Model Representation II

To re-iterate, the following is an example of a neural network:

In this section we'll do a vectorized implementation of the above functions. We're going to define a new variable?z_k^(j)??that encompasses the parameters inside our g function. In our previous example if we replaced by the variable z for all the parameters we would get:

In other words, for layer j=2 and node k, the variable z will be:

The vector representation of x and?z^(j) is:

Setting?x = a^(1), we can rewrite the equation as:

We are multiplying our matrix?Θ(j?1)?with dimensions sj?×(n+1)?(where?s_j is the number of our activation nodes) by our vector?a^(j-1) with height (n+1). This gives us our vector?z^(j) with height?s_j. Now we can get a vector of our activation nodes for layer j as follows:

Where our function g can be applied element-wise to our vector?z^(j).

We can then add a bias unit (equal to 1) to layer j after we have computed?a^(j). This will be element?a_0^(j)??and will be equal to 1. To compute our final hypothesis, let's first compute another z vector:

We get this final z vector by multiplying the next theta matrix after?Θ^(j?1)?with the values of all the activation nodes we just got. This last theta matrix?Θ^(j)?will have only?one row?which is multiplied by one column?a^(j) so that our result is a single number. We then get our final result with:

Notice that in this?last step, between layer j and layer j+1, we are doing?exactly the same thing?as we did in logistic regression. Adding all these intermediate layers in neural networks allows us to more elegantly produce interesting and more complex non-linear hypotheses.

來源：coursera 斯坦福 吳恩達(dá) 機(jī)器學(xué)習(xí)