Time, h

Figure 6.34. Four variables (out of 13) of 55 good performance batches after synchronization.

where is a small residual relative to x, and roughly centered about 0 [495]. The curve registration task is to determine the time warping functions hi so that the de-warped trajectories x,[/i,(£fc)] can be interpreted more accurately. The hi can be determined by using a smooth monotone transformation family consisting of functions that are strictly increasing (monotone) and have an integrable second derivative [494]:

A strictly increasing function has a nonzero derivative and consequently the weight function q = D2h/Dh or the curvature of h. The differential equation 6.88 has the general solution h(t) = Co + C\ / exp / q(v)dv Jo I Jo

where D~l is the integration operator and Co and C\ are arbitrary constants [495]. Imposing the constraints h(0) = 0 and h(To) — Ti: Cq = 0 and C\ = Ti/[{D_1exp(D_1g)}(To)]. Hence, h depends on q. The time warping function h can be estimated by minimizing a measure of the fit T,, of Xi[hi{tj)} to y. A penalty term in T,, based on q permits the adjustment of the smoothness of hi [495]. To estimate the warping function hi, one minimizes

T„(i),x\h) = ¿/^(t) \\D'y{t)—D^x [ft(i)] \\2 dt + rj f q2(t)dt (6.90) j=0J J

where aj(i)'s are weight functions and

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