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Support Vector Data Description (SVDD) describes data by using a hyper-sphere. In this paper, we propose an extended SVDD (ESVDD) which describes data by using a hyper-ellipse. Clearly, ESVDD can describe data better than SVDD in the input space. Both hyper-sphere and hyper-ellipse are very rigid for data description. The kernel ESVDD which will be proposed in this paper and the kernel SVDD enhance the ability of ESVDD and SVDD for data description, respectively. The formulation of SVDD/ESVDD contains a penalty term C which controls the tradeoff between the volume of hyper-sphere/hyper-ellipse and the training errors. We show that the ESVDD can control this tradeoff better than the SVDD.

M-distance based recommendation system (MBR) is a nearest neighbor based recommendation method which uses the average of ratings given to an item as the attribute of that item. This attribute is used to determine similar items. Then, the average of the rating given to the similar items to an item of the active user determines the rating of that item. In this paper, to decrease the error of MBR, by combining the following ideas, eight MBR-based recommendation systems are proposed: (a) Using the variance of item ratings in addition to the average of item ratings, as two attributes of an item, for determining similar items in an item-based nearest neighbor method; (b) Using the variance of user ratings in addition to the average of user ratings, as two attributes of a user, for determining similar users in a user-based nearest neighbor method; (c) Using a weighted average method for combining the ratings of similar items or similar users; (d) Using ensemble learning. Experimental results on real datasets show that our proposed EVMBR and EWVMBR which use ensemble learning have the least error. The error of the suggested EWVMBR is at-least 20% lower than that of MBR, Slope-One, P-kNN, and C-kNN.