835
The apriori values for each crop type p(X_^) can be obtained from recent
agricultural statistics pertaining to the local area of interest. (It is
almost unique to agriculture to have sources of published statistics from
which to obtain the respective values of p(X.) in order to construct the
key for local situations).
Summarizing to this Point : The Bayesian Formula (1) revises the
apriori probability p(X^) to the posterior conditional probability P(X^jT)
in accordance to the observed photo-density values t^ and t£* Thus
P(XjJT) expresses the probability that the two measured photo-density
values, one value from each of two image dates, came from a specified crop
type. By choosing to call each unknown field in accordance to the highest
posterior probability, the probability of error is mathematically minimized
(Duda and Hart, 1973). If one desired to minimize the risk of mis-identifi-
cation (assuming a zero-one loss function) he should choose to identify the
crop in accordance to greatest value of P(X i |T).
The P(tjJXj_) P(X-j^) values for all t and x possibilities in this example
are shown in Figure 6. Values from this matrix are subsequently used to
calculate P(X^jT) values for each crop type t^, t 2 combination. A working
version of a probabilistic key would pre-calculate all possible values of
P(X^|T) and arrange the answers in a matrix whereby they could be "located"
using the observed values of t-^ and t£ as pointers. The internal structure
of a probabilistic key is illustrated by completing the calculation for only
one set of hypothetically "observed" photo-densities.
RESULTS
Example of Calculations
Suppose that a value of .38 was obtained from
densitométrie measurement of a field on a 1000 degree day image and that
later in the growing season (at 2000 degree days) a densitométrie value of
.20 was obtained for the same field. Applying the hypothetical probabilistic
key valid for the region, a set of probabilities is obtained (Figure 7 )
The P(XjJT) values indicate that the identification involving the least
error would be to call the unknown field spring wheat. The alternative
identifications are presented also in Figure 7 along with their respective
probabilities in this example.
Probabilistic Key Style : A section of a simple probabilistic key is
shown in Figure 8. Only two photo-derived pieces of information are required
in order to use this particular key. When more than two inputs are used, the
key mechanism would be more complex than the simple matrix. Using a computer
to pre-calculate all possible posterior probabilities, the arranging of these
values in a practical manner is a technical problem where several alternatives
can usually be found, such as using a series of look-up matrices.