Multiyear, Through-the-Season Crop Acreage Estimation ...

MULTIYEAR, THROUGH-THE-SEASON CROP ACREAGE ESTIMATION USING ESTIMATED ACREAGE IN SAMPLE SEGMENTS

Robert L. Sielken, Jr., Texas A&M University

Abstract. Using remote sensing technology and = +b +6£ +e t= 1 ...... T
knowledge of crop growth behavior, NASA can esti- Y (Pts£) ~ t s ts£
mate a crop's proportional at harvest acreage in
a 5x6 nautical mile segment. Such estimates s = i,..., S,
are determined at a few times during the crop's
growing season for a small number of segments £=i,..., L
sampled from a large homogeneous area (stratum).
Similar estimates are obtained for consecutive (2)
years on possibly different yearly samples of
segments from the same stratum. A mixed effect where
analysis of variance model for such a multiyear
data set is used as the basis for a weighted Pts£ = the estimated proportion of the s-th
least squares estimate of the crop's proportional segment's acreage that will contain
at harvest acreage in the stratum for the current the crop at harvest time in the t-th
year. Potentially useful weighting procedures year when the estimate is made at crop
and transformations of the segment estimates are calendar time £ (for example, £=i could
also briefly discussed. denote early season, £=2 mid-season,
and £=3 at harvest time);
i. INTRODUCTION
y(Pts£) = a variate transformation of Pts£;
= the stratum's transformed crop acreage

t
proportion for the t-th year;

NASA seeks to use its remote sensing capabili- b = the s-th sampled segment's departure
ties to estimate a particular crop's proportional s from the stratum's transformed crop
acreage in a large homogeneous area (stratum). acreage proportion; the b's are
Since it is not practical to process all of the s
satellite information on the entire stratum, a random variables with expectations
sample of relatively small segments is selected zero and variance o~;
each year over a multiyear period. (In the past
a segment has been a 5x6 nautical mile rectangle.) 6£ = the systematic difference between the
An estimate of the crop's at harvest acreage non-harvest time estimates of the crop's
proportion is made for each sample segment at a transformed at harvest acreage propor-
few times during the crop's growing season. This tion and the corresponding estimate
paper focuses on the procedure for estimating the made at harvest time (6L = 0);
crop's current year at harvest acreage proportion
in the stratum on the basis of the through-the- ets £ = the aggregate of sampling and classi-
season segment level estimates collected over a fications errors in the transformed
multiyear period. data.

Since the same segments do not have to be in The primary objective is to estimate the
the sample every year, there is an interesting
associated problem of determining an optimal crop's at harvest proportion of the stratum
multiyear sampling design. Although this problem
is not specifically discussed herein, some tech- acreage in the current year, T; that is, estimate
nical reports on this topic are included in the
references. the inverse transformation of sT denoted by

H. O. Hartley, during his years (1963-1979) as PT = y-l(~T). Secondary objectives could be im-
Distinguished Professor of Statistics at the
Institute of Statistics, Texas A&M University, proved estimates of at harvest acreages in pre-
contributed greatly to NASA's research efforts
pertaining to crop acreage estimation and stimu- vious years or estimates of changes in the stratum's
lated his co-workers' efforts. Several of the
more recent technical reports on crop acreage crop at harvest acreage proportion from year to
estimation which he either wrote or encouraged
are listed among the references. year.
Estimates of the stratum's crop at harvest
2. THE BASIC MODEL FOR MULTIYEAR ESTIMATION
acreage proportion are often needed throughout
The basic model relating the stratum's at
harvest crop acreage to the crop's estimated at the current year as well as at harvest time. For
harvest acreage in the sample segments has the
general form example, an early season estimate based on obser-

y(observation) = year effect + segment effect vations for £=I,...,L for t=l,...,T-I and only

(1) £=i for t=T is often desired. ^

+ season bias + noise Of course, even though the estimate PT=y -I(~T ) ,

where y(.) is an appropriate transformation. The of the stratum's crop at harvest acreage proportion
specific form of model (I) is
for the current year involves only ~T, the ~T

depends on the entire multiyear data set and the

estimates of the segment effects and the systematic

biases which are assumed to be constant from year

to year.

