公式推导

含义	公式	维度
输入（矩阵形式）	X=[−x(1)T−−x(2)T−⋯−x(i)T−⋯−x(m)T−]\mathbf X= \begin{bmatrix}-\mathbf {x^{(1)}}^T - \\-\mathbf {x^{(2)}}^T- \\\cdots\\-\mathbf {x^{(i)}}^T-\\\cdots\\-\mathbf {x^{(m)}}^T-\end{bmatrix}X=⎣⎡−x(1)T−−x(2)T−⋯−x(i)T−⋯−x(m)T−⎦⎤	m×nm\times nm×n
输入	x(i)=[x1(i)x2(i)⋯xj(i)⋯xn(i)]T\mathbf x^{(i)}=\begin{bmatrix} x_{1}^{(i)} & x_{2}^{(i)} & \cdots & x_{j}^{(i)} & \cdots & x_{n}^{(i)}\end{bmatrix}^Tx(i)=[x1(i)x2(i)⋯xj(i)⋯xn(i)]T	n×1n\times 1n×1
标签	y=[y(1)y(2)⋯y(i)⋯y(m)]T\mathbf y={\begin{bmatrix} y^{(1)} & y^{(2)} & \cdots & y^{(i)} &\cdots &y^{(m)}\end{bmatrix}}^Ty=[y(1)y(2)⋯y(i)⋯y(m)]T	m×1m\times 1m×1
参数	w=[w1w2⋯wj⋯wn]T\mathbf w={\begin{bmatrix}w_{1} & w_{2} & \cdots & w_{j} & \cdots & w_{n}\end{bmatrix}}^Tw=[w1w2⋯wj⋯wn]T	n×1n\times 1n×1
输出	fw,b(x(i))=g(wTx(i)+b)g(z)=11+e−z\begin{aligned}f_{\mathbf w,b}(\mathbf x^{(i)}) &=g({\mathbf w}^T{\mathbf x}^{(i)} + b) \\ g(z) &= \frac{1}{1+e^{-z}} \end{aligned}fw,b(x(i))g(z)=g(wTx(i)+b)=1+e−z1	标量
输出（矩阵形式）	fw,b(X)=g(Xw+b)f_{\mathbf w,b}(\mathbf X) = g(\mathbf X \mathbf w+ b)fw,b(X)=g(Xw+b)	m×1m\times 1m×1
预测	y^(i)={1if fw,b(x(i))≥0.50if fw,b(x(i))<0.5\hat{y}^{(i)}= \begin{cases} 1 & \text{if }f_{\mathbf w,b}(\mathbf x^{(i)})\ge 0.5\\ 0 & \text{if }f_{\mathbf w,b}(\mathbf x^{(i)}) <0.5\end{cases}y^(i)={10if fw,b(x(i))≥0.5if fw,b(x(i))<0.5	标量
损失函数	cost(i)={−log⁡(fw,b(x(i)))if y(i)=1−log⁡(1−fw,b(x(i)))if y(i)=0=−y(i)log⁡(fw,b(x(i)))−(1−y(i))log⁡(1−fw,b(x(i)))\begin{aligned}cost^{(i)} &= \begin{cases} -\log\left(f_{\mathbf{w},b}\left( \mathbf{x}^{(i)} \right) \right) & \text{if }y^{(i)}=1\\ -\log \left( 1 - f_{\mathbf{w},b}\left( \mathbf{x}^{(i)} \right) \right)&\text{if }y^{(i)}=0\end{cases} \\ &=-y^{(i)} \log\left(f_{\mathbf{w},b}\left( \mathbf{x}^{(i)} \right) \right) - \left( 1 - y^{(i)}\right) \log \left( 1 - f_{\mathbf{w},b}\left( \mathbf{x}^{(i)} \right) \right)\end{aligned}cost(i)={−log(fw,b(x(i)))−log(1−fw,b(x(i)))if y(i)=1if y(i)=0=−y(i)log(fw,b(x(i)))−(1−y(i))log(1−fw,b(x(i)))	标量
