7 Classification by use of minimum distance

The normalized spectra were standardized and a metric was introduced in vector space (Murtagh & Heck [1984]; Vieira & Ponz [1995]). After standardization, we obtain

$${D_{ij}^{{\rm new}}}=\frac{(D_{ij}^{{\rm old}}-\bar{D_i})}{\sigma_i}$$ (3)

with $\bar{D_i}$ being the mean value (over j variable) and $\sigma_i$ the standard deviation of the $i^{\rm th}$ spectrum; i=1,..., 426. The spectra have zero mean and unit standard deviation. The metric is the standard unweighted Euclidean distance between two real-valued vectors d_ik given by

$${d_{ij}^2}=\displaystyle \sum_{j=1}^{128}(D_{ij}-S_{kj})^2.$$ (4)

The minimum distance d_ik for the $i^{\rm th}$ spectrum for the class k was calculated with displacement $\pm3$ pixels to predict a possible displacement from the detection algorithm caused by the local background. The final result was given by

$${d}_i = \mbox{arg}~(\mbox{min}~~d_{ik}),~~k=1,...~6.$$ (5)

Figure 4: A characteristic $1 \times 128$ OB spectrum of our sample

Figure 5: A characteristic $1 \times 128$ A spectrum of our sample

Figure 6: A characteristic $1 \times 128$ F spectrum of our sample

Figure 7: A characteristic $1 \times 128$ G spectrum of our sample

Figure 8: A characteristic $1 \times 128$ K spectrum of our sample

Figure 9: A characteristic $1 \times 128$ M spectrum of our sample