Artificial neural network
Artificial neural network (artificial neural network, or ANN, for short),hereinafter referred to as neural network, neural network, the abbreviation NN) or artificial neural network, is a kind of imitate biological neural network (animal central nervous system, especially the brain) structure and function of the mathematical model or computational model. Neural network by a large number of artificial neurons connected to calculate. Most cases of artificial neural network can change internal structure on the basis of outside information,is a kind of adaptive system. Modern neural network is a kind of nonlinear statistical data modeling tool, is used to modeling of complex relationship between the input and output, or to explore data model.
Below to illustrate the perceptron model.
In "input" position to add the MP model neuron nodes, mark it as "input unit". The rest remains the same, so we have below: starting from this picture, we will weight w1, w2, w3 wrote, in the middle of the "line".
In the "sensors", there are two levels. Is the input layer and output layer respectively. "Input unit" of the input layer is responsible for data transmission, do not do calculation. The "output units" in the output layer requires the input of the previous layer. We call the level of the need to compute the computational layer, and calculate have a layer of the network is called "single layer neural network". There are some documents will be named according to the network with the number of layers, for example, called the "sensors" two layers of neural network. But in this article, we named according to calculate the amount of layers. If we want to predict the target is no longer a value, but it is a vector, such as [2, 3]. You can in the output layer to add a "output units". The figure below shows the single neural network, with two output units of output unit z1 calculation formula of the diagram below.
Figure 1 single layer neural network
Figure 2 single layer neural network (Z1)
As you can see, the z1 calculation with the original z and no difference. We know the output of a neuron can be passed to multiple neurons, thus z2 calculation formula of the following figure
As you can see, the calculation of z2 in addition to three new weights, w4, w5, w6, the other is the same with z1.The output of the entire network below
Figure 3 single layer neural network (Z2)
Figure 4 single layer neural network (Z1 and Z2)


Figure 5 single-layer neural network (extension)
If we look at the output calculation formula, we will find that the two formulas are linear algebraic equations. Therefore, the two formulas can be expressed by matrix multiplication.For example, the input variables are [a1, a2, a3] T (represented by a1, a2, a3) columns, represented by vector a; the left side of the equation is [z1, z2] T, represented by vector z.The coefficient is matrix W (the matrix with 2 lines 3 columns; the arrangement forms is same with the formula).As a result, the output formula can be rewritten as:
G = z (W * a);
This formula is the matrix manipulation of the neural network for calculating the later from the former layer.