6 Classification by use of linear correlation

The most widely used method to measure the dependence on two variables, is the linear correlation coefficient r. The spectra are normalized and they are considered as vectors in R^N with N=128.

Let $D_{ij}=D_{i}(\lambda_{j})$; j=1,...,128 be the normalized density value for the $i^{\rm th}$stellar spectrum and $S_{kj}=S_{k}(\lambda_{j})$; j=1,...,128 be the normalized density value of the $k^{\rm th}$ class standard stellar spectrum with k=1,...,6. For k=1,...,6 the standard stellar spectra are OB,...,M of Figs. 4-9. The correlation coefficient for the $i^{\rm th}$ stellar spectrum for the $k^{\rm th}$ class is

$${r}_{ik}=\frac{\displaystyle\sum_{j=1}^{128}(D_{ij}-\bar{D_i}... ...\bar{D_i})^2}\sqrt{\sum_{j=1}^{128}(S_{kj}-\bar{S_k})^2}}\;\;,$$ (1)

with $\bar{D_i}$ being the mean value (over the j variable) of the $i^{\rm th}$ spectrum, i=1,...,426and $\bar{S_k}$ the mean value of the $k^{\rm th}$ class standard spectrum.

The correlation coefficient r_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. For these seven correlation coefficients for every class k, the maximum value was choosen. The final classification was given by the maximum value of the coefficient r_i of all the r_ikcoefficients as

$${r}_i = \mbox{arg}~(\mbox{max}~~r_{ik}),~~k=1,...~6.$$ (2)