3. TRANSFOR~MATIONS OF THE ESTIMATED S E G ~ N T
PROPORTIONS

The simplest transformation, y(p), of the
estimated segment crop acreage proportion, p, to
use in (2) is the identity transformation

y(p) = p •

490

However, it is very doubtful that the additive unweighted least squares analysis.
model (2) would hold for y(p)=p particularly if The first step in the weighted least squares
the p's exhibit a large variation within the
stratum. On-the-other-hand a multiplicative model analysis of model (2) proceeds as follows:
for p may be more reasonable. For instance if
(i) The data is the segment estimates Pts£"

(ii) For all Pts/ satisfying ly(Pts£)l <~,

(i) 30% of the stratum's acreage is planted define Yts£ = Y(Pts£)" (This is all
in wheat at the time wheat is harvested
in year t; Ptsf if y(p) = p but only Ptsf > 0 if
y(p~ = In(p) and only 0 < Ptsi <i if
(ii) the s-th segment's wheat acreage averages y(p) = 1/2 £n[p/(1-p)]).
only 80% of the stratum's wheat acreage
at harvest time; (iii) Using only Y+s~ with ly(Pts£)l <~,
determine ~t, ~s, ~£ for the fixed
(iii) the at harvest acreage estimate made at effects model
mid-season is only 70% of the at harvest
estimate made at harvest time; and Yts£ = ~t + b s + 6£ + ets/
from a weighted least squares analysis of
(iv) the sampling and classification errors
cause the estimated at harvest acreage
to be 110% of what it would be without Wts£ Yts£ = Wts£~t + Wts£bs + Wts£ 6£ + ets£
these errors,

then

Pts/ = (.30)(.80)(.70)(I.i0) . with

Here a logarithmic transformation, y(p) = /n(p), ~b - 0 6L-0 '
would be appropriate and
S'
S

and

Y()P't"s£ = ~t + b s + 6£ + ets£ Wts£ = {Var[y(Pts£)]}-½

= In(.30) + ln(.80) + In(.70) + /n(l.lO) . (iv) For all t, s, £ calculate

The logit transformation, Y.ts£ = .~t + b.s + ~£. and Pts£ = y l(Yts£) •

y(p) = (1/2) £n[p/(l-p)] , (v) (a) If all Pts£ satisfy ly(Pts£) I < ~,
then redefine Pts£ = Pts£ and either
is another useful transformation which approximately return to (iii) or stop this first
converts a multiplicative model for p into an step after a "sufficient" number of
additive model for y(p). In addition, the logit iterations.
transformation has the property that
(b) If all Pts/ do not satisfy ly(Pts/)l
0 <__y T ) <_i < ~ , then use a Taylor s^ eries expan-
sion of y(p) about p = p,
whereas the logarithmic transformation guarantees
only Fay

y -i (~r ) ~,0 y(p) = y(p) + (p-p) Edp Z = ; '

and the identity transformation makes no guarantees. to create "working y's." For the
All three of the above transformations are logarithmic transformation, define

considered in Sielken and Dahm (1981). There Yts/ = Yts/ + (Pts/ - Pts/)/Ptsl ;
approximate expressions are derived for
for the logit transformation
(i) the variance of y(p) ,
(ii) -i ^ Yts/ = Yts/ + (Pts/ - Pts/)/[2ptsl(l-Ptsl) ]"
(iii) the bias of y (~T) ,
-i ^ !
the mean squared error of y (~T) , and
Using these Yts/ s redefine
(iv) confidence intervals for PT
-i
under the assumption that p arises from a binomial Pys/ = y (Yts/)
random variable.
for all ts/ and return to (iii).
4. THE WEIGHTED LEAST SQUARES ANALYSIS OF
THE SEGMENT ESTIMATES ^

When estimating the parameters (~t, bs,6£) in The final estimate ~T, which is of primary
model (2), it is not particularly reasonable to interest, is obtained in the next step from a
assume that the variance of y(Pts£) is the same mixed effects model weighted least squares analysis
for all t,s,£. Hence a weighted least squares of the final Yts/'S determined in the first step,
analysis is called for as opposed to the usual namely the iterative weighted least squares analysis

491

using a fixed effects model• The mixed model The estimation of the variance components o~ and
weighted analysis assumes the b s is a random o~ and their ratio y can be based upon equa~ing
certain sums of squares from the fixed effects
factor with mean 0 and variance o2 and is outlined model with their expectations under the mixed model
below• and then solving the equations for o~ and o~.
Basically this is Henderson's Method~3. In par-
When the basic mixed effects model ticular, if the segment effects bl,...,b S are
treated as fixed effects in
Yts£ = ~t + b s + 6£ + ets£
is analyzed, it is considered in the form W y : W X (~6) + WlPo + Ie,

