Interpolation {x,y} Data with Suavity. Peter K. Ott Forest Analysis & Inventory Branch BC Ministry of FLNRO Victoria, BC

Transcription

1 Interpolation {x,y} Data with Suavity Peter K. Ott Forest Analysis & Inventory Branch BC Ministry of FLNRO Victoria, BC 1

2 The Goal Given a set of points: x i, y i, i = 1,2,, n find a function that passes through the points affording the prediction of y i at new x i Regression or smoothing is a related but different problem 2

3 Outline Example Data Linear Interpolation Thin Plate Splines (TPS) Ordinary Kriging (OK) Two implementations Conclusion 3

4 data fake; input x y; cards; ; ods html; ods graphics on; proc sgplot data=fake noautolegend; scatter y=y x=x / markerattrs=(symbol=circle size=4pt color=blue); title 'Y versus X'; ods graphics off; ods html close; 4

5 Our Data 5

6 Linear Interpolation (~200 yrs BC) Form a straight line between pairs of known points What is y 0 x 0 where x 0 lies between x 1 and x +1? Slope must be constant, so Solve for y 0 : y 0 y 1 x 0 x 1 = y +1 y 0 x +1 x 0 y 0 = y 1 x +1 x 0 + y +1 x 0 x 1 x +1 x 1 6

7 data interp0; *denser range of x to be interpolated; do x=0 to 8 by 0.1; output; end; proc sql; create table lin_pred(drop=x0) as select * from interp0 left join fake(rename=(x=x0)) on put(interp0.x, 6.3) = put(fake.x0, 6.3) ; quit; proc print data=lin_pred(obs=34) noobs; 7

8 x y

9 proc expand data=lin_pred(keep=x y) out=lin_interp; convert y=linear / method=join; id x; *data must be sorted by x; ods html; ods graphics on; proc sgplot data=lin_interp noautolegend; series y=linear x=x / lineattrs=(pattern=2 thickness=1pt color=red) lineattrs=graphprediction; scatter y=y x=x / markerattrs=(symbol=circle size=4pt color=blue); title 'Linear interpolated values'; ods graphics off; ods html close; 9

10 Linear Interpolation 10

11 Thin Plate Splines (1970s) Want a function to minimize: n L = y i f x i 2 + λ f x 2 dx i=1 or more generally n L = y i f x i 2 + λ J f where, for d = 2 i=1 J f = 2 f x f x 1 x f x dx 1 dx 2 Where λ 0 is an unknown parameter that controls the wiggliness 11

12 Solution to this problem is a function that relies on radial basis functions and it passes through data without knots One dimension example: y 0 x 0 = α 0 + α 1 x i=1 n β i x 0 x i 3 Two dimension example: y 0 x 01, x 02 = α 0 + α 1 x 01 + α 2 x π i=1 where z i = x 01 x 1i 2 + x 02 x 2i 2 n β i z i 2 log z i 12

13 proc tpspline data=fake; model y =(x); */ lambda0=1e-15; *setting lambda0 to zero is necessary for interpolation; score data=interp0 out=tps_pred pred; *this will yield the interpolated points and more; output out=tps_coef pred coef; proc print data=tps_coef noobs; *output are alpha[0], alpha[1], beta[1],..., beta[n], with the beta aligned with sorted (unique) x[i]; *Note also that sum(beta[i])=0 and sum(beta[i]*x[i])=0; 13

14 x y P_y Coef_y

15 proc sql; create table tps_pred2(drop=x0) as select * from tps_pred left join fake(rename=(x=x0)) on put(tps_pred.x, 6.3) = put(fake.x0, 6.3) ; quit; ods html; ods graphics on; proc sgplot data=tps_pred2 noautolegend; series y=p_y x=x / lineattrs=(pattern=2 thickness=1pt color=red) lineattrs=graphprediction; scatter y=y x=x / markerattrs=(symbol=circle size=4pt color=blue); title 'Interpolated values'; ods graphics off; ods html close; 15

16 Thin Plate Spline 16

17 Ordinary Kriging (1960s) Consider y i x i process: y i x i as a multivariate Gaussian = y ~ N n μ1, R Find the estimator y 0 x 0 = n i=1 such that: w i y i = w y E y 0 x 0 = μ (unbiased), and Prediction error, Var(y 0 y 0 ) is minimized 17

