will vary, preventing the use of an assumption of a constant time shift r between calculated and reference landmarks.
For illustration, consider the example in Figure 6.35, where the solid curve represents the mean pH trajectory calculated from an aligned reference set and the dashed curve in the upper figure represents a different test trajectory. The estimated landmarks (denoted by hollow circles) are mixed in leading or lagging the reference landmarks (indicated by dotted vertical lines and numbers) and they have varying time shift magnitudes. The first estimated landmark is advanced with respect to the first mean-value landmark, the second and third estimated landmarks are delayed with respect to their mean values. The advance between the first estimated and mean-value landmark is not equal to the delay between the second and the third estimated and their mean-value counterparts. When the test trajectory is warped with respect to the reference trajectory (as illustrated in the lower portion of Figure 6.35), the warped time-values (the solid line) is a curved pattern that crosses the center dashed-curve that represents linear mapping. This pattern suggests that when warping the test pattern (to align similar events to the reference pattern), the test trajectory will be stretched and compressed, respectively. In this example, the warping pattern lies slightly below the center line until the first landmark, so test values before it will be stretched to align the estimated landmark. After the first landmark, the warped curve lies above the center dashed-line, indicating that the values after the first landmark location need to be compressed to align the second and third landmarks with respect to the mean-value landmarks. This makes intuitive sense, because an estimated landmark that is advanced before a mean-value landmark must shift the data in a direction that will align similar process events.
The on-line optimization procedure sequentially searches for the optimal location of the landmarks of the test data with respect to the reference mean landmarks. The following procedure is given for the mixed landmarks case and can be modified to implement in an adaptive hierarchical PCA framework for online SPM:
1. Initialize estimated landmarks vector Lj.
2. For i — 1,..., m (m, number of landmarks in the reference set).
3. Collect values of test trajectory that contains landmark information up to time K. Choose time A', for zth landmark so that it will span the reference landmarks range as
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