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Preprocessing the input data

An efficient preprocessing of the data is necessary to input it into the net. All information must be normalized to fit into the interval $\left[0, 1\right]$ . 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 a 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)

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

The sale of article i within week t ($sal$it ) 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 and the i-th net adapts the sale behaviour of article i. Therefore we have a training set with the following pairs (see figure 4):
 $\left(INPUTt, sal$it) with n ≤t ≤T

To forecast the unkown sale $sal$iT+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 $sal$iT+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}

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