Change-in-velocity detection for multidimensional data

TL;DR

This work addresses the challenge of detecting changes in velocity in multidimensional time series, notably intracellular transport trajectories, where continuity constraints undermine traditional changepoint methods. It introduces CPLASS, an MCMC-based framework that fits a continuous piecewise-linear trajectory and optimizes a penalty-augmented likelihood, augmented by a biophysically informed speed penalty and a Cumulative Speed Allocation (CSA) statistic. A consistency theorem for the penalized MLE under a strengthened SIC-like penalty with gamma>1 is established, and the method is shown to outperform change-in-mean approaches in detecting short or slow motile segments while producing more realistic speed estimates in simulated and real data (lysosomal and quantum-dot transport). The approach delivers practical tools for single-particle tracking analysis, enabling robust, multidimensional changepoint inference and biologically plausible interpretations of motor-driven transport, albeit with computational cost typical of MCMC methods. Potential extensions include faster search strategies, improved CSA inference, and broader applicability to multidimensional trajectory data beyond intracellular transport.

Abstract

In this work, we introduce CPLASS (Continuous Piecewise-Linear Approximation via Stochastic Search), an algorithm for detecting changes in velocity within multidimensional data. The one-dimensional version of this problem is known as the change-in-slope problem (see Fearnhead & Grose, 2022; Baranowski et al., 2019). Unlike traditional changepoint detection methods that focus on changes in mean, detecting changes in velocity requires a specialized approach due to continuity constraints and parameter dependencies, which frustrate popular algorithms like binary segmentation and dynamic programming. To overcome these difficulties, we introduce a specialized penalty function to balance improvements in likelihood due to model complexity, and a Markov Chain Monte Carlo (MCMC)-based approach with tailored proposal mechanisms for efficient parameter exploration. Our method is particularly suited for analyzing intracellular transport data, where the multidimensional trajectories of microscale cargo are driven by teams of molecular motors that undergo complex biophysical transitions. To ensure biophysical realism in the results, we introduce a speed penalty that discourages overfitted of short noisy segments while maintaining consistency in the large-sample limit. Additionally, we introduce a summary statistic called the Cumulative Speed Allocation, which is robust with respect to idiosyncracies of changepoint detection while maintaining the ability to discriminate between biophysically distinct populations.

Figures (14)

• Figure 1: Binary segmentation failure in a change-in-velocity setting. A 2D trajectory was simulated at 20Hz (the position was observed every $1/20 = 0.05$ seconds) for 30 s with Gaussian noise ($\sigma = 0.1$). Two true changepoints occur at 10 s and 20 s, generated with velocity vectors $v_x = (0.1, 0, -0.1),\mu$m/s and $v_y = (-0.1, 0, 0.1),\mu$m/s. Left panel: CPLASS criterion values, with the null model (green dashed) and one-changepoint models (red solid). Higher values indicate a better fit. The figure illustrates how Binary segmentation incorrectly introduces an extra changepoint between the true ones. Right panel: The data on which the criterion function was calculated. Time series for $x$ and $y$ positions (gray), true changepoint times (dashed), and segmentation fit (red).
• Figure 2: Comparison between discontinuous and continuous piecewise linear approximations. The simulated lysosomal movement trajectory in 2D at 20Hz for a duration of 20.3 seconds and the two actual changepoints, 9 seconds and 10.35 seconds, are represented by the t-vs-x and t-vs-y time series. The dashed lines represent the detected changepoints. The corresponding segmentation is overlaid in red. The red column in the CPLASS output table indicates a significant active segment that BCP missed.
• Figure 3: Example showing why gradient ascent is unsuitable for the change-in-speed problem. Panel (A) shows a simulated 2D lysosomal trajectory at 100Hz for 6s, containing two true changepoints at 3s and 3.5s. The corresponding segment speeds are $(0, 0.2, 0)$$\mu$m/s. The noisy $t$–$x$ and $t$–$y$ time series are plotted in gray, with the true changepoints marked by dashed lines and the true segmentation overlaid in red. Panel (B) shows a contour plot of CPLASS criterion value differences between two-changepoint models, $\Phi(r_2)$, and the no-changepoint model, $\Phi(r_0)$. The global maximum corresponds to the true changepoints, but a gradient ascent search may get trapped in local maxima and fail to recover them.
• Figure 4: Necessity of the new proposal function, summaries of results from Numerical Experiment \ref{['numex: why_prop_type_3']}. Panel (A), the simulated lysosomal movement trajectory in 2D at 100Hz for a duration of 6 seconds and the two actual changepoints, 3 seconds and 3.5 seconds, is represented by the t-vs-x and t-vs-y time series. The dashed lines represent the real changepoints. The corresponding segmentation is overlaid in red. Panel (B) shows different models with changepoints and their corresponding log-likelihood, the algorithm criterion value, and the decision in the MH algorithm \ref{['alg:Gibbs']}.
• Figure 5: Investigating CPLASS on detecting motile short segments while varying the $\gamma$ values (Section \ref{['sec:vary_linear_pen']}).Panel (A) 200 simulation paths over $2.65$ seconds at 20Hz with $s=0$, $\sigma = 0.01$ for the null hypothesis. 200 simulation paths over $2.65$ seconds at 20Hz ($n = 53$) with two actual changes at $t = 1.1$s and $t = 1.55$s, three segments speeds $(s_1,s_2,s_3) = (0,0.1,0) \mu$m/s, $\sigma = 0.01$ for the alternative hypothesis. Panel (B) 200 simulation paths over $10.15$ seconds at 20Hz ($n = 203$) with $s=0$, $\sigma = 0.01$ for the null hypothesis. 200 simulation paths over $10.15$ seconds at 20Hz with two actual changes at $t = 5$s and $t = 5.15$s, three segments speeds $(s_1,s_2,s_3) = (0,0.15,0)\mu$m/s, $\sigma = 0.01$ for the alternative hypothesis. The red colors show the simulation results under the null hypotheses, and the blue colors show the simulation results under the alternative hypotheses. Panel (C) and (D) The simulated paths under the null model with no changepoints and the alternative model with two stationary segments and the motile short segment at the middle.

Theorems & Definitions (4)

• Lemma 1
• proof : Proof of Lemma \ref{['lem:bound-normal-distance']}
• proof
• proof : Proof of Theorem \ref{['thm:consistency-pen-MLE_1']}