代价函数	J(w,b)=1m∑i=1mcost(i)+λ2m∑j=1nwj2=1m(−yTlog(fw,b(X))−(I−y)Tlog(I−fw,b(X)))+λ2mwTw\begin{aligned}J(\mathbf w,b) &= \frac{1}{m} \sum\limits_{i = 1}^{m} cost^{(i)}+\frac{\lambda}{2m}\sum\limits_{j = 1}^{n} w_{j}^2\\&=\frac{1}{m}\left(-\mathbf y^Tlog(f_{\mathbf w,b}(\mathbf X))-(\Iota-\mathbf y)^Tlog(\Iota-f_{\mathbf w,b}(\mathbf X))\right)+\frac{\lambda}{2m}\mathbf w^T\mathbf w\end{aligned}J(w,b)=m1i=1∑mcost(i)+2mλj=1∑nwj2=m1(−yTlog(fw,b(X))−(I−y)Tlog(I−fw,b(X)))+2mλwTw	标量
梯度下降	wj:=wj−α∂J(w,b)∂wjb:=b−α∂J(w,b)∂b∂J(w,b)∂wj=1m∑i=1m(fw,b(x(i))−y(i))xj(i)+λmwj∂J(w,b)∂b=1m∑i=1m(fw,b(x(i))−y(i))\begin{aligned}w_j :&= w_j - \alpha \frac{\partial J(\mathbf{w},b)}{\partial w_j}\\ b :&= b - \alpha \frac{\partial J(\mathbf{w},b)}{\partial b}\\\frac{\partial J(\mathbf{w},b)}{\partial w_j} &= \frac{1}{m} \sum\limits_{i = 1}^{m} (f_{\mathbf{w},b}(\mathbf{x}^{(i)}) - y^{(i)})x_{j}^{(i)} + \frac{\lambda}{m} w_j \\ \frac{\partial J(\mathbf{w},b)}{\partial b} &= \frac{1}{m} \sum\limits_{i = 1}^{m} (f_{\mathbf{w},b}(\mathbf{x}^{(i)}) - y^{(i)}) \end{aligned}wj:b:∂wj∂J(w,b)∂b∂J(w,b)=wj−α∂wj∂J(w,b)=b−α∂b∂J(w,b)=m1i=1∑m(fw,b(x(i))−y(i))xj(i)+mλwj=m1i=1∑m(fw,b(x(i))−y(i))	标量
梯度下降（矩阵形式）	w:=w−α∂J(w,b)∂w∂J(w,b)∂w=1mXT(fw,b(X)−y)+λmw\begin{aligned}\mathbf w:&=\mathbf w-\alpha\frac{\partial J(\mathbf{w},b)}{\partial \mathbf{w}}\\ \frac{\partial J(\mathbf{w},b)}{\partial \mathbf{w}}&=\frac{1}{m}\mathbf X^T(f_{\mathbf w,b}(\mathbf X) -\mathbf y)+\frac{\lambda}{m} \mathbf w\end{aligned}w:∂w∂J(w,b)=w−α∂w∂J(w,b)=m1XT(fw,b(X)−y)+mλw	n×1n\times 1n×1

