In data mining area, weight based sequential pattern mining has been suggested to find important sequential patterns by considering the weights of sequential patterns. More extensions with weight constraints have been proposed such as mining weighted association rules, weighted sequential patterns, weighted closed patterns, frequent patterns with dynamic weights, weighted graphs, weighted sub-trees or sub structures, and so on. In previous approach of weighted frequent sequential pattern mining, weighted supports of sequential patterns are exactly matched to prune weighted infrequent sequential patterns. However, in the noisy environment, the small change in weights or supports of items affects the result sets seriously. This may make the weighted sequential patterns less useful in the noisy environment. In this paper, we propose the robust concept of mining weighted approximate sequential patterns. Based on the framework of weight based sequential pattern mining, an approximate factor is defined to relax the requirement for exact equality between weighted supports of sequential patterns and a minimum threshold. After then, we address the concept of mining weighted approximate sequential frequent patterns to find important sequential patterns with/without the noisy data. We analyze the characteristics of weighted approximate sequential patterns and run extensive performance tests.