k representative skyline query is a type of query derived from traditional skyline query. Given a set of d-dimensional dataset D, a skyline query finds all objects in D that are not dominated by other ones, which helps users to select high-quality objects based on their preference. However, the scale of skyline objects may be large in many cases, users have to choose target objects from a large number of objects, leading that both the selection speed and quality cannot be guaranteed. Compared with traditional skyline query, k representative skyline query chooses the most "representative" k objects from all skyline objects, which effectively solves such problem causes by traditional skyline query. Given the sliding window W and a continuous query q, q monitors objects in the window. When the window slides, q returns k skyline objects with the largest group dominance size in the window. The key behind existing algorithms is to monitor skyline objects in the current window. When the skyline set is updated, the algorithm updates k representative skyline set. However, the cost of monitoring skyline set is usually high. When the skyline set scale is large, the computational cost of choosing k representative skyline objects is also high. Thus, existing algorithms cannot efficiently work under high-speed stream environment. This study proposes a query named r-approximate k representative skyline query. In order to support this type of queries, a novel framework is proposed named PAKRS (predict-based approximate k representative skyline). Firstly, PAKRS partitions the current window into a group of sub-windows. Next, the predicted result sets of a few future windows are constructed according to the partition result. In this way, the earliest moment can be predicted when new arrived objects may become skyline objects. Secondly, an index is proposed named r-GRID, which can help PAKRS to select r-approximate k representative skyline with O(k/s+k/m) computational cost under 2-dimensional space, and O(2^Ld/m+2^Ld/s) computational cost under d-dimensional space (d>2), where L is a little integer smaller than k. Theoretical analysis shows that the computational complexity of PAKRS is lower than the state-of-the-art efforts. Extensive experiments have been conducted to confirm the efficiency and effectiveness of the proposed algorithms. Experimental results show that the running time of PAKRS is about 1/4 times of PBA (prefix-based algorithm), algorithm 1/6 times of GA (greedy algorithm) and about 1/3 times of e-GA (e-constraint greedy algorithm).