Bernal S D_2010.pdf - University of Plymouth

4.3. LEARNING
2. The list of selected prototypes. Fjc/rn.'^. will initially contain no prototypes. A parameter
called the minimum distance, Dmin- '^^ initialized to a relatively high starting value.
3. The algorithm loops through all die potential prototypes /*,,n,„v,;. Prototypes are added to
the selected prototype li,^t. l^^hried- if ihe Euclidean distance to all previously stored pro­
totypes is above the minimum distance, i.e. if d{Ppo,eni,ai.i- Pseimrdj) i l^mtn j ^ {\--N}
then Psfierredjj+i — Pporeniiai.i' wHcrc N is the number of selected prototypes.
4. Lower D,„,„ and repeat step .1 until N - K^i- The initial value of D,„i„ and the decreasing
step size in each iteration dictate the dissimilarity between the final selection.'; of proto­
types,
The S2 prototypes represent the weight matrix between a parent S2 node and all of its CI
afferent nodes. The corresponding CI'T for each C1 node, i.e. the weight matrix between each
n node and all of its S2 parents P{C\\S2). is calculated in an analogous way lo thai described
for the Sl-Cl CRTs. Note that the CPT elements for the CI features within the same group
are set to the same value. This is because the CPTs are derived from the weight matrices that
contain a single value for each CI group.
Figure 4,9 follows from the previous toy example where now the number of CI units is « =
3 • 3 — 9, the number o!' CI slates is Kn = 12 and the number of S2 stales (prototypes) is
Ks2 = 10. The equation to calculate Aci(52), which sends the bottom-up evidence to S2, and
A (S2), which combines the botlom-up evidence, are shown in the diagram,
The toy example illustrates how the weighls are learned for each fixed S2 state=( as a function
of the n afferent CI ntMJes and the ./ CI .states per node. This yields a weight matrix (shown
on the botlom left) for each of the S2 i states. However, the CPTs of a Bayesian network are
defined as a function of the child and the parent states, j and i respectively, for each fixed CI
*Capiiiin for Figure 4.9, Toy example illusirFiliiiy liow ID Lipprojiimaie ihe sekniiviiy operalion using Ihf CRTs
tviweenC'l and S2 nodes. The weighis between eafh uf the S2 ilaie^ aiidlhe fl rwdts are shown in the boilom-lett
tables. These are ihen eonverted in ihe conespimtlinp CF1\ l>eiween each CI node ami iis pureiil S2 node as shown
in ihe tables of ihe bottom-right. The belief propagation equaiions ui ihe lop-lelt square show how the CI^Ts are
u.M'd io generate ilic output A messages from each CI node loiis S2 parem niKie. These messages are then combined
muliiplicaiively by ihe S2 likcliho(xl runciion k{S2) ami used lo generate the output X me-ssases from each CI node
10 its S2 ptireni node, .Note that the Cl'T elemenw for iho CI fcaiures within the same group are sei lu ihc same
value, as only one value is learned per CI ptoup.
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