目錄
- 概述
- MSE
- 線性回歸
- 線性回歸實(shí)現(xiàn)
- 計(jì)算 MSE
- 梯度下降
- 迭代訓(xùn)練
- 主函數(shù)
- 完整代碼
概述
線性回歸 (Linear Regression) 是利用回歸分析來確定兩種或兩種以上變量間相互依賴的定量關(guān)系.
對(duì)線性回歸還不是很了解的同學(xué)可以看一下這篇文章:
python深度總結(jié)線性回歸
MSE
均方誤差 (Mean Square Error): 是用來描述連續(xù)誤差的一種方法. 公式:
y_predict: 我們預(yù)測(cè)的值y_real: 真實(shí)值
線性回歸
公式
w: weight, 權(quán)重系數(shù)
b: bias, 偏置頂
x: 特征值
y: 預(yù)測(cè)值
梯度下降
梯度下降 (Gradient Descent) 是一種優(yōu)化算法. 參數(shù)會(huì)沿著梯度相反的方向前進(jìn), 以實(shí)現(xiàn)損失函數(shù) (loss function) 的最小化.
計(jì)算公式:
w: weight, 權(quán)重參數(shù)
w': 更新后的 weight
lr : learning rate, 學(xué)習(xí)率
dloss/dw: 損失函數(shù)對(duì) w 求導(dǎo)
w: weight, 權(quán)重參數(shù)
w': 更新后的 weight
lr : learning rate, 學(xué)習(xí)率
dloss/dw: 損失函數(shù)對(duì) b 求導(dǎo)
線性回歸實(shí)現(xiàn)
計(jì)算 MSE
def calculate_MSE(w, b, points):
"""
計(jì)算誤差MSE
:param w: weight, 權(quán)重
:param b: bias, 偏置頂
:param points: 數(shù)據(jù)
:return: 返回MSE (Mean Square Error)
"""
total_error = 0 # 存放總誤差, 初始化為0
# 遍歷數(shù)據(jù)
for i in range(len(points)):
# 取出x, y
x = points.iloc[i, 0] # 第一列
y = points.iloc[i, 1] # 第二列
# 計(jì)算MSE
total_error += (y - (w * x + b)) ** 2 # 計(jì)總誤差
MSE = total_error / len(points) # 計(jì)算平均誤差
# 返回MSE
return MSE
梯度下降
def step_gradient(index, w_current, b_current, points, learning_rate=0.0001):
"""
計(jì)算梯度下降, 跟新權(quán)重
:param index: 現(xiàn)行迭代編號(hào)
:param w_current: weight, 權(quán)重
:param b_current: bias, 偏置頂
:param points: 數(shù)據(jù)
:param learning_rate: lr, 學(xué)習(xí)率 (默認(rèn)值: 0.0001)
:return: 返回跟新過后的參數(shù)數(shù)組
"""
b_gradient = 0 # b的導(dǎo), 初始化為0
w_gradient = 0 # w的導(dǎo), 初始化為0
N = len(points) # 數(shù)據(jù)長度
# 遍歷數(shù)據(jù)
for i in range(len(points)):
# 取出x, y
x = points.iloc[i, 0] # 第一列
y = points.iloc[i, 1] # 第二列
# 計(jì)算w的導(dǎo), w的導(dǎo) = 2x(wx+b-y)
w_gradient += (2 / N) * x * ((w_current * x + b_current) - y)
# 計(jì)算b的導(dǎo), b的導(dǎo) = 2(wx+b-y)
b_gradient += (2 / N) * ((w_current * x + b_current) - y)
# 跟新w和b
w_new = w_current - (learning_rate * w_gradient) # 下降導(dǎo)數(shù)*學(xué)習(xí)率
b_new = b_current - (learning_rate * b_gradient) # 下降導(dǎo)數(shù)*學(xué)習(xí)率
# 每迭代10次, 調(diào)試輸出
if index % 10 == 0:
print("This is the {}th iterations w = {}, b = {}, error = {}"
.format(index, w_new, b_new,
calculate_MSE(w_new, b_new, points)))
# 返回更新后的權(quán)重和偏置頂
return [w_new, b_new]
迭代訓(xùn)練
def runner(w_start, b_start, points, learning_rate, num_iterations):
"""
迭代訓(xùn)練
:param w_start: 初始weight
:param b_start: 初始bias
:param points: 數(shù)據(jù)
:param learning_rate: 學(xué)習(xí)率
:param num_iterations: 迭代次數(shù)
:return: 訓(xùn)練好的權(quán)重和偏執(zhí)頂
"""
# 定義w_end, b_end, 存放返回權(quán)重
w_end = w_start
b_end = b_start
# 更新權(quán)重
for i in range(1, num_iterations + 1):
w_end, b_end = step_gradient(i, w_end, b_end, points, learning_rate)
# 返回訓(xùn)練好的b, w
return [w_end, b_end]
主函數(shù)
def run():
"""
主函數(shù)
:return: 無返回值
"""
# 讀取數(shù)據(jù)
data = pd.read_csv("data.csv")
# 定義超參數(shù)
learning_rate = 0.00001 # 學(xué)習(xí)率
w_initial = 0 # 權(quán)重初始化
b_initial = 0 # 偏置頂初始化
w_end = 0 # 存放返回結(jié)果
b_end = 0 # 存放返回結(jié)果
num_interations = 200 # 迭代次數(shù)
# 調(diào)試輸出初始誤差
print("Starting gradient descent at w = {}, b = {}, error = {}"
.format(w_initial, b_initial, calculate_MSE(w_initial, b_initial, data)))
print("Running...")
