On the Interpolation Algorithm Ranking

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10th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil. On the Interpolation Algorithm Ranking. Carlos López-Vázquez LatinGEO – Lab SGM+Universidad ORT del Uruguay.

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10th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil.

### On the Interpolation Algorithm Ranking

Carlos López-Vázquez

LatinGEO – Lab

What is algorithm ranking?
• There exist many interpolation algorithms
• Which is the best?
• Is there a general answer?
• Is there an answer for my particular dataset?
• How to define the better-than relation between two given methods?
• How confident should I be regarding such answer?
What has been done?
• {A}
• {B}
• Many papers so far
• Permanent interest
• How is a typical paper?
• Takes a dataset as an example
• N points sampled somewhere
• Subdivide N in two sets: Training Set {A} and Test Set {B}
• A∩B=Ø; N=#{A}+#{B}
• Repeat for all available algorithms:
• Define interpolant using {A};

blindly interpolate at locations of {B}

• Compare known values at {B}with those interpolated ones
• Better-Than is equivalent to lower-RMSE
Is RMSE/MAD/etc. suitable as a metric?
• Different interpolation algorithms lead to different look
• RMSE might not be representative. Why?
• Let’s consider spectral properties

Images from www.spatialanalysisonline.com

Some spectral metric of agreement
• For example, ESAM metric
• U=fft2d(measured error field), U(i,j)≥0
• V=fft2d(interpolated error field), V(i,j)≥0
• ideally, U=V
• 0≤ESAM(U,V)≤1
• ESAM(W,W)=1

Hint!: There might be better options than ESAM

How confident should I be regarding such answer?
• Given {A} and {B}a deterministic answer
• How to attach a confidence level? Or just some uncertainty?
• Perform Cross Validation (Falivene et al., 2010)
• Set #{B}=1, and leave the rest with {A}
• N possible choices (events) to select B
• Evaluate RMSE for each method and event
• Average for each method over N cases
• Better-than is now Average-run-better-than
• Simulate
• Sample {A} from N, #{A}=m, m<N
• Evaluate RMSE for each method and event, and create rank(i)
• Select confidence level, and apply Friedman’s Test to all rank(i)

n wines judges each rank k different wines

The experiment
• DEM of Montagne Sainte Victoire (France)
• Sample {B}, 20 points, held fixed

Apply six algorithms

Evaluate ranking(i)

• Evaluate ranking of means over i
• Apply Friedman’s Test and compare
• Do 250 times:

Sample {A} points

Results
• Ranking using mean of simulated values might be different from Friedman’s test
• Ranking using spectral properties might disagree with that of RMSE/MAD
• Friedman’s Test has a sound statistical basis
• Spectral properties of the interpolated field might be important for some applications

Thank you!

Questions?

Results
• Other results, valid for this particular dataset
• Ranking using ESAM varies with #{A}
• According to ESAM criteria, Inverse Distance Weighting (IDW) quality degrades as #{A} increases
• According to RMSE criteria, IDW is the best
• With a significative difference w.r.t. the second
• With 95% confidence level
• Irrespective of #{A}
• According to ESAM criteria, IDW is NOT the best