神经网络的编程基础

1、神经网络的计算通常包括前向传播(foward propagation)步骤和反向传播(backward propagation)的步骤；

2、数据预处理：

• X.shape:[nx,m]--->nx 是特征数，m为样本数
• Y.shape:[1,m]
• W.shape:[nx,1]
• b---->是实数 
• X 数据标准化

3、激励函数：

---

 sigmoid(x)=1/(1+e^-x)函数：

import numpy as np
def sigmoid(x):
    """
    Compute the sigmoid of x

    Arguments:
    x -- A scalar or numpy array of any size

    Return:
    s -- sigmoid(x)
    """
    
   
    s = 1.0/(1+np.exp(-x))

    
    return s

 softmax函数：

def softmax(x):
    """Calculates the softmax for each row of the input x.

    Your code should work for a row vector and also for matrices of shape (n, m).

    Argument:
    x -- A numpy matrix of shape (n,m)

    Returns:
    s -- A numpy matrix equal to the softmax of x, of shape (n,m)
    """
    
 
    # Apply exp() element-wise to x. Use np.exp(...).

    x_exp = np.exp(x)

    # Create a vector x_sum that sums each row of x_exp. Use np.sum(..., axis = 1, keepdims = True).

    x_sum = np.sum(x_exp,axis=1,keepdims=True)
    
    # Compute softmax(x) by dividing x_exp by x_sum. It should automatically use numpy broadcasting.

    s = x_exp / x_sum

    
    return s

4、二分类（Binary Classification)-Logistic Regression：

---

Building the parts of our algorithm:

• Define the model structure(such as number of input features)
• Initialize the model's parameters
• Loop:
  • Calculate current loss(forward propagation)
  • Calculate current gradient(backward propagation)
  • Update parameters(gradient descent)

Forward and Backward propagation:

• You get X
• You compute  A=σ(w^TX+b)=(a⁽⁰⁾,a⁽¹⁾,...,a^(m−1),
  
• You calculate the cost function:J=−1/m∑^m_i=1y⁽ⁱ⁾log(a⁽ⁱ⁾)+(1−y⁽ⁱ⁾)log(1−a⁽ⁱ⁾)
• Backward propagation:
  • ∂J/∂w=(1/m)X(A−Y)^T
    
  •  
    ∂J/∂b=1m∑_i=1^m(a(i)−y(i))
    

import numpy as np

#define sigmoid

def sigmoid(z):
    s=1.0/(1+np.exp(z))

#Initializing parameters

def initialize_with_zeros(dim):
    w=np.zeros((dim,1))
    b=0
    assert(w.shape==(dim, 1))
    assert(isinstance(b,float) or isinstance(b,int))
    return w,b

# forward and backward propagate

def propagate(w,b,X,Y):
        """
    Implement the cost function and its gradient for the propagation explained above

    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat) of size (1, number of examples)

    Return:
    cost -- negative log-likelihood cost for logistic regression
    dw -- gradient of the loss with respect to w, thus same shape as w
    db -- gradient of the loss with respect to b, thus same shape as b
    """
    m=X.shape[1]
    A=sigmoid(np.dot(w.T,X))
    cost=(-1.0/m)*np.sum(Y*np.log(A)+(1-Y)*np.log((1-A)))
    dw=(1/m)*np.dot(X,(A-Y).T)
    db=(1/m)*np.sum(A-Y)
    assert(dw.shape==w.shape)
    assert(db.type==b.type)
    cost=np.squeeze(cost)
    assert(cost.shape==())
    grads={"dw":dw,
                "db":db} 
    return grads,cost

#using gradient descent,The goal is to learn  ww  and  bb  by minimizing the cost function  JJ . For a parameter  θθ , the update rule is  θ=θ−α dθ , where  αα  is the learning rate.

def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):
    costs=[]
    for i in range(num_iterations):
        grads, cost = propagate(w,b,X,Y)
        dw = grads["dw"]
        db = grads["db"]
        w = w - learning_rate * dw
        b = b - learning_rate * db
        if i%100==0:
            costs.append(cost)
        if print_cost and i%100==0:
            print ("Cost after iteration %i: %f" %(i, cost))
    params = {"w": w,
              "b": b}
    
    grads = {"dw": dw,
             "db": db}
    
    return params, grads, costs

#predict
def predict(w,b,X):
    m=X.shape[1]
    Y_prediction=np.zeros((1,m))
    w=w.reshape(X.shape[0],1)
    A=sigmoid(np.dot(w.T,X)+b)
    for i in range(A.shape[1]):
        Y_prediction[0,i]=np.where(A[0,i]>=0.5,1,0)
    assert(Y_prediction.shape==(1,m))
    return Y_prediction

#model

def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
     w, b = initialize_with_zeros(X_train.shape[0])
     parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost)
    Y_prediction_test = predict(w, b, X_test)
    Y_prediction_train = predict(w, b, X_train)
    print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100))
    print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100))
    d = {"costs": costs,
         "Y_prediction_test": Y_prediction_test, 
         "Y_prediction_train" : Y_prediction_train, 
         "w" : w, 
         "b" : b,
         "learning_rate" : learning_rate,
         "num_iterations": num_iterations}
    
    return d