Nonparametric Small Area Estimation Using Penalized Spline Regression

Transcription

1 Nonparametric Small Area Estimation Using Penalized Spline Regression

2 0verview Spline-based nonparametric regression Nonparametric small area estimation Prediction mean squared error Bootstrapping small area and local effects

3 Penalized spline Regression, often referred to as P-splines popularized by Eilers and Marx (1996) P-splines; attractive smoothing method because of their flexibility also a natural candidate for constructing nonparametric small area estimation Approach; shown consistency of the estimator provided tests for small area effects with bootstrap methods

4 Spline-based nonparametric regression Spline based nonparametric regression model and estimator, consider the simple model; where y i = m(x i ) + e i e i are independent random variables with mean zero and variance σ 2 e m(x i ) is unkown but to be estimated using P-spline

5 It is assumed that the unkown function can be approximated suffeciently well by; where m(x; β, γ) = β 0 + β 1 x β p x p + p is the degree of spline K γ k (x κ k ) p + (1) k=1 (x) n + denotes the function x n I {x>0} β = (β 0,..., β p ) and γ = (γ 1,..., γ K ) are the coefficients for the parametric and the spline portions of the model

6 In P-spline regression; K is typically taken to be large e.g. with 1 knot every 4 or 5 observations Model 1 is over-parameterized which is avoided by a penalty. Define the resgression estimators as the minimizers over β and γ of n (y i m(x i ; β, γ)) 2 + λ γ γ γ i=1 where λ γ is a fixed penalty parameter.

7 It is to interest to treat λ γ as an unkown parameter, Since, different values of λ γ result in different estimates of β and γ treat γ as a random effect possible to estimate jointly λ γ, β and γ by ML methods both the P-spline and the SAE models can be viewed as random effects models combine both into a nonparametric SAE framework based on linear mixed model regression

8 Nonparametric small area estimation Specifically, suppose there are T small areas, U 1,..., U T define d it = I {i Ut} let d i = (d i1,..., d it ) and D = (d 1,..., d n) let Y = (y 1,..., y n ), 1 x 1 x p 1 (x 1 κ 1 ) p + (x 1 κ K ) p + X =..., Z =.. 1 x n xn p (x n κ 1 ) p + (x n κ K ) p +

9 Assume that the data follow the model Y = Xβ + Zγ + Du + e (2) where γ (0, Σ γ ) with Σ γ = σγi 2 K u (0, Σ u ) with Σ u = σui 2 T e (0, Σ e ) with Σ e = σei 2 n V (Y) V = ZΣ γ Z + DΣ u D + Σ e

10 Given the model (2) and the spesifications, the GLS estimator ˆβ = (X VX) 1 X V 1 Y and the predictors ˆγ = Σ γ Z V 1 (Y X ˆβ) û = Σ u D V 1 (Y X ˆβ) are optimal among all linear estimators/predictors

11 For a given small area U t, interested, for example, in predicting where ȳ t = x t β + z t γ + u t x t and z t are the true means of the powers of x i and are assumed known u t is the small area effect and d t u = u t = b t u b t is a vector with 1 in the t-th position and 0 else. As a predictor of ȳ t, ŷ t = x t ˆβ + z t ˆγ + b t û is used which is a linear combination of the GLS estimator and the BLUPs

12 Prediction mean squared error For simplicity, consider the prediction error ŷ ȳ in the case of estimated variance components W = [Z D], w = (γ, u ), w t = ( z t, b t ) Then it is obtained with c t = x t w t Σ w W V 1 X that; ŷ ȳ = c t (ˆβ β) + w t (Σ w W V 1 (Y Xβ) w) (3) The EBLUP ỹ ȳ = ĉ t (ˆβ β) + w t (ˆΣ w W ˆV 1 (Y Xβ) w) The prediction mean squared error of EBLUP predictor can be estimated.

13 Bootstrapping small area and local effects Need of using bootstrap-based procedure; a likelihood ratio test for testing the presence of small area random effects can be constructed. likelihood ratio statistic is often poorly approximated by the asymptotics useful to supplement bootstrap-based procedure Bootstrap replicate observations are generated as Y = Xˆβ + Zγ + Du + e where γ, u and e are bootstrap replicates of the random components in the model.

