Content deleted Content added
→top: complain about confusion of math and phys notions |
→Basic mathematical definition: complain about nonsensical definition |
||
Line 7:
== Basic mathematical definition ==
[[Image:Chow-liu.png|thumb|400 px|An example of a sparsely connected dependency structure. Direction of causal flow is upward.]]
For a multivariate function <math>y = f(x_1,x_2,\dots,x_n)</math>, functional decomposition generally refers to a process of identifying a set of functions <math>\{g_1, g_2, \dots g_m\}</math> such that :<math>f(x_1,x_2,\dots,x_n) = \phi(g_1(x_1,x_2,\dots,x_n), g_2(x_1,x_2,\dots,x_n), \dots g_m(x_1,x_2,\dots,x_n))</math>
where <math>\phi</math> is some other function.{{clarify|reason=This definition is horribly underspecified, similar to e.g. decomposing a number into the sum of serveral others. Unless the set of admitted g_i or/and phi is given, it does not make sense at all. One trivial solution is m=n, g_i(x_1,...,x_n)=x_i, and phi=f; another one is m=1, g_1=f, and phi(x)=x.}} Thus, we would say that the function <math>f</math> is decomposed into functions <math>\{g_1, g_2, \dots g_m\}</math>. This process is intrinsically hierarchical in the sense that we can (and often do) seek to further decompose the functions <math>g_i</math> into a collection of constituent functions <math>\{h_1, h_2, \dots h_p\}
:<math>g_i(x_1,x_2,\dots,x_n) = \gamma(h_1(x_1,x_2,\dots,x_n), h_2(x_1,x_2,\dots,x_n), \dots h_p(x_1,x_2,\dots,x_n))</math>
| |||