numpy实现

def zscore_normalize_features(X):mu=np.mean(X,axis=0)                 sigma=np.std(X,axis=0)           X_norm=(X-mu)/sigma      return X_norm,mu,sigma
# SIGMOID
def sig(z):return 1/(1+np.exp(-z))
# f_wb
def compute_f_wb(X,w,b):f_wb=sig(np.dot(X,w)+b)# (m,1)return f_wb
# j_wb
def compute_cost(X,y,w,b,lambda_,f_wb_function):m,n=X.shapef_wb=f_wb_function(X,w,b) # m*1j_wb=1/m*(-np.dot(y.T, np.log(f_wb))-np.dot((1-y).T,np.log(1-f_wb)))+(lambda_/2*m)*np.dot(w.T,w)# (1,1)j_wb=j_wb[0,0] # scalar  return j_wb
# dj_dw,dj_db
def compute_gradient(X, y, w, b, lambda_,f_wb_function): m,n=X.shapef_wb=f_wb_function(X,w,b) # m*1dj_dw=(1/m)*np.dot(X.T,(f_wb-y))+(lambda_/n)*w # n*1dj_db=(1/m)*np.sum(f_wb-y) # scalarreturn dj_dw,dj_db
# w,b,j_history,w_history
def gradient_descent(X, y, w, b, cost_function, gradient_function, f_wb_function,alpha, num_iters,lambda_): J_history = []w_history = []w_temp = copy.deepcopy(w)  b_temp = bfor i in range(num_iters):dj_dw, dj_db = gradient_function(X, y, w_temp,b_temp,lambda_,f_wb_function)    w_temp = w_temp  - alpha * dj_dw               b_temp  = b_temp  - alpha * dj_db      cost =  cost_function(X, y, w_temp, b_temp,lambda_,f_wb_function)J_history.append(cost)  return w_temp, b_temp, J_history, w_history

样本点

x_train, y_train = load_data("data/ex2data2.txt")
y_train=y_train.reshape(-1,1)
x_train.shape,y_train.shape
fig=go.Figure()
fig.add_trace(go.Scatter(x=x_train[np.where(y_train==0)[0],0],y=x_train[np.where(y_train==0)[0],1],mode="markers",name="第一类")
)
fig.add_trace(go.Scatter(x=x_train[np.where(y_train==1)[0],0],y=x_train[np.where(y_train==1)[0],1],mode="markers",name="第一类")
)
fig.update_layout(width=1000,height=618)
fig.show()

特征

def map_feature(X1, X2,degree):"""Feature mapping function to polynomial features    """X1 = np.atleast_1d(X1)X2 = np.atleast_1d(X2)out = []for i in range(1, degree+1):for j in range(i + 1):out.append((X1**(i-j) * (X2**j)))return np.stack(out, axis=1)feature_power=3
features=map_feature(x_train[:, 0], x_train[:, 1],feature_power)
features.shape
x_,mu,sigma=zscore_normalize_features(features)
y_=y_train
x_.shape,y_.shape

梯度下降

m,n=x_.shape
initial_w = np.zeros((n,1))
initial_b = 0
iterations = 1500
alpha = 0.3
lambda_=0
w,b,J_history,w_history = gradient_descent(x_ ,y_, initial_w, initial_b, compute_cost, compute_gradient, compute_f_wb,alpha, iterations,lambda_)fig=go.Figure()
fig.update_layout(width=1000,height=618)
fig.add_trace(go.Scatter(x=np.arange(1,iterations+1),y=J_history,name="学习曲线",mode="markers+lines")
)
fig.update_layout(xaxis_title="迭代次数",yaxis_title="J_wb"
)
fig.show()

决策边界

u=np.linspace(-1.5,1.5,100)
v=np.linspace(-1.5,1.5,100)
z = np.zeros((len(u), len(v)))for i in range(len(u)):for j in range(len(v)):temp=(map_feature(u[i], v[j],feature_power).T-mu.reshape(-1,1))/sigma.reshape(-1,1)z[i,j] = sig(np.dot(w.T,temp) + b)[0,0]fig=go.Figure()
fig.update_layout(width=1000,height=618)
fig.add_trace(go.Contour(z=z,contours_coloring='lines',x=u,y=v,)
)
fig.add_trace(go.Scatter(x=x_train[np.where(y_train==0)[0],0],y=x_train[np.where(y_train==0)[0],1],mode="markers",name="第一类")
)
fig.add_trace(go.Scatter(x=x_train[np.where(y_train==1)[0],0],y=x_train[np.where(y_train==1)[0],1],mode="markers",name="第一类")
)
fig.show()

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逻辑回归（公式推导+numpy实现）

公式推导

numpy实现

样本点

特征

梯度下降

决策边界

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