一：建立一個邏輯回歸模型來預測一個學生是否被大學錄取。
假設你是一個大學系的管理員，你想根據兩次考試的結果來決定每個申請人的錄取機會。
你有以前的申請人的歷史數據，你可以用它作為邏輯回歸的訓練集。
對於每一個培訓例子，你有兩個考試的申請人的分數和錄取決定。
為了做到這一點，我們將建立一個分類模型，根據考試成績估計入學概率。
（一）導入庫，並且讀取數據集
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import pandas as pd import numpy as np import matplotlib.pyplot as plt import seaborn as sns path = 'ex2data1.txt' data = pd.read_csv(path,header=None,names=['exam1','exam2','admitted'])
print(data.head()) #數據展示
print(data.describe()) #數據描述
（二）原始數據可視化
positive = data[data['admitted']==1] #獲取正樣本 negative = data[data['admitted']==0] #獲取負樣本 fig,ax = plt.subplots(figsize=(12,8)) ax.scatter(positive['exam1'],positive['exam2'],s=30,c='g',marker='o',label='Admitted') ax.scatter(negative['exam1'],negative['exam2'],s=30,c='r',marker='x',label='Admitted') ax.legend(loc=4) #顯示標簽位置 ax.set_xlabel("Exam1 Score") ax.set_ylabel("Exam2 Score") plt.show()
（三）數據預處理
在訓練模型之前經常要對數據進行數組轉化
當然很多時候在提取完數據后其自身就是數組形式（<class ‘numpy.ndarray’>），這只是習慣性的謹慎。很多時候取得的數據是DataFrame的形式，這個時候要記得轉換成數組
def get_X(df): #獲取特征，並添加x_0列向量（全為1） ones = pd.DataFrame({'ones':np.ones(df.shape[0])}) data = pd.concat([ones,df],axis=1) #按axis=1列合並連接 return data.iloc[:,:-1].as_matrix() #按相對位置，獲取全部特征值數組 或者我們直接使用.values代替 def get_y(df): #讀取標簽值 return np.array(df.iloc[:,-1]) #這樣可以轉換為數組 def normalize_feature(df): #進行歸一化處理 return df.apply(lambda column:(column-column.mean())/column.std())#標准差標准化
X = get_X(data)
y = get_y(data)
print(data.head())
print(X)
print(y)
或者使用：
data = (data-data.mean())/data.std()
進行歸一化。這里數據差距不大，所以不進行歸一化。而且歸一化，我們只對特征值進行處理，標簽值可以不進行處理。
（四）實現sigmoid函數
def sigmoid(z): #實現sigmoid函數 return 1/(1+np.exp(-z))
測試sigmoid函數，繪制圖形：
fig,ax = plt.subplots(figsize=(12,8)) ax.plot(np.arange(-10,10,step=0.01), sigmoid(np.arange(-10,10,step=0.01))) #設置x,y軸各自數據，繪制圖形 ax.set_xlabel('z',fontsize=18) ax.set_ylabel('g(z)',fontsize=18) ax.set_title('sigmoid function',fontsize=18) plt.show()
（五）實現代價函數
def cost(theta, X, y): theta = np.matrix(theta) X = np.matrix(X) y = np.matrix(y) first = np.multiply(-y, np.log(sigmoid(X* theta.T))) second = np.multiply((1 - y), np.log(1 - sigmoid(X* theta.T))) return np.sum(first - second) / (len(X))
注意：我們要使用*實現矩陣乘法，那么我們必須使得兩邊參數類型為matrix類型
補充：numpy中的matrix矩陣處理---numpy似乎並不推薦使用matrix矩陣類型
對於上面的X*theta.T，我們使用了“*”運算符，進行了矩陣乘法操作。但是我們如果將*當作矩陣乘法，那么我們必須保證兩邊操作數類型是matrix矩陣類型。另外dot也可以實現矩陣乘法，但是它要求傳參是ndarray類型，並且兩個數組保持第一個矩陣列數等於第二個矩陣行數。
def cost(theta, X, y): #實現代價函數 ----直接使用array類型 first = np.multiply(-y, np.log(sigmoid(X.dot(theta.T)))) second = np.multiply((1 - y), np.log(1 - sigmoid(X.dot(theta.T)))) return np.sum(first - second) / (len(X))
測試代價函數：
X = get_X(data) y = get_y(data) theta = np.zeros(X.shape[1]) #返回numpy.ndarray類型 print(cost(theta,X,y))
（六）梯度下降法
1.計算梯度：推導
def gradient(theta,X,y): #實現求梯度 return X.T@(sigmoid(X@theta)-y)/len(X)
補充：numpy矩陣相乘matmul可以用@來代替
