例項：梯度下降求解邏輯迴歸

此例項是關於評估打算留學的學生的兩個考試成績後，是否被錄取的資料。話不多說，直接上乾貨，例項用到的資料檔案看尾部的百度雲連結

# -*- coding: utf-8 -*-
"""
Created on Wed Oct  3 23:03:33 2018

@author: xxx
"""
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

#由於LogiReg_data.txt檔案第一行不是列名，所以不讓header=None，自己定義列名names
pdData = pd.read_csv("LogiReg_data.txt", header=None, names=['Exam 1','Exam 2','Admitted'])

"""#可以把資料畫在座標軸上看一下
positive = pdData[pdData['Admitted'] == 1]
negative = pdData[pdData['Admitted'] == 0]

fig, ax = plt.subplots(figsize=(10,5))
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='Admitted')
ax.legend()
ax.set_xlabel('Exam 1 Score')
ax.set_ylabel('Exam 2 Score')
"""
#sigmoid函式
def sigmoid(z):
    return 1/(1 + np.exp(-z))

"""可以看一下這些資料的sigmoid函式影象
nums = np.arange(-10, 10, step=1)
fig, ax = plt.subplots(figsize=(12,4))
ax.plot(nums, sigmoid(nums), 'r')
"""
def model(X, theta): #向量版sigmoid函式
    return sigmoid(np.dot(X, theta.T)) 

pdData.insert(0, 'Ones', 1) #往X中第一列插入1
#print(pdData)
orig_data = pdData.as_matrix() #除去了表名與索引
#print(orig_data)
cols = orig_data.shape[1] #檢視orig_data的列數，返回cols=4
#print(cols)
X = orig_data[:,0:cols-1]
y = orig_data[:, cols-1:cols]
theta = np.zeros([1, 3]) #構造全零theta

#損失函式
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))
#print(cost(X, y, theta))
#print(len(X))
#print(X)

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

#不同梯度下降方法，停止策略
STOP_ITER = 0 #按照次數進行停止
STOP_COST = 1 #根據前後差異，若很小就可以停止
STOP_GRAD = 2
def stopCriterion(type, value, threshold):   #threshold是閾值
    if type == STOP_ITER:    return value > threshold 
    elif type == STOP_COST:    return abs(value[-1]-value[-2]) < threshold #value[-2]表示倒數第二個元素
    elif type == STOP_GRAD:    return np.linalg.norm(value) < threshold #求範數
        
import numpy.random
#洗牌,重新打亂資料
def shuffleData(data):
    np.random.shuffle(data)
    cols = data.shape[1]
    X = data[:, 0:cols-1]
    y = data[:, cols-1:cols] #####
    return X, y

import time
#梯度下降求解
def descent(data, theta, batchSize, stopType, thresh, alpha):
    init_time = time.time()
    i = 0 #迭代次數
    k = 0 #batch
    X, y = shuffleData(data)
    grad = np.zeros(theta.shape) #計算梯度
    costs = [cost(X, y, theta)] #損失值
    print('1')
    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 #引數更新
        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
    print("10")
    return theta, i-1, costs, grad, time.time() - init_time

#計算並畫圖
def runExpe(data, theta, batchSize, stopType, thresh, alpha):
    theta, iter, costs, grad, dur = descent(data, theta, batchSize, stopType, thresh, alpha) #初始化
    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. Iterations')
    return theta

n=100 #基於所有樣本，即全部100個數據
#根據迭代次數停止，這裡設定了5000次
#runExpe(orig_data, theta, n, STOP_ITER, thresh=5000, alpha=0.000001)

#根據損失值停止
runExpe(orig_data, theta, n, STOP_COST, thresh=0.000001, alpha=0.001)

#根據梯度變化停止
#runExpe(orig_data, theta, n, STOP_GRAD, thresh=0.05, alpha=0.001)

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

predictions = predict(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))

#如果再加上資料預處理，精度會好很多，90%左右





 

結果圖：

用到的LogiReg_data.txt檔案：

連結：https://pan.baidu.com/s/1gHj8M5HcHjZ9QnFdiw6E9g
提取碼：kwux