Hierarchical area level model approach to small area
estimation incorporating auxiliary information
Jae-Kwang Kim
1
Iowa State University
SAE meeting @ Santiago
Aug 4, 2015
1
Joint work with Zhonglei Wang, Zhengyuan Zhu (ISU) and Nathan Cruz (NASS)
Introduction
1
Introduction
2
Hierarchical Structural model
3
Application to NASS project
4
Conclusion
Kim (ISU)
Small area estimation
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Introduction
Motivation
Cooperative agreement with National Agricultural Statistical Service
(NASS) in USDA.
Want to incorporate information from several sources
1
2
3
JAS (June Area Survey): Survey estimates obtained from an area
frame sample.
FSA (Farm Service Agency): Self-reported administrative data.
CDL (Cropland Data Layer): Classification of satellite imagery data.
Summary
Source
JAS
FSA
CDL
Kim (ISU)
Observation
direct measurement
Self-reported data
Satellite image classification
Small area estimation
Main source of Error
Sampling error
Coverage error
Measurement error
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Introduction
Figure : The 2009 cropland data layer products. The legend identifies aggregated
agricultural and non-agricultural land cover categories by decreasing acreage.
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Small area estimation
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Introduction
Area level model
Area level model is used to combine information from the three sources.
We observe three estimates for each area.
Measurement: crop acres (for each crop)
Yi : true crop acres for area i (Unobserved)
Ŷi : estimate from JAS (subject to sampling error)
X1i : estimate from FSA data (subject to coverage error)
X2i : estimate from CDL data (subject to measurement error, or
classification error)
The analysis unit is a sub-state area, called a district.
Three estimates are correlated.
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2500000
Introduction
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JAS
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CDL
Figure : Relationship between JAS and CDL.
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Small area estimation
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Introduction
Area level model (Cont’d)
The goal is to predict Yi (=true crop acres) using the observation of
Ŷi (=JAS), Xi1 (=FSA), and Xi2 (=CDL).
Area level model is a useful tool for combining information from
different sources by making an area level matching.
Area level model consists of two parts:
1
2
Sampling error model: relationship between Ŷi and Yi .
Structural error model: relationship between Yi and (Xi1 , Xi2 ).
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Small area estimation
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Introduction
Area level model: Fay-Herriot model approach
Figure : A Directed Acyclic Graph (DAG) for classical area level models
(X = (X1 , X2 )).
Y
(1)
(2)
X
Ŷ
(1): Sampling error model (known),
(2): Structural error model (known up to θ).
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Small area estimation
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Introduction
Combining two models
Prediction model = sampling error model + structural error model
Bayes formula for prediction model
p(Yi | Ŷi , Xi1 , Xi2 ) ∝ g (Ŷi | Yi )f (Yi | Xi1 , Xi2 ),
where g (·) is the sampling error model and f (·) is the structural error
model.
g (·): assumed to be known.
f (·): known up to parameter θ.
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Small area estimation
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Introduction
Parameter estimation
Obtain the prediction model using Bayes formula
EM algorithm: Update the parameters
X
θ̂(t+1) = argθ max
E {log f (Yi | Xi1 , Xi2 ; θ) | Ŷi , Xi1 , Xi2 ; θ̂(t) }
i
where the conditional expectation is with respect to the prediction
model evaluated at the current parameter θ̂(t) .
Kim (ISU)
Small area estimation
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Introduction
Prediction vs Parameter estimation
Figure : EM algorithm
E-step
Y
θ̂
M-step
X
Ŷ
Kim (ISU)
Small area estimation
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Introduction
Prediction (frequentist approach)
Best prediction: Expectation from the prediction model at θ = θ̂
Ŷi∗ = E {Yi | Ŷi , Xi1 , Xi2 ; θ̂}
If f (Yi | Xi1 , Xi2 ) is a normal distribution then
Ŷi∗ = αi Ŷi + (1 − αi )E (Yi | Xi1 , Xi2 ; θ̂)
for some αi where
αi =
V (Yi | Xi1 , Xi2 ; θ̂)
V (Ŷi ) + V (Yi | Xi1 , Xi2 ; θ̂)
.
This is often called composite estimator.
Kim (ISU)
Small area estimation
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Hierarchical Structural model
1
Introduction
2
Hierarchical Structural model
3
Application to NASS project
4
Conclusion
Kim (ISU)
Small area estimation
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Hierarchical Structural model
Hierarchical structural model
Use a two-level structural error model.
Level 1 (Within-state model):
Yhi = Xhi βh + uhi
where Xhi = (1, Xhi1 , Xhi2 ) and βh = (βh0 , βh1 , βh2 )0 .
Level 2 (between-state model):
βh ∼ N(β2 , Σ2 )
i.e.