或者：
def gradient(theta,X,y): #實現求梯度 return (X.T).dot(np.array([(sigmoid(X.dot(theta.T))-y)]).T).flatten()/len(X)
補充：對於numpy.ndarray類型數據，如果要進行矩陣運算，那么數組必須是二維數組[[]]
print(gradient(theta,X,y))
2.參數擬合---梯度下降法
def gradientDescennt(theta,X,y,alpha,iters): #進行梯度下降法擬合參數 costs = np.zeros(iters) temp = np.ones(len(theta)) for i in range(iters): #進行迭代 temp = theta - alpha*gradient(theta,X,y) theta = temp costs[i] = cost(theta,X,y) return theta,costs
可視化操作：
theta,costs =gradientDescennt(theta,X,y,0.001,1000) #注意：這里我們選取的步長不易太大 print(theta) print(costs[-1]) fig, ax = plt.subplots(figsize=(12,8)) ax.plot(np.arange(1000),costs,'r') ax.set_xlabel("Iterations") ax.set_ylabel("Cost") ax.set_title("Error vs. Iteration") plt.show()
可以發現，當步長太大，導致代價值發散。
而且：梯度下降法得到的擬合參數。和下面高級優化算法參數不一致....決策邊界也不對....???
難道是因為數據沒有進行歸一化處理？？
以后進行回顧補充！！！
3.參數擬合---高級優化算法
import scipy.optimize as opt #使用高級優化算法
res = opt.minimize(fun=cost,x0=theta,args=(X,y),jac=gradient,method='TNC') print(res)
https://www.cnblogs.com/tongtong123/p/10634716.html
其中：x就是我們要的θ值
根據訓練的模型參數對數據進行預測:
def predict(X,theta): #對數據進行預測 return (sigmoid(X @ theta.T)>=0.5).astype(int) #實現變量類型轉換
res = opt.minimize(fun=cost,x0=theta,args=(X,y),jac=gradient,method='TNC') theta_res = res.x #獲取擬合的θ參數 y_pred = predict(X,theta_res) print(classification_report(y,y_pred))
（七）可視化決策邊界
這里已經得到了θ的參數，將Xθ取0即可得到對應的決策邊界函數。
#先繪制原來數據 positive = data[data['admitted'] == 1] # 挑選出錄取的數據 negative = data[data['admitted'] == 0] # 挑選出未被錄取的數據 fig, ax = plt.subplots(figsize=(10, 5)) # 獲取繪圖對象 # 對錄取的數據根據兩次考試成績繪制散點圖 ax.scatter(positive['exam1'], positive['exam2'], s=30, c='b', marker='o', label='Admitted') # 對未被錄取的數據根據兩次考試成績繪制散點圖 ax.scatter(negative['exam1'], negative['exam2'], s=30, c='r', marker='x', label='Not Admitted') # 添加圖例 ax.legend() # 設置x,y軸的名稱 ax.set_xlabel('Exam1 Score') ax.set_ylabel('Exam2 Score') plt.title("fitted curve vs sample") #繪制決策邊界 print(theta_res) exam_x = np.arange(X[:,1].min(),X[:,1].max(),0.01) theta_res = - theta_res/theta_res[2] #獲取函數系數θ_0/θ_2 θ_0/θ_2 print(theta_res) exam_y = theta_res[0]+theta_res[1]*exam_x plt.plot(exam_x,exam_y) plt.show()
import pandas as pd import numpy as np import matplotlib.pyplot as plt import seaborn as sns from sklearn.metrics import classification_report import scipy.optimize as opt def get_X(df): #獲取特征，並添加x_0列向量（全為1） ones = pd.DataFrame({'ones':np.ones(df.shape[0])}) data = pd.concat([ones,df],axis=1) #按axis=1列合並連接 return data.iloc[:,:-1].values #按相對位置，獲取全部特征值數組 def get_y(df): #讀取標簽值 return np.array(df.iloc[:,-1]) #這樣可以轉換為數組 def normalize_feature(df): #進行歸一化處理 return df.apply(lambda column: (column-column.mean())/column.std())#標准差標准化 def sigmoid(z): #實現sigmoid函數 return 1/(1+np.exp(-z)) def cost(theta, X, y): #實現代價函數 first = np.multiply(-y, np.log(sigmoid(X.dot(theta.T)))) second = np.multiply((1 - y), np.log(1 - sigmoid(X.dot(theta.T)))) return np.sum(first - second) / (len(X)) def gradient(theta,X,y): #實現求梯度 return X.T@(sigmoid(X@theta)-y)/len(X) def gradientDescennt(theta,X,y,alpha,iters): #進行梯度下降法擬合參數 costs = np.zeros(iters) temp = np.ones(len(theta)) for i in range(iters): #進行迭代 temp = theta - alpha*gradient(theta,X,y) theta = temp costs[i] = cost(theta,X,y) return theta,costs def predict(X,theta): #對數據進行預測 return (sigmoid(X @ theta.T)>=0.5).astype(int) #實現變量類型轉換 path = 'ex2data1.txt' data = pd.read_csv(path,header=None,names=['exam1','exam2','admitted']) X = get_X(data) y = get_y(data) theta = np.zeros(X.shape[1]) #返回numpy.ndarray類型 res = opt.minimize(fun=cost,x0=theta,args=(X,y),jac=gradient,method='TNC') theta_res = res.x #獲取擬合的θ參數 #theta_res,costs =gradientDescennt(theta,X,y,0.00001,500000) #先繪制原來數據 positive = data[data['admitted'] == 1] # 挑選出錄取的數據 negative = data[data['admitted'] == 0] # 挑選出未被錄取的數據 fig, ax = plt.subplots(figsize=(10, 5)) # 獲取繪圖對象 # 對錄取的數據根據兩次考試成績繪制散點圖 ax.scatter(positive['exam1'], positive['exam2'], s=30, c='b', marker='o', label='Admitted') # 對未被錄取的數據根據兩次考試成績繪制散點圖 ax.scatter(negative['exam1'], negative['exam2'], s=30, c='r', marker='x', label='Not Admitted') # 添加圖例 ax.legend() # 設置x,y軸的名稱 ax.set_xlabel('Exam1 Score') ax.set_ylabel('Exam2 Score') plt.title("fitted curve vs sample") #繪制決策邊界 print(theta_res) exam_x = np.arange(X[:,1].min(),X[:,1].max(),0.01) theta_res = - theta_res/theta_res[2] #獲取函數系數θ_0/θ_2 θ_0/θ_2 print(theta_res) exam_y = theta_res[0]+theta_res[1]*exam_x plt.plot(exam_x,exam_y) plt.show()
二：正則化邏輯回歸
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（一）數據讀取和展示
path = 'ex2data2.txt' data = pd.read_csv(path,header=None,names=['test1','test2','accepted']) #先繪制原來數據 positive = data[data['accepted'] == 1] # 挑選出錄取的數據 negative = data[data['accepted'] == 0] # 挑選出未被錄取的數據 fig, ax = plt.subplots(figsize=(10, 5)) # 獲取繪圖對象 # 對錄取的數據根據兩次考試成績繪制散點圖 ax.scatter(positive['test1'], positive['test2'], s=30, c='b', marker='o', label='Accepted') # 對未被錄取的數據根據兩次考試成績繪制散點圖 ax.scatter(negative['test1'], negative['test2'], s=30, c='r', marker='x', label='Not Accepted') # 添加圖例 ax.legend() # 設置x,y軸的名稱 ax.set_xlabel('test1') ax.set_ylabel('test2') plt.title("fitted curve vs sample") plt.show()
（二）特征映射
對於較復雜的邊界，我們需要使用多項式來表示，所以，我們需要通過特征映射，來找到合適的多項式。
其中i決定了最高次項。
主要功能，是將原來的2個特征值（不含accepted),變為新的28為多項式特征值。----n(n+1)/2其中n等於7（從0到x^6)
def feature_mapping(x,y,power,as_ndarray=False): data = {"f{}{}".format(i - p, p): np.power(x, i - p) * np.power(y, p) for i in np.arange(power+1) for p in np.arange(i+1)} if as_ndarray: return pd.DataFrame(data).as_matrix() else: return pd.DataFrame(data)
path = 'ex2data2.txt' data = pd.read_csv(path,header=None,names=['test1','test2','accepted']) print(data.shape) print(data.head()) data = feature_mapping(np.array(data.test1),np.array(data.test2),power=6) print(data.shape) print(data.head())
