Within-class scatter matrixIt is used to represent the distribution of sample points around the mean, and its definition is as follows:

Suppose there are $latex {M}$latexM categories, $latex {Ω\mathop{{}}\nolimits\_{{i}},…,Ω\mathop{{}}\nolimits\_{{M}}}$latexΩ_i,…,Ω_M , $latex {Ω\mathop{{}}\nolimits\_{{i}}}$latexΩ_i class sample set$latex { \left\{ {X\mathop{{}}\nolimits\_{{1}}^{{{ \left( {i} \right) }}},X\mathop{{}}\nolimits\_{{2}}^{{{ \left( {i} \right) }}},…,X\mathop{{}}\nolimits\_{{N\mathop{{}}\nolimits\_{{i}}}}^{{{ \left( {i} \right) }}}} \right\} }$latex{X_1(i),X_2(i),…,X_N_i(i)} , $latex {Ω\mathop{{}}\nolimits\_{{i}}}$latexΩ_i The divergence matrix of the class is defined as:

latex {S\mathop{{}}\nolimits\_{{w}}^{{{ \left( {i} \right) }}}\text{ }=\text{ }\frac{{1}}{{N\mathop{{}}\nolimits\_{{i}}}}{\mathop{ \sum }\limits\_{{k=1}}^{{N\mathop{{}}\nolimits\_{{i}}}}{{ \left( { {X\mathop{{}}\nolimits\_{{k}}^{{{ \left( {i} \right) }}}-m\mathop{{}}\nolimits^{{{ \left( {i} \right) }}}} \right) }\mathop{{}}\nolimits^{{T}}}}}

Among them, $latex {S\mathop{{}}\nolimits\_{{{w}}}^{{ \left( {i} \right) }}}$latexS_w(i) is the covariance matrix of the class $latex {Ω\mathop{{}}\nolimits\_{{i}}}$latexΩ_i.

The total intra-class scatter matrix is:

latex {S\mathop{{}}\nolimits\_{{w}}\text{ }=\text{ }{\mathop{ \sum }\limits\_{{i=1}}^{{M}}{P{ \left( {Ω\mathop{{}}\nolimits\_{{i}}} \right) }S\mathop{{}}\nolimits\_{{w}}^{{{ \left( {i} \right) }}}}}\text{ }=\text{ }{\mathop{ \sum }\limits\_{{i=1}}^{{M}}{P{ \left( {Ω\mathop{{}}\nolimits\_{{i}}} \right) }\frac{{1}}{{N\mathop{{}}\nolimits\_{{i}}}}{\mathop{ \sum }\limits\_{{k=1}}^{{N\mathop{{}}\nolimits\_{{i}}}}{{ \left( { {X\mathop{{}}\nolimits\_{{k}}^{{{ \left( {i} \right) }}}-m\mathop{{}}\nolimits^{{{ \left( {i} \right) }}}} \right) }\mathop{{}}\nolimits^{{T}}}}}}}

Then: trace$latex { \left\{ {S\mathop{{}}\nolimits\_{{w}}} \right\} }$latex{S_w} is the average measure of feature variance of all classes.

Regarding the results of feature selection and extraction, the smaller the product of the within-class scatter matrix, the better.

Related words: between-class scatter matrix