Skip to main page content
Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
. 2015 Jan;25(1):5-24.

A MULTIVARIATE GAUSSIAN PROCESS FACTOR MODEL FOR HAND SHAPE DURING REACH-TO-GRASP MOVEMENTS

Affiliations
Free PMC article

A MULTIVARIATE GAUSSIAN PROCESS FACTOR MODEL FOR HAND SHAPE DURING REACH-TO-GRASP MOVEMENTS

Lucia Castellanos et al. Stat Sin. .
Free PMC article

Abstract

We propose a Multivariate Gaussian Process Factor Model to estimate low dimensional spatio-temporal patterns of finger motion in repeated reach-to-grasp movements. Our model decomposes and reduces the dimensionality of variation of the multivariate functional data. We first account for time variability through multivariate functional registration, then decompose finger motion into a term that is shared among replications and a term that encodes the variation per replication. We discuss variants of our model, estimation algorithms, and we evaluate its performance in simulations and in data collected from a non-human primate executing a reach-to-grasp task. We show that by taking advantage of the repeated trial structure of the experiments, our model yields an intuitive way to interpret the time and replication variation in our kinematic dataset.

Keywords: Dynamical factor analysis; experiment structure; multivariate Gaussian process; reach-to-grasp; registration; variance decomposition.

Figures

Figure 1
Figure 1
Example of energy profiles raw and aligned (small cone, 45° adduction). On the x-axis we show time and on the y-axis the value of the energy functions.
Figure 2
Figure 2
MGPFM: Average MISE in the test set as a function of number of training samples. The figure shows the results on a single test set of 500 samples using ten independent training sets of size 80. We report the average mean error in the test set for the ten simulations ((110)i=110ESi) and its standard error. The dimensionality of the observed data was p = 50 and the latent dimension dtrue = 4. We modelled Σ as unconstrained, the μ with splines and unconstrained, and initialized the learning algorithm with the down-projection.
Figure 3
Figure 3
MGPFM: Average MISE in the test set, and BIC for different values of the latent dimensionality when varying the initialization regimes and the ways of modelling μ. The true latent dimension d = 4 is most obviously recovered when μ is modelled with splines and the initialization is through the projection procedure. When μ is modelled with splines the initialization regime has a large impact on performance.
Figure 4
Figure 4
MGPFM: we show 10-fold cross validated MISE in grasping data for two conditions (small cone flexion and adduction), comparing the baseline model considering only the mean, PCA, MGPFM on raw pre-processed data and MGPFM on aligned data. Experiments were run for various sizes of latent dimensionality d. The MGPFM was applied modelling μ with splines, Σ constrained, initializing with the MLE of the matrix normal distribution, and 50 iterations of the learning procedure. Observe that the MGPFM applied on aligned data achieves better results than other methods, but its advantage decreases as the size of the latent dimension increases.
Figure 5
Figure 5
Error profiles in data (small cone, 45° abduction). In the upper panel, each line corresponds to a replication and the dashed line is the mean value. The lower panel displays the mean integrated square error per replication. The MGPFM reduces the error significantly, as compared to the baseline of modelling only the mean (by approximately an order of magnitude).
Figure 6
Figure 6
The trajectory μ^ (projected onto the position space) represents what is shared by all replications of a condition. Here we show the visualization of μ^ for the small cone at 45° of abduction. The trajectory presents five epochs corresponding to hand postures: all plots are in the same scale and the third epoch looks very similar to the first and last epochs (neutral position), so we omit it. The trajectory begins at a neutral position, followed by a slight opening of the grasp through a synchronized movement of fingers and a slight rotation, then back to the neutral hand configuration after which the fingers spread slightly (after the subject releases the object) and back to a neutral position. All replications for this condition followed this pattern; they differentiated among themselves with the movement modelled through the loading matrix and the MGP term.
Figure 7
Figure 7
Visualization of loadings encoded in for the small cone presented at 45° abduction. The first loading corresponds to synchronized opening-closing of the hand; the second loading to curling of the fingers wrapping around the cone. The estimation of is explained in the Supplementary Material Section S1.2 and it can be reduced to solving a specific linear regression problem (Equation (S1.9)).
Figure 8
Figure 8
Learned factors X.^ for condition: small cone, 45° abduction. In the top panel we show (in green) the distribution of learned factors in the velocity space (left) and their integrated version on the positional space (right). This figure depicts differences between trials in the space of learned factors. On this plot we overlap two exemplary trials. In the middle and lower panel we show details of these replications: the shape and values they display are different. The starting point of the trial is denoted by an open circle, the end position, by a star. There are arrows along the trajectory show the direction of movement. Arrows marked with different symbols represent time and allow for comparison between trajectories: arrow with circle (33%), arrow with star (40%), arrow with spade (50%), arrow with double spade (54%), arrow with cross (60%). In Figure 9 we show how difference between the integrated learned factors in these two trials manifest on hand posture.
Figure 9
Figure 9
Interpretation of latent factors showing differences between replications. On the left we plot (X.^1(t),X.^2(t)) as a function of t. The start of the trial is at the open circle, the solid dot corresponds to t = 50, the triangle to t = 58 and the star to the end of the trial. Middle and right panels: hand configurations corresponding to those time points. The interaction between the first latent factor (moving negatively) and the corresponding loading (Figure 9 panel 1) corresponds to an opening of the fingers in a synchronized manner – this movement differs between the two replications and leads to an exaggerated opening of the hand in replication 1 (top panel).

Similar articles

See all similar articles

Cited by 3 articles

LinkOut - more resources

Feedback