Mining Linguistic Trends from Time Series - Data Mining: Foundations and Practice - page 53

Tabl e 1 . The time series used in this example

Time series
3, 4, 7, 9, 8, 3, 2, 4, 8, 10, 8, 4, 2

Membership value

Low LowMiddle Middle MiddleHigh High

1

0 20 40 45 50 70 Angle

Fig. 1. The membership functions for angles

Tabl e 2 . The transformed angular series from Table 1

Angular sequence
45, 71.56, 63.43, − 45, − 78 . 96, − 45, 63.43, 75.96, 63.43, − 63 . 43, − 75 . 96, − 63 . 43

90 0 and 90 0 . Assume the member-
ship functions for the angular values are defined as shown in Fig. 1. There
are five fuzzy membership functions, represented as linguistic terms, Low ,
LowMiddle , Middle , MiddleHigh and High , for positive angles. There are
another five membership functions, represented as low , lowmiddle , middle ,
middlehigh and high , for negative angles. Thus, the uppercase initial letter
means the angle is positive, and the lowercase initial letter means the angle
is negative.
For the time series given in Table 1, the proposed fuzzy mining algorithm
proceeds as follows.
STEP 1: Every two adjacent data points in Table 1 are transformed into
an angle. The results are shown in Table 2.
STEP 2: The angular series is then used to generate a set of subsequences
according to the predefined window size. Assume the given window size is 6.
There are totally 7 (=13

The range of the angles is between −

− 6) subsequences to be obtained. The results are

shown in Table 3.
STEP 3: The angular values in each subsequence are then transformed
into fuzzy sets according to the membership functions given in Fig. 1. Take
the first value v 11 (=45) in the subsequence s 1 as an example. The value “45”
is converted into the fuzzy set (1.0/ P 1 .m ), where P i .term is a fuzzy region of
the i -th data in the subsequences and is called a fuzzy term. For example, P 1 .m
represents the fuzzy region middle of the first data point in each subsequence.