Why parameter choice can flip a distance verdict
A gamma distribution can be written in several equivalent coordinate systems, such as shape and scale, or mean and variance. Euclidean distance measured in those coordinates depends on which one is chosen, so two identical distributions can be ranked differently depending on the parameterization. The paper makes this concrete with a reference distribution R and two candidates: under shape-scale coordinates, P looks closer to R (1.46 versus 2.59), but under mean-variance coordinates the ranking reverses and Q looks closer (0.34 versus 1.50).
The Fisher-Rao distance avoids that problem. It treats the two-parameter gamma family as a curved two-dimensional surface where each point is one distribution, and local distances are set by the Fisher information matrix. The distance between two distributions is the length of the shortest path, or geodesic, across that surface. Unlike Kullback-Leibler divergence, it is a true metric, symmetric and obeying the triangle inequality, and it is invariant to reparameterization. In the same example it returns one answer, Q closer at 0.75 versus 3.35, matching the distributions' actual overlap.
That invariance is the reason the paper needs a numerical solver. Closed-form expressions exist only in special cases, such as when two distributions share a shape parameter, so the general case must be integrated. The authors re-express each distribution in coordinates (α, η), where α is the shape and η is the logarithm of the mean, which puts the metric in a diagonal form and simplifies the geodesic equations. Their three-stage solver uses an initial guess from a hyperbolic approximation, computes the endpoint sensitivity exactly through variational equations rather than finite differences, and converges across the full range of shape parameters in the data, including the large-α regime where a naive shooting method fails.
How a drawing trial becomes a gamma point
The empirical setting is a digitized Trail Making Test run on a tablet. Nineteen healthy adults (mean age 28.8 ± 8.6, 9 female and 10 male) connected 12 numbered targets in one continuous pen stroke, with no time limit and no constraint on path shape. Cognitive load was manipulated within subject by changing the target sequence: a purely numeric order (1-2-3-4-5…) for low load, and an alternating alphanumeric order (1-a-2-b-3-c…) for high load, which requires holding and switching between two ordered sets. This mirrors the standard Trail Making A/B contrast in executive demand.
Pen-tip position was sampled at 240 Hz. The analysts extracted each sub-trail between consecutive targets, estimated velocity by backward finite difference, smoothed it with a 13-sample Gaussian window, and identified the peaks of the speed profile. Those peak amplitudes were fitted with a gamma distribution by maximum likelihood, yielding a shape and a scale with 95% confidence intervals. Each participant's condition therefore becomes a point on the gamma plane, and the Fisher-Rao distance between a participant's low- and high-load points summarizes how far load moved that person's fluctuation signature.
The paper also builds a second, windowed representation. Because a single static shape parameter mixes how often motor noise enters a heavy-tailed regime with how deep that regime runs, the authors split the windowed shape sequence at a threshold τ of about 11.8, the pooled 25th percentile of α. This yields two axes per participant per condition: frequency, the fraction of windows below the threshold, and severity, the mean log α over those heavy-tailed windows. The static and windowed representations are not interchangeable, and the paper reports unique results within each.
Load moved every signature, with no shared direction
The static comparison gives the headline number. The Fisher-Rao distance between low- and high-load fits had a median of 0.59, with a range of 0.05 to 1.22, so every participant's signature moved. But the movement had no common direction. The shape parameter α increased under load in only 9 of 18 participants, a sign test with p = 1.000, and paired t-tests on log α and log β were both null (p = 0.85 and 0.88). The cohort Fréchet means were nearly coincident, at (11.09, 0.055) under low load and (10.54, 0.058) under high load.
The cloud of participants did tend to contract. The Fisher-Rao Fréchet radius fell by a factor of 1.30, from 0.524 to 0.402, and the parameter-coordinate spread agreed, with the standard deviation of log α shrinking by 1.55×. That trend did not reach significance, however: a one-sided signed-rank test gave p = 0.075, and a bootstrap 95% confidence interval on the ratio was [0.94, 1.96], which contains one. Split-half reliability was identical across conditions (p = 0.998), so the low-load cloud is not simply noisier. The radius nonetheless sits near the within-participant noise bound, with radius-to-noise ratios of 0.82 under low load and 0.59 under high load, which means the pooled static estimate is at its resolving limit.
Pooling also overstated convergence. The pooled static shape sat below the geometric mean of the windowed sequence (paired t, p = 0.01 under low load and p < 0.001 under high load), and the between-participant contraction measured from windowed summaries was weaker, 1.23×, than the 1.55× seen in static fits. The authors read this as evidence that a pooled fit represents almost none of the data accurately, because shapes span a symmetric and a heavy-tailed regime and splitting the difference loses each person's distinct signature.
Baseline distance predicts how far load moves a person
The windowed decomposition carries the paper's strongest result. A participant's baseline level strongly predicted their load-induced change on both axes: r = -0.89 for frequency and r = -0.81 for severity, with the frequency correlation staying between -0.79 and -0.91 across the threshold sweep. Participants who began farthest from the attractor moved most toward it, and the high-load cloud settled near (f*, v*) ≈ (0.22, 1.78).