  
βh0
β0
σ00
βh1  ∼ N β1  , 
β2
βh2

σ01
σ11

σ02
σ12  .
σ22
Sampling error model remains the same. (Ŷhi ∼ N(Yhi , Vhi ).)
Kim (ISU)
Small area estimation
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Hierarchical Structural model
Two-level structural error model
Level 1 model
Yhi ∼ f1 (Yhi | Xhi ; θ h )
and Ŷhi is a measurement for Yhi .
Level 2 model
θh ∼ f2 (θh | Zh ; ζ)
where Zh is the state-specific covariate. An example of Zh is a
classification of states into groups (major crop states / minor crop
states).
Kim (ISU)
Small area estimation
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Hierarchical Structural model
Figure : A DAG for two-level Fay-Harriott model.
Yhi
(1)
Ŷhi
θh
(3)
(2)
Xhi
Zh
(1): Sampling error model,
(2): level-one structural error model,
(3): level-two structural error model
Kim (ISU)
Small area estimation
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Hierarchical Structural model
Why two-level models?
1
To allow for state-specific models
2
To borrow strength from other states
Kim (ISU)
Small area estimation
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Hierarchical Structural model
Frequentist approach to multi-level model estimation
Three steps for parameter estimation in each level
1
2
3
Summarization: Find a measurement for latent variable to obtain the
sampling error model.
Combine: Find a prediction model for latent variable by combining the
sampling error model and the structural error model.
Learning: Estimate the parameters from data.
Apply the three steps in level one model and then do these in level
two model.
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Small area estimation
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Hierarchical Structural model
Parameter estimation for Level 1 model
θh : parameter in f1 (Yhi | Xhi ; θh )
Summarization: Find a measurement Ŷhi for Yhi and obtain the
sampling error model
Ŷhi | Yhi ∼ g1 (Ŷhi | Yhi )
Combine the two models using Bayes formula
p1 (Yhi | Xhi , Ŷhi ; θh ) ∝ g1 (Ŷhi | Yhi )f1 (Yhi | Xhi ; θh )
Learning: Parameter is estimated using EM algorithm
X
(t+1)
(t)
θ̂h
= arg max
E {log f1 (Yhi | Xhi ; θh ) | Xhi , Ŷhi ; θ̂h }.
θh
Kim (ISU)
i
Small area estimation
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Hierarchical Structural model
Parameter estimation for Level 1 model (Cont’d)
Figure : EM algorithm for level 1 model
E-step
Yhi
θ̂h
M-step
Xhi
Ŷhi
Kim (ISU)
Small area estimation
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Hierarchical Structural model
Structures in Two level model
Model
Level one
Level two
Measurement
(Data summary)
Ŷh = (Ŷh1 , · · · , Ŷhnh )
θ̂ = (θ̂ 1 , · · · , θ̂ H )
Kim (ISU)
Parameter
Latent variable
θh
ζ
Yh = (Yh1 , · · · , Yhnh )
θ = (θ 1 , · · · , θ H )
Small area estimation
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Hierarchical Structural model
Parameter estimation for two-level models
E-step
θh
ζ̂
M-step
E-step
Yhi
θ̂h
M-step
Ŷhi
Kim (ISU)
Xhi
Small area estimation
Zh
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Hierarchical Structural model
Best Prediction
Prediction under Level 1 model
Latent variable: Yhi
Best prediction
Ỹhi∗ (θh ) = E {Yhi | Ŷhi , Xhi ; θh }
Under single level model, we would use Ŷhi∗ = Ỹhi∗ (θ̂h ), where θ̂h is the
MLE of θh under the level one model.
Prediction under two level model: compute
Ỹhi∗∗ (ζ) = E {Ỹhi∗ (θh ) | Zh , θ̂ h ; ζ}
and use
Ŷhi∗∗ (ζ) = Ỹhi∗∗ (ζ̂).
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Small area estimation
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Hierarchical Structural model
Estimation of MSPE
For Ŷh∗∗ =
Ŷhi∗∗ , we can show that


