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The discriminative functional mixture model for a comparative analysis of bike sharing systems

Charles Bouveyron, Etienne Côme, Julien Jacques

arXiv:1601.07999v1stat.AP

TL;DR

Large bike-sharing datasets need methods that can identify operating patterns and compare systems across cities, while prior approaches were limited to one system and did not explicitly model functional data. The paper introduces FunFEM, a functional mixture clustering method, and applies it to eight European systems, identifying ten general patterns, including pathological ones.

  • Problem

    Prior BSS clustering studies were limited to one city and did not explicitly model the functional nature of station data, despite the need to summarize large datasets and compare service levels across cities.

    Source: §1 · p. 3

  • Method

    FunFEM clusters functional data in a discriminative subspace using a functional mixture model, with an algorithm for model inference.

    Source: §1 · p. 3, §3.3 · p. 9, §3.5 · p. 13

  • Results

    FunFEM’s comparison of eight European BSS identified ten general operating patterns, including pathological ones, and supported practical improvement strategies.

    Source: §1 · p. 3, §6 · p. 31

  • Takeaways & Limitations

    The discriminative subspace offers an interpretable way to compare systems by operating patterns, and clustering results support recommendations to BSS operators.

    Source: §6 · p. 31

    AIC and BIC may be inappropriate for mixture models because their required regularity conditions are not fully satisfied in that framework.

    Source: §3.6 · p. 14

Abstract

from arXiv · show

Bike sharing systems (BSSs) have become a means of sustainable intermodal transport and are now proposed in many cities worldwide. Most BSSs also provide open access to their data, particularly to real-time status reports on their bike stations. The analysis of the mass of data generated by such systems is of particular interest to BSS providers to update system structures and policies. This work was motivated by interest in analyzing and comparing several European BSSs to identify common operating patterns in BSSs and to propose practical solutions to avoid potential issues. Our approach relies on the identification of common patterns between and within systems. To this end, a model-based clustering method, called FunFEM, for time series (or more generally functional data) is developed. It is based on a functional mixture model that allows the clustering of the data in a discriminative functional subspace. This model presents the advantage in this context to be parsimonious and to allow the visualization of the clustered systems. Numerical experiments confirm the good behavior of FunFEM, particularly compared to state-of-the-art methods. The application of FunFEM to BSS data from JCDecaux and the Transport for London Initiative allows us to identify 10 general patterns, including pathological ones, and to propose practical improvement strategies based on the system comparison. The visualization of the clustered data within the discriminative subspace turns out to be particularly informative regarding the system efficiency. The proposed methodology is implemented in a package for the R software, named funFEM, which is available on the CRAN. The package also provides a subset of the data analyzed in this work.

1. Introduction.

The paper addresses how to extract useful patterns from large bike-sharing data and compare systems across cities. It introduces FunFEM, a functional-data clustering method, and applies it to eight European systems.

1. Introduction.

  • Analyzing BSS data can inform system design, pricing policies, and service improvements such as redistributing bikes among stations.The introduction describes these as potential benefits of statistical analysis.Source: §1 · p. 2
  • Large volumes of BSS data motivate automatic clustering to summarize station behavior and compare service levels through cluster proportions across cities.Station clusters have also been used to study network imbalance and inform reallocation strategies.Source: §1 · p. 2
  • FunFEM’s comparison of eight European systems identifies healthy and pathological operating patterns and supports practical improvement strategies.The method models data in a discriminative functional subspace that also supports visualization and system comparison.Source: §1 · p. 3
  • FunFEM models time series in a discriminative functional subspace; its 12-model family accommodates varied settings, with BIC or the slope heuristic for selection and optional sparsity.Sparsity is introduced through an ℓ1-type penalization to select discriminative basis functions.Source: §1 · p. 3
  • Earlier studies were limited to one city and did not explicitly model functional data, often relying on distances or time-period summary features.Prior methods included feature-based and hierarchical clustering, as well as Poisson-mixture clustering of hourly usage profiles.Source: §1 · pp. 2–3

2. The BSS data.

The study analyzes station occupancy from eight European bike-sharing systems over about one month. It normalizes station availability, smooths the profiles using a weekly Fourier basis, and uses functional clustering to compare systems.

