Periodicity Analysis For Time-related Sequence Data

Introduction: Periodicity analysis is the mining of periodic patterns, that is, the search for recurring patterns in time-related sequence data. Periodicity analysis can be applied to many important areas. For example, seasons, tides, planet trajectories, daily power consumptions, daily traffic patterns, and weekly TV programs all present certain periodic patterns. Periodicity analysis is often performed over time-series data, which consists of sequences of values or events typically measured at equal time intervals (e.g., hourly, daily, weekly). It can also be applied to other time-related sequence data where the value or event may occur at a non equal time interval or at any time (e.g., on-line transactions). Moreover, the items to be analyzed can be numerical data, such as daily temperature or power consumption fluctuations, or categorical data (events), such as purchasing a product or watching a game.

The problem of mining periodic patterns can be viewed from different perspectives. Based on the coverage of the pattern, we can categorize periodic patterns into full versus partial periodic patterns:

  • A full periodic pattern is a pattern where every point in time contributes (precisely or approximately) to the cyclic behavior of a time-related sequence. For example, all of the days in the year approximately contribute to the season cycle of the year.
  • A partial periodic pattern specifies the periodic behavior of a time-related sequence at some but not all of the points in time. For example, Sandy reads the New York Times from 7:00 to 7:30 every weekday morning, but her activities at other times do not have much regularity. Partial periodicity is a looser form of periodicity than full periodicity and occurs more commonly in the real world.

Based on the precision of the periodicity, a pattern can be either synchronous or asynchronous, where the former requires that an event occur at a relatively fixed offset in each “stable” period, such as 3 p.m. every day, whereas the latter allows that the event fluctuates in a somewhat loosely defined period. A pattern can also be either precise or approximate, depending on the data value or the offset within a period. For example, if

Sandy reads the newspaper at 7:00 on some days, but at 7:10 or 7:15 on others, this is an approximate periodic pattern. Techniques for full periodicity analysis for numerical values have been studied in signal analysis and statistics. Methods like FFT (Fast Fourier Transformation) are commonly used to transform data from the time domain to the frequency domain in order to facilitate such analysis.

Mining partial, categorical, and asynchronous periodic patterns poses more challenging problems in regards to the development of efficient data mining solutions. This is because most statistical methods or those relying on time-to-frequency domain transformations are either inapplicable or expensive at handling such problems.

Take mining partial periodicity as an example. Because partial periodicity mixes periodic events and non periodic events together in the same period, a time-to-frequency transformation method, such as FFT, becomes ineffective because it treats the time series as an inseparable flow of values. Certain periodicity detection methods can uncover some partial periodic patterns, but only if the period, length, and timing of the segment (subsequence of interest) in the partial patterns have certain behaviors and are explicitly specified. For the newspaper reading example, we need to explicitly specify details such as “Find the regular activities of Sandy during the half-hour after 7:00 for a period of 24 hours.” A naïve adaptation of such methods to the partial periodic pattern mining problem would be prohibitively expensive, requiring their application to a huge number of possible combinations of the three parameters of period, length, and timing.

Most of the studies on mining partial periodic patterns apply the A priori property heuristic and adopt some variations of A priori-like mining methods. Constraints can also be pushed deep into the mining process. Studies have also been performed on the efficient mining of partially periodic event patterns or asynchronous periodic patterns with unknown or with approximate periods.

Mining partial periodicity may lead to the discovery of cyclic or periodic association rules, which are rules that associate a set of events that occur periodically. An example of a periodic association rule is “Based on day-to-day transactions, if afternoon tea is well received between 3:00 to 5:00 p.m., dinner will sell well between 7:00 to 9:00 p.m. on weekends.”

Due to the diversity of applications of time-related sequence data, further development of efficient algorithms for mining various kinds of periodic patterns in sequence databases is desired.