The Earth Mover's Distance

The EMD is based on the minimal cost that must be paid to transform one distribution into the other.Intuitively,given two distributions,one can be seen as a mass of earth properly spread in space,the other as a collection of holes in that same space.Then,the EMD measures the least amount of work needed to fill the holes with earth.Here,a unit of work corresponds to transporting a unit of earth by a unit of ground distance.

　　This can be formalized as the following linear programming problem:

　　　　Let P={(p1,w_p1),...,(p_m,w_pm)}

be the first signture with m clusters,where p_i is the cluster representative and w_pi is the weight of the cluster; 

　　　　Q={(q1,w_q1),...,(q_n,w_qn)}

the second signature with n cluster; and 

　　　　D=[d_ij] 

the ground distance matrix where d_{ij is the ground distance between cluster} p_i and q_{j .}

　　We want to find a flow

　　　　F=[f_ij]

with f_{ij the flow between pi and qj}, _{that minimizes the overall cost} 

　　

subject to the following constranits:

Constraint (1) allows moving "supplis" from P to Q and not vice versa. Constraint (2) limits the amount of supplies that can be sent by the clusters in P to their weights.Constaint (3) limits the clusters in Q to receive no more supplies than their weights; and constraint (4) forces to move the maximum amount of supplies possible. We call this amount the total flow. Once the transportation problem is solved, and we hve found the optimal flow F, the earth mover's distance is defined as the resulting work normalied by the total flow:

The normalization factor is the total weight of the smaller signature, because of constraint (4). This factor is needed when the two signatures have different total weight, in order to avoid favoring smaller signatures. In general, the ground distance d_{ij can be any distance and will be chosen according to the problem at hand.}