Cluster analysis is a technique that divides a given data set into a set of clusters in such a way that two objects from the same cluster are as similar as possible and the objects from different clusters are as dissimilar as possible. In this background, different rough-fuzzy clustering algorithms have been shown to be successful for finding overlapping and vaguely defined clusters. However, the crisp lower approximation of a cluster in existing rough-fuzzy clustering algorithms is usually assumed to be spherical in shape, which restricts to find arbitrary shapes of clusters. In this regard, this paper presents a new rough-fuzzy clustering algorithm, termed as robust rough-fuzzy c-means. Each cluster in the proposed clustering algorithm is represented by a set of three parameters, namely, cluster prototype, a possibilistic fuzzy lower approximation, and a probabilistic fuzzy boundary. The possibilistic lower approximation helps in discovering clusters of various shapes. The cluster prototype depends on the weighting average of the possibilistic lower approximation and probabilistic boundary. The proposed algorithm is robust in the sense that it can find overlapping and vaguely defined clusters with arbitrary shapes in noisy environment. An efficient method is presented, based on Pearson's correlation coefficient, to select initial prototypes of different clusters. A method is also introduced based on cluster validity index to identify optimum values of different parameters of the initialization method and the proposed clustering algorithm. The effectiveness of the proposed algorithm, along with a comparison with other clustering algorithms, is demonstrated on synthetic as well as coding and non-coding RNA expression data sets using some cluster validity indices.