12. Neural network model

primary coverage ：

1. The reason why neural networks are introduced
2. Introduction to neural network model
3. Computational vectorization of Neural Networks
4. The relationship between neural network and logistic regression algorithm

One 、 The reason why neural networks are introduced

• Both linear regression and logistic regression have such a disadvantage , namely ： When there are too many features , The calculated load will be very large .
• Most machine learning involves many features , For nonlinear classification problems , Often need to construct polynomial To represent the relationship between data , Ordinary logistic regression model , Can't handle so many features effectively , Now we need neural networks

Two 、 Introduction to neural network model

2.1 A single neuron model

• In the neural network , The small yellow circles in the figure are called with sigmoid perhaps logistic Activate the Artificial neuron
• x_0 node , It is called bias unit or bias neuron ,x_0 Always =1, You can draw or not draw , It mainly depends on how to be more convenient in specific examples
• Another term of activation function in neural network refers to nonlinear function g(z)
• θ The parameter is called the weight of the model in the neural network

2.2 Neural network model

• A neural network is a group of neurons , Connected collections

  

  Neural network has three layers :

  • The first input layer —— Input characteristics x_1,x_2…
  • The second hidden layer —— I can't see it in the training set , Its value is not x Neither y , There may be more than one hidden layer , Any non input layer and non output layer are called hidden layers
  • The third output layer —— Output the final calculation result of the hypothesis h_θ(x)

• Calculation method of neural network model ：

  

  • Activation term ： It refers to the value calculated and output by a specific neuron

  • θ^(j): Weight matrices , Control from j Layer to j+1 Mapping of layers

    For example, in the figure ：θ^(1) The parameter matrix that controls the mapping from three input units to three hidden units , It's a 3 x 4 Matrix

  • If a neural network is in the j Layer has a s_j A unit , stay j+1 Layer has a s_j +1 A unit , that θ^(j) That is, it controls the j layer To j+1 Layer mapping ,θ^(j) For one s_（j+1） X （s_j + 1） Matrix

• The mathematical definition of the hypothetical function of neural network ：

  Neural networks define functions h_θ(x) From input x To output y Mapping , These hypothetical functions are parameterized , The parameter is recorded as θ, So just change θ, You can get different hypothetical functions

3、 ... and 、 Computational vectorization of Neural Networks

• For the convenience of representation and calculation , Input in the input layer x , It can be regarded as the activation value of the first layer , That is to say a^(1) = x
• Input from a particular neuron x_1,x_2,x_3 The weighted linear combination of the other is z
• a_0^(2) —— It is called a bias unit , be equal to 1

Four 、 The relationship between neural network and logistic regression algorithm

• In the neural network , In fact, the latter part is a logistic regression algorithm
• But neural network does not directly use the original x_1,x_2, … x_n As an input feature , Instead, it uses the learned function input value a1,a2,…an As an input feature
  But neural network does not directly use the original x_1,x_2, … x_n As an input feature , Instead, it uses the learned function input value a1,a2,…an As an input feature
• Neural networks do not use input features x_1,x_2, … x_n To train logical regression , Instead, use what you learned a1,a2,…an To train the logical model by yourself , Thus, we can get complex and diverse nonlinear hypothetical functions

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