Keywords: class-average

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### Resampling

Let's consider the input spectrum . We assume that the intensity at channel is a random variable which follows the normal distribution . Its probability density function is:

| (8)

where is the mean and the standard deviation. The variance of such a distribution is:

| (9)

Choosing a weight of the following form:

| (10)

ensure that the weighted mean is the maximum likelihood estimator of the mean, under the assumption of independent (i.e. uncorrelated at first order) and normally distributed channel intensities with the same mean.

Let's call the resampled version of such as it is aligned with the desired output spectrum . The intensity of at channel can be written (see fig. ):

where:

- and are the channel range of input spectrum which are overlapped at least partially by the output channel number ,
- is the used fraction of the different input channels. It quantifies their contribution in the selected range:

  | if the input channel is out of the range,
  | if the output channel fully overlaps the input channel,
  | otherwise (proportionally to the overlap).

We can also define the normalization factor and the weight :

|  |  | (11)
|  |  | (12)

The normalization factor ensures that the total integrated intensity is preserved during the resampling process. With these, can be written:

| (13)

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