1. High-order Spatial Direct and Cross-
statistics for Categorical Attributes
David F. Machuca-Mory
COSMO – Stochastic Mine Planning Laboratory
Department of Mining and Materials Engineering
1

2. Presentation Outline
• Introduction
• High-order Spatial Indicator Statistics
– Indicator spatial cumulants
– Multiple-point transition probabilities
• Implementation and Case Studies
– 2D channels
– 3D gold deposit
– 3D Kimberlite diamond pipe
• Discussion and Conclusions
© David F. Machuca-Mory, 2012 2

3. Introduction
SISim Realization
• Traditional indicator approach:
– Based on 2-point statistics
– Produce patchy, broken images
– Does not allow for complex patterns
• Multiple-point approach Training Image SNESim Realization
– Able to produce complex
patterns
– But, they rely too much
on training images
© David F. Machuca-Mory, 2012 3

4. Introduction
• Proposal: to obtain more high-order spatial
information from categorical hard data
• Objectives
– To extend the spatial high-order statistics to
categorical variables:
• High-order spatial indicator moments
• High-order spatial indicator cumulants
• Multiple-point transition probabilities
– To show how these statistics can be extracted from
scattered datasets
© David F. Machuca-Mory, 2012 4

5. High-order Spatial Indicator
Statistics
© David F. Machuca-Mory, 2012 5

6. High-order Spatial Indicator Moments
1 if category sk is present in u
• Indicator transform i(u; sk )  
0 if it is not
• Spatial indicator moment of order   j0  j1  ...  jn
0...n; j0 ... jn  E  I j0 (u0 ; s0 )  I j1 (u1; s1)   I jn (u n ; sn ) 
 
 E  I (u0 ; s0 )  I (u1; s1)   I (u n ; sn ) 
• Spatial n-point template u1
Heads
h1
un hn u0
h3 Tail
h3 u2
u3
© David F. Machuca-Mory, 2012 6

7. High-order Spatial Indicator Moments
• They express joint probabilities
E  I (u0 ; s0 )    I (u n ; sn )  Pr  Z (u0 )  s0   Z (u n )  sn 
 ps0 ,...sn (h1,..., hn ),
• Direct: s0  s j , j  1,..., n
• Cross otherwise
• Experimental form:
Nh1 ,...,hn
1
E  I (u0 ; s0 )  ...  I (u n ; sn )  
ˆ  i(u0 ; s0 )  ...  i(u n ; sn )
Nh1 ,...,hn k 1
© David F. Machuca-Mory, 2012 7

8. Spatial Indicator Cumulants
• They are combinations of indicator moments
CI (u0 ;s0 )  ps0 u1
h1
CI (h1; s0 , s1)  ps0 s1 (h1)  ps0  ps1 un hn u0
h2
h3 u2
u3
CI (h1, h 2 ; s0 , s1, s2 )  ps0 s1s2 (h1, h 2 )
 ps0  ps1s2 (h 2  h1)  ps1  ps0 s2 (h 2 )
 ps2  ps0 s1 (h1)  2 ps0  ps1  ps2
• And so on …
8

9. Spatial Indicator Cumulants
• Indicator direct cumulants
Data C(s1,s1,s1) C(s0,s0,s0)
uY
hY
u0
hX uX
uY 
hY 
u0
hX 
uX 
© David F. Machuca-Mory, 2012 9

10. Spatial Indicator Cumulants
• Indicator cross-cumulants
C(s1,s0,s1) C(s1,s1,s0) C(s1,s0,s0)
uY
hY
u0
hX uX
uY 
hY 
u0
hX 
uX 
© David F. Machuca-Mory, 2012 10

11. Multiple-point Transition Probabilities
• 1-point: ps0 (u0 )  Pr  Z (u0 )  s0   E  I (u0 ; s0 )
• 2-points:
Pr  Z (u0 )  s0  Z (u1)  s1 
ts0 / s1 (h1)  Pr  Z (u0 )  s0 | Z (u1)  s1  
Pr  Z (u1)  s1 
E  I (u0 ; s0 )  I (u1; s1)

E  I (u1; s1)
• N-points:
Pr  Z (u0 )  s0   Z (u0  h n )  sn 
ts0 / s1 ...sn (h1,, hn ) 
Pr  Z (u0  h1)  s1   Z (u0  h n )  sn 
u1
h1 E  I (u0 ; s0 )  I (u1; s1)  ...  I (u n ; sn ) 
un hn u0 
h3 E  I (u1; s1)  ...  I (u n ; sn ) 
h3 u2
u3 © David F. Machuca-Mory, 2012 11

12. Multiple-point Transition Probabilities
Direct Cross
ts1/s1,s1) ts1/s1,s0) ts1/s0,s1) ts1/s0,s0)
uY
hY
u0
hX uX
uY 
hY 
u0
hX 
uX 
Data
12

13. Case Studies
13

14. 2D Channels: Data
• Exhaustive and scattered datasets:
© David F. Machuca-Mory, 2012 14

15. 2D Channels: Indicator Direct Cumulants
3rd-order Exhaustive data Scattered data
u2
h2
u0 h1 u1
4th-order
u2
h2
u3 h 3 u 0 h1 u1
© David F. Machuca-Mory, 2012 15

16. 2D Channels: Direct Transition Probabilities
3rd-order Exhaustive data Scattered data
u2
h2
u0 h1 u1
4th-order
u2
h2
u3 h 3 u 0 h1 u1
© David F. Machuca-Mory, 2012 16

17. 3D Case – A Structurally Complex Gold
Deposit
• The Apensu dataset
Fault
Family 1 of
subsidiary structures
© David F. Machuca-Mory, 2012 17

18. 3D Case – A Structurally Complex Gold
Deposit
3-point direct and
cross-transition
probabilities
h3
h2
h1
© David F. Machuca-Mory, 2012 18

19. 3D Case – A Structurally Complex Gold
Deposit
4-point direct and
cross-transition
probabilities
• 25% probability
isosurfaces
© David F. Machuca-Mory, 2012 19

20. 3D Case – A Kimberlitic Diamond Pipe
Geological model Hard samples
Host rock
hX hY
Crater
Diatreme hZ
Xenoliths © David F. Machuca-Mory, 2012 20

21. 3D Case – A Kimberlitic Diamond Pipe
Geological model
• 4th-point direct and
cross transition
probabilities
• 25% probability
isosurfaces
hX hY
hZ
© David F. Machuca-Mory, 2012 21

22. 3D Case – A Kimberlitic Diamond Pipe
Drill-hole samples
• 4th-point direct and
cross transition
probabilities
• 25% probability
isosurfaces
hX hY
hZ
© David F. Machuca-Mory, 2012 22

23. Conclusions
• Indicator high-order statistics can be obtained
from hard data.
• Indicator cumulants incorporate multiple joint
probabilities.
• Transition probabilities are simpler and more
straightforward to interpret.
• Transition probabilities can be used directly to
build conditional distributions.
• Dimensionality issues for multiple categories.
© David F. Machuca-Mory, 2012 23

24. Future Work
• High-order spatial indicator statistics from hard
and soft data.
• Simulation based on multiple-point transition
probabilities.
• SGeMS integration.
© David F. Machuca-Mory, 2012 24