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Two or more of the neurons shown earlier can be combined in a layer, and a particular network could contain one or more such layers. First consider a single layer of neurons.
A one-layer network with R input elements and S neurons follows.
In this network, each element of the input vector p is connected to each neuron input through the weight matrix W. The ith neuron has a summer that gathers its weighted inputs and bias to form its own scalar output n(i). The various n(i) taken together form an S-element net input vector n. Finally, the neuron layer outputs form a column vector a. The expression for a is shown at the bottom of the figure.
Note that it is common for the number of inputs to a layer to be different from the number of neurons (i.e., R is not necessarily equal to S). A layer is not constrained to have the number of its inputs equal to the number of its neurons.
You can create a single (composite) layer of neurons having different transfer functions simply by putting two of the networks shown earlier in parallel. Both networks would have the same inputs, and each network would create some of the outputs.
The input vector elements enter the network through the weight matrix W.
$$W=\left[\begin{array}{cccc}{w}_{1,1}& {w}_{1,2}& \dots & {w}_{1,R}\\ {w}_{2,1}& {w}_{2,2}& \dots & {w}_{2,R}\\ & & & \\ {w}_{S,1}& {w}_{S,2}& \dots & {w}_{S,R}\end{array}\right]$$
Note that the row indices on the elements of matrix W indicate the destination neuron of the weight, and the column indices indicate which source is the input for that weight. Thus, the indices in w_{1,2} say that the strength of the signal from the second input element to the first (and only) neuron is w_{1,2}.
The S neuron R-input one-layer network also can be drawn in abbreviated notation.
Here p is an R-length input vector, W is an S × R matrix, a and b are S-length vectors. As defined previously, the neuron layer includes the weight matrix, the multiplication operations, the bias vector b, the summer, and the transfer function blocks.
To describe networks having multiple layers, the notation must be extended. Specifically, it needs to make a distinction between weight matrices that are connected to inputs and weight matrices that are connected between layers. It also needs to identify the source and destination for the weight matrices.
We will call weight matrices connected to inputs input weights; we will call weight matrices connected to layer outputs layer weights. Further, superscripts are used to identify the source (second index) and the destination (first index) for the various weights and other elements of the network. To illustrate, the one-layer multiple input network shown earlier is redrawn in abbreviated form here.
As you can see, the weight matrix connected to the input vector p is labeled as an input weight matrix (IW^{1,1}) having a source 1 (second index) and a destination 1 (first index). Elements of layer 1, such as its bias, net input, and output have a superscript 1 to say that they are associated with the first layer.
Multiple Layers of Neurons uses layer weight (LW) matrices as well as input weight (IW) matrices.
A network can have several layers. Each layer has a weight matrix W, a bias vector b, and an output vector a. To distinguish between the weight matrices, output vectors, etc., for each of these layers in the figures, the number of the layer is appended as a superscript to the variable of interest. You can see the use of this layer notation in the three-layer network shown next, and in the equations at the bottom of the figure.
The network shown above has R^{1} inputs, S^{1} neurons in the first layer, S^{2} neurons in the second layer, etc. It is common for different layers to have different numbers of neurons. A constant input 1 is fed to the bias for each neuron.
Note that the outputs of each intermediate layer are the inputs to the following layer. Thus layer 2 can be analyzed as a one-layer network with S^{1} inputs, S^{2} neurons, and an S^{2} × S^{1} weight matrix W^{2}. The input to layer 2 is a^{1}; the output is a^{2}. Now that all the vectors and matrices of layer 2 have been identified, it can be treated as a single-layer network on its own. This approach can be taken with any layer of the network.
The layers of a multilayer network play different roles. A layer that produces the network output is called an output layer. All other layers are called hidden layers. The three-layer network shown earlier has one output layer (layer 3) and two hidden layers (layer 1 and layer 2). Some authors refer to the inputs as a fourth layer. This toolbox does not use that designation.
The architecture of a multilayer network with a single input vector can be specified with the notation R − S^{1} − S^{2} −...− S^{M}, where the number of elements of the input vector and the number of neurons in each layer are specified.
The same three-layer network can also be drawn using abbreviated notation.
Multiple-layer networks are quite powerful. For instance, a network of two layers, where the first layer is sigmoid and the second layer is linear, can be trained to approximate any function (with a finite number of discontinuities) arbitrarily well. This kind of two-layer network is used extensively in Multilayer Neural Networks and Backpropagation Training.
Here it is assumed that the output of the third layer, a^{3}, is the network output of interest, and this output is labeled as y. This notation is used to specify the output of multilayer networks.
Network inputs might have associated processing functions. Processing functions transform user input data to a form that is easier or more efficient for a network.
For instance, mapminmax transforms input data so that all values fall into the interval [−1, 1]. This can speed up learning for many networks. removeconstantrows removes the rows of the input vector that correspond to input elements that always have the same value, because these input elements are not providing any useful information to the network. The third common processing function is fixunknowns, which recodes unknown data (represented in the user's data with NaN values) into a numerical form for the network. fixunknowns preserves information about which values are known and which are unknown.
Similarly, network outputs can also have associated processing functions. Output processing functions are used to transform user-provided target vectors for network use. Then, network outputs are reverse-processed using the same functions to produce output data with the same characteristics as the original user-provided targets.
Both mapminmax and removeconstantrows are often associated with network outputs. However, fixunknowns is not. Unknown values in targets (represented by NaN values) do not need to be altered for network use.
Processing functions are described in more detail in Choose Neural Network Input-Output Processing Functions.