Correlating a baseline against its own change is mechanically biased negative, so the authors tested the finding against an exchangeability null. Shuffling alone produced only r ≈ -0.31 for frequency and -0.40 for severity, far short of the observed values, with permutation p = 0.0002 and 0.0024. Oldham's coupling-free correlation also stayed negative, at -0.51 and -0.43. On this evidence the convergence is not a regression-to-the-mean artifact.
The mean shifts were consistent with this picture but secondary. Among heavy-tailed windows, mean log α fell from 1.92 under low load to 1.77 under high load (p = 0.03), a significant deepening across τ between 10 and 11.5. The frequency of entering that regime dropped from 0.30 to 0.22, but only marginally at this threshold (one-sided p ≈ 0.07), strengthening to p = 0.03 for α ≥ 15. The authors therefore describe the effect as a contraction of between-participant variability onto a shared operating point, accompanied by a significant deepening of the tail, rather than a uniform population-wide shift.
A contraction reading, not an added-noise reading
The interpretation the authors favor is that a fluctuation signature is something spare capacity affords. When executive resources are committed to maintaining and switching between two ordered sets, motor output falls back toward a common operating regime, and individuality in the fluctuation signature is among the first things given up. This is a different claim from the familiar one that load adds noise. The authors note that an added-noise reading fits novel and kinematically difficult tasks better, where execution itself is a limiting factor, unlike a simple drawing task.
The paper also positions its method against prior use of the gamma plane. Earlier applications relied on non-parametric tests such as the Wilcoxon rank-sum test to detect differences. Those tests deliberately ignore the intrinsic curvature of the space the parameters lie on, and the authors argue an unstated consequence is that they can detect pairwise differences within each person while remaining agnostic to population-level compression of the dataset variance. The Fisher-Rao ruler makes the population-level question, whether a set of distributions has moved or tightened, answerable on the manifold rather than in flat space.
The authors are explicit that the Trail Making application is a concrete demonstration of the ruler rather than the principal claim, and that it is informative precisely because it is the regime where summary statistics and geometry-agnostic methods fail. The measurement instrument and the numerical solver are presented as the main contributions, with the solver to be released in MATLAB R2025b and Python 3.11 and verified against reference tables and theoretical invariants.
Where the convergence claim stops
The study is a proof of concept in 18 healthy adults under an experimental load manipulation, and the authors make no clinical claims. Only a subset of 17 participants entered the windowed heavy-tail decomposition. The sample's ages, motor capability, and cognitive capability are not representative. Condition order was not randomized, so the contraction finding could be limited to this specific ordering of the task. The authors checked for within-session drift under a time-ordered split and found none (p = 0.27 and 0.84), but they state this is not a substitute for counterbalancing.
The static contraction is the weaker of the two results and should be read that way. Its Fréchet radius sits at or below a deliberately conservative within-participant noise bound in both conditions, and its confidence interval on the contraction ratio contains one. The windowed baseline-versus-change correlations, tested against both a permutation null and Oldham's method, are what carry the convergence claim.
Nothing in the ruler is specific to stylus kinematics. The same construction applies to any positive-valued fluctuation series adequately described by a gamma family, such as inter-stride intervals, tremor amplitude, or speech envelope peaks, and the authors argue the invariance becomes more valuable as cohorts grow more heterogeneous in body size and anatomy, because it removes dependence on units and coordinates. Two questions follow directly and are testable with the same apparatus: whether a participant's baseline distance from the common operating point indexes something like reserve and predicts how much load it takes to move them, and whether populations with reduced executive capacity already sit near that operating point at baseline.
The windowed axes, not the pooled fit, carry the signal
For readers deciding what to take from this paper, the priority is the split between the two representations. The pooled static fit produces a contraction that is directionless, near the noise floor, and not significant on its own. The windowed frequency and severity axes produce strong, threshold-robust negative correlations between baseline and change, and those survive the regression-to-the-mean checks. If a future study reports only a pooled gamma fit, it is reporting the representation the authors themselves found least informative.
The second priority is the direction of the claim. The paper's result is convergence onto a shared operating regime near (f*, v*) ≈ (0.22, 1.78), with the largest movements coming from those farthest away at baseline, not a uniform shift of the population. That distinction matters for anyone using the Fisher-Rao distance as an outcome measure: a near-zero cohort mean displacement can coexist with substantial individual movement, and the informative quantity is the change in spread and its relation to baseline distance.
The third is the boundary the authors set themselves. The convergence evidence comes from 18 healthy adults in a non-randomized order, and the clinical extension is framed as a future question rather than a finding. The specific testable version of that question is whether reduced executive capacity shows up as a baseline already close to the operating point, which would make the absence of individuality under low load, rather than any shift under high load, the measurement of interest.
Sources used in this answer
Fisher-Rao Distance Detects Shifts in Kinematic Profiles under Cognitive Load
Using the Fisher-Rao distance on the gamma manifold, cognitive load displaced every one of 18 healthy adults' micromovement signatures (median distance 0.59, range 0.05-1.22) with no shared direction; participants instead converged toward a common operating regime, with baseline distance predicting the size of the shift (r = -0.89 for heavy-tail frequency, -0.81 for severity).