Constructing Time Series Shape Association Measures: Minkowski Distance and Data Standardization
This addresses a gap in time series data mining for analyzing relationships in domains like finance and environmental science, but it is incremental as it builds on existing similarity measures.
The paper tackles the lack of measures for direct and inverse relationships between time series, proposing a theoretical basis and methods to construct shape association measures, with Minkowski distance and data standardization as examples, deriving cosine similarity and Pearson's correlation as specific cases.
It is surprising that last two decades many works in time series data mining and clustering were concerned with measures of similarity of time series but not with measures of association that can be used for measuring possible direct and inverse relationships between time series. Inverse relationships can exist between dynamics of prices and sell volumes, between growth patterns of competitive companies, between well production data in oilfields, between wind velocity and air pollution concentration etc. The paper develops a theoretical basis for analysis and construction of time series shape association measures. Starting from the axioms of time series shape association measures it studies the methods of construction of measures satisfying these axioms. Several general methods of construction of such measures suitable for measuring time series shape similarity and shape association are proposed. Time series shape association measures based on Minkowski distance and data standardization methods are considered. The cosine similarity and the Pearsons correlation coefficient are obtained as particular cases of the proposed general methods that can be used also for construction of new association measures in data analysis.