifies each trajectory into one of the five canonical anomalous-diffusion models: ATTM, CTRW, FBM, LW, SBM (see Muñoz-Gil et al. (2021) for more details). The predictions are done through an ML model, and shown as a grouped bar chart of the percentage of trajectories per experiment assigned to each model.

Turning Angles

A polar rose histogram of turning angles between successive displacement vectors, pooled per experiment and normalised to a fraction, so that experiments with different track counts are comparable.

PSD

Ensemble-averaged power spectral density of the trajectories, based on andi_datasets.analysis.psd (which wraps scipy.signal.periodogram with its defaults: unit sampling, boxcar window, constant detrend, one-sided, density scaling).

• Each shown periodogram is averaged over all trajectories of an experiment on a shared log-frequency grid (from 1/max-length up to the Nyquist frequency 0.5). Only the global average is shown — no per-track curves — with independent 2D / X / Y selectors per experiment.
• Freq. points sets the number of frequency bins; Log–log toggles the axes.
• A Draw line tool (as in the tMSD tab) reports the slope of a reference line — useful since the PSD of an anomalous process scales as a power law in frequency (the spectral slope is related to the anomalous exponent).

To learn about the PSD for anomalous diffusion analysis, we recommend reading the following publications:

• Krapf, D., et al. (2018). Power spectral density of a single Brownian trajectory: what one can and cannot learn from it. New Journal of Physics, 20(2), 023029.
• Krapf, D., Lukat, N., Marinari, E., Metzler, R., Oshanin, G., Selhuber-Unkel, C., … & Xu, X. (2019). Spectral content of a single non-Brownian trajectory. Physical Review X, 9(1), 011019.
• Sposini, V., Krapf, D., Marinari, E., Sunyer, R., Ritort, F., Taheri, F., … & Oshanin, G. (2022). Towards a robust criterion of anomalous diffusion. Communications Physics, 5(1), 305.

Summary

A table with one row per experiment and values aggregated over its trajectories. Click the ▸ next to an experiment’s name to show its X and Y rows. Click a column header marked ⓘ for a description of the value; the descriptions are listed below.

• Values from the other tabs (D, α, eMSD) appear when the tab opens and are updated each time it is shown, so they follow the parameters chosen in those tabs.
• Compute fills the trajectory statistics of every experiment: α₂, VACF, mean speed, step anisotropy, FMI, centre of mass and directness.
• α (ML) and Diffusion model come from the ML model of the Anomalous exponent or Diffusion Model tab. Run it there to fill or update these values; Compute does not run it.
• Yellow cells are not computed yet; hover over one to see where to compute it. Empty cells are values not defined for that row (for example FMI in 2D).
• Units sets the units of D, mean speed and centre of mass. Save table… saves a csv with one row per experiment and component (2D, X, Y); values not defined or not computed are left empty.

In the formulas, r(t) = (x(t), y(t)) is a trajectory sampled every Δt (one frame), and ⟨…⟩ is the average over all times and trajectories of the experiment.