Wts£Yts£ = Wts£ ~t + Wts£b s + Wts£6£ + e ts£ then the residual sum of squares is
-½
(Wy)' [I- (WX,WU) { (WX,WU) '(WX,WU) }-i (WX,WU) '] (Wy)
with wts£ again proportional to [Var(Yts£)] •
In matrix notation (3)
which under the mixed effects model has expectation
Wy : WX (6) + WU b + Ie

where

[n-(T + L + S -2)] o 2 (4)

= . ..)T g

Y (YlII ,YlI2, where n is the number of observations• Equating
(3) to (4) provides a nonnegative estimate, ~2 ,
= (~i ..... ~T )T ' of o 2 . Similarly, the regression sums of squares
due to the b's given the ~'s and 6's, namely
6 : (61,62 ..... 6L_I)T ,(6 - o)
L

b = (bl, b2, )T

W = matrix Containing the Wts£'S , ( W y ) ' (WX,WU) { (WX,WIJ) ' (WX,WU) } - I ( w x , w u ) ' (Wy)

X = matrix of O's and l's corresponding to -(Wy), (wx){ (wx)' (wx) }-i (wx)' (Wy) , (5)
the ~'s and 6's,

U = sampling design matrix of 0's and l's under the mixed effects model has expectation
corresponding to which b s's appear in
which years, and o2 trace{ (WU) '[I- (WX) { (W-X) '(WX) }-I (WX)' ] (WU) }

D

I = identity matrix•

+ o 2 (s - l) . (6)

The random portion of Wy is WUb + I~ which has ^
covariance
Substituting o~ for 02 in^ (6) and equating (5) to
Vo2 = Io 2 + WIFuTwTo~
g (6) nrovides an estimate, 02 of o 2 The estimate
of Y is y~ = o~~2/o^2 as long ba's o^~ >b" 0 Since oh2
= (I + wuuTwTy) o2
- U ~^ -- "
> O, an estimate o2 < 0 strongly suggests that~y

is-in reality a smaIl--positive number.

where 5. COMPUTER IMPLEMENTATION: ACRE

o2 : Var (ets£) ' A self-contained computer implementation of the
above weighted least squares estimation procedure
O~ : Var (b) , has been given to Lockeed, NASA, and ERIM.

S The computer program has been nicknamed ACRE.
ACRE will accept up to 5 years of data (T < 5) on
22 up to 20 segments (S < 20) with up to 5 observations
y = Ob/O s per segment per year (--L< 5). These dimension
restrictions could of course be easily increased•
o" With the current dimensions ACRE requires 768K
bytes of core memory on an AMDAHL 470/V6. The
The weighted least squares estimator of (~ , 6) T IS program is written in straightforward FORTRAN and
is extensively internally documented•
^
ACRE will allow the user to choose any one of
(~) = (xTwTv-Iwx)-IxTwTv-I Wy the three transformations

and

^

Var [(6) ] : (xTwTv-~qX)-I O2

In particular (i) y(p) = p , .

^ (ii) y(p) = £n(p) , or
(iii) y(p) = .5 £n[p/(1-p)]
Var (~T) : (xTwTv-~x) -I o 2
T,T e ACRE regularly provides the following output:

where ( )-I denotes the T-th element on the i. A listing of the observations that have
T,T been input•

diagonal of the matrix inverse. 2. An indication of the transformation y(p) that
Of course, since y = 02/02 is unknown, an

estimate of y must be substmbtuted in the weighted
least squares analysis of the mixed effects model.

492

the user has requested. growing season,

^ (viii) the composition of the segment, etc.

3. The estimated effects (~, b, ~) in the final The derivation of appropriate weights under this
iteration of the weighted least squares latter scenario is being investigated.
analysis of the fixed effects model.
REFERENCES
4. An analysis of the residuals in the fixed
Technical reports prepared for NASA:
effects model.
(i) Hartley, H. O. June, 1974. The Sample Design
^^ for the ERTS Satellite.

5. The variance component estimates ~ o 2 (2) Hartley, H. O. July, 1974. The "ERSATZ"
^ ~' ~, Sampling Plan for the ERTS Satellite II (repre-
senting a revision of an earlier report under the
and Y. same title)

^^ (3) Feiveson, A.H. and H. O. Hartley. August, 1974.
Proposed LACIE Sampling Plan.
6. The estimates of the fixed effects (~, 6)
(4) Hartley, H. O. and D. A. Lamb. April, 1975.
in the weighted least squares amlysis of Formulas for Variance Estimation in Proposed LACIE
Sampling Plan.
the mixed model.
(5) Hartley, H. O. April 1975. The Treatment of
7. An analysis of the residuals in the mixed the Defective Area Segments in the LACIE Survey
effects model. for the U.S.A.

