Ž respectively. Note also that the lower diagonal portion of the matrix has been omitted as
. it is symmetric.
4. Empirical Bayes analysis
The philosophy underlying empirical Bayes analysis is to combine data from previous years with current, year-to-date data in producing forecasts which reflect the overall
form of historical patterns but are adapted for current conditions.
ˆ ˆ
2
Recall that b,C and s are the historical components used in forecasting. Suppose
ˆ
we have collected 8 weeks of data for a particular pond in the current year. Organizing Ž .
this data as in Eq. 7 , let 1
t s
1 t
s y
1 1
1 1
1
2 2 t
2 s 4
4 t 4 s
y
2 2
2 2
2
3 3t
3s 9
9t 9 s
y
3 3
3 3
3
4 4 t
4 s 16
16 t 16 s
y
4 4
4 4
4
y s and X s
5 5t
5s 25
25t 25s
y
5 5
5 5
5
6 6 t
6 s 36
36 t 36 s
y
6 6
6 6
6
7 7t
7s 49
49t 49 s
y
7 7
7 7
7
y 8
8t 8 s
64 64 t
64 s
8 8
8 8
8
be a column vector containing weight measurements and a matrix of independent variables, respectively. To forecast weight for a future week, say week 10, we construct
a row vector x
X
by entering values for that week in the same order as the values in the rows of X . Thus,
x
X
s 10,10 t ,10 s,100,100 t ,100 s .
Ž .
Since temperature and salinity are unknown for the week we are predicting, we can either enter average values based on the past few weeks or enter an outside forecasted
value. For example, if a cold front is expected, one could place the forecasted temperature value into t. Additionally, if our interest is to predict for a specific pond, we
ˆ ˆ
may use the pond-adjusted values of b. Note that the covariance matrix C would remain the same.
The matrix equation used to perform an empirical Bayes forecast of shrimp yield is given by
y1 X
X X
2
ˆ ˆ
ˆ ˆ
ˆ
Y s x b q x C X X C X q s I
y y X b , 8
Ž . ˆ
Ž .
ž
ˆ
where Y is the forecasted yield and I is an identity matrix of appropriate dimension. In the context of this problem, the empirical Bayes estimator is identical to the estimated
Ž .
Ž best linear unbiased predictor BLUP from linear model theory. See Christensen, 1987
Ž . for the derivation of Eq. 8 and Maritz and Lwin, 1989 or Carlin and Louis, 1996 for
. further information on empirical Bayes estimation.
Table 1 Values for 1-, 2- and 3-week ahead forecasts for a pond from summer 1997
Week Actual
Lag 1 Lag 2
Lag 3 7
3.8 3.28
– –
8 4.3
4.21 3.90
– 9
4.7 4.74
4.87 4.56
10 5.6
5.35 5.42
5.56 11
5.2 6.09
6.07 6.12
12 6.6
6.83 6.84
6.82 13
7.7 7.79
7.59 7.62
14 8.5
8.73 8.55
8.38 15
9.4 9.62
9.48 9.33
16 10.9
10.54 10.36
10.24 17
11.0 11.46
11.23 11.10
18 12.5
12.13 12.08
11.92 19
13.0 12.70
12.69 12.70
20 13.6
13.53 13.33
13.25 21
14.3 14.66
14.30 13.97
22 14.6
15.68 15.52
15.08 23
15.4 16.25
16.61 16.39
24 16.2
16.30 17.16
17.55 25
17.0 16.59
17.03 18.09
26 16.9
16.74 17.20
17.76 27
16.8 17.32
17.25 17.80
Ž . Ž .
In summary, the empirical Bayes Eq. 8 has the following components: 1 historical
ˆ ˆ
2
Ž . data gives us values for b,C and s , 2 year-to-date data gives us values for y and
ˆ
Ž .
X
X and 3 x is created by plugging in the week number as well as average or predicted values for the covariates temperature and salinity.
Fig. 4. Forecasted values for a pond from summer 1997.
5. Forecasting example
