Small Area Modeling of County Estimates for Corn and Soybean Yields in the US
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1 Small Area Modeling of County Estimates for Corn and Soybean Yields in the US Matt Williams National Agricultural Statistics Service United States Department of Agriculture Federal Committee on Statistical Methodology Research Conference Nov. 4, providing timely, accurate, and useful statistics in service to U.S. agriculture. 1
2 Background County Agricultural Production Survey (CAPS) Program In the County Estimates Pilot Study was initiated in 5 States. New approach for selecting and processing county estimates, consistent with procedures used for the Agricultural Survey programs County estimates data are merged with year-end survey data. From 2011 onward, the probability design was expanded to cover all States. FCSM
3 Motivation CAPS results are used by field offices and HQ to establish county estimates for acreage and yield (production per acre). Other administrative data are examined (Farm Service Agency, remote sensing, etc.) Centralized Database for examining inputs and setting estimates Our GOAL is to add value to this process. Efficiency: increased automation? Added functionality: SE s and CV s for estimates? We examine Small Area Models as a possible solution. FCSM
4 Methods The methods described in this presentation are considered model-based (Rao, 2003). Nested error (mixed model) regression Combine survey and covariate information. Induce a correlation between small areas, pooling information across the areas. Referred to as borrowing strength, a phrase ubiquitous in the literature. Methods are related to shrinkage estimates. Models can focus on the County (area) or Record (unit). Inference can be conducted with Frequentist and Bayesian approaches. FCSM
5 Shrinkage Estimates Mean Square Error (E[Y ] = θ) ] ( [ 2 ( ] 2 E [(θ ˆθ) 2 = θ E[ˆθ]) + E E[ˆθ] ˆθ) MSE = Bias 2 + Variance Stein s Paradox (Efron and Morris, 1977) [ ] d For d > 2, total MSE i=1 E (θ i ˆθ i ) 2 is not minimzed by the Least Squares estimator ˆθ LS. A class of shrinkage estimates ˆθ Sh can be created which dominate ˆθ LS FCSM
6 Shrinkage Estimates Shrinkage Estimate ˆθSh = ȳ + c(ˆθ LS ȳ) c can be chosen several ways. Composite Estimate ˆθC = φˆθ A + (1 φ)ˆθ B φ can be chosen several ways. Small Area Models Combine two estimators: Survey Indications and Regression Model Use variance components from the Survey and the Model to estimate φ. FCSM
7 Shrinkage Estimates: Example θ N(100, 10) Y N(θ, 5) ˆθ Bias RMSE MAD Raw Y Model FCSM
8 County vs. Record Level Models County (Area) Level Covariates are available at the county level Survey estimates and standard errors (SE s) used directly Composite parameter φ determined by survey SE s and model estimate of between-county variance Record (Unit) Level Covariates may be used for the record level Survey estimates are modeled via record responses and weights Composite parameter φ determined by model estimate of between-county variance and residual variance of the model (with weights) FCSM
9 Parameter Estimation Empirical Bayes Frequentist approach Maximum Likelihood and Restricted ML Estimation for Parameters Inference uses plug-in estimates of parameters. Hierarchical Bayes Full Bayesian approach Constructs a posterior using a likelihood and a prior distribution for parameters. Inference uses distributions (or samples from them) of the parameters. Considerations for computation and prior selection FCSM
10 Results: 2010 Pilot States Models (14 results) Area Level (EB & HB) Unit Level (EB & HB) Unit Level with Sampling Weights (10) EB (2), HB (3) With & with/out modifications (x2) FCSM
11 Results: 2010 Pilot States Models (14 results) Area Level (EB & HB) Unit Level (EB & HB) Unit Level with Sampling Weights (10) EB (2), HB (3) With & with/out modifications (x2) Each model used the same area-level covariates National Commodity Crop Productivity Index (NCCPI) Normalized Difference Vegetation Index (NDVI) based FCSM
12 Results: 2010 Pilot States Models (14 results) Area Level (EB & HB) Unit Level (EB & HB) Unit Level with Sampling Weights (10) EB (2), HB (3) With & with/out modifications (x2) Each model used the same area-level covariates National Commodity Crop Productivity Index (NCCPI) Normalized Difference Vegetation Index (NDVI) based Survey Indications generally best predictor of final published values Area Level models consistently second best EB and HB models generally consistent tighter agreement for area-level models FCSM
13 Results: Selected States Similar to 2010 Results A few states showed modest gains using Models over Survey Indications Some states broke even Many state Models had no improvement over the Survey Indications FCSM
14 Example: Modest Gain ˆθ Bias RMSE MAD Direct EB HB CN 31 Board Yield Estimates Survey Ind Area EB Area HB FCSM
15 Example: Break Even ˆθ Bias RMSE MAD Direct EB HB SB 27 Board Yield Estimates Survey Ind Area EB Area HB FCSM
16 Example: Adding Noise ˆθ Bias RMSE MAD Direct EB HB CN 46 Board Yield Estimates Survey Ind Area EB Area HB FCSM
17 Comments Final published values considered the true mean. Direction of causality is reversed. Published values did not generate the survey responses. Benchmarks to state values and other considerations (weather, growing conditions, etc). A few large deviations (changes) and some systematic (benchmarking) Shrinkage estimates Small amounts of change spread out over all counties No decision to leave some unchanged. External Validation How is the current county estimates program being evaluated? Best available given timeliness requirement. What is available later to compare? (Census 2012) FCSM
18 Related Research Benchmarking County yield is a ratio. Benchmarking to state yield (ratio) is therefore nonlinear (Williams and Berg, 2013). Multi-stage benchmarks and penalties (districts or variances) are also possible. Nonlinear relationships Strong relationships between covariates and responses may be nonlinear. General additive models (GAM s) have been applied to unit level models (Opsomer et al., 2008). Similar methods can be used for area level models. FCSM
19 References Efron, B. and Morris, C. (1977). Stein s paradox in statistics. Scientific American, 236: Opsomer, J., Claeskens, G., Ranalli, M., Kauermann, G., and Breidt, F. (2008). Non-parametric small area estimation using penalized spline regression. Journal of the Royal Statistical Society: Series B, 70: Rao, J. (2003). Small Area Estimation. John Wiley & Sons, New Jersey. Williams, M. and Berg, E. (2013). Incorporating user input into optimal constraining procedures for survey estimates. Journal of Official Statistics, 29: FCSM
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Introduction to Machine Learning and Cross-Validation Jonathan Hersh 1 February 27, 2019 J.Hersh (Chapman ) Intro & CV February 27, 2019 1 / 29 Plan 1 Introduction 2 Preliminary Terminology 3 Bias-Variance
Now consider the case where E(Y) = µ = Xβ and V (Y) = σ 2 G, where G is diagonal, but unknown.
Weighting We have seen that if E(Y) = Xβ and V (Y) = σ 2 G, where G is known, the model can be rewritten as a linear model. This is known as generalized least squares or, if G is diagonal, with trace(g)