示例一：梯度下降求解邏輯回歸

一、The data 

我們將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。

#三大件
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

import os
path = 'data' + os.sep + 'LogiReg_data.txt'
pdData = pd.read_csv(path, header=None, names=['Exam 1', 'Exam 2', 'Admitted'])
pdData.head()

	Exam 1	Exam 2	Admitted
0	34.623660	78.024693	0
1	30.286711	43.894998	0
2	35.847409	72.902198	0
3	60.182599	86.308552	1
4	79.032736	75.344376	1

 

 

 

 

 

 

pdData.shape  #(100, 3)

 展示圖形：

positive = pdData[pdData['Admitted'] == 1] # 被錄取的學生的數據
negative = pdData[pdData['Admitted'] == 0] # 未被錄取的學生的數據

fig, ax = plt.subplots(figsize=(10,5))  #定義一個畫圖區域
# 2個散點圖：ax.scatter(x軸數據，y軸數據, s=顏色飽和度，c=顏色，maker=散點的標記圖形，label=label導航標題)
ax.scatter(positive['Exam 1'], positive['Exam 2'], s=30, c='b', marker='o', label='Admitted')
ax.scatter(negative['Exam 1'], negative['Exam 2'], s=30, c='r', marker='x', label='Not Admitted')
ax.legend()
ax.set_xlabel('Exam 1 Score')  #設置x軸的標題
ax.set_ylabel('Exam 2 Score')

 

 

 取得以上兩個散點圖的分界線

二、The logistic regression：邏輯回歸

 

目標：建立分類器（求解出三個參數 θ0θ1θ2）,θ0偏置項θ1第一次測試的成績θ2第二次測試的成績

設定閾值，根據閾值判斷錄取結果

要完成的模塊

• sigmoid : 映射到概率的函數

• model : 返回預測結果值

• cost : 根據參數計算損失

• gradient : 計算每個參數的梯度方向

• descent : 進行參數更新

accuracy: 計算精度

 三、實現邏輯回歸的各模塊

1.sigmoid函數：

def sigmoid(z):
    return 1 / (1 + np.exp(-z))

 測試sigmoid函數：

nums = np.arange(-10, 10, step=1) # -10到10之間，步長為1的數字列表
fig, ax = plt.subplots(figsize=(12,4))
ax.plot(nums, sigmoid(nums), 'r')

 

 2.預測函數

# 預測函數h；np.dot(X, theta.T)兩個矩陣相乘，如圖所示
def model(X, theta):
    
    return sigmoid(np.dot(X, theta.T))

查看數據：

pdData.insert(0, 'Ones', 1) # in a try / except structure so as not to return an error if the block si executed several times


# set X (training data) and y (target variable)
orig_data = pdData.as_matrix() # convert the Pandas representation of the data to an array useful for further computations
cols = orig_data.shape[1]
X = orig_data[:,0:cols-1]
y = orig_data[:,cols-1:cols]

# convert to numpy arrays and initalize the parameter array theta
#X = np.matrix(X.values)
#y = np.matrix(data.iloc[:,3:4].values) #np.array(y.values)
# 構造theta矩陣結構
theta = np.zeros([1, 3])

X[:5]
Out[10]:
array([[  1.        ,  34.62365962,  78.02469282],
       [  1.        ,  30.28671077,  43.89499752],
       [  1.        ,  35.84740877,  72.90219803],
       [  1.        ,  60.18259939,  86.3085521 ],
       [  1.        ,  79.03273605,  75.34437644]])

y[:5]
Out[11]:
array([[ 0.],
       [ 0.],
       [ 0.],
       [ 1.],
       [ 1.]])

theta
# array([[ 0.,  0.,  0.]])

X.shape, y.shape, theta.shape
# ((100, 3), (100, 1), (1, 3))

 3.損失函數

# 損失函數
def cost(X, y, theta):
    left = np.multiply(-y, np.log(model(X, theta)))
    right = np.multiply(1 - y, np.log(1 - model(X, theta)))
    return np.sum(left - right) / (len(X))  # 求平均損失

cost(X, y, theta)
# 0.6931471805599453

 

4.計算梯度

# 計算梯度，求偏導
def gradient(X, y, theta):
    grad = np.zeros(theta.shape)
    error = (model(X, theta)- y).ravel()
    for j in range(len(theta.ravel())): #for each parmeter
        term = np.multiply(error, X[:,j])
        grad[0, j] = np.sum(term) / len(X)
    
    return grad

 

