affected by each input dimension and how each output dimension is affected by a hidden layer unit. In other words, there is no method that can reveal whether an increase in the input dimension value has a positive or negative effect on the output value of a hidden layer unit, or whether an increase in the output value of a hidden layer unit has a positive or negative effect on the output dimension value. To obtain such sign information regarding the roles of each hidden layer unit, we use the following definition based on the correlation.

We define the effect of the $i$-th input dimension on the $k$-th hidden layer unit as $v^{\mathrm{in}}_{ik}$, where $$\begin{aligned}
  v^{\mathrm{in}}_{ik}=\frac{E\Bigl[ \Bigl( X^{(n)}_i-E[X^{(n)}_i] \Bigr) \Bigl( o^{(n)}_k-E[o^{(n)}_k] \Bigr) \Bigr]}{\sqrt{E\Bigl[ \Bigl( X^{(n)}_i-E[X^{(n)}_i] \Bigr)^2 \Bigr]E\Bigl[ \Bigl( o^{(n)}_k-E[o^{(n)}_k] \Bigr)^2 \Bigr]}}.\end{aligned}$$ Here, $E[\cdot]$ represents the mean for all the data samples, $X^{(n)}_i$ is the $i$-th input dimension value of the $n$-th data sample, and $o^{(n)}_k$ is the output of the $k$-th hidden layer unit for the $n$-th input data sample.

We define the effect of the $k$-th hidden layer unit on the $j$-th output dimension as $v^{\mathrm{out}}_{kj}$, where $$\begin{aligned}
  v^{\mathrm{out}}_{kj}=\frac{E\Bigl[ \Bigl( o^{(n)}_k-E[o^{(n)}_k] \Bigr) \Bigl( y^{(n)}_j-E[y^{(n)}_j] \Bigr) \Bigr]}{\sqrt{E\Bigl[ \Bigl( o^{(n)}_k-E[o^{(n)}_k] \Bigr)^2 