  $K$-means clustering is one of the most widely-used partitioning algorithm in
cluster analysis due to its simplicity and computational efficiency. However,
$K$-means does not provide an appropriate clustering result when applying to
data with non-spherically shaped clusters. We propose a novel partitioning
clustering algorithm based on expectiles. The cluster centers are defined as
multivariate expectiles and clusters are searched via a greedy algorithm by
minimizing the within cluster '$\tau$ -variance'. We suggest two schemes: fixed
$\tau$ clustering, and adaptive $\tau$ clustering. Validated by simulation
results, this method beats both $K$-means and spectral clustering on data with
asymmetric shaped clusters, or clusters with a complicated structure, including
asymmetric normal, beta, skewed $t$ and $F$ distributed clusters. Applications
of adaptive $\tau$ clustering on crypto-currency (CC) market data are provided.
One finds that the expectiles clusters of CC markets show the phenomena of an
institutional investors dominated market. The second application is on image
segmentation. compared to other center based clustering methods, the adaptive
$\tau$ cluster centers of pixel data can better capture and describe the
features of an image. The fixed $\tau$ clustering brings more flexibility on
segmentation with a decent accuracy.
