Fuzzy Time Series

Modelling of Fuzzy Time Series

When speaking of time series, one usually has in mind a series of data points with a constant temporal interval. An example is the series of daily closing prices, where the interval between two data points of the time series is exactly one day. In the analysis of time-dependent data, however, it is not always possible to fit the measurements into such a uniform grid.

Let us consider as an example the intraday price development of a stock. Every concluded contract is contained in the intraday history as a so-called tick. The date, the time of day and the price at which the relevant security was traded are all known. The intervals between the concluded transactions, however, are not constant.

The modelling of such a time series requires the use of interpolation methods. Thus, for example, the average price of the past 30 minutes can constitute a data point for the time series. Such procedures, however, are not without problems. Within the said 30 minutes, two, 200 or no transaction might be concluded. Aside from the fact that in such a time series the price following a single transaction would weigh just as much as the average value of 200 transactions in another 30 minute time period, much information regarding the actual price movement is lost in this type of interpolation. In order to conserve all of this information, a modelling method is required, which is able to represent "fuzzy" time series with varying intervals between the data points.

This type of modelling is indeed possible - with the use of a universal metric time measurement. Independently of location and time zones, this time measurement would have to indicate a particular point in time in a real number that can be processed by a neural analysis system.

Especially in the field of databases, various so-called time stamps are already in use, which encode the date and the time of day as an integer. One of these time measurements is the Julian date. Independently of the circumstances of our calendar, which with its leap years and varying lengths of months is rather unsuitable for analytical purposes, Julian dates can be added and subtracted without difficulty. A Julian "tick" comprises exactly 24 hours. The algorithm, however, does not provide for a more precise subdivision. This measure is thus not suited for use with neural analysis systems, since these systems are based exclusively on floating point arithmetic.

An inspiration is the universal time measure used in the science fiction series Startrek: stardate. The stardate is a real number that can be as precise as one likes. Like the Julian date, stardates can be calculated from a normal date/time of day statement. Yet, the algorithm is much more effective than that for the calculation of the Julian date.

The origin in science fiction might at first not strike one as very scientific. The basis, however, is a mathematical algorithm. Thus, as universal, time zone-independent time stamps, stardates can perform valuable service in the modelling of fuzzy time series. By means of such real stardates, for example, a time series such as the above-mentioned intraday price development can be interpolated by a neural system without loss of information.

The "Stardate FAQ", written by Andrew Main for fans of the Startrek series and films, contains all of the information that a developer needs as inspiration for corresponding implementations. For this reason, we have made this source accessible, providing you with the source code as well as a Stardate ActiveX Control.