Sequential pattern mining aims to discover interesting sequential patterns in a sequence database, and it is one of the essential data mining tasks widely used in various application fields such as Web access pattern analysis, customer purchase pattern analysis, and DNA sequence analysis. In general sequential pattern mining, only the generation order of data element in a sequence is considered, so that it can easily find simple sequential patterns, but has a limit to find more interesting sequential patterns being widely used in real world applications. One of the essential research topics to compensate the limit is a topic of weighted sequential pattern mining. In weighted sequential pattern mining, not only the generation order of data element but also its weight is considered to get more interesting sequential patterns. In recent, data has been increasingly taking the form of continuous data streams rather than finite stored data sets in various application fields, the database research community has begun focusing its attention on processing over data streams. The data stream is a massive unbounded sequence of data elements continuously generated at a rapid rate. In data stream processing, each data element should be examined at most once to analyze the data stream, and the memory usage for data stream analysis should be restricted finitely although new data elements are continuously generated in a data stream. Moreover, newly generated data elements should be processed as fast as possible to produce the up-to-date analysis result of a data stream, so that it can be instantly utilized upon request. To satisfy these requirements, data stream processing sacrifices the correctness of its analysis result by allowing some error. Considering the changes in the form of data generated in real world application fields, many researches have been actively performed to find various kinds of knowledge embedded in data streams. They mainly focus on efficient mining of frequent itemsets and sequential patterns over data streams, which have been proven to be useful in conventional data mining for a finite data set. In addition, mining algorithms have also been proposed to efficiently reflect the changes of data streams over time into their mining results. However, they have been targeting on finding naively interesting patterns such as frequent patterns and simple sequential patterns, which are found intuitively, taking no interest in mining novel interesting patterns that express the characteristics of target data streams better. Therefore, it can be a valuable research topic in the field of mining data streams to define novel interesting patterns and develop a mining method finding the novel patterns, which will be effectively used to analyze recent data streams. This paper proposes a gap-based weighting approach for a sequential pattern and amining method of weighted sequential patterns over sequence data streams via the weighting approach. A gap-based weight of a sequential pattern can be computed from the gaps of data elements in the sequential pattern without any pre-defined weight information. That is, in the approach, the gaps of data elements in each sequential pattern as well as their generation orders are used to get the weight of the sequential pattern, therefore it can help to get more interesting and useful sequential patterns. Recently most of computer application fields generate data as a form of data streams rather than a finite data set. Considering the change of data, the proposed method is mainly focus on sequence data streams.