Warped K-Means: An algorithm to cluster sequentially-distributed data

Abstract

Many devices generate large amounts of data that follow some sort of sequentiality, e.g., motion sensors, e-pens, eye trackers, etc. and often these data need to be compressed for classification, storage, and/or retrieval tasks. Traditional clustering algorithms can be used for this purpose, but unfortunately they do not cope with the sequential information implicitly embedded in such data. Thus, we revisit the wellknown K-means algorithm and provide a general method to properly cluster sequentially-distributed data. We present Warped K-Means (WKM), a multi-purpose partitional clustering procedure that minimizes the Sum of Squared Error criterion, while imposing a hard sequentiality constraint in the classification step. We illustrate the properties of WKM in three applications, one being the segmentation and classification of human activity. WKM outperformed five state-of-the-art clustering techniques to simplify data trajectories, achieving a recognition accuracy of near 97%, which is an improvement of around 66% over their peers. Moreover, such an improvement came with a reduction in the computational cost of more than one order of magnitude.

Trajectory‐based phase partition and multiphase multilinear models for monitoring and quality prediction of multiphase batch processes

“…Based on this idea, a trajectory‐based phase partition method is developed. It divides a batch process into different operation phases by clustering the trajectory data of reference batches using the WKM algorithm . This phase partition method implements phase partition for all reference batches simultaneously, and thus, it can identify common phase division points of batches.…”

“…These time slices constitute a sequential dataset { X 1 ( I × J c ), …, X K ( I × J c )} that represents the reference trajectories of normal batches. As shown in Figure , the WKM clustering algorithm is applied on the dataset { X 1 ( I × J c ), …, X K ( I × J c )} to divide all time slices into different clusters, with each cluster representing an operation phase. The number of operation phases (ie, clusters) is needed before using the WKM algorithm.…”

“…where b x new;l JK l ×1 ð Þis a vector unfolded from X new,l (J × K l ) and b Q l , b B l , and c W l are defined in Equation 15. Figure 5 shows offline modeling and online prediction procedures for MP batch processes.…”

“…Recently, Luo et al proposed a trajectory‐based phase partition method for MP batch processes with uneven durations. This method divides a batch into different operation phases by clustering the trajectory data (ie, measurement data at all sampling time) of phase‐sensitive variables using the warped K‐means (WKM) clustering algorithm . This trajectory‐based phase partition method has 2 advantages over the correlation‐based methods.…”

“…This method divides a batch into different operation phases by clustering the trajectory data (ie, measurement data at all sampling time) of phase-sensitive variables using the warped K-means (WKM) clustering algorithm. 15 This trajectory-based phase partition method has 2 advantages over the correlation-based methods. First, it has the potential advantages of being easier to implement in industrial practice and being applicable for various MP batch processes, because the phase partition is accomplished by directly clustering variable trajectory data, with no need for data modeling.…”