A partir de uma solução inicial o algoritmo FCM com distâncias adaptativas (ou seja tanto FCMA I quanto FCMA II) busca por soluções que vão decrescendo iteração a iteração do algoritmo. Para tanto a cada iteração é executado um passo de representação e em seguida um de alocação. O passo de representação é dividido agora em duas etapas, na primeira, atualiza-se os protótipos e na segunda, atualiza-se os pesos utilizados no cálculo das distâncias.
Proposição C.0.3. A funçãoJ(U, G, d)é decrescente.
Demonstração:
A descendência é demonstrada se é possível verificar a seguinte desigualdade: J(Ut, Gt, dt)≥ J(Ut, Gt+1, dt)≥ J(Ut, Gt+1, dt+1)≥ J(Ut+1, Gt+1, dt+1)
Seja Gt+1 = r(Ut, dt), onde r(Ut, dt) é uma função de representação, dt+1 = e(Ut, Gt+1), onde e(Ut, Gt+1) é uma função de atualização dos pesos utilizados para o cálculo da distância entre um indivíduo e um protótipo e Ut+1 = a(Gt+1, dt+1), onde a(Gt+1, dt+1) é uma função de alocação.
A primeira desigualdade é verificada, pois, fixados Ut e dt, a função de representação r(Ut, dt) (fórmula 3.3 ou 3.8) é tal que minimiza J(Ut, Gt+1, dt) conforme demonstrado pela proposição 3.2.1 e pela proposição 3.3.1.
A segunda desigualdade J(Ut, Gt+1, dt)≥ J(Ut, Gt+1, dt+1) é verificada, pois, fixados Ut e Gt+1, a função de atualização dos pesos e(Ut, Gt+1) (fórmula 3.4 ou 3.9) é tal que minimiza J(Ut, Gt+1, dt+1) conforme demonstrado tanto na proposição 3.2.2 quanto na proposição 3.3.2. A terceira desigualdade J(Ut, Gt+1, dt+1)≥ J(Ut+1, Gt+1, dt+1) é verificada, pois fixados Gt+1 e dt+1 foi demonstrado pela proposição 3.2.3 e pela proposição 3.3.3 que Ut+1 atu- alizado pela função de alocação a(Gt+1, dt+1) (fórmula 3.5 ou 3.10) minimiza a função J(Ut+1, Gt+1, dt+1).
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