18 It turns out: w = R 1 c 1 R R 1 11 R 1 c + 1 R R 1 1 (ugly) y 0 x 0 = μ + c 0 R 1 y μ1 (better) where μ = 1 R R 1 y and c 0 = Cov y 1, y 0 Cov y 2, y 0 Cov y n, y 0 18

19 How do we determine c 0? We ll need to model the covariance structure as a function of distance, say h Tradition is to use semivariances (semivariogram) instead of covariances (covariogram) or correlations (correlogram): γ ij = σ 2 σ ij = σ 2 1 ρ ij γ h = 1 n h 2 n h i=1 y i x i + h y i x i 2 19

20 Semivariogram 20

21 Features of the (Semi)variogam Nugget: discontinuity at the origin. Can t have this for interpolation with kriging! Range: distance it takes for the variogram to level off (reach asymptote) Sill: value of variogram at asymptote (= σ 2 = var y 0 ). When a nugget is present, sill = partial sill + nugget 21

22 Ordinary Kriging Implementation - two options: 1. Use both proc variogram & proc krige2d need to create a second variable (x2) with constant values 2. Use a mixed model procedure (proc mixed) not provided empirical and fitted variograms automatically 22

23 data fake2; set fake; x2=1; *constant value; ods html; ods graphics on; proc variogram data=fake2 outvar=look; store out=semivar_store; directions 90(0); *not really needed; compute lagdist=1 maxlag=10; *lagdist should be ~ 2*min norm and maxlag should be ~ max norm among xs; coordinates xc=x yc=x2; var y; model nugget=0 form=auto(mlist=(gau,pow,she) nest=2) choose=(aic SSE STATUS); *important that nugget is zero for interpolation; proc krige2d data=fake2 outest=kr_pred(rename=(gxc=x estimate=y_est)); restore in=semivar_store; coordinates xc=x yc=x2; predict var=y; model storeselect; grid x=0 to 8 by 0.01 y=1 to 1 by 1; 23

24 The VARIOGRAM Procedure Dependent Variable: y Lag Class Empirical Semivariogram at Angle=90 Pair Count Average Distance Semivariance

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26 proc sql; create table kr_pred2(drop=x0) as select *, (y_est+1.96*stderr) as cl_upp, (y_est-1.96*stderr) as cl_low from kr_pred(keep=x y_est stderr) left join fake2(keep=x y rename=(x=x0)) on put(kr_pred.x, 6.3) = put(fake2.x0, 6.3) ; quit; proc sgplot data=kr_pred2 noautolegend; series y=y_est x=x / lineattrs=(pattern=2 thickness=1pt color=red) lineattrs=graphprediction; scatter y=y x=x / markerattrs=(symbol=circle size=4pt color=blue); series y=cl_upp x=x / lineattrs=(pattern=2 thickness=1pt color=green) lineattrs=graphprediction; series y=cl_low x=x / lineattrs=(pattern=2 thickness=1pt color=green) lineattrs=graphprediction; title 'Interpolated values'; ods graphics off; ods html close; 26

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28 Getting setup for proc mixed data fake_fmixed; set fake end=last; output; if last then do x=0 to 8 by 0.01; y=.; output; end; proc print data=fake_fmixed(obs=34) noobs; 28

29 x y

30 proc mixed data=fake_fmixed; model y = / outp=ok_preds; *outputing predictions; repeated / subject=intercept type=sp(matern)(x); title 'Ordinary Kriging in Proc Mixed'; proc print data=ok_preds(obs=34) noobs; ods html; ods graphics on; proc sgplot data=ok_preds(where=(resid=.)) noautolegend; series y=pred x=x / lineattrs=(pattern=2 thickness=1pt color=red) lineattrs=graphprediction; series y=lower x=x / lineattrs=(pattern=2 thickness=1pt color=green) lineattrs=graphprediction; series y=upper x=x / lineattrs=(pattern=2 thickness=1pt color=green) lineattrs=graphprediction; title 'Kriged values via Mixed Model'; ods graphics off; ods html close; 30

31 StdErr x y Pred Pred DF Alpha Lower Upper Resid

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33 Comparison of all 4 approaches 33

34 Conclusions TPSs and OK are both capable of interpolation and smoothing TPSs require no distributional assumptions but predictions can be overly wiggly when λ = 0 OK takes a bit more effort/practice but is powerful when a suitable model is available for the empirical variogram Consider TPS and OK over linear interpolation! 34

35 Thanks! 35