Time Normalization of Time Series using Their Wavelet Coefficients. Applications to Action Time Series of a Bunraku Puppet.

Observe same phenomena K times, then one obtain K sample paths x1(·), x2(·), …, xK(·)of the time series. The graphs of these K sample paths are similar, but they do not same exactly. Although the state x1(t1)at time t1 of the 1 st sample path and the state x2(t2) at time t2 of the 2 nd sample path mean same events, t1nt2 and x1(t1)nx2(t2)in general. Time normalization between two sample paths x1(·)and x2(·)is the process of finding the time t1 of the 1 st sample and the time t2 of the 2 nd sample path, such that x1(t1) and x2(t2)show same events. Consider graphs of two sample paths. If the local form of the graph of the sample x1 at time t1 is similar to the one of the sample x2 at time t2, we can think that x1(t1)and x2(t2)mean same events. In order to evaluate the similarity of the local forms of the graphs, the wavelet coefficients of the sample paths are used. If one know x1(t1), x2(t2), …, xK(tK)mean same events, one can normalize these time fluctuations and obtain new average time series (x^-)(t^-) where (t^-)=(t1+…+tK)/K and (x^-)(t^-)=(x1(t1)+…+xK(tK))/K. This time normalization method is applied to action time series of a Bunraku puppet, for generation of humanoid robots' actions with fertile emotions.