Generic function that computes the sparseness of an object, as defined by Hoyer (2004). The sparseness quantifies how much energy of a vector is packed into only few components.
sparseness(x, ...)usually a single numeric value -- in [0,1], or a numeric vector. See each method for more details.
an object whose sparseness is computed.
extra arguments to allow extension
signature(x = "numeric"): Base
method that computes the sparseness of a numeric vector.
It returns a single numeric value, computed following the definition given in section Description.
signature(x = "matrix"):
Computes the sparseness of a matrix as the mean
sparseness of its column vectors. It returns a single
numeric value.
signature(x = "NMF"): Compute
the sparseness of an object of class NMF, as the
sparseness of the basis and coefficient matrices computed
separately.
It returns the two values in a numeric vector with names ‘basis’ and ‘coef’.
In Hoyer (2004), the sparseness is defined for a real vector \(x\) as: $$Sparseness(x) = \frac{\sqrt{n} - \frac{\sum |x_i|}{\sqrt{\sum x_i^2}}}{\sqrt{n}-1}$$
, where \(n\) is the length of \(x\).
The sparseness is a real number in \([0,1]\). It is
equal to 1 if and only if x contains a single
nonzero component, and is equal to 0 if and only if all
components of x are equal. It interpolates
smoothly between these two extreme values. The closer to
1 is the sparseness the sparser is the vector.
The basic definition is for a numeric vector, and
is extended for matrices as the mean sparseness of its
column vectors.
Hoyer P (2004). "Non-negative matrix factorization with sparseness constraints." _The Journal of Machine Learning Research_, *5*, pp. 1457-1469. <URL: http://portal.acm.org/citation.cfm?id=1044709>.
Other assess: entropy, purity