Estado Actual de la Construcción de Marca Universitaria

maximizing the logarithm of the marginal likelihood in equation (2.20), where the elements of the matrix Kf,f are computed using expression (3.13) with k_fq

d,f q d0(t, t

0₎ given by (3.14). For prediction we use expression (2.23) and the posterior distri- bution is found using expression (2.24), where the elements of the matrix Kf,u, kfd,uq(t, t

0_{) = k} fq

d,uq(t, t

0_{), are computed using}

Sd,q Z T Gd(t − τ)kuq,uq(τ, t 0_{)dτ. (3.15)}

3.2.3 Multidimensional inputs

In the sections above we have introduced latent force models for which the input variable is one-dimensional. For higher-dimensional inputs, x ∈ <p_{, we can use} linear partial differential equations to establish the dependency between the latent forces and the outputs. The initial conditions turn into boundary conditions, specified by a set of functions that are linear combinations of yd(x) and its lower derivatives, evaluated at a set of specific points of the input space. Inference and learning is done in a similar way to the one-input dimensional latent force model. Once the Green’s function associated to the linear partial differential operator has been established, we employ similar equations to (3.14) and (3.15) to compute kfd,fd0(x, x

0_{) and k}

fd,uq(x, x

0_{) and the hyperparameters appearing in the covariance} function are estimated by maximizing the marginal likelihood.

3.3 A Latent Force Model for Motion Capture

Data

In section 3.1 we introduced the analogy of a marionette’s motion being controlled by a reduced number of forces. Human motion capture data consists of a skeleton and multivariate time courses of angles which summarize the motion. This motion can be modelled with a set of second order differential equations which, due to variations in the centers of mass induced by the movement, are non-linear. The simplification we consider for the latent force model is to linearize these differential equations, resulting in the following second order system,

mdd 2_y d(t) dt2 + υd dyd(t) dt + κdyd(t) = Q X q=1 Sd,quq(t) + ˆed(t). (3.16)

Whilst (3.16) is not the correct physical model for our system, it will still be helpful when extrapolating predictions across different motions, as we shall see later. Note also that, although similar to (3.4), the dynamic behavior of this system is much richer than that of the first order system, since it can exhibit inertia and resonance. In what follows, we will assume without loss of generality that the masses are equal to one.

For the motion capture data, yd(t) corresponds to a given observed angle over time, and its derivatives represent angular velocity and acceleration. The system is summarized by the undamped natural frequency, ω0d = √κd, and the damping ratio, ζd = 1₂υd/√κd. Systems with a damping ratio greater than one are said to be overdamped, whereas underdamped systems exhibit resonance and have a damping ratio less than one. For critically damped systems ζd = 1, and finally, for undamped systems (this is no friction) ζd= 0.

Ignoring the initial conditions, the solution of (3.16) is given by the integral operator of equation (3.11), with Green’s function

Gd(t, s) = _ω1

dexp(−αd(t − s)) sin(ωd(t − s)), (3.17) where ωd= p4κd− υd2/2 and αd= υd/2.

According to the general framework described in section 3.2.2, the covariance function between the outputs is obtained by solving expression (3.14), where kuq,uq(t, t

0_{) follows the SE form in equation (3.7). Solution for k} fq

d,f q d0(t, t

0_{) is then} given as (´Alvarez et al., 2009)

K0hq(eγd0, γ_d, t, t0) + h_q(γ_d, e γd0, t0, t) + h_q(γ_d0, e γd, t, t0) + hq(eγd, γd0, t0, t) − hq(eγd0, e γd, t, t0) − hq(eγd,eγd0, t 0_{, t) − h} q(γd0, γ_d, t, t0) − h_q(γ_d, γ_d0, t0, t), where K0 = `q√π/8ωdωd0, γ_d = α_d+ jω_d and e

γd = αd− jωd, and the functions hq(eγd0, γ_d, t, t0) follow

hq(γd0, γ_d, t, t0) = Υq(γd

0, t0, t) − e−γdtΥ_q(γ_d, t0, 0)

