Change-in-velocity detection in multidimensional data

Representative lysosome trajectory in the periphery of a cell from ^[25]. The gray curves show the observed trajectory in the $ x $–$ y $ plane and the corresponding coordinate-wise time series. The green and red curves denote the inferred anchor positions, colored by inferred state: green indicates motile segments, and red indicates stationary segments. Green and red shaded regions indicate motile and stationary segments, respectively. In the segment-duration versus speed plot, point colors correspond to the inferred segment state

Failure of binary segmentation in a change-in-velocity setting. A 2D trajectory was simulated for 30 s at 20 Hz ($ \sigma = 0.1 $), with true changepoints at 10 and 20 s and segment velocities $ (v_x, v_y) = (0.1, -0.1) $, $ (0, 0) $, and $ (-0.1, 0.1)\, \mu $m/s. Left: CPLASS criterion values for the null (green dashed) and one-changepoint models (red solid), showing that binary segmentation introduces a spurious changepoint (higher value is better). Right: Observed $ x $ and $ y $ positions (gray), true changepoints (dashed), and fitted segmentation (red)

Discontinuous versus continuous piecewise-linear approximations. A simulated 2D lysosomal trajectory is shown via $ t $–$ x $ and $ t $–$ y $ time series, with true changepoints at 9 and 10.35 s. Detected changepoints are shown as dashed lines, with the fitted segmentation overlaid in red. The red column in the CPLASS output table indicates a significant active segment missed by BCP

Example showing why gradient ascent is unsuitable for the change-in-speed problem. (A) A simulated 2D lysosomal trajectory (100 Hz, 6 s) with true changepoints at 3 and 3.5 s and segment speeds $ (0, 0.2, 0)\, \mu $m/s. Noisy $ t $–$ x $ and $ t $–$ y $ time series are shown in gray, with true changepoints (dashed) and the true segmentation (red). (B) Contours of CPLASS score differences between the two-changepoint model $ \Phi(r_2) $ and the no-changepoint model $ \Phi(r_0) $. Although the global maximum occurs at the true changepoints, gradient ascent can become trapped in local maxima and fail to recover them

Necessity of the new proposal function (Numerical Experiment 2.4.2). (A) A simulated 2D lysosomal trajectory (100 Hz, 6 s) with true changepoints at 3 and 3.5 s, shown in $ t $–$ x $ and $ t $–$ y $ coordinates. True changepoints are dashed; the fitted segmentation is overlaid in red. (B) Candidate changepoint models with their log-likelihoods, criterion values, and corresponding MH decisions (Algorithm 1)

Sensitivity of CPLASS to $ \gamma $. (A) Detection probabilities for $ n = 53 $. (B) Corresponding results for $ n = 203 $. Blue curves show $ \Pr(\widehat{|r|} = 2\mid H_1) $, and red curves show $ \Pr(\widehat{|r|} > 0\mid H_0) $. Dark solid and light dashed curves denote CPLASS with and without the speed penalty, respectively. The vertical line marks $ \gamma = 1.01 $; horizontal lines mark probabilities 0.80 and 0.05. (C, D) Representative trajectories (path 1) for $ n = 53 $ and $ n = 203 $. Gray curves show observations, red dashed curves show the generating trajectories, and blue curves show CPLASS fits. Gray long-dashed and black double-dashed vertical lines mark inferred and true changepoints, respectively

Necessity of the speed penalty (Section 3.1.2). We simulated 250 trajectories from the parameter sets in Table 1 of ^[29], sampled at 25 Hz. (A) CSA: theoretical CSA (black), CPLASS with speed penalty (green), CPLASS without speed penalty (blue), and BCP ^[25] (orange). (B) Empirical cumulative distributions of maximum sustained segment speeds (minimum duration $ 0.6 $ s) for BCP (orange), CPLASS with (green) and without (blue) the speed penalty, compared with the true values (black). (C, D) A representative trajectory after running CPLASS without/with speed penalty