X X

X
E {(Ŷh∗∗ − Yh )2 } =
αhi Vhi +
(1 − αhi )(1 − αhj )qhij


P
i
i
i
j
where Vhi = V̂ (Ŷhi )
αhi =
and
Kim (ISU)
V (Yhi | Xhi )
Vhi + V (Yhi | Xhi )
n
o−1
−1
+
V
(
β̂
)
qhij = Xhi Σ−1
X0hi .
h
2
Small area estimation
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Application to NASS project
1
Introduction
2
Hierarchical Structural model
3
Application to NASS project
4
Conclusion
Kim (ISU)
Small area estimation
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Application to NASS project
Logistic regression model
Mhi : total crop acres in district (hi). (This is available to us.)
Since Ȳhi = Yhi /Mhi is a proportion, we may use
Ȳhi = p(βh0 + βh1 X̄hi1 + βh2 X̄hi2 ) + ehi
(1)
where p(x) = {1 + exp(−x)}−1 and
ehi ∼ (0, ψh phi (1 − phi ))
for some ψh > 0 and phi = p(βh0 + βh1 X̄hi1 + βh2 X̄hi2 ).
Berg and Fuller (2014) also considered model (1) in the single-level
model approach.
Thus, we extend Berg and Fuller (2014) approach to two-level model.
Kim (ISU)
Small area estimation
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Application to NASS project
Application to NASS data
We are interested in predicting the acres of soybean for each state.
14 major states: KY, WI, IL, SD, AR, KS, ND, MI, OH, MO, IN, IA,
MN, NE
13 minor states: AL, NC, NY, GA, LA, TX, MX, TN, PA, OK, MD,
SC, VA
Two-level FHM approach
1
2
Level One model: Logistic regression model
Level Two model: normal model within major /
 
  
βh0
β0
σ00 σ01
βh1  ∼ N β1  , 
σ11
βh2
β2
Kim (ISU)
Small area estimation
minor groups

σ02
σ12  .
σ22
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Application to NASS project
Results—2013 soybeans (Major Crop States)
Estimation results for soybeans of year 2013 based on unadjusted FSA (Major case)
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Estimates
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AR
IA
IL
IN
KS
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KY
MI
MN
MO
ND
NE
OH
SD
WI
State
Method:
Kim (ISU)
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JAS
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FSA
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CDL
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Logis_S
Small area estimation
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Application to NASS project
Results—2013 soybeans (Minor Crop States)
Estimation results for soybeans of year 2013 based on unadjusted FSA (Minor case)
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Estimates
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AL
GA
LA
MD
MS
NC
NY
OK
PA
TN
VA
State
Method:
Kim (ISU)
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JAS
●
FSA
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CDL
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Logis_S
Small area estimation
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Application to NASS project
Summary statistics
The following summaries are obtained.
Mean of the standard errors.
Mean of relative efficiency:
smodel /sJAS ,
where smodel and sJAS are the standard errors of the model estimator
and JAS, respectively.
Kim (ISU)
Small area estimation
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Application to NASS project
Summary statistics for the two models (Cont’d)
Year
2011
2012
2013
Kim (ISU)
Crop
corn
soybeans
winterwheat
corn
soybeans
winterwheat
corn
soybeans
winterwheat
s.e
1.687
1.567
1.086
1.827
1.611
1.157
2.058
1.508
1.143
Small area estimation
R.E. .
0.768
0.816
0.733
0.819
0.741
0.754
0.750
0.685
0.770
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Conclusion
1
Introduction
2
Hierarchical Structural model
3
Application to NASS project
4
Conclusion
Kim (ISU)
Small area estimation
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Conclusion
Discussion
Two-level structural model for “borrowing strength” from other
states.
Model specification
Parameter estimation
Prediction
MSPE estimation
Frequentist approach to hierarchical Fay-Herriot model.
Extension of the proposed method to Measurement error model is
under development.
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Small area estimation
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Conclusion
The end
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