1. Dr. Hesam Izakian October 2014
Cluster-Centric Anomaly Detection and Characterization in Spatial Time Series
The slides can be downloaded from
Dr. Hesam Izakian
October 2014

2. Outline Spatial time series Problem formulation
Anomaly detection in spatial time series- questions
Overall scheme of the proposed method
Time series segmentation
Spatial time series clustering
Assigning anomaly scores to clusters
Visualizing the propagation of anomalies
An outbreak detection scenario
Application
Conclusions

3. Spatial time series Structure of data Examples
A set of spatial coordinates
One or more time series for each point
Examples
Daily average temperature in different climate stations
Stock market indexes in different countries
Number of absent students in different schools
Number emergency department visits in different hospitals
Measured signals in different parts of brain
Alberta Health Services is recording
Number of ED visits in different hospitals
Absenteeism in different schools etc.
For outbreak detection in the province

4. Problem formulation There are N spatial time series
Objective: Find a spatial neighborhood of data
In a time interval
Containing a high level of unexpected changes
N : number of spatial time series
r : number of features in spatial part of data
n : length of time series
xi(s): spatial part of data
xi(t): time series part of data
l :length of time interval

5. Anomaly detection in spatial time series- questions
Spatial neighborhood of data
Size of neighborhood
Overlapping neighborhoods
Unexpected changes (anomalies)
What kind of changes are expected/not expected
How to evaluate the level of unexpected changes
Anomaly visualization
Anomaly characterization
What was the source of anomaly
How the anomaly is propagated over time

6. Overall scheme of the proposed method
Revealing the structure of data in various time intervals
Comparing the revealed structures
Spatial time series data
Spatial time series data
Sliding window
Anomaly scores
Spatial time series clustering
Fuzzy relations

7. Time series part segmentation
Sliding window
Spatio-temporal subsequences
Local view of time series part
The spatial part of data is always fixed

8. Overall scheme of the proposed method
Revealing the structure of data in various time intervals
Comparing the revealed structures
Spatial time series data
Sliding window
Anomaly scores
Spatial time series clustering
Fuzzy relations

9. Fuzzy C-Means clustering- visual illustration
Clustering: Grouping objects (data) so that objects in the same group are similar and objects inside different groups are dis-similar
K-means is one of the well-known clustering techniques with Boolean membership assignment

10. Fuzzy C-Means clustering- visual illustration
In fuzzy clustering instead of Boolean assignment of data to clusters, membership degrees are employed

11. Fuzzy C-Means clustering…
Partitions N data
Into clusters
Result:
Objective function:
Minimization:
N data
c cluster
uik indicates the membership degree of xk to vi
m controls the overlap between clusters (fuzziness)
FCM algorithm:
1- Generate a partition matrix U randomly
2- Calculate cluster centers
3- Update partition matrix
4- If not converged go to 2

12. Spatial time series clustering
Reveals available structure within data
In form of partition matrices
Challenges
Different sources: Spatial part vs. temporal part
Different dimensionality in each part
Different structure within each part
A partition matrix is able to express the structure of data in terms of a set of membership degrees

13. Spatial time series clustering…
In spatial time series, we define
Adopted FCM objective function
Characteristics
When λ=0: Only spatial part of data in clustering
A higher value of λ : a higher impact of time series part in clustering
Optimal value of λ: Optimal impact of each part in clustering X

14. Spatial-time series clustering- Optimal value of λ
Reconstruction criterion evaluates the quality of clusters in terms of data granularization and de-granularization

15. Overall scheme of the proposed method
Revealing the structure of data in various time intervals
Comparing the revealed structures
Spatial time series data
Sliding window
Anomaly scores
Spatial time series clustering
Fuzzy relations

16. Assigning anomaly scores to clusters in different time windows
Assign an anomaly score to each single subsequence based on historical data
Aggregating anomaly scores inside revealed clusters
fk is the anomaly score calculated for the subsequence corresponding to xk in time window

17. Overall scheme of the proposed method
Revealing the structure of data in various time intervals
Comparing the revealed structures
Spatial time series data
Sliding window
Anomaly scores
Spatial time series clustering
Fuzzy relations

18. Visualizing the propagation of anomalies- Fuzzy relations
Objective: quantifying relations between clusters
Each data in time interval Wi is expressed in terms of a set of membership degrees in Ui
So, each spatial time series xk is expressed in Ui as a set of membership degrees

19. Visualizing the propagation of anomalies…
Objective function to construct relation
Optimization
To construct a fuzzy relation R, we try to estimate the elements of U1 through the elements of U2
o is a sup-t composition (e.g., max-min operator)
c1: number of clusters in U1 that is correspond to W1
c2: number of clusters in U2 that is correspond to W2
R is a matrix is size c1 * c2 and its elements are in range [0, 1]
alpha: learning rate
rij=1 means that ith jth cluster in U2 has a strong relation with ith cluster in U1
rij=0 indicates no relation

20. Example An outbreak In southern part of Alberta
Using NAADSM for 100 days
NAADSM: North American Animal Disease Spread Model
For each station we will have its x-y coordinates and a time series in length 100 measuring the rate of infected herds in 100 days

21. Example… A sliding window is used
Length : 20
Movement: 10
Generated spatio-temporal subsequences:

23. Example…
The order of clusters are different in different time windows (why?)

24. Example…
Calculated anomaly scores for each cluster

25. Example…
Only strong fuzzy relations corresponding to anomalous clusters are considered in this figure

26. Example…
One may represent the structure of data in different time intervals using a graph-based representation
Nodes are clusters
Edges are relations between clusters
The numbers reported above nodes are anomaly scores

27. Application
Implemented for Agriculture and Rural Development (Government of Alberta)
Using KNIME (Konstanz Information Miner)
Animal health surveillance in Alberta
Anomaly detection
Data visualization

28. Conclusions
A framework for anomaly detection and characterization in spatial time series is developed
A sliding window to generate a set of spatio-temporal subsequences is considered
Clustering is used to discover the available structure within the spatio-temporal subsequences
An anomaly score assigned to each revealed spatio-temporal cluster
A fuzzy relation technique is proposed to quantify the relations between clusters in successive time steps
For more information please see
1. Hesam Izakian and Witold Pedrycz, Anomaly Detection and Characterization in Spatial Time Series Data: A Cluster-Centric Approach, IEEE Transactions on Fuzzy Systems, DOI: /TFUZZ , 2014.
2. Hesam Izakian and Witold Pedrycz, Agreement-Based Fuzzy C-Means for Clustering Data with Blocks of Features, Neurocomputing, vol. 127, pp. 266–280, 2014.
3. Hesam Izakian, Witold Pedrycz, and Iqbal Jamal, Clustering Spatio–temporal Data: An Augmented Fuzzy C–Means, IEEE Transactions on Fuzzy Systems, vol. 21, no. 5, pp. 855 – 868, 2013.

29. Thank you