Power analysis comparing CPLASS and BCP (Section 3.2). A total of 20,000 trajectories were simulated at 20 Hz with two true changepoints ($ \sigma = 0.01 $). The first and third segments have fixed durations of 2 s, while the middle segment varies across 20 durations and 20 speeds ($ \mu $m/s). For each speed–duration pair, 50 trajectories were generated. The detection rate $ {\bf P}_{\text{correct}} $ is the percentage of paths for which the correct number of changepoints is identified. Trajectories follow the same simulation design as Figure 6(C), (D)

Comparison of CPLASS with alternative changepoint methods. Results are based on 100 frozen trajectories per method and simulation regime. (A) Precision, defined as the proportion of inferred changepoints matched to a generating changepoint. (B) Recall, defined as the proportion of generating changepoints successfully localized. In panels A and B, matches were determined one-to-one within five observations ($0.20$ s). (C) Ratio of the inferred to the true number of changepoints. The dotted line denotes the correct ratio of one, and the vertical axis is logarithmic. (D) Type Ⅰ error, defined as the proportion of null trajectories for which at least one changepoint was selected. The dotted line denotes $5\%$. Scalar methods were applied separately to the $x$- and $y$-coordinates, followed by exact union of the reported changepoints. Error bars in panels A–C show normal-approximation $95\%$ intervals for the trajectory-level means; error bars in panel D show $95\%$ Wilson intervals

CSA plots for different group comparisons (Section 3.4). (Left) The CSA bootstrap ensemble curves compare lysosomal transport in perinuclear and periphery regions. (Right) The CSA bootstrap ensemble curves compare the sucrose-treated groups restricted to the periphery region of the cell from ^[25]

CSA plot for different motor families (Section 3.5). (Left) The CSA bootstrap ensemble curves compare cargo trajectories among three groups of motor transports: kinesin-1 (Kin1), dynein-dynactin-BicD2 (DDB), and kinesin-1/DDB pairs. (Right) WKDE of inferred segment speeds, duration-weighted, for the three groups

CSA and WKDE plots for EB1-GFP microtubule comet trajectories (Section 3.6). (A) CSA bootstrap ensemble curves compare inferred speeds across two larvae of the same cell type (top 2: 14 trajectories, gray; top 3: 16 trajectories, teal). (B) Duration-weighted kernel density estimates (WKDEs) of inferred segment speeds for the same two animals. Vertical lines indicate the duration-weighted mean speed for each animal (top 2: $ 6.32\, \mu\mathrm{m}/\mathrm{min} $, gray; top 3: $ 5.92\, \mu\mathrm{m}/\mathrm{min} $, teal). Both panels report speeds in $ \mu\mathrm{m}/\mathrm{min} $. The similar distributions across the two animals suggest that CPLASS provides a stable summary of EB1-GFP comet speed in this preliminary dataset

EB1-GFP microtubule comet imaging and TrackMate trajectory output (Section 3.6). (A, B) Representative first frames from two imaging videos of the same cell type (class Ⅰ da sensory neuron) from two different larvae of the same genotype (top 2 and top 3, respectively). Each video is a zoomed-in field of view of the dendritic arbor; the neuron's cell body (soma) lies outside the imaged region, above the field shown, and dendrite direction is defined relative to it ("top" denotes the side nearer the soma, used to establish comet travel direction). EB1-GFP labels the growing plus ends of microtubules within these dendrites. The bright branching structures illustrate the curved and spatially complex geometry along which comets were tracked. (C) Comet trajectories extracted from the top 2 video using TrackMate, with each color representing a distinct tracked comet. (D) Comet trajectories for the top 3 video. These trajectories serve as input to CPLASS, which segments each path into piecewise-linear velocity segments as summarized in Figure 12. The branched and curved trajectory shapes in panels C and D reflect the underlying dendritic geometry visible in panels A and B