Constructing Time Series Shape Association Measures: Minkowski Distance and Data Standardization

Abstract

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 Pearson's 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.

Bipolar Rating Scales: A Survey and Novel Correlation Measures Based on Nonlinear Bipolar Scoring Functions

“…The reasons to use nonlinear bipolar utility functions in correlation measure introduced in the paper are discussed in Section 7. The results on association measures considered in Section 7 are based on the papers [5][6][7]. Here, we use the terms association measure and correlation measure as interchangeable.…”

“…We extend the property of Cseparability from association measure on [0,1] considered in [6] on the set of utility profiles. The general formula for association measure on bipolar utility profiles is based on Minkowski distance and on general results considered in [5] for time series. The formulas (37) and (39) for centered bipolar utility functions are specific for C-separable association measures on bipolar utility profiles.…”

“…Consider a method for construction of correlation (association) measures on the set of bipolar utility profiles based on the general methods discussed in [5]. …”

On definition and construction of association measures

“…In [6], it was considered another set of properties similar to the properties of Pearson's correlation coefficient and defining the time series shape association measures. In [7], the general methods of construction of such association measures have been proposed and the sample Pearson's correlation coefficient was obtained as a particular case of the general approach. In [8], the methods proposed in [7] have been extended on the general case of functions A : X × X → [−1, 1] defined on a set X with involutive operation N (called reflection) and satisfying the properties similar to the properties of the Pearson's correlation coefficient.…”

“…Different association and correlation measures have been introduced in statistics, data mining, fuzzy set theory etc. [1,7,12,13,17] for different types of data. The Pearson's correlation coefficient [12] corr(x, y) =…”