5.Gradient descent 

比較3中不同梯度下降方法：批量梯度下降、隨機梯度下降、小批量梯度下降

STOP_ITER = 0  #根據迭代次數停止迭代
STOP_COST = 1  #根據損失值停止迭代
STOP_GRAD = 2  #根據梯度變化停止迭代

def stopCriterion(type, value, threshold):
    #設定三種不同的停止策略
    if type == STOP_ITER:        return value > threshold
    elif type == STOP_COST:      return abs(value[-1]-value[-2]) < threshold
    elif type == STOP_GRAD:      return np.linalg.norm(value) < threshold

import numpy.random
#對數據進行洗牌，打亂原有數據的規律
def shuffleData(data):
    np.random.shuffle(data)
    cols = data.shape[1]
    X = data[:, 0:cols-1]
    y = data[:, cols-1:]
    return X, y

import time

# stopType停止策略；thresh預值；alpha學習率
def descent(data, theta, batchSize, stopType, thresh, alpha):
    #梯度下降求解
    
    init_time = time.time()
    i = 0 # 迭代次數
    k = 0 # batch
    X, y = shuffleData(data)
    grad = np.zeros(theta.shape) # 計算的梯度
    costs = [cost(X, y, theta)] # 損失值

    
    while True:
        grad = gradient(X[k:k+batchSize], y[k:k+batchSize], theta)
        k += batchSize #取batch數量個數據
        if k >= n: 
            k = 0 
            X, y = shuffleData(data) #重新洗牌
        theta = theta - alpha*grad # 參數更新
        costs.append(cost(X, y, theta)) # 計算新的損失
        i += 1 

        if stopType == STOP_ITER:       value = i
        elif stopType == STOP_COST:     value = costs
        elif stopType == STOP_GRAD:     value = grad
        if stopCriterion(stopType, value, thresh): break
    
    return theta, i-1, costs, grad, time.time() - init_time

def runExpe(data, theta, batchSize, stopType, thresh, alpha):
    #import pdb; pdb.set_trace();
    theta, iter, costs, grad, dur = descent(data, theta, batchSize, stopType, thresh, alpha)
    name = "Original" if (data[:,1]>2).sum() > 1 else "Scaled"
    name += " data - learning rate: {} - ".format(alpha)
    if batchSize==n: strDescType = "Gradient"
    elif batchSize==1:  strDescType = "Stochastic"
    else: strDescType = "Mini-batch ({})".format(batchSize)
    name += strDescType + " descent - Stop: "
    if stopType == STOP_ITER: strStop = "{} iterations".format(thresh)
    elif stopType == STOP_COST: strStop = "costs change < {}".format(thresh)
    else: strStop = "gradient norm < {}".format(thresh)
    name += strStop
    print ("***{}\nTheta: {} - Iter: {} - Last cost: {:03.2f} - Duration: {:03.2f}s".format(
        name, theta, iter, costs[-1], dur))
    fig, ax = plt.subplots(figsize=(12,4))
    ax.plot(np.arange(len(costs)), costs, 'r')
    ax.set_xlabel('Iterations')
    ax.set_ylabel('Cost')
    ax.set_title(name.upper() + ' - Error vs. Iteration')
    return theta

 

6.不同的停止策略

1).設定迭代次數

#選擇的梯度下降方法是基於所有樣本的
n=100
runExpe(orig_data, theta, n, STOP_ITER, thresh=5000, alpha=0.000001)

2).根據損失值停止

設定閾值 1E-6, 差不多需要110 000次迭代

runExpe(orig_data, theta, n, STOP_COST, thresh=0.000001, alpha=0.001)

 

 3).根據梯度變化停止 

設定閾值 0.05,差不多需要40 000次迭代

runExpe(orig_data, theta, n, STOP_GRAD, thresh=0.05, alpha=0.001)

 

 7.對比不同的梯度下降方法

 1).Stochastic descent

runExpe(orig_data, theta, 1, STOP_ITER, thresh=5000, alpha=0.001)

 

2).以上的結果：有點爆炸。。。很不穩定,再來試試把學習率調小一些

runExpe(orig_data, theta, 1, STOP_ITER, thresh=15000, alpha=0.000002)

速度快，但穩定性差，需要很小的學習率

3).Mini-batch descent

runExpe(orig_data, theta, 16, STOP_ITER, thresh=15000, alpha=0.001)

4).使用skleaning，進行數據預處理

from sklearn import preprocessing as pp

scaled_data = orig_data.copy()
scaled_data[:, 1:3] = pp.scale(orig_data[:, 1:3])

runExpe(scaled_data, theta, n, STOP_ITER, thresh=5000, alpha=0.001)

5).

runExpe(scaled_data, theta, n, STOP_GRAD, thresh=0.02, alpha=0.001)

6).

theta = runExpe(scaled_data, theta, 1, STOP_GRAD, thresh=0.002/5, alpha=0.001)

7).

runExpe(scaled_data, theta, 16, STOP_GRAD, thresh=0.002*2, alpha=0.001)

8.得到精度

#設定閾值
def predict(X, theta):
    return [1 if x >= 0.5 else 0 for x in model(X, theta)]

scaled_X = scaled_data[:, :3]
y = scaled_data[:, 3]
predictions = predict(scaled_X, theta)
correct = [1 if ((a == 1 and b == 1) or (a == 0 and b == 0)) else 0 for (a, b) in zip(predictions, y)]
accuracy = (sum(map(int, correct)) % len(correct))
print ('accuracy = {0}%'.format(accuracy))

accuracy = 89%