Kalman filter, and Logical revision: comparing them on GI

1. Kalman filter, and Logical revision: comparing them on GI Geoffrey Edwards (Séminaires du CRG - Mars 2001)

2. Outline • Fusion and revision – definitions • Topographic fusion – the problem • Framing the problem formally • Kalman filtering • Comparison with Logical Revision • Conclusion

3. Fusion and Revision - definitions • Many terms • Fusion • Integration • Revision • Updating • What is the difference between fusion and integration? • Fusion is a subset of integration, a kind of « total » integration • What is the difference between revision and update • Revision is a subset of updating

4. Fusion and Revision - definitions • What is the difference between revision and fusion? • The size of the respective databases? • I.e. revision occurs if one database is substantially larger than the other – one revises the larger database with the smaller one • Many current « fusion » techniques are actually « revision » techniques

5. Fusion and Revision - definitions • Fusion • An integration of two knowledge bases (or data sets) where the identity of the product (i.e. the set of characteristic properties) is different than the simple combination of the earlier identities (and their associated properties) • Revision • An integration of two knowledge bases where the identify of the product is the same as one of the earlier identities, although some (non-essential) properties might be different

6. Topographic fusion – the problem (CITS – roads & rivers)

7. Topographic fusion – the problem (CITS – roads & streams)

8. Topographic fusion – the problem (CITS – streams & contours)

9. Topographic fusion – the problem • Incompatabilities • Roads which pass across water bodies • Roads which cross streams with no bridges • Lakes on the flanks of mountains • Streams which flow uphill • Etc.

10. Framing the problem formally • Variables and constraints • Slope < x • No intersections between contours and lake boundaries • Contours have errors associated with them, which are different from errors for lake boundaries, roads, etc. • Upward watershed matched to flow rate (i.e. watershed area must remain roughly constant) • Water bodies must be connected • Streams follow the maximum slope • Roads cross important streams at bridges • Roads cannot intersect with the interior of lakes

11. Framing the problem formally • Define contours as closed forms, each embedded in the next • Partition space « azimuthally » to formalize stream flows • Express 2, 3, 4, 8 as intersections between closed contours and other elements • Slope < x • No intersections between contours and lake boundaries • Contours have errors associated with them, which are different from errors for lake boundaries, roads, etc. • Upward watershed matched to flow rate (i.e. watershed area must remain roughly constant) • Water bodies must be connected • Streams follow the maximum slope • Roads cross important streams at bridges • Roads cannot intersect with the interior of lakes

12. Kalman filtering • Principle: • given several predicates x(k), x(k+1), … linked by some dependance model (dynamic model: F) • given a series z(k), z(k+1) infered from respective x(.) through an observation model: H, • let ’s try to reduce the uncertainty (that any one model may causes) • Classical mathematical representation: • Between step k and k+1, the evolution (dependance) is: • x(k+1) = F(k).x(k) + u(k) + v(k) (1) • Between a predicate x and its « observed » counterpart z: • z(k) = H(k).x(k) + w(k) (2) • (v is a « state noise » and w an « observation noise ») • (u is a « deterministic -certain- predicate, may be T, i.e. null)

13. Kalman filtering • Where the « revision » takes place: • according to (1) and (2): z(k+1) can be infered twice:

14. Kalman filtering • Where the « dependance » takes place: possible locations (x) are ruled by topo, geology, … (model F) • Where the « revision » takes place: the observations (z) may be the amount of incorrect surface and may decrease or increase depending on trajectory F B A

15. Kalman filtering • Analogy Revision - Kalmnan Filtering • One cause of uncertainty: the knowledge of some « target variable » x comes through the observation of z « observed variable », then applying some inference to derive x from z. We need « link » rules between x and z (set L) as well as rules, or « constraints » that govern the x according to our knowledge of this « target » (set C). • In the « revisionscheme » we consider two sets of formulas: one large uncertain set A and a second B, smaller and trustworthy. Hence we use B to « revise » A, trying to restore a possibly damaged global (A and B) consistency. • The « dissymetry » of this scheme tells us to split C and L into strong (F and H) and weak parts (v and w), and put them into set A and B respectively.

16. Kalman filtering • From the Kalmanfilter viewpoint: • The x are the « target variables » while the z are the « observed » ones • matrix F (evolution rules) is what we try to improve (revise), and H (observation) matrix plays the role of the rules L. • According to the evolution of z, we will revise F: by choosing a model with the shortest distance.

17. Comparison • Modification of one fact at a time • Bayesian networks • Global update which minimizes errors • Kalman filtering • Recursive updating which minimizes errors • Modification of a collection of facts • Logical revision • Cumulative updating which rejects or minimizes incompatability • Kalman • The minimization is also global

18. Conclusion • a Phd thesis subject to start • … next talk will be given by Gilles Cotteret