Clustering of time series using a hierarchical linear dynamical system

Abstract

The auditory cortex in the brain does effortlessly a better job of extracting information from the acoustic world than our current generation of signal processing algorithms. Abstracting the principles of the auditory cortex, the proposed architecture is based on Kalman filters with hierarchically coupled state models that stabilize the input dynamics and provide a representation space. This approach extracts information from the input and self-organizes it in the higher layers leading to an algorithm capable of clustering time series in an unsupervised manner. An important characteristic of the methodology is that it is adaptive and self-organizing, i.e. previous exposure to the acoustic input is the only requirement for learning and recognition, so there is no need of selecting the number of clusters.

Hierarchical linear dynamical systems for unsupervised musical note recognition

“…Therefore, similar time structure (same musical note) presented at the observation layer will drive the top layer state to similar locations in state space, while differences in the input time structure will push the top layer state mean values to different points in the space, creating invariant representations (clusters) for musical data as we have illustrated in previous works [17], [18]. This multilayer state model is still linear, and can be trained using recursive state estimators since it is a special case of the system model defined in the Kalman Filter [19], which further exploits computational efficiency.…”

“…In every sample of the internal clock, a temporary musical note is assigned using the minimum Euclidean distance to the states found in the training set. Since HLDS is a dynamical system, a given note is said to occur in the input musical stream after 4 consecutive internal assignments to the same label [17]. In a trained HLDS, when the states leave a cluster the same rule is used to declare a new note.…”

“…Once the system is trained, the mean value of the top layer state (cluster) of each note is assigned to the corresponding musical note, as shown in [17], and the procedure is repeated to all notes. This step assigns the limit cycles created in the top layer of HLDS to musical notes, and only has to be done once.…”