14 Robust way to draw replicates; to resample from the emprical distributions of the fitted residuals and predictions assume the variances σ 2 are known start with the predictors ˆγ and û obtained earlier denote H = VP where P = V 1 V 1 X(X V 1 X) 1 X V 1 Var(ˆγ) = σ 4 γz V 1 (I H)Z

15 adjust ˆγ to obtain predictors with the correct second moment by letting γ = ˆγ(Z V 1 (I H)Z) 1/2 /σ γ (4) ũ = û(d V 1 (I H)D) 1/2 /σ u (5) to generate estimated errors start from; ê = Y Xˆβ Zˆγ Dû ẽ = ê(v 1 (I H)) 1/2 /σ e (6) bootstrap sampling can be done form γ, ũ and ẽ

16 Some conclusions on current part of the approach; it would create bootstrap random effects and errors that of estimating β the error of the parameter estimation is often very small relative to the prediction error bootstrap approach that ignores estimation is likely perform almost as well as a more complicated procedure used the expressions (4), (5) and (6) after replacing the variance components by their REML estimators similarly found that the matrix H which accounts for the estimation of β had no effect on the adjustments, set H = 0

17 Given B bootstrap samples, the prediction mean squared error for the small areas can be obtained by 1 B B (ỹt b b=1 ȳ b t ) where ȳt b ỹt b is the mean for small area t in the b-th bootstrap sample represents its REML predictor

18 Survey of lakes in the Northeastern states of U.S. Between 1991 and 1996, the Environmental Monitoring and Assessment Program (EMAP) of the U.S. Environmental Protection Agency conducted a survey of lakes in the Northeastern states of U.S. It is considered the estimation of the mean acid neutralizing capacity (ANC) for each of 113 small areas defined by 8-digit Hydrologic Unit Codes(HUC) within the region of interest. ANC measures the buffering capacity of water against negative changes in ph HUCs represent a nested subdivision of all U.S. land based on hydrological features

19 Figure: Locations of sampled lakes in Northeastern U.S.,the region of interest and the locations of the sampled lakes on a population of 21,026 lakes from which 334 lakes were surveyed.

20 Summary for the analysis; the aim to identify HUCs of concern within the region, based on the results from the survey lakes in close geographical proximity but located in different HUCs are expected to be less similar than two lakes in the same HUC factors affecting ANC such as acid deposition and soil characteristics cut across HUCs a HUC prediction model has the potential to capture most of the interesting patterns in the data

21 Figure: Hydrologic Unit Code (HUC) small areas within Northeastern U.S. region, with average ANC computed in all small areas containing sample observations.

22 The variables that can be used in the construction of a SAE model in the application are the geographical coordinates of the centroid of each lake (in the UTM coordinate system) and its elevation. Construction of the small area estimator using a transformed radial basis defined as; [ Z = [C(x i κ k )] 1 k K C(κk κ k ) ] 1/2 1 k,k, i = (1,..., n) K where C(r) = r 2 log r and x i = (x 1i, x 2i ) denotes the geographical coordinates for observation i and κ k, k = 1,..., K are spline knots. The locations of the K = 80 knots are selected by the space-filling algorithm.

23 Figure: Shows the locations of the knots selected by the space-filling algorithm. Lake locations (open circles) and knot locations of the bivariate radial spline function on the UTM coordinates(solid circles).

24 The ANC small area model can now be written as in (2); That model includes Y for the ANC observations Y = Xβ + Zγ + Du + e X a matrix containing an intercept and the linear elevation term Z for the spatial locations D a matrix of indicators for the HUCs Then the model can be fitted using REML.

25 Figure: Parameter estimates for penalized spline small area estimation model for Northeastern Lakes data.the P-values are computed using the bootstrap procedure described earlier.

26 Figure: Map of model predicted mean ANC for all HUCs. The small area estimation map is smoother and also contains values in all HUCs, offsetting some of the limitations of the original data.

27 Figure: Plot of estimated root EMSE for the small area predictions obtained by the bootstrap method against the estimated root EMSE.

28 Figure: Comparison of AIC values and correlations between HUC model predictions and averages of the sample observations in the HUCs, for inclusion and exclusion of random effect terms in model; Left number in each cell is AIC, right number is correlation.

29 Figure: (a)comparison of HUC predictions for model with both random effects and model with HUC random effect only. (b)comparison of HUC predictions for model with both random effects and model with spline random effect only.

30 Some conclusions; inclusion of a spatial spline can improve the fit relative to a model which only uses a random effect for the small areas this would be done in traditional small area estimation

31 Thank You..

32 References Eilers, P. H. C. and B. D. Marx (1996). Flexible smoothing with B-splines and penalties. Statistical Science 11 (2), Jiang, J. and P. Lahiri (2006). Mixed model prediction and small area estimation. Test 15, 196. Opsomer, J.D., Claeskens, G., Ranalli, M.G., Kauermann, G., Breidt, F. J. (2006). Nonparametric Small Area Estimation Using Penalized Spline Regression Journal of the Royal Stat. Society, Series B, 70, Ruppert, R., M. Wand, and R. Carroll (2003). Semiparametric Regression. Cambridge University Press.