2. The BSS data.

  • The data cover eight European systems and 3,230 station loading profiles sampled at 1,448 time points over five weeks in 2014.The cities span different geographic positions and system sizes.Source: §2 · pp. 4–5
  • Available bikes and docks were downloaded hourly from open-data APIs, and available-bike counts were normalized by station size to create each station’s loading profile.The sources were JCDecaux APIs and the Transport for London initiative.Source: §2 · pp. 4–5
  • Because station profiles have a natural weekly period, the study projected them onto 41 Fourier basis functions to smooth the curves.The plotted examples show observations alongside smoothed profiles, with some stations displaying less clear patterns.Source: §2 · p. 5, Figure 2 · p. 6
  • The study aims to identify patterns in station loading profiles through functional clustering and use them to compare the eight systems.The map marks the study locations, with dot size indicating system size.Source: Figure 1 · p. 5, §2 · p. 6

3. The discriminative functional mixture model.

The section develops functional mixture clustering by representing curves in a discriminative subspace, then presents FunFEM inference, model selection, and sparse basis selection. Its model family is parsimonious, with complexity linear in the number of basis functions.

3.1. Related work in functional clustering.

  • Prior functional-clustering approaches span spline and wavelet representations, regression mixtures, parametric time series, and model-based clustering of functional coefficients.Source: §3.1 · p. 7

3.2. Transformation of the observed curves.

  • Curves observed at discrete times are reconstructed in a finite basis, using interpolation without noise or least-squares smoothing when observations contain error.Source: §3.2 · pp. 7–8

3.3. The model.

  • The model represents curves in a low-dimensional discriminative subspace, with group-specific Gaussian structure inside it and noise variance β outside it.Source: §3.3 · pp. 8–9

3.4. A family of discriminative functional model.

  • The DFM family contains 12 covariance models whose complexity grows linearly with the number of basis functions, rather than quadratically as in many model-based approaches.Source: §3.4 · p. 10

3.5. Model inference: The FunFEM algorithm.

  • FunFEM alternates inference steps and estimates the discriminative subspace by seeking directions that reduce within-group variation relative to between-group separation.The discriminative functions are obtained recursively from generalized eigenproblems, taking the eigenvector associated with the largest eigenvalue at each step.Source: §3.5 · pp. 10, 11, 13

3.6. Model selection.

  • Model selection jointly chooses the DFM parametrization, cluster count K, and dimension d; the slope heuristic supplements AIC and BIC, which may be inappropriate for mixtures.Source: §3.6 · pp. 14–15

3.7. Selection of discriminative basis functions.

  • Discriminative basis functions are selected by constraining the loading matrix U to be sparse through ℓ1-penalized optimization, rather than simply thresholding large loadings.Source: §3.7 · p. 15

4. Numerical experimentations.

Simulated experiments assess model selection and sparse discriminative-basis selection, followed by comparisons on four real datasets. FunFEM is competitive with six state-of-the-art methods, while sparse selection identifies relevant periodicities and time intervals in the simulations.

4.1. Model selection.

  • In one simulated run, both BIC and the slope heuristic selected the actual cluster count, K = 4; the slope heuristic's penalty was close to BIC's.Figures 5 and 6 show the criteria for that run.Source: §4.1 · p. 16, Figure 5 · p. 17, Figure 6 · p. 17
  • Across 100 simulations of 12 DFM models, the slope heuristic was more consistent, while BIC could be unsatisfactory in difficult inference settings but efficient when the model was appropriate.Tables 3 and 4 report the selected cluster counts; the actual number of clusters was four.Source: §4.1 · p. 17, Table 3 · p. 18, Table 4 · p. 18

4.2. Selection of discriminative basis functions.

  • Sparse FunFEM selected two of 25 Fourier basis functions, corresponding to the two periodicities in the simulated data and retaining its main features.The selected functions represent the most discriminative periodicities.Source: §4.2 · p. 19, Figure 8 · p. 20
  • With spline smoothing, sparse FunFEM selected three basis functions to identify time intervals that distinguish the simulated clusters.The selected intervals distinguish the green cluster, black and green clusters, and black cluster from the others.Source: §4.2 · p. 20, Figure 9 · p. 20

4.3. Comparison with state-of-the-art methods.

  • FunFEM outperforms the other methods on three of four datasets and is second-best on the remaining dataset; gains are significant on Kneading, ECG, and Wafer.The accuracies are reported in Table 5.Source: Table 5 · p. 21, §4.3 · p. 21

5. Analysis of bike sharing systems.

The section applies FunFEM to stock data from eight European bike-sharing systems, using functional clustering and a discriminative subspace to compare station patterns and identify operational issues.

