two new approaches to smoothing over complex regions · outline smoothing over complex regions...

Two new approaches to smoothing overcomplex regions

David Lawrence Miller

Mathematical SciencesUniversity of Bath

useR! 2009, Rennes

Outline

Smoothing over complex regionsIntroSolutions

Schwarz-Christoffel transform

Multidimensional ScalingDetailsSimulation Results

Conclusions

Smoothing in 2 dimensions

I Have some geographical region and wish to find outsomething about the biological population in it.

I Response is eg. animal distribution, wish to predict basedon (x , y) and other covariates eg. habitat, size, sex, etc.

I This problem is relatively easy if the domain is simple.

Smoothing over complex domains

I Smoothing of complex domains makes this a lot moredifficult.

I Problem of leakage.I Euclidean distance doesn’t always make sense.I Models need to incorporate information about the intrinsic

structure of the domain.

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(modified) Ramsay test function Thin plate spline fit

Smoothing with penalties

I Objective function takes the form:

n∑i=1

(zi − f (xi , yi ; θ))2 + λ

∫Ω

Pf (x , y ; θ)dΩ

I f is the function you want to estimate, made up of somecombination of basis functions.

I P is some squared derivative penalty operator, usuallyP = ( ∂2

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∂y2 )2.

I This can be generalized to an additive model or GAM.

Possible solutions to leakage problems

I FELSPLINE (Ramsay, (2002).)I Domain morphing (Eilers, (2006).)I Within-area distance (Wang and Ranalli, (2007).)I Soap film smoothers (Wood et al, (2008).)

Why morph the domain?

I Takes into account within-area distance.I Gives a known domain that is easier to smooth over.I Potentially less computationally intensive.

However:

I Don’t maintain isotropy - distribution of points odd.I Not clear what this does to the smoothness penalty.

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Outline




Conclusions

The Schwarz-Christoffel transform

I Take a polygon in some domain W and morph it to a newdomain W ∗.

I We then have a function for the mapping, ϕ(x , y).I ϕ(x , y) is a conformal mapping.I Do this by starting at the new domain and working back to

the polygon.I Can draw a polygonal bounding box around some arbitrary

shape.

φ(x)

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The mappingI Use a bounding box around the horseshoe.

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I Morphing the horseshoe shape still gives a slightly odddomain however, we are still doing better than before.

Horseshoe plots

Truth

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SC+PS

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SC+TPRS

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Soap film

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Problems

I Small:I Implementation is Matlab+R. (YUCK!)

I BIG:I Weird artifacts.I Morphing of domain appears to cause features to be

smoothed over.I Arbitrary selection of vertices.

A more realistic domain

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sc+tprs prediction

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A more realistic domain - what’s happening?

I Weird “crowding” effect.I Different with each vertex choice. All bad.

Outline




Conclusions

Multidimensional scaling and within-area distances

I Idea: use MDS to to arrange points in the domainaccording to their “within-domain distance.”

Scheme:

I First need to find the within-area distances.I Perform MDS on the matrix of within-area distances.I Smooth over the new points.

Multidimensional scaling refresher

I Double centre matrix of between point distances, D,(subtract row and column means) then find DDT .

I Finds a configuration of points such that Euclideandistance between points in new arrangement isapproximately the same as distance in the domain.

I Already implemented in R by cmdscale.

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Finding within-area distancesI Use a new algorithm to find the within area distances.

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Ramsay simulations

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Outline




Conclusions

Conclusions

I Seems that the S-C transform does not have much utility.I MDS shows more promise, easier to transfer to higher

dimensions.I MDS does not impose strict boundary conditions so

leakage still possible.I Pushing the data into more dimensions might be useful to

separate points.I After initial “transform” calculation, both methods only use

the same computational time as a thin plate regressionspline. (Soap is expensive.)

References

I S.N. Wood, M.V. Bravington, and S.L. Hedley. Soap filmsmoothing. JRSSB, 2008

I H. Wang and M.G. Ranalli. Low-rank smoothing splines oncomplicated domains. Biometrics, 2007

I T.A. Driscoll and L.N. Trefethen. Schwarz-ChristoffelMapping. Cambridge, 2002

I T. Ramsay. Spline smoothing over difficult regions. JRSSB,2001

I P.H.C. Eilers. P-spline smoothing on difficult domains.University of Munich seminar, 2006

I J.C. Gower. Adding a point to vector diagrams inmultivariate analysis. Biometrika, 1968.

Slides available at http://people.bath.ac.uk/dlm27

http://people.bath.ac.uk/dlm27

two new approaches to smoothing over complex regions · outline smoothing over complex regions...

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