1. Spatial Temporal Data Mining
Wei Wang
Data Mining Lab, UCLA
January 21, 1999

2. Outline Introduction Statistical Clustering User-defined Trigger
Spatial Index Structure for High Dimensional Point Data
Temporal Spatial Pattern Detection
ongoing research

3. Introduction
Spatial data mining has been an active research area during recent years.
For some well know problem, e.g., clustering, many existing algorithms are not efficient enough.
There is still room for improvement.
There are a lot of interesting problems remaining uninvestigated.
We classify a subset of problems and try to solve them efficiently.

4. Outline
STING: a statistical information grid approach to spatial data mining
STING+: an approach to active spatial data mining
PK-tree: a spatial index structure for high dimensional point data
Temporal spatial pattern detection

5. STING Spatial database is usually huge.
Efficiency of the data mining algorithm is crucial.
Example: each person is an object
Query: Find high income area within California
high income: salary > $50,000
area > 4 square miles
Traditional Method
Step 1: Select out all person whose salary are high.
Step 2: Do clustering analysis on those persons selected out.
Step 3: Form the region that each cluster occupies.
Step 4: Return those regions larger than 4 square miles.
If “high income” is defined as: 80% persons have salary > $50,000
then the previous method can not even answer the query.
STING was proposed to solve such problem efficiently.

6. STING Region Query Example:
Select the maximal regions that have at least 100 houses per unit area and at least 70% of the house prices are above $400K and with total area at least 100 units with 90% confidence.
SELECT REGION
FROM house-map
WHERE DENSITY IN (100, )
AND price RANGE (400000, )
WITH PERCENT (0.7, 1)
AND AREA (100, )
AND WITH CONFIDENCE 0.9

7. STING
Objects are represented by points, each of which has associated spatial attributes, its location, and non-spatial (numerical) attributes.
Space is recursively divided into smaller rectangular cells until certain level is reached.
A hierarchical structure is employed.
The average number of objects in a leaf cell is in the range from several dozens to several thousands.
Preprocess data
capture the statistical information

8. STING 1st layer (root) (i-1)th layer ith layer (i+1)th layer
(leaf layer)

9. STING For each cell, we have
attribute-independent parameter
n: number of objects
attribute-dependent parameters (for each numerical attribute)
mean: mean value of the attribute
std: standard deviation of the attribute value
min: the minimum value of the attribute
max: the maximum value of the attribute
distribution: the type of distribution that best fits the attribute value (can be NONE)
Bottom-up generation when the data is loaded into the database.
Linear compilation time
Only has to be done once  not for each query.

10. STING Take advantage of the statistical information captured.
Only go through relevant cells at each level.
Root is relevant.
For each relevant cell, we exam its children at next level by statistical test and label them as relevant or not relevant.
Form regions from relevant leaf level cells.
Do not need to access full database.
It is very efficient.

11. STING
The computational complexity of STING is linearly proportional to the number of leaf cells.
We used the SEQUOIA 2000 benchmark as the data set to compare the performance of STING with other approaches.
DBSCAN
STING

12. STING STING is a query-independent approach.
The statistical information exists independently of queries.
STING has a much smaller response time compared to other approaches
The computational complexity is linearly proportional to the number of leaves.
I/O cost is low.
STING can support different resolution of query result.
Regions returned by STING approach that returned by DBSCAN when the granularity approaches zero.
Parameters in the hierarchical structure can be maintained efficiently by incremental update.

13. Outline
STING: a statistical information grid approach to spatial data mining
STING+: an approach to active spatial data mining
PK-tree: a spatial index structure for high dimensional point data
Temporal spatial pattern detection

14. STING+
Moreover, since objects evolve, interesting patterns may emerge or disappear over time.
Example:
Trigger: Do bandwidth reallocation when the average call length is greater than 10 minutes within the region where at least 10 cellular phones are in use per squared mile.
This can not be supported by traditional database triggers efficiently
due to the fact that the class membership of an object is not only determined by its non-spatial attributes but also by the attributes of objects in its neighborhood.
Naïve approach: re-evaluate condition periodically.
Not efficient.

15. STING+
STING+ was an extension of STING to support user-defined trigger.
In spatial databases, object insertion, deletion, and update are primitive events.
Observation: Usually, only the cumulative effect of a set of primitive events may cause the trigger condition to be true.
We refer such set of primitive events to as a composite events. . +

16. STING+ Condition-Action paradigm
In general, it is difficult or even impossible for user to specify all possible composite events that may cause the trigger condition to be true.
In general, evaluating a user-defined trigger T usually involves two aspects:
Find a set of composite events E(s) that may cause the trigger condition CT to become true.
Each time some composite event in E(s) occurs, check the status (false or true) of CT (given that CT was false previously).

17. STING+
Observation: As a side effect of the occurrence of some composite event, the set of composite events E(s) that could cause CT to transition from false to true might also evolve over time.
Two set of composite events we need to consider:
the set of composite events E(s) that can cause CT to become true
need to re-evaluate CT
the set of composite events F(s) that can cause a change to E(s)
need to update E(s) . + . +

18. STING+
Observation: In spatial databases, the effect of an event is usually local to its neighborhood.
STING+ decomposes the user-defined trigger into a set of sub-triggers associated with individual cells in the hierarchical structure.
These sub-triggers are used to monitor composite events in E(s) and F(s) and change accordingly when E(s) and F(s) evolves. . x o1 o2 . .
Level 4
Level 3

19. STING+ . .
Updates are suspended at some level in the hierarchy until such time that the cumulative effect of these updates might cause the trigger condition to become satisfied.
Level 2
Level 1 . +
Level 1 . +
Level 1 . + . +
Level 2
Level 3

