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线性回归（公式推导+numpy实现）

2022-07-22 07:34:00 【西红柿爱喝水】

公式推导

含义	公式	维度
输入（矩阵形式）	$\mathbf X= \begin{bmatrix}-\mathbf {x^{(1)}}^T - \\-\mathbf {x^{(2)}}^T- \\\cdots\\-\mathbf {x^{(i)}}^T-\\\cdots\\-\mathbf {x^{(m)}}^T-\end{bmatrix}$	$m\times n$
输入	$\mathbf x^{(i)}=\begin{bmatrix} x_{1}^{(i)} & x_{2}^{(i)} & \cdots & x_{j}^{(i)} & \cdots & x_{n}^{(i)}\end{bmatrix}^T$	$n\times 1$
标签	$\mathbf y={\begin{bmatrix} y^{(1)} & y^{(2)} & \cdots & y^{(i)} &\cdots &y^{(m)}\end{bmatrix}}^T$	$m\times 1$
参数	$\mathbf w={\begin{bmatrix}w_{1} & w_{2} & \cdots & w_{j} & \cdots & w_{n}\end{bmatrix}}^T$	$n\times 1$
输出	$f_{w,b}(\mathbf x^{(i)}) = {\mathbf w}^T{\mathbf x}^{(i)} + b$	标量
输出（矩阵形式）	$f_{w,b}(\mathbf X) = \mathbf X \mathbf w+ b$	$m\times 1$
损失函数	$cost^{(i)} = (f_{w,b}(\mathbf x^{(i)}) - y^{(i)})^2$	标量
代价函数	$\begin{aligned}J(\mathbf w,b) &= \frac{1}{2m} \sum\limits_{i = 1}^{m} cost^{(i)}+\frac{\lambda}{2m}\sum\limits_{j = 1}^{n} w_{j}^2\\&=\frac{1}{2m}(f_{w,b}(\mathbf X)-\mathbf y)^T(f_{w,b}(\mathbf X)-\mathbf y)+\frac{\lambda}{2m}\mathbf w^T\mathbf w\end{aligned}$	标量
梯度下降	$\begin{aligned}w_j :&= w_j - \alpha \frac{\partial J(\mathbf{w},b)}{\partial w_j}\\ b :&= b - \alpha \frac{\partial J(\mathbf{w},b)}{\partial b}\\\frac{\partial J(\mathbf{w},b)}{\partial w_j} &= \frac{1}{m} \sum\limits_{i = 1}^{m} (f_{\mathbf{w},b}(\mathbf{x}^{(i)}) - y^{(i)})x_{j}^{(i)} + \frac{\lambda}{m} w_j \\ \frac{\partial J(\mathbf{w},b)}{\partial b} &= \frac{1}{m} \sum\limits_{i = 1}^{m} (f_{\mathbf{w},b}(\mathbf{x}^{(i)}) - y^{(i)}) \end{aligned}$	标量
梯度下降（矩阵形式）	$\begin{aligned}\mathbf w:&=\mathbf w-\alpha\frac{\partial J(\mathbf{w},b)}{\partial \mathbf{w}}\\ \frac{\partial J(\mathbf{w},b)}{\partial \mathbf{w}}&=\frac{1}{m}\mathbf X^T(f_{w,b}(\mathbf X) -\mathbf y)+\frac{\lambda}{m} \mathbf w\end{aligned}$	$n\times 1$

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
# f_wb
def compute_f_wb(X,w,b):
    f_wb=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.shape
    f_wb=f_wb_function(X,w,b) # m*1
    j_wb=1/(2*m)*np.dot((f_wb-y).T, f_wb-y)+(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.shape
    f_wb=f_wb_function(X,w,b) # m*1
    dj_dw=(1/m)*np.dot(X.T,(f_wb-y))+(lambda_/n)*w # n*1
    dj_db=(1/m)*np.sum(f_wb-y) # scalar
    return 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 = b
    for 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 = np.arange(0, 20, 1)
y = 1+x+2*pow(x,2)+3*pow(x,3)
fig=go.Figure()
fig.update_layout(width=1000,height=618)
fig.add_trace(
    go.Scatter(
        x=x,
        y=y,
        name="样本点",
        mode="markers"
    )
)
fig.show()

在这里插入图片描述

特征

feature_1=x # (m,)
feature_2=pow(x,2) # (m,)
feature_3=pow(x,3) # (m,)
x_=zscore_normalize_features(np.transpose(np.array([
    feature_1,
    feature_2,
    feature_3
    ])))
y_=y.reshape(-1,1)
x_.shape,y_.shape

梯度下降

m,n=x_.shape
initial_w = np.zeros((n,1))
initial_b = 0
iterations = 150
alpha = 0.5
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()

在这里插入图片描述

预测

y_hat=compute_f_wb(x_,w,b) # m*1
fig=go.Figure()
fig.update_layout(width=1000,height=618)
fig.add_trace(
    go.Scatter(
        x=x,
        y=y,
        name="y",
        mode="markers"
    )
)
fig.add_trace(
    go.Scatter(
        x=x,
        y=y_hat.ravel(),
        name="y_hat"
        
    )
)
fig.update_layout(
    xaxis_title="x",
    yaxis_title="y"
)
fig.show()