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On the Interpolation Algorithm RankingPowerPoint Presentation

On the Interpolation Algorithm Ranking

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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 in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil. 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? in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil.

- {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? in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil.

- 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 in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil.

- 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? in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil.

- 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

- Perform Cross Validation (Falivene et al., 2010)
- 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 in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil.

- 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 in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil.

- 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! in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil.

Questions?

Results in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil.

- 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

Other possible spectral metrics (to be developed) in Natural Resources and Environmental Sciences from 10th to 13th July 2012, Florianópolis, SC, Brazil.

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