3.3. A LATENT FORCE MODEL FOR MOTION CAPTURE DATA with Υq(γd0, t, t0) 2e „ `2<sub>q γ</sub>2 d0 4 « e−γ<sub>d0</sub>(t−t0<sub>)</sub> − e „ −(t−t0)2 `2_q « w(jzd0_,q(t)) − e „ −(t0)2 `2q « e(−γd0t)w(−jz d0_,q(0)), (3.18)

and zd0_,q(t) = (t − t0)/`<sub>q</sub>− (`_qγ_d0)/2. Note that z_d0_,q(t) ∈ C, and w(jz) in (3.18), for z ∈ C, denotes Faddeeva’s function w(jz) = exp(z2_{)erfc(z), where erfc(z) is} the complex version of the complementary error function, erfc(z) = 1 − erf(z) =

2 √ π

R∞

z exp(−v2)dv. Faddeeva’s function is usually considered the complex equiv- alent of the error function, since |w(jz)| is bounded whenever the imaginary part of jz is greater or equal than zero, and is the key to achieving a good numerical stability when computing (3.18) and its gradients.

Similarly, the cross-covariance between latent functions and outputs in equation (3.15) is given by kfq d,uq(t, t 0_{) =} `qSd,q √ π j4ωd [Υq(eγd, t, t 0_{) − Υ} q(γd, t, t0)].

Motion Capture data

Our motion capture data set is from the CMU motion capture data base.1 _We considered 3 balancing motions (18, 19, 20) from subject 49. The subject starts in a standing position with arms raised, then, over about 10 seconds, he raises one leg in the air and lowers his arms to an outstretched position. Of interest to us was the fact that, whilst motions 18 and 19 are relatively similar, motion 20 contains more dramatic movements. We were interested in training on motions 18 and 19 and testing on the more dramatic movement to assess the model’s ability to extrapolate. The data was down-sampled by 32 (from 120 frames per second to just 3.75) and we focused on the subject’s left arm. For the left arm, we chose D = 9 outputs including the humerus (X, Y and Z rotations), the radius (X rotation), the wrist (the Y rotation), the hand (X and Z rotations), and the thumb (X and Z rotations). The number of latent forces is fixed to Q = 2. Our objective was to reconstruct the movement of this arm for motion 20 given the angles of the humerus and the parameters learned from motions 18 and 19

1_{The CMU Graphics Lab Motion Capture Database was created with funding from NSF}

using two latent functions. First, we train the second order differential equation latent force model on motions 18 and 19, treating the sequences as independent, but sharing parameters (this is, the damping coefficients and natural frequencies of the two differential equations associated with each angle were constrained to be the same). The training is done by maximizing the log-marginal likelihood in equation (2.20). Once the parameters are learned, we use them for testing the extrapolation ability of the model over movement 20. For the test data, we condition on the observations of the humerus orientation to make predictions for the rest of the arm’s angles.

For comparison, we considered a regression model that directly predicts the angles of the arm given the orientation of the humerus using standard independent GPs with SE covariance functions. A similar setup is used, this is, we learn hyperparameters for 6 independent GPs, having as inputs the humerus’ angle rotations (three rotations) of motions 18 and 19. For testing, we use the three angles of humerus for motion 20, and predict over the 6 other outputs. Results are summarized in table 3.1, with some example plots of the tracks of the angles given in figure 3.1.

Latent Force Regression

Angle Error Error

Radius 4.11 4.02 Wrist 6.55 6.65 Hand X rotation 1.82 3.21 Hand Z rotation 2.76 6.14 Thumb X rotation 1.77 3.10 Thumb Z rotation 2.73 6.09

Table 3.1: Root mean squared (RMS) angle error for prediction of the left arm’s configuration in the motion capture data. Prediction with the latent force model outperforms the prediction with regression for all apart from the radius’s angle.

In the next section, we present related work of differential equations in statistics and machine learning.