20. STING+
Example: Trigger bandwidth reallocation when the total area occupied by those regions in California where at least 10 cellular phones are in use per squared mile and the average length of phone calls is at least 15 minutes with total area at least 50 squared miles increases by at least 10 squared miles.
DEFINE TRIGGER example
ON cellular-phone
WHEN SELECT SIZE(REGION) INCREASE RANGE (10, )
WHERE DENSITY IN RANGE (10, )
AND AVERAGE(length) IN RANGE (15, )
AND AREA IN RANGE (50, )
LOCATION California
DO bandwidth-reallocation

21. STING+
Observation: Trigger condition CT is a conjunction of predicates P1  P2  …  Pn and can not be true if one predicate is false.
They can be evaluated in a certain order: the ith predicate is tested when all previous i -1 predicates are true.
The evaluation order should be chosen in such a way that the total cost is minimum.
STING+ evaluates CT in the order {location, density condition, attribute condition}, each of which is evaluated in a different phase.
Location only needs to be evaluated once and the cost can be regarded as constant in the trigger evaluation process.
If the location is fixed, unnecessary sub-triggers set on cells outside the location can be avoided and hence save the evaluation cost of other predicates.
Sub-triggers set during an earlier phase will exist longer than those set in a later phase.
It is better to first evaluate the predicate that takes less time to handle.
cost(density) < cost(attribute)

22. STING+
Average CPU cycles for handling each type of sub-trigger

23. STING+

24. Outline
STING: a statistical information grid approach to spatial data mining
STING+: an approach to active spatial data mining
PK-tree: a spatial index structure for high dimensional point data
Temporal spatial pattern detection

25. PK-tree
As both the number of objects and the number of attributes are very large, it is essential to organize the set of objects by some dynamic indexing structure.
Point index methods

26. PK-tree .
Level 0 .
Level 1 .
Level 2 .
Level 3
Spatial decomposition: Space is recursively divided until a level LD such that each cell contains at most one point.

27. PK-tree . . . Example of a PR Quad-tree:b4 c3 e1 e2 f3 g2 h2 g3 a7 g5 f6 f5 e5 d8 c8 d7 b8 b7 .
Level 3 (LD) 1 2 3 4 5 6 7 8 a b c d e f g h
16 intermediate nodes, height = 3 1 2 3 4 5 6 7 8 .
Level 2 a b c d e f g h K M N A B C D E F G H L 1 2 3 4 5 6 7 8 .
Level 1 a b c d e f g h Q R T S
root

28. Example of a PK-tree of rank 3:d1 b4 d2 c3 e1 e2 f3 g2 h2 g3 a7 g5 f6 f5 e5 d8 c8 d7 b8 b7 .
Level 3 (LD) 1 2 3 4 5 6 7 8 a b c d e f g h
5 intermediate nodes, height = 2 1 2 3 4 5 6 7 8 .
Level 1 a b c d e f g h Q R K N M M N K 1 2 3 4 5 6 7 8 .
Level 2 a b c d e f g h
root

29. PK-tree
PK-tree employs a concept of K-instantiable cell to eliminate unnecessary nodes.
Point cell: a non-empty cell at level LD
A cell C is K-instantiable iff
C is a point cell, or
there does not exist (K-1) or less K-instantiable sub-cells to cover all non-empty space in C
Only K-instantiable cells serve as nodes in the PK-tree (expect the root).
The parent-child relationship follows naturally from the cell-subcell relationship.

30. PK-tree
Properties:
Bounds on node’s outdegree
allows allocating one node to a page
Bounded storage space
Existence and Uniqueness
enables us to analyze the behavior of a PK-tree easier.
Expected height
log(N) under some general condition
guarantees efficiency of retrieval and update.
No overlapping among sibling nodes
efficient retrieval
Empirical studies shown that the PK-tree outperforms SR-tree and X-tree by a wide margin.

31. PK-tree Height of generated trees on 100,000 points
Size of index in MB

32. PK-tree
KNN query on clustered data distribution

33. PK-tree Real data set: NASA Sky Telescope Data
200,000 two-dimensional points (they are the coordinates of crater locations on the surface of Mars)

34. Outline
STING: a statistical information grid approach to spatial data mining
STING+: an approach to active spatial data mining
PK-tree: a spatial index structure for high dimensional point data
Temporal spatial pattern detection

35. Temporal Spatial Pattern Detection
When the number of attributes is large and/or the value of attributes evolve frequently, the complexity of patterns and the number of potential patterns increase.
It is not desirable or even feasible to ask the user specify interesting patterns.
E.g., the user wants to know any possible patterns involving certain attributes such as salary, rent, cellular phone usage, etc.
Existing association rule algorithm can not be applied.
Continuous attribute domain
Temporal evolution
Prior knowledge about relationships among attributes and objects

36. Temporal Spatial Pattern Detection
Object  represented by point
primitive attributes
spatial attributes, i.e., coordinates of its position
non-spatial attributes, e.g., name, weight, height, salary, rent
derived attributes  derived from primitive attribute(s)
environment attributes, e.g., distance to a hospital, average income in the neighborhood area
Consider a sequence of snapshots S1, S2, …, Sn
Temporal Spatial Pattern
describes a possible relationship among evolution of attributes
E.g., if the user want to know patterns involving salary and distance to big city, then one interesting pattern would be “people receiving a raise tends to move further away from the big city from 1987 to 1993.”.

37. Temporal Spatial Pattern Detection
More complicated patterns
Patterns on clustering evolution
Patterns of high order
Patterns whose cause and consequence do not happen together
There is a delay for the consequence to show up.
Patterns involving relationships among objects
e.g., people who live far away from any doctor tend to move to a place closer to some doctor.
Environment variables evolve independently over time.