• N, Length — number of trajectories, and their length in frames as median (min–max).
• D — mean over the trajectories of D from the Diffusion coefficient tab.
• Dₓ / Dᵧ — ratio of the diffusion coefficients along x and y, computed from the X and Y values of D. A value of 1 means isotropic diffusion; values far from 1 mean faster motion along one axis.
• α (tMSD) — mean of the anomalous exponents of the trajectories, each from a log–log linear fit of the trajectory’s tMSD with the time lags of the Anomalous exponent tab. α < 1: subdiffusion · α = 1: normal diffusion · α > 1: superdiffusion.
• α (eMSD) — anomalous exponent from a log–log linear fit of the ensemble MSD, eMSD(Δ) ∝ Δ^α, over the lags of the tMSD Visualization tab. The eMSD averages the squared displacement from the first frame of each trajectory. If α (eMSD) ≠ α (tMSD), the time and ensemble averages differ (see Ergodicity gap). Reference: R. Metzler, J.-H. Jeon, A. G. Cherstvy, E. Barkai, Phys. Chem. Chem. Phys. 16, 24128 (2014).
• α (ML) — mean of the α predicted by the ML model for each trajectory.
• Diffusion model — the most frequent diffusion model among the trajectories, with the percentage of trajectories assigned to it.
• Ergodicity gap — ratio of the time-averaged and ensemble-averaged MSDs, averaged over the lags of the tMSD Visualization tab: G = mean over Δ of ⟨tMSD(Δ)⟩ / eMSD(Δ). G ≈ 1: time and ensemble averages agree, as for an ergodic process such as Brownian motion or fractional Brownian motion. G ≠ 1: weak ergodicity breaking, as for continuous-time random walks (CTRW), annealed transient time motion (ATTM) or ageing systems. References: Y. He, S. Burov, R. Metzler, E. Barkai, Phys. Rev. Lett. 101, 058101 (2008); R. Metzler et al. (2014), above.
• α₂ (non-Gaussian) — measures whether the displacements over one frame are Gaussian: α₂ = ⟨|r|⁴⟩ / ((1 + 2/d) ⟨|r|²⟩²) − 1, where d is the number of dimensions: 2 for 2D, 1 for X or Y. In 1D this is α₂ = ⟨Δx⁴⟩ / (3⟨Δx²⟩²) − 1. α₂ = 0 for Gaussian displacements. α₂ > 0 means heavier tails than a Gaussian. This typically comes from heterogeneity: particles, or parts of trajectories, with different diffusion coefficients (“Fickian yet non-Gaussian” diffusion). Reference: B. Wang, J. Kuo, S. C. Bae, S. Granick, Nat. Mater. 11, 481 (2012).
• VACF — correlation between consecutive steps, normalised by the step variance: C = ⟨v(t) · v(t+Δt)⟩ / ⟨|v(t)|²⟩, with v(t) = r(t+Δt) − r(t). C ≈ 0: uncorrelated steps (Brownian motion, continuous-time random walk). C < 0: anti-persistent steps (fractional Brownian motion with α < 1 gives C = 2^α−1 − 1, confinement). C > 0: persistent steps (directed or superdiffusive motion). Static localisation noise pushes C towards −0.5. A strongly negative C can therefore also mean noisy data. Reference: S. Burov, J.-H. Jeon, R. Metzler, E. Barkai, Phys. Chem. Chem. Phys. 13, 1800 (2011).
• Mean speed — accumulated distance of each trajectory divided by its duration, averaged over the trajectories: v = ⟨ Σ |r(t+Δt) − r(t)| / (T Δt) ⟩. For X and Y, |Δx| or |Δy| replaces the step length. The value depends on the frame rate and on localisation noise, which adds apparent motion, so compare it only between experiments recorded the same way.
• Step anisotropy — orientation order of the steps. With φ = atan2(Δy, Δx) the direction of each one-frame step: S = ⟨cos 2φ⟩. S = 1: all steps along x. S = −1: all steps along y. S = 0: no preferred axis. Opposite steps (φ and φ + 180°) count the same, so S measures alignment along an axis, not drift in one direction (see FMI and Centre of mass).
• FMI (forward migration index) — net displacement along one axis relative to the path length, averaged over the trajectories: FMI_x = ⟨ (x_end − x_start) / d_acc ⟩, with d_acc = Σ |r(t+Δt) − r(t)|. FMI_y is the same with y. It ranges from −1 to 1. Values near 0 mean no drift along that axis. In a chemotaxis assay with the gradient along one axis, FMI along the gradient measures how efficiently the cells follow it.
• Centre of mass — mean end point of the trajectories, each shifted to start at the origin: M_end = ⟨ r_end − r_start ⟩. M_end close to (0, 0) means no net drift. Its x and y components give the direction of the drift; the 2D value is its distance from the origin.
• Directness — straight-line distance between start and end divided by the path length, averaged over the trajectories: Directness = ⟨ |r_end − r_start| / d_acc ⟩. It ranges from 0 to 1, where 1 is a straight line. It decreases with trajectory length: for Brownian motion, Directness ≈ 1/√T for trajectories of T frames, since the net displacement grows as √T and the path length as T. Localisation noise also lowers it. Compare it only between experiments with similar lengths and frame rates, for example by setting the same Max. length for all files in the Data Manager.

Reference for mean speed, FMI, centre of mass and directness: P. Zengel et al., BMC Cell Biol. 12, 21 (2011).