A frequent goal is the minimization of the expected quantization
(or distortion) error. In the case of a continuous input signal
distribution 
this amounts to finding values for the
reference vectors 
such that the error
is minimized (
is the Voronoi region of unit c).
Correspondingly, in the case of a finite data set 
the error
has to be minimized with 
being the Voronoi set of the unit c.
A typical application where error minimization is important is vector quantization (Gray, 1984; Linde et al., 1980). In vector quantization data is transmitted over limited bandwidth communication channels by transmitting for each data vector only the index of the nearest reference vector. The set of reference vectors (which is called codebook in this context) is assumed to be known both to sender and receiver. Therefore, the receiver can use the transmitted indexes to retrieve the corresponding reference vector. There is an information loss in this case which is equal to the distance of current data vector and nearest reference vector. The expectation value of this error is described by equations (3.1) and (3.2). In particular if the data distribution is clustered (contains subregions of high probability density), dramatic compression rates can be achieved with vector quantization with relatively little distortion.