Simultaneous estimation

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Simultaneous estimation using the k-Nearest
Neighbors method
Ronald E. McRoberts Erkki O. Tomppo Andrew O.
Finley Juha Heikkinen
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Objective Derive variance estimators for
the k-Nearest Neighbor method Outline 1. A
brief introduction to the k-NN method 2. A few
superpopulation issues 3. Assumptions 4. Superpo
pulation estimators 5. Results
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Motivation ? FIA 1 plot per 6,000 acres
(2,400 ha) ? Plot data with perturbed locations
released to public - satisfactory for large
area analyses - unsatisfactory for small and
irregular areas ? Construct Internet-accessible
maps - multivariate - unbiased - provide
areal estimates with variance estimates
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FIA plot configuration
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FIA subplots relative to TM pixels
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The k Nearest Neighbors (k-NN) technique
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Implementing k-NN
For an arbitrary pixel, i
where yji j1,,k is the set of k pixels with
subplots nearest to pixel i in spectral space
with respect to a distance metric,
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Possibly correlated neighbors
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Possibly correlated neighbors
Same neighbor
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Superpopulation parameter estimates
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Superpopulation parameter estimates
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Areal estimates
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Data ? 3 dates of Landsat TM/ETM
imagery ? 2266 FIA plots (9064
subplots) ? Match FIA subplots to Landsat
pixels ? 15 circles of radius 10-km ( 76,000
ac) ? 4 variables - proportion forest
area - volume (m3/hectare) - basal area
(m2/hectare) - tree count (count/hectare)
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Proportion forest
Volume
20 km
Basal area
Tree count
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Proportion forest area
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Volume (m3/ha)
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Basal area (m2/ha)
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Tree count (count/ha)
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AOI Design-based Model-based 1
nbr/plot cluster 1 nbr/plot
cluster n Mean SE Mean SE1 SE2 Mean SE1 SE2 Pro
portion forest area 5 20 0.963 0.021 0.907 0.0069
1 0.00696 0.904 0.00692 0.00695 8 24 0.812 0.050
0.810 0.00494 0.00504 0.813 0.00458 0.00469 13 24
0.615 0.075 0.643 0.00648 0.00661 0.639 0.00620 0
.00629 Volume 5 20 81.4 13.5 74.6 2.50 2.51 73.2
2.41 2.42 8 24 52.9 6.5 49.1 1.20 1.22 48.8 1.17
1.18 13 24 35.0 6.9 39.8 1.20 1.21 39.8 1.14 1.1
5 Tree count 5 20 313.8 38.3 382.6 10.50 10.55 3
77.2 9.84 9.88 8 24 322.4 49.6 253.8 5.15 5.21 25
3.2 4.99 5.05 13 24 187.2 32.6 188.7 5.11 5.16 18
7.9 5.04 5.08
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Conclusions ? Model-based approach yields maps
as by-products ? k-NN multivariate and
non-parametric ? Estimates comparable to
design-based approach ? From practical
perspective, k-NN areal variances are no larger
than sample-based variance estimates
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http//knn.gis.umn.edu/meeting/