5. Analysis of bike sharing systems.

  • The study applies FunFEM to one month of stock data from eight European systems to compare operating patterns and visualize potential system issues.The analysis uses the method's parsimonious modeling to handle time series with 1,448 time points.Source: §5 · pp. 21–22

5.1. Clustering results for Paris stations.

  • Paris yielded ten roughly balanced clusters spanning commute-phase patterns, weekend differences, stable fullness levels, nighttime rebalancing, and nearly empty stations.The cluster profiles and proportions are shown with sampled station curves.Source: §5.1 · pp. 22–23, Figure 11 · p. 24
  • Paris clusters also grouped spatially: Morning stations occupied employment-dense centers, while Afternoon stations were in residential areas and Full stations were central.Balanced stations were primarily peripheral; Night rebalancing stations often occupied uphill locations.Source: §5.1 · pp. 25–26, Figure 12 · p. 25

5.2. Clustering results on several cities.

  • Across eight cities, Paris had more Night rebalancing and less Balanced activity, while Marseille and Bruxelles had many Full and Empty stations and low shares of active clusters.Valencia, Lyon, and London appeared more active and balanced in the comparison.Source: §5.2 · pp. 26–27, Figure 13 · p. 27
  • FunFEM's subspace separates fullness on axis 1 from commute phase on axis 2, and its city densities distinguish Marseille and Bruxelles from other systems.The third and fourth axes were less interpretable and appeared primarily linked with Night rebalancing.Source: §5.2 · pp. 27–29, Figure 14 · p. 28, Figure 16 · p. 29

5.3. Recommendations for BSS operators.

  • For Marseille and Bruxelles, bonus status had limited effect on Full-station distributions, prompting recommendations to revise incentives or increase nighttime rebalancing.The paper also recommends commute-linking bike paths modeled on London's Cycle Superhighways to improve system commutes.Source: §5.3 · pp. 29–30, Figure 17 · p. 30

6. Conclusion.

The conclusion presents FunFEM as a parsimonious way to model functional BSS data and compare operating patterns across systems. Applied to eight systems, it produced interpretable results and a discriminative subspace that may also help track station behavior under policy changes.

6. Conclusion.

  • The work aims to compare European BSSs to identify shared operating patterns and propose practical solutions to potential issues.The DFM model was proposed to model the functional data generated by these systems.Source: §6 · p. 30
  • Numerical experiments found FunFEM efficient on simulated and benchmark data, a challenger to leading methods, and supported the slope heuristic for model selection.The DFM models functional data in a discriminative functional subspace; sparsity can be introduced afterward through an ℓ1-type penalty for selecting discriminative basis functions.Source: §6 · p. 31
  • Applied to one month of usage statistics from 8 bike sharing systems, FunFEM produced interpretable results and recommendations for operators.Its parsimonious model and discriminative subspace provide a compact representation of system behaviors and a way to compare systems by identified operating patterns.Source: §6 · p. 31
  • Sliding-window projections of station descriptions into the discriminative subspace can form trajectories that may help detect behavior changes during pricing or bonus policy trials.The proposed operational use is to track station behavior and check the effects of new policies.Source: §6 · p. 31

APPENDIX A: ADDITIONAL INFORMATION ABOUT THE BENCHMARK DATA SETS

The appendix describes the Kneading benchmark and the ECG, Face, and Wafer benchmarks, including their sampling and curve-reconstruction choices.

APPENDIX A: ADDITIONAL INFORMATION ABOUT THE BENCHMARK DATA SETS

  • The Kneading benchmark contains 115 flour curves sampled at 241 instants over 480 seconds and reconstructed with 18-knot cubic B-splines.The flours produced 50 good-quality, 25 medium-quality, and 40 low-quality cookies.Source: Appendix A · p. 31
  • ECG, Face, and Wafer contain 200, 112, and 7,174 curves, respectively, sampled at 96, 350, and 152 instants; all use 20 cubic B-splines.The datasets comprise 2 ECG patient groups, 4 Face groups, and 2 Wafer groups.Source: Appendix A · p. 32

APPENDIX B: DETAILED CLUSTERING RESULTS ON THE 8 BSS

The appendix provides cluster-mean profiles with sampled curves and maps of clustering results for the eight bike sharing systems.

APPENDIX B: DETAILED CLUSTERING RESULTS ON THE 8 BSS

  • Cluster-mean profiles are shown with 1,000 randomly sampled curves for the full dataset across seven named systems.The caption lists Paris, London, Bruxelles, Lyon, Valencia, Sevilla, and Nantes.Source: Figure B.1 · p. 33
  • Maps show clustering results for Paris, London, Bruxelles, Lyon, Valencia, Sevilla, and Nantes.The caption specifies their order from left to right and top to bottom.Source: Figure B.2 · p. 34
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