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Mining Time-Series Databases

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Mining Time-Series Databases

Mohamed G. Elfeky

- A Time-Series Database is a database that contains data for each point in time.
- Examples:
- Weather Data
- Stock Prices

- Full Periodic Patterns
- Every point in time contributes to the cyclic behavior of the time-series for each period.
- e.g., describing the weekly stock prices pattern considering all the days of the week.

- Partial Periodic Patterns
- Describing the behavior of the time-series at some but not all points in time.
- e.g., discovering that the stock prices are high every Saturday and small every Tuesday.

- Problem Definition
- Methods
- Apriori
- Max-Subpattern Hit Set
Jiawei Han, Guozhu Dong, and Yiwen Yin – ICDE98

- The time-series is: S = D1 D2 … Dn
- A pattern is: s = s1 … sp over the set of features L and the letter *.
- |s| = p is the period of the pattern s.
- L-length of s is the number of si that is not *.
- If s has L-length j, it is called a j-pattern.
- A subpattern of s is: s’ = s’1 … s’psuch that for each position i: s’iis a * or subset of si.

- Each segment of the form Di|s|+1 … Di|s|+|s|is called a period segment.
- A period segment matchess if for each position j, either sjis * or subset of Di|s|+j.
- The frequency count of s in a time-series S is the number of period segments of S that matches s.
- The confidence of s is defined as the division of its frequency count by the maximum number of periods of length |s| in S.
- A pattern is called frequent if its confidence not less than a minimum threshold.

- The pattern: a*{a,c}de is of length 5 and of L-length 4 and so it is called 4-pattern.
- The patterns: a*{a,c}** and **cde are subpatterns of the above pattern.
- In the series a{b,c}baebaced, the pattern: a*b, whose period is 3, has frequency count 2. Its confidence is 2/3 where 3 is the maximum number of periods of length 3.

- Apriori Property:
Each subpattern of a frequent pattern of period p is itself a frequent pattern of period p.

- Method:
- Find F1, the set of frequent 1-patterns of period p.
- Find all frequent i-patterns of period p, for i from 2 to p, based on the idea of Apriori, and terminate when the candidate i-pattern set is empty.

- Definitions
- Algorithm
- Implementation Data Structure

- A candidate max-patternCmax is the maximal pattern which can be generated from F1 (the set of frequent 1-patterns).
- Example:
- If F1 = {a***, *b** , *c** , **d*},
- Then Cmax = a{b,c}d*

- A subpattern of Cmax is hit in a period segment Si if it is the maximal subpattern of Cmaxin Si.
- Example:
- For Cmax = a{b,c}d* and Si = a{b,c}ce,
- The hit subpattern is: a{b,c}**

- The hit setH is the set of all hit subpatterns of Cmax in S.

- Scan S once to find F1 and form the candidate max-pattern Cmax.
- Scan S again, and for each period segment, add its max-subpattern to the hit set setting its count to 1 if it is not exist, or increase its count by 1.
- Derive the frequent patterns from the hit set.

Max-Subpattern Tree

- The root node is: Cmax.
- A child node is a subpattern of the parent node with one non-* letter missing. The link is labeled by this letter.
- A node containing only 2 non-* letters have no children since they are already in F1.
- Each node has a count field which registers its number of hits.

10

d

a

b

c

0

50

40

32

acd*

abd*

a{b,c}**

*{b,c}d*

a

d

a

d

b

b

c

b

b

c

d

a

2

18

8

0

5

19

*bd*

*{b,c}**

a*d*

ac**

ab**

*cd*

a{b,c}d*

- Finding w the max-subpattern in the current segment.
- Search for w in the tree, starting from the root and following the path corresponds to the missing non-* letters in order.
- If the node w is found, increase its count by 1. Otherwise, create a new node w (with count 1) and its missing ancestors in the followed path (with count 0).

*cd*

0

a{b,c}d*

a

0

*{b,c}d*

b

1

*cd*

- After the second scan, the tree will contain all the max subpatterns of the time-series.
- Now the tree must be traversed to compute the confidence value of each subpattern.

- The frequency count of each node is the sum of its count and those of all its reachable ancestors.
- For Example:
- The frequency count of *cd* is 78.
- The frequency count of a*d* is 105.

10

d

a

b

c

0

50

40

32

acd*

abd*

a{b,c}**

*{b,c}d*

a

d

a

d

b

b

c

b

b

c

d

a

2

18

8

0

5

19

*bd*

*{b,c}**

a*d*

ac**

ab**

*cd*

a{b,c}d*