Page 226 - Geochemical Anomaly and Mineral Prospectivity Mapping in GIS

228 Chapter 7

Fig. 7-18. Schematic relationships of evidential belief functions (EBFs). See text for further
explanation.

assessment. The Unc represents ‘ignorance’ or ‘doubt’ that a given piece of spatial
evidence supports the proposition. The value of Unc is the difference between Bel and
Pls. The Dis represents evaluation that a given piece of spatial evidence does not support
the proposition.
The four EBFs are inter-related (Fig. 7-18). The sum of Bel+Unc+Dis of a piece of
spatial evidence is equal to 1. Likewise, the sum of Pls+Dis of a piece of spatial
evidence is equal to 1. From these two equalities, therefore, Pls = Bel+Unc or Bel = Pls–
Unc. The degree of Unc influences the relation between Bel and Dis. If Unc = 0 (i.e.,
there is complete knowledge about a given piece of spatial evidence), then Bel+Dis = 1
and the relation between Bel and Dis for a given piece of evidence is binary (i.e., Bel =
1–Dis or Dis = 1–Bel), as in the theory of probability. If Unc = 1 (i.e., there is complete
ignorance or doubt about a given piece of spatial evidence), then Bel and Dis for a given
piece of evidence are both equal to zero. That is, if there is complete uncertainty, then
there can be neither belief nor disbelief. Usually, however, Unc is neither equal to zero
nor equal to one. Therefore, in the case that 0<Unc<1, then Bel = 1–Dis–Unc or Dis = 1–
Bel–Unc. This means that, because uncertainty is usually present, the relation between
Bel and Dis for a given piece of evidence is usually not binary. This means further that,
for a piece of evidence used to evaluate a proposition, one should estimate not only Bel
but also Dis and Unc.
In the application of EBFs to knowledge-driven mineral prospectivity mapping, two
of the three EBFs – Bel, Dis and Unc – are usually first estimated together in order to
represent not only degree of support (or lack of support) to a proposition by a piece of
spatial evidence but also the degree of uncertainty about this evidence. The Pls is then
simply derived from estimates of any of two EBFs and, as shown below, the Pls is not
used in Dempster’s (1968) rule of combination. Estimating Bel and Dis together is
usually the most difficult, because one tends to think of the binary relation between these
two EBFs and then neglect Unc. Estimating Dis and Unc together is cumbersome,
because of confusion between disbelieving and doubting. So, estimating Bel and Unc