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### On the Interpolation Algorithm Ranking

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

Carlos López-Vázquez

LatinGEO – Lab

SGM+Universidad ORT del Uruguay

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

- Compare? Typically through RMSE/MAD
- 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 RMSE, MAD, ESAM, etc.

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

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

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