8. The stratum's estimated at harvest crop (6) Hartley, H. O. and D. A. Lamb. September, 1975.
acreage proportion, P t = Y-l(~t)' for each Formulas for Variance Estimation in Proposed LACIE
year t = i, ..., T. Sampling Plan II.

9. Indications of the precision of the esti- (7) Book, D., Hartley, H. O.,Lamb, J., Larsen, G.
mated at harvest crop acreage proportions and Michael Trenchard. March, 1976. Small Area
for all years. Namely, approximate values Estimates of Historical Wheat Acreages for Russia
for the bias and mean squared error of P , and China.
t
and an approximate 90% confidence interval (8) Hartley, H. O. March, 1976. An Outline of
on Pt for each year t = i, ...,T. Muitiyear Estimates in the LACIE Sampling Plan.

6. CURRENT RESEARCH (9) Hartley, H. O. and David Lamb. August, 1976.
Estimation of Wheat Acreage Via Small Grain Crop
The sensitivity of the estimate, P^T = y- 1 (_~T) Acreage.
of the stratum's at harvest crop acreage proportion
to such things as the transformation used, the (i0) Hartley, H. O. March, 1977. The Use of the
accuracy of the weights, and the reliability of 1974 Ag Census for LACIE Estimates.
the estimate of y = 02/02 is under study. The
empirical behavior of~th$ approximate expressions (ii) Hartley, H. O. October, 1976. Bias Through
for the bias of PT and the mean squared error of Engineering Restrictions.
PT as well as the approximate confidence
intervals on PT is also being evaluated. The (12) Hartley, H. O. and Ly Cong Thuan.
extension of the basic model (i) to include year- Multiyear Estimates for the LACIE Sampling Plans.
segment interactions and segment-season interactions
is being considered. Another possibility is to (13) Hartley, H. O. Improvements in Multiyear
replace the seasonal bias term in (I) by a covariate Estimates of Wheat Acreage.
in terms of something like the number of "crop
calendar days" passed by the date of the last (14) Hartley, H. O. November 1978. Control of
satellite imagery used in determining the estimated Classification Bias Through Ground Truth Data Bank.
segment at harvest crop acreage proportion.
(15) Hartley, H. O. February, 1979. A Multicrop-
Another important line of research concerns Multicountry Sampling Strategy.
the nature of the weights themselves. If the true
segment at harvest crop acreage proportion were p* (16) Freund, R.J.,Hartley, H. O. and T. Lee. May
and the estimated p's were binomial in nature, then 1979. Gains of Precision Achievable by Multi-Year
the variance of a segment estimate p would be Es t imat ion.
proportional to p*(l-p*). Furthermore, the variance
of y(p) could be derived for a given y, and the (17) Hartley, H.O., Hughes, T.H. and R. L. Sielken,
appropriate weight in the weighted least squares Jr. May, 1979. A Computer Implementation of the
procedure could be straight-forwardly approximated
using the estimated p. However, the variance of
the estimate of the segment's at harvest crop
acreage proportion may not be binomial in nature
but rather depend on such things as

(i) the satellite being used,

(ii) the sharpness of the satellite imagery,

(iii) the amount of satellite imagery available
at the time of the segment estimate,

(iv) the nearness of the segment's observed
behavior to classical crop profiles,

(v) the season during which the estimate is
being made,

(vi) the weather conditions during the crop's

493

Multicrop Sampling Strategy.
(18) Gbur, E.E. and R. L. Sielken, Jr. December
1980. Optimal Rotation Designs for Multiyear
Estimation, I. Unweighted Estimation.
(19) Gbur, E.E. and R. L. Sielken, Jr. December
1980. Optimal Rotation Designs for Multiyear
Estimation, II. Weighted Estimation.
(20) Dahm, P.F. and Robert L. Sielken, Jr. March
1981. Multiyear Estimation of the At Harvest Crop
Acreage Proportion: Methodology and Implementation.
(21) Sielken, Robert L. Jr., December 1981. Incor-
porating Partially Identified Sample Segments Into
Acreage Estimation Procedures: Estimates Using
Only Observations From the Current Year.
(22) Gbur, E. E. and R. L. Sielken, Jr. December
1981. Missing Observations in Multiyear Rotation
Sampling Designs.

494