The derivative [dD?x(h)/dh) must be estimated with a smoothing technique to ensure monotonie increase .
One can use either an a priori local time stamp indicator or an optimization procedure for determining the landmarks in a reference set. The challenge of implementing multivariate landmarking is that landmarks are different (in placement and number) for different process variables. Critical issues are the selection of the process variable(s) for determining the landmarks, the number of landmarks and their locations to define clearly the progress of the batch.
One alternative is to select a process variable based on process knowledge and implement landmarking by using the trajectory of that variable. Another alternative is to use an iterative approach which will reconcile the identification of process landmarks with respect to particular trajectory landmarks:
1. Find the landmarks of the most important variable trajectories Lmi. Align all other variable trajectories with respect to the landmarks Lm\.
2. Calculate the principal components of the aligned set of process variables. Determine the landmarks of the first principal component LmpcA-
3. Realign the process trajectories with respect to LmpcA-
4. Recalculate the principal components of the realigned set of process variables. Determine the landmarks of the first principal component
Lm PC Anew-
5. Determine if LmpcAnew are reasonably close to LmpcA■ If so, the process landmarks are defined by LmpcAnew- If not, return to Step 3.
The outcome of this procedure may be interpreted using several alternatives. Once LmpcAnew has converged, one may proceed with statistical analysis using the data warped with respect to LmpcAnew- As an alternative, only the data identified as "most significant" (either by user or principal components) may be warped with respect to LmpcAnew, and other process data may be warped with respect to its own optimal landmarks.
When landmarking a test trajectory with respect to a reference trajectory, two distinct cases may be considered. The first case is a simple idealized situation where all the landmarks are delayed (or advanced) by a constant time r and is called uniform landmark case. The second is the mixed landmark case that represents a general framework where some landmarks of the new batch are delayed and others are advanced with respect to the landmarks of the reference trajectory, yielding a more challenging landmark detection problem. Furthermore, the time shifts of the landmarks reference test
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