似然函数的定义:给定联合样本值X下关于(未知)参数 的函数
似然函数:什么样的参数跟我们的数据组合后恰好是真实值
对数似然:
(误差的表达式,我们的目的就是使得真实值与预测值之前的误差最小)
(导数为0取得极值,得到函数的参数)
逻辑回归是在线性回归的结果外加一层Sigmoid函数
前提数据服从伯努利分布
对数似然:
引入 转变为梯度下降任务,逻辑回归目标函数
我的理解就是求导更新参数,达到一定条件后停止,得到近似最优解
Sigmoid函数
def sigmoid(z): return 1 / (1 + np.exp(-z))
预测函数
def model(X, theta): return sigmoid(np.dot(X, theta.T))
目标函数
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))
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())): #for each parmeter term = np.multiply(error, X[:,j]) grad[0, j] = np.sum(term) / len(X) return grad
梯度下降停止策略
STOP_ITER = 0 STOP_COST = 1 STOP_GRAD = 2 def stopCriterion(type, value, threshold): # 设定三种不同的停止策略 if type == STOP_ITER: # 设定迭代次数 return value > threshold elif type == STOP_COST: # 根据损失值停止 return abs(value[-1] - value[-2]) < threshold elif type == STOP_GRAD: # 根据梯度变化停止 return np.linalg.norm(value) < threshold
样本重新洗牌
import numpy.random #洗牌 def shuffleData(data): np.random.shuffle(data) cols = data.shape[1] X = data[:, 0:cols-1] y = data[:, cols-1:] return X, y
梯度下降求解
def descent(data, theta, batchSize, stopType, thresh, alpha): # 梯度下降求解 init_time = time.time() i = 0 # 迭代次数 k = 0 # batch X, y = shuffleData(data) grad = np.zeros(theta.shape) # 计算的梯度 costs = [cost(X, y, theta)] # 损失值 while True: grad = gradient(X[k:k + batchSize], y[k:k + batchSize], theta) k += batchSize # 取batch数量个数据 if k >= n: k = 0 X, y = shuffleData(data) # 重新洗牌 theta = theta - alpha * grad # 参数更新 costs.append(cost(X, y, theta)) # 计算新的损失 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
import numpy as np import pandas as pd import matplotlib.pyplot as plt import os import numpy.random import time def sigmoid(z): return 1 / (1 + np.exp(-z)) def model(X, theta): return sigmoid(np.dot(X, theta.T)) 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)) 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 STOP_ITER = 0 STOP_COST = 1 STOP_GRAD = 2 def stopCriterion(type, value, threshold): # 设定三种不同的停止策略 if type == STOP_ITER: # 设定迭代次数 return value > threshold elif type == STOP_COST: # 根据损失值停止 return abs(value[-1] - value[-2]) < threshold elif type == STOP_GRAD: # 根据梯度变化停止 return np.linalg.norm(value) < threshold # 洗牌 def shuffleData(data): np.random.shuffle(data) cols = data.shape[1] X = data[:, 0:cols - 1] y = data[:, cols - 1:] return X, y def descent(data, theta, batchSize, stopType, thresh, alpha): # 梯度下降求解 init_time = time.time() i = 0 # 迭代次数 k = 0 # batch X, y = shuffleData(data) grad = np.zeros(theta.shape) # 计算的梯度 costs = [cost(X, y, theta)] # 损失值 while True: grad = gradient(X[k:k + batchSize], y[k:k + batchSize], theta) k += batchSize # 取batch数量个数据 if k >= n: k = 0 X, y = shuffleData(data) # 重新洗牌 theta = theta - alpha * grad # 参数更新 costs.append(cost(X, y, theta)) # 计算新的损失 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" # 批量梯度下降 elif batchSize == 1: strDescType = "Stochastic" # 随机梯度下降 else: strDescType = "Mini-batch ({})".format(batchSize) # 小批量梯度下降 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 path = 'data' + os.sep + 'LogiReg_data.txt' pdData = pd.read_csv(path, header=None, names=['Exam 1', 'Exam 2', 'Admitted']) positive = pdData[pdData['Admitted'] == 1] negative = pdData[pdData['Admitted'] == 0] # 画图观察样本情况 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='Not Admitted') ax.legend() ax.set_xlabel('Exam 1 Score') ax.set_ylabel('Exam 2 Score') pdData.insert(0, 'Ones', 1) # 划分训练数据与标签 orig_data = pdData.values cols = orig_data.shape[1] X = orig_data[:, 0:cols - 1] y = orig_data[:, cols - 1:cols] # 设置初始参数0 theta = np.zeros([1, 3]) # 选择的梯度下降方法是基于所有样本的 n = 100 runExpe(orig_data, theta, n, STOP_ITER, thresh=5000, alpha=0.000001) runExpe(orig_data, theta, n, STOP_COST, thresh=0.000001, alpha=0.001) runExpe(orig_data, theta, n, STOP_GRAD, thresh=0.05, alpha=0.001) runExpe(orig_data, theta, 1, STOP_ITER, thresh=5000, alpha=0.001) runExpe(orig_data, theta, 1, STOP_ITER, thresh=15000, alpha=0.000002) runExpe(orig_data, theta, 16, STOP_ITER, thresh=15000, alpha=0.001) from sklearn import preprocessing as pp # 数据预处理 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) runExpe(scaled_data, theta, n, STOP_GRAD, thresh=0.02, alpha=0.001) theta = runExpe(scaled_data, theta, 1, STOP_GRAD, thresh=0.002 / 5, alpha=0.001) runExpe(scaled_data, theta, 16, STOP_GRAD, thresh=0.002 * 2, alpha=0.001) # 设定阈值 def predict(X, theta): return [1 if x >= 0.5 else 0 for x in model(X, theta)] # 计算精度 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))
形式简单,模型的可解释性非常好。从特征的权重可以看到不同的特征对最后结果的影响,某个特征的权重值比较高,那么这个特征最后对结果的影响会比较大。
模型效果不错。在工程上是可以接受的(作为baseline),如果特征工程做的好,效果不会太差,并且特征工程可以大家并行开发,大大加快开发的速度。
训练速度较快。分类的时候,计算量仅仅只和特征的数目相关。并且逻辑回归的分布式优化sgd发展比较成熟,训练的速度可以通过堆机器进一步提高,这样我们可以在短时间内迭代好几个版本的模型。
资源占用小,尤其是内存。因为只需要存储各个维度的特征值。
方便输出结果调整。逻辑回归可以很方便的得到最后的分类结果,因为输出的是每个样本的概率分数,我们可以很容易的对这些概率分数进行cutoff,也就是划分阈值(大于某个阈值的是一类,小于某个阈值的是一类)。
准确率并不是很高。因为形式非常的简单(非常类似线性模型),很难去拟合数据的真实分布。
很难处理数据不平衡的问题。举个例子:如果我们对于一个正负样本非常不平衡的问题比如正负样本比 10000:1.我们把所有样本都预测为正也能使损失函数的值比较小。但是作为一个分类器,它对正负样本的区分能力不会很好。
处理非线性数据较麻烦。逻辑回归在不引入其他方法的情况下,只能处理线性可分的数据,或者进一步说,处理二分类的问题 。
逻辑回归本身无法筛选特征。有时候,我们会用gbdt来筛选特征,然后再上逻辑回归。
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