Preprocessing the input data

An efficient preprocessing of the data is necessary to input it into the net. Due to our implementation of the back-propagation algorithm all information must be scaled to $[0,\; 1]$ . We assume that the necessary information is given for T weeks in the past. With the following definitions

$ADV$it	$:=$	$number\; of\; advertising\; days$
		$for\; article\; i\; within\; week\; t$
$SAL$it	$:=$	$sale\; of\; article\; i\; within\; week\; t$
$MAXSAL$i	$:=$	$max$1 ≤t ≤TSALit

we have decided to use the following inputs for each article i and week t:

For each article i and recent week t we use the three-dimensional vector:

$vec$it:= ( advit, priit, salit)

For a week t in the future the vector is reduced by the unknown sale:

$vec$it:= ( advit, priit)

To predict the sale for one article within a week t, we use a window of the last n weeks. So we have the following input vector for each article i:

$input$it:= ( vecit-n, vecit-n+1, ... , vecit-1, vecit)

We have quite a constant sales volume of all products, because all the considered articles belong to one product group. An increasing sale of one article in general leads to a decrease of other sales. Due to this, we concatenate the input vectors of all p articles to get the vector given to the input layer:

$INPUTt:=\; (input$1t, input2t, ... , inputpt)

The sale of article i within week t ($sal$_i^t ) is the requested nominal value in the output layer that has to be learned by one net for this $INPUTt$ vector. So we have p nets where the i-th net adapts the sale behaviour of article i. Therefore we have a training set with the following pairs (see figure 1):

$(INPUTt,\; sal$it) with n ≤t ≤T

To forecast the unknown sale $sal$_i^T+1 for any article i within a future week T+1 we give the following input vector to the trained i-th net:

$INPUTT+1$

The output value of this net is expected to be the value $sal$_i^T+1 , which has to be re-scaled to the value for the sale of article i within week T+1:

$SAL$iT+1= {saliT+1.MAXSALi0.8}