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Stratification Skill Assessment Based on Average Discrete Fréchet Distance (ADFD)
Wen Long (UMCES) Raleigh Hood (UMCES)
Kevin Sellner (CRC)SURA Testbed Estuarine Hypoxia Group Meeting
Oct 1, 2010
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Assessing the model skill of salinity stratification is like walking a dog …..
---anonymous
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Introduction to Fréchet Distance
• Fréchet Distance (Fréchet, 1906) is the minimum leash length required for a dog-man’s problem:
• Dog walks on curve L1 from start to end
• Man walks on curve L2 from start to end
• No backtracking is allowed in the walk
• when FD=0, i.e. leash length =0, L1 and L2 must be the same
Slide 3
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Typical Applications of Fréchet Distance
• Signature matching for online bank service
• Handwriting based web search
• GIS route recognition
Slide 4
References:
[1] Eiter, T. and Mannila, H., Computing Discrete Frechet Distance, Apr 25, 1994, Report # CD-TR 94/64
[2] Mascret, A., Devogele, T., Le Berre, I., Henaff, A., Coastline Matching Process based on the discrete Frechet distance, 2009
[3] Fréchet, M., Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Mathematico di Palermo, 22:1-74, 1906
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FD is the minimum leash length required for the man and dog to finish their respective continuous route from start to end without backtracking
Slide 5
Fréchet Distance (FD)
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Properties of FD
• FD(L1, L2) = FD (L2,L1), i.e. exchanging routes between dog and man makes NO difference
• When FD=0, L1, L2 must collapse
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Discrete Fréchet Distance (DFD)
DFD is the minimum leash length required if L1 (dog) and L2 (man) are discretized as piecewise linear curves
Slide 7
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Average Discrete Fréchet Distance (ADFD)Slide 8
ADFD is the average distance of the “minimum path”
ADFD=(d1+d2+…+d9)/9
“Minimum Path” is a series of point-pairs that the dog and the man have to follow to finish their routes while staying as close to each other as possible. (See next slide)
ADFD is an optimal measure of the discrepancy of the two curves L1 and L2 !!! Hence ADFD can be used for skill assessment.
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How the dog and man walk through with only DFD leash length available
Slide 9
Step 1
Step 2
Step 3
Step 4
…….
Step 9
dfd happens to be d2
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Example of CBay CB5.1 Salt ProfileSlide 10
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CB5.1 Salt Profile (with normalized axis)Slide 11
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DFD and ADFD (top down walk)Slide 12
Start
End
DFD=0.51355
ADFD=0.2138
Minimum Path has 31 steps
d26
dh
dv
(d26 )^2=dh^2+dv^2
α
d26 is contributed by dh and dv
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Analyze Error (ADFD) ContributionADFD=( d1+d2 + …+d26+…+ d31)/31 mean length of minimum path pairs
• d26 is contributed by dh and dv
• dh is associated with salinity error on the same depth level
• dv is associated with mislocation of isopycnals• Angle α may be used to define ADFS (Average Discrete Frechet Slope) as a measure of model-obs stratification
d26
dh
dv
(d26 )^2=dh^2+dv^2
α
Slide 13
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DFD and ADFD (bottom up walk)Slide 14
End
Start
DFD=0.51355
ADFD=0.25069
Minimum Path has 41 steps
When reversing the direction of walk, DFD is the same, ADFD can be DIFFERENT. In our skill assessment we should use mean value of the ADFD’s of the two opposite directions
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DFD and ADFD of CH3D for CB5.1 (top down walk)
Slide 15
Start
End
DFD=0.21312
ADFD=0.104
Minimum Path has 29 steps
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DFD and ADFD of CH3D for CB5.1 (bottom up walk)
Slide 16
End
Start
DFD=0.21312
ADFD=0.10206
Minimum Path has 29 steps
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Modified Target DiagramSlide 17
Y axis: normalized model-data bias (B*)
X axis: ADFD
Sign of X axis:
sign of (model std – obs std)
standard deviation
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Conclusion
• ADFD is an optimal measure of curve mismatch and can be used for skill assessment of salinity stratification.
• ADFS (Average Discrete Frechet Slope) could be introduced and be used for analyzing error due to same stratification strength but with isopycnal shift.
• Target Diagram can be modified to incorporate ADFD.
Slide 18