Intravariance

The intravariance (from the Latin intra , “within” and variantia for “difference”) or dispersion ( variance ) within the classes is a measure of how much the objects differ within a class. The lower the intravariance, the more similar the objects are.

The intravariance, together with the intervariance, is of particular importance in the classification , where objects are arranged in classes. There the guideline applies: the lower the intravariance and the higher the intervariance, the easier the classification.

Formal representation

Let be any feature space that includes the classes . There are two ways of specifying the intravariance, depending on whether the mean values or the a priori probabilities and covariance matrices of the classes are (in each case ) known: ${\displaystyle M}$ ${\displaystyle c}$ ${\displaystyle C_{1},\ldots ,C_{c}\subseteq M}$ ${\displaystyle \mu _{i}}$ ${\displaystyle P_{i}}$ ${\displaystyle \Sigma _{i}}$ ${\displaystyle i=1,\ldots ,c}$

${\displaystyle S_{W}:=\sum _{i=1}^{c}\sum _{x\in C_{i}}(x-\mu ^{i})(x-\mu ^{i})^{T}=\sum _{i=1}^{c}P_{i}\Sigma _{i}}$

If has dimensions, it is a square matrix with rows and columns. Since covariance matrices are always symmetrical, the intravariance is also a symmetrical matrix due to the second form . ${\displaystyle M}$ ${\displaystyle d}$ ${\displaystyle S_{W}}$ ${\displaystyle d}$ ${\displaystyle d}$

Individual evidence

1. ↑ Matthias Michelsburg: Material classification in optical inspection systems using hyperspectral data. , P. 50.