（三）正則化代價函數
def regularized_cost(theta, X, y, learningRate=1): #實現代價函數
first = np.multiply(-y, np.log(sigmoid(X.dot(theta.T))))
second = np.multiply((1 - y), np.log(1 - sigmoid(X.dot(theta.T))))
reg = (learningRate/(2*len(X)))*np.sum(np.power(theta[1:theta.shape[0]],2))
return np.sum(first - second) / (len(X)) + reg
path = 'ex2data2.txt' data = pd.read_csv(path,header=None,names=['test1','test2','accepted']) X = feature_mapping(np.array(data.test1),np.array(data.test2),power=6) #X = get_X(data) y = get_y(data) theta = np.zeros(X.shape[1]) #返回numpy.ndarray類型 print(regularized_cost(theta,X,y))
因為我們設置的θ為0，所以這個正則化代價函數與代價函數的值應該相同。
（四）正則化梯度
正則化不包含θ_0！！！
def gradient(theta,X,y): #實現求梯度 return X.T@(sigmoid(X@theta)-y)/len(X) def regularized_gradient(theta,X,y,learningRate=1): theta_new = theta[1:] #不加θ_0 regularized_theta = (learningRate/len(X))*theta_new regularized_term = np.concatenate([np.array([0]),regularized_theta]) #前面加上0，是為了加給θ_0 return gradient(theta,X,y)+regularized_term
path = 'ex2data2.txt' data = pd.read_csv(path,header=None,names=['test1','test2','accepted']) X = feature_mapping(np.array(data.test1),np.array(data.test2),power=6) #X = get_X(data) y = get_y(data) theta = np.zeros(X.shape[1]) #返回numpy.ndarray類型 print(regularized_gradient(theta,X,y))
（五）高級優化算法擬合參數θ
res = opt.minimize(fun=regularized_cost,x0=theta,args=(X,y),jac=regularized_gradient,method='Newton-CG') print(res.x)
（六）畫出決策邊界
#先繪制原來數據 positive = data[data['accepted'] == 1] # 挑選出錄取的數據 negative = data[data['accepted'] == 0] # 挑選出未被錄取的數據 fig, ax = plt.subplots(figsize=(12, 8)) # 獲取繪圖對象 # 對錄取的數據根據兩次考試成績繪制散點圖 ax.scatter(positive['test1'], positive['test2'], s=30, c='b', marker='o', label='Accepted') # 對未被錄取的數據根據兩次考試成績繪制散點圖 ax.scatter(negative['test1'], negative['test2'], s=30, c='r', marker='x', label='Not Accepted') # 添加圖例 ax.legend() # 設置x,y軸的名稱 ax.set_xlabel('test1') ax.set_ylabel('test2') plt.title("Regularized Logistic Regression") #繪制決策邊界 x = np.linspace(-1, 1.5, 250) xx, yy = np.meshgrid(x, x) #生成網格點坐標矩陣 https://blog.csdn.net/lllxxq141592654/article/details/81532855 z = feature_mapping(xx.ravel(), yy.ravel(), 6) #xx,yy都是（250，250）矩陣，經過ravel后，全是(62500,1)矩陣。之后進行特征變換獲取(62500,28)的矩陣 z = z @ theta_res #將每一行多項式與系數θ相乘，獲取每一行的值=θ_0+θ_1*X_1+...+θ_n*X_n z = np.array([z]).reshape(xx.shape) #將最終值轉成250行和250列坐標 print(z) plt.contour(xx, yy, z, 0) #傳入的xx,yy需要同zz相同維度。 # 其實xx每一行都是相同的，yy每一列都是相同的，所以本質上來說xx是決定了橫坐標，yy決定了縱坐標. # 而z中的每一個坐標都是對於(x,y)的值---即輪廓高度。 # 第四個參數如果是int值，則是設置等高線條數為n+1---會自動選取最全面的等高線。 如何選取等高線？具體原理還要考慮：是因為從所有數據中找到數據一樣的點？ # https://blog.csdn.net/lanchunhui/article/details/70495353 # https://matplotlib.org/api/_as_gen/matplotlib.pyplot.contour.html#matplotlib.pyplot.contour plt.show()
可以通過改變入的值，來調整參數。