機器學習----線性迴歸原理---最下二乘法和梯度下降怎麼來的-----專案預測大學生是否被錄取程式碼案例
阿新 • • 發佈：2018-12-24
這節課說明了 最下二乘法 是怎麼來的。
接下來是面試需要問的
誤差，（機器學習是建立在獨立同分布的基礎上，事實上，根本無法證明獨立同分布而且是正態分佈，我們假設的，只要模型可用，就可以）
獨立： 每個人的誤差是獨立同分布的，如何不獨立那就說明有關係了，黑關係，不能考慮這個，
同分布：假定是同一家銀行，如果不是同一家銀行，不成立。
高斯分佈：就是正態分佈 大塊面積處在-1 ，1 之間。
似然函式（就是根據資料樣本預測引數值的），這裡就是讓我的預測值與接近真實值，換句話說誤差的可能性越小越好
接下來推到最下二乘法：（矩陣求導怎麼做？）
梯度下降法：
ligistic regression
現實中，往往先拿邏輯迴歸試試效果怎樣，在去試試其他的演算法，優先選擇的演算法、
大學生按成績是否被錄取案例，程式碼分析
''' ''' # %matplotlib inline ：表示在notebook中不需要plt.show()即可顯示影象，在pycharm中不行，報錯 pdData = pd.read_csv('LogiReg_data.txt', header=None, names=['Exam 1', 'Exam 2', 'Admitted']) ''' 我們發現讀取這個的檔案，第一行竟然是資料，沒有列名，因此讓header=none，自己指定列名 names=['Exam 1', 'Exam 2', 'Admitted']，第三列位標籤項，1：錄取；0：沒有被錄取 ''' positive = pdData[pdData['Admitted'] == 1] negative = pdData[pdData['Admitted'] == 0] ''' pdData['Admitted'] == 1 ：表示返回一個100x1的矩陣，元素為true或者false positive = pdData[pdData['Admitted'] == 1] ：則是讓true為索引的行全部拿出來 negative 同理 ''' fig, ax = plt.subplots(figsize=(10,5)) ax.scatter(positive['Exam 1'], positive['Exam 2'], s=30, c='b', marker='o', label='Admitted') ''' subplots：建立一個子圖，大小（10,5），接下來畫錄取和沒有被錄取的散點圖，做好標記 '''
def model(X, theta):
return sigmoid(np.dot(X, theta.T))
'''
np.dot(X, theta.T) X:為100x3 的矩陣，theta為1x3的矩陣，沒毛病
'''pdData.insert(0, 'Ones', 1)
'''
在第一列的前面插上一列，列名為one 值為1
'''
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]
'''
他這麼寫，感覺太麻煩了不就是把x和y分離嗎，
'''
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))
'''
multiply(a,b)就是個乘法，如果a,b是兩個陣列，那麼對應元素相乘
'''
def gradient(X, y, theta):
grad = np.zeros(theta.shape) #1x3的矩陣
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)
'''
我們要求 theta的偏導數，這裡面有三個theta，因此求三個
j ：代表每一列
----------------------------------------------------------
>>> x.ravel()
array([1, 2, 3, 4])
兩者預設均是行序優先
>>> x.flatten('F')
array([1, 3, 2, 4])
>>> x.ravel('F')
array([1, 3, 2, 4])
---------------------------------------------------------------
np.sum(term) #這裡term是矩陣怎麼求和呢，預設是把矩陣所有元素相加
np.sum 和Python自帶的sum不一樣，真可氣
np.sum(X,axis=1) 每一行相加 axis=0就是縱向相加
'''
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 # 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
return theta, i - 1, costs, grad, time.time() - init_time
'''
核心程式碼就這
theta, iter, costs, grad, dur = descent(data, theta, batchSize, stopType, thresh, alpha)
其他都是修飾
'''
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))
#--------------------------上面是列印結果，下面是畫圖
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')
plt.show()
return theta'''
開始執行程式
'''
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)全部程式碼
'''’
我們將建立一個邏輯迴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資料，
你可以用它作為邏輯迴歸的訓練集。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
'''
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
# %matplotlib inline
import os
path = 'data' + os.sep + 'LogiReg_data.txt'
pdData = pd.read_csv('LogiReg_data.txt', header=None, names=['Exam 1', 'Exam 2', 'Admitted'])
# print(pdData.shape)
# pdData.head()
positive = pdData[pdData['Admitted'] == 1] # returns the subset of rows such Admitted = 1, i.e. the set of *positive* examples
negative = pdData[pdData['Admitted'] == 0] # returns the subset of rows such Admitted = 0, i.e. the set of *negative* examples
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')
# plt.show()
def sigmoid(z):
return 1 / (1 + np.exp(-z))
nums = np.arange(-10, 10, step=1) #creates a vector containing 20 equally spaced values from -10 to 10
fig, ax = plt.subplots(figsize=(12,4))
ax.plot(nums, sigmoid(nums), 'r')
def model(X, theta):
return sigmoid(np.dot(X, theta.T))
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)
theta = np.zeros([1, 3])
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(X,y,theta)
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):
#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
#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
import time
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 # 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
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))
#--------------------------上面是列印結果，下面是畫圖
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')
plt.show()
return theta
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)
#