**2.4.1 The advantage and disadvantage of symmetric and asymmetric DTW**

As mentioned above, DTW works with pairs of patterns. Therefore, the problem of whether symmetric or asymmetric is suitable for synchronization.

Let *Bi*(*bi*×*N*), *i*=1,2,…,I be a training set of good quality batches for MPCA/MICA models, where *bi* is the number of observations and *N* is number of measured variables, and one defined reference batch trajectories *BREF*, the objective is to synchronize each *Bi* with *BREF*  (*bREF*×*N*).

Symmetric DTW algorithms include all points in the original trajectories, but expanded trajectories of various lengths, because the length is determined by DTW. After synchronization, each *Bi* will be individually synchronized with *BREF*, but not with each other unfortunately.

Although asymmetric may eliminate some points, they will produce synchronized trajectories of equal length, because each time axis of *Bi* will be mapped with the one of *BREF* so that they all are synchronized with reference trajectories *BREF* and synchronized with each other.

Unavoidably, the asymmetric algorithms have to skip some points in the optimal path, so the characteristics of some segments may be left out after synchronization to construct incomplete MPCA/MICA model from 'trimmed' trajectories to cause miss/false alarm.
