Distance measure for probability distribution function of mixture type

摘要

In accordance with our invention, for two mixture-type probability distribution functions (PDF's), G, H, $\mathop{\mathrm{image}}G\⁡\left(x\right)=\underset{i=1}{\overset{N}{\∑}}\⁢{\mathrm{\μ}}_i\⁢g_i\⁡\left(x\right),\text{\ }\⁢H\⁡\left(x\right)=\underset{k=1}{\overset{K}{\∑}}\⁢{\mathrm{\γ}}_k\⁢h_k\⁡\left(x\right),$;where G is a mixture of N component PDF's g_i (x), H is a mixture of K component PDF's h_k (x), _i and _k are corresponding weights that satisfy $\mathop{\mathrm{image}}\underset{i=1}{\overset{N}{\∑}}\⁢{\mathrm{\μ}}_i=1\⁢\text{\ }\⁢\mathrm{and}\⁢\text{\ }\⁢\underset{k=1}{\overset{K}{\∑}}\⁢{\mathrm{\γ}}_k=1;$;we define their distance, D_M(G, H), as $\mathop{\mathrm{image}}{D}_M\⁡\left(G,H\right)=\underset{w=\left[{\mathrm{\ω}}_{\mathrm{ik}}\right]}{\min }\⁢\underset{i=1}{\overset{N}{\∑}}\⁢\underset{k=1}{\overset{K}{\∑}}\⁢{\mathrm{\ω}}_{\mathrm{ik}}\⁢d\⁡\left(g_i,h_k\right)$;where d(g_I, h_k is the element distance between component PDF's g_i and h_k and w satisfie ;and $\mathop{\mathrm{image}}\underset{k=1}{\overset{K}{\∑}}\⁢{\mathrm{\ω}}_{\mathrm{ik}}={\mathrm{\μ}}_i,1\≤i\≤N,\underset{i=1}{\overset{N}{\∑}}\⁢{\mathrm{\ω}}_{\mathrm{ik}}={\mathrm{\γ}}_k,1\≤k\≤K.$;The application of this definition of distance to various sets of real world data is demonstrated.