phys-gmm

This package contains the inference implementation (Gibbs Sampler) for the "Physically Consistent Bayesian Non-Parametric Mixture Model" (PC-GMM) proposed in [1]. This approach is used to automatically (no model selection!) fit GMM on trajectory data while ensuring that the points clustered in each Gaussian represent/follow a linear dynamics model, in other words the points assigned to each Gaussian should be close in "position"-space and follow the same direction in "velocity"-space.

Dependencies

[Necessary] LightSpeed Matlab Toolbox: Tom Minka's library which includes highly optimized versions of mathematical functions. Please download/install it in the thirdparty/lightspeed directory.
[Optional] crp: Frank Wood's Infinite Gaussian Mixture Model (IGMM) / Dirichlet process (DP) mixture model Matlab implementation. If you want to use/test option 2 of the given GMM fitting function you must download/install it in the thirdparty/crp directory. If you are not interested in this, it is not necessary.

Instructions and Content

This package offers the physically-consistent GMM fitting approach, as well as examples and code for fitting GMM with standard EM approach and the Bayesian non-parametric approach following the Chinese Restaurant Process construction through the [Mu, Priors, Sigma] = fit_gmm() function by filling its options as follows:

%%%%%%%%%%%%%%%%%% GMM Estimation Algorithm %%%%%%%%%%%%%%%%%%%%%%
% 0: Physically-Consistent Non-Parametric (Collapsed Gibbs Sampler)
% 1: GMM-EM Model Selection via BIC
% 2: CRP-GMM (Collapsed Gibbs Sampler)
est_options = [];
est_options.type             = 1;   % GMM Estimation Algorithm Type   

% If algo 1 selected:
est_options.maxK             = 15;  % Maximum Gaussians for Type 1
est_options.fixed_K          = [];  % Fix K and estimate with EM for Type 1

% If algo 0 or 2 selected:
est_options.samplerIter      = 20;  % Maximum Sampler Iterations
                                    % For type 0: 20-50 iter is sufficient
                                    % For type 2: >100 iter are needed
                                    
est_options.do_plots         = 1;   % Plot Estimation Statistics
est_options.sub_sample       = 1;   % Size of sub-sampling of trajectories

% Metric Hyper-parameters
est_options.estimate_l       = 1;   % Estimate the lengthscale, if set to 1
est_options.l_sensitivity    = 2;   % lengthscale sensitivity [1-10->>100]
                                    % Default value is set to '2' as in the
                                    % paper, for very messy, close to
                                    % self-interescting trajectories, we
                                    % recommend a higher value
est_options.length_scale     = [];  % if estimate_l=0 you can define your own
                                    % l, when setting l=0 only
                                    % directionality is taken into account

% Fit GMM to Trajectory Data
[Priors, Mu, Sigma] = fit_gmm(Xi_ref, Xi_dot_ref, est_options);

To test the function, you can either draw 2D data by running the demo script:

demo_drawData.m

or you can load pre-drawn 2D or real 3D datasets with the following script:

demo_loadData.m

Example Datasets

These examples + more datasets are provided in ./datasets folder. Following we show some notably challenging trajectory datasets that cannot be correctly clustered with the standard GMM either through MLE via the EM algorithm (center) or the CRP-GMM via collapsed Gibbs sampling (right), but are correctly clustered through our proposed approach (left).

GMM fit on 2D Concentric Circles Dataset

GMM fit on 2D Opposing Motions (Different Targets) Dataset

GMM fit on 2D Multiple Motions (Different Targets) Dataset

GMM fit on 2D Multiple Motions (Same Target) Dataset

GMM fit on 2D Messy Snake Dataset

Estimation Statistics

By setting est_options.do_plots= 1; the function will plot the corresponding estimation statistics for each algorithm.

For the PC-GMM we show the values of the posterior distribution p(C|...) and the estimated clusters at each iteration:

For the EM-based Model Selection approach we show the BIC curve computed with increasing K=1,...,15. The 1st and 2nd order numerical derivative of this curve is also plotted and the 'optimal' K is selected as the inflection point:

For the CRP-GMM we show the values of the posterior distribution p(Z|...) and the estimated clusters at each iteration:

Known Issues and Limitations

The only limitation of the proposed approach is its computational complexity. Although the collapsed gibbs sampler needs significantly less iterations than that for the CRP-GMM, it is computationally taxing as all customer assignments have to be sampled in each iteration. Specifically, the first iteration is slow as the sampler begins with N clusters....more discussion here.
Installation issues: If you are having trouble compiling the mex files for the lightspeed library on a Mac, follow the instructions here

Usage

Such physically-consistent clustering is particularly useful for learning Dynamical Systems (DS) that are formulated as Linear Parameter Varying (LPV) systems, as introduced in [1]. To use this approach for DS learning, download the ds-opt package.

References

[1] Figueroa, N. and Billard, A. (2018) "A Physically-Consistent Bayesian Non-Parametric Mixture Model for Dynamical System Learning". In Proceedings of the 2nd Conference on Robot Learning (CoRL).

Contact: Nadia Figueroa (nadia.figueroafernandez AT epfl dot ch)

Acknowledgments This work was supported by the EU project Cogimon H2020-ICT-23-2014.

nbfigueroa / phys-gmm