# 得到訓(xùn)練好的值
w_end, b_end = runner(w_initial, b_initial, data, learning_rate, num_interations, )
# 調(diào)試輸出訓(xùn)練后的誤差
print("\nAfter {} iterations w = {}, b = {}, error = {}"
.format(num_interations, w_end, b_end, calculate_MSE(w_end, b_end, data)))
完整代碼
import pandas as pd
import tensorflow as tf
def run():
"""
主函數(shù)
:return: 無返回值
"""
# 讀取數(shù)據(jù)
data = pd.read_csv("data.csv")
# 定義超參數(shù)
learning_rate = 0.00001 # 學(xué)習(xí)率
w_initial = 0 # 權(quán)重初始化
b_initial = 0 # 偏置頂初始化
w_end = 0 # 存放返回結(jié)果
b_end = 0 # 存放返回結(jié)果
num_interations = 200 # 迭代次數(shù)
# 調(diào)試輸出初始誤差
print("Starting gradient descent at w = {}, b = {}, error = {}"
.format(w_initial, b_initial, calculate_MSE(w_initial, b_initial, data)))
print("Running...")
# 得到訓(xùn)練好的值
w_end, b_end = runner(w_initial, b_initial, data, learning_rate, num_interations, )
# 調(diào)試輸出訓(xùn)練后的誤差
print("\nAfter {} iterations w = {}, b = {}, error = {}"
.format(num_interations, w_end, b_end, calculate_MSE(w_end, b_end, data)))
def calculate_MSE(w, b, points):
"""
計(jì)算誤差MSE
:param w: weight, 權(quán)重
:param b: bias, 偏置頂
:param points: 數(shù)據(jù)
:return: 返回MSE (Mean Square Error)
"""
total_error = 0 # 存放總誤差, 初始化為0
# 遍歷數(shù)據(jù)
for i in range(len(points)):
# 取出x, y
x = points.iloc[i, 0] # 第一列
y = points.iloc[i, 1] # 第二列
# 計(jì)算MSE
total_error += (y - (w * x + b)) ** 2 # 計(jì)總誤差
MSE = total_error / len(points) # 計(jì)算平均誤差
# 返回MSE
return MSE
def step_gradient(index, w_current, b_current, points, learning_rate=0.0001):
"""
計(jì)算梯度下降, 跟新權(quán)重
:param index: 現(xiàn)行迭代編號(hào)
:param w_current: weight, 權(quán)重
:param b_current: bias, 偏置頂
:param points: 數(shù)據(jù)
:param learning_rate: lr, 學(xué)習(xí)率 (默認(rèn)值: 0.0001)
:return: 返回跟新過后的參數(shù)數(shù)組
"""
b_gradient = 0 # b的導(dǎo), 初始化為0
w_gradient = 0 # w的導(dǎo), 初始化為0
N = len(points) # 數(shù)據(jù)長度
# 遍歷數(shù)據(jù)
for i in range(len(points)):
# 取出x, y
x = points.iloc[i, 0] # 第一列
y = points.iloc[i, 1] # 第二列
# 計(jì)算w的導(dǎo), w的導(dǎo) = 2x(wx+b-y)
w_gradient += (2 / N) * x * ((w_current * x + b_current) - y)
# 計(jì)算b的導(dǎo), b的導(dǎo) = 2(wx+b-y)
b_gradient += (2 / N) * ((w_current * x + b_current) - y)
# 跟新w和b
w_new = w_current - (learning_rate * w_gradient) # 下降導(dǎo)數(shù)*學(xué)習(xí)率
b_new = b_current - (learning_rate * b_gradient) # 下降導(dǎo)數(shù)*學(xué)習(xí)率
# 每迭代10次, 調(diào)試輸出
if index % 10 == 0:
print("This is the {}th iterations w = {}, b = {}, error = {}"
.format(index, w_new, b_new,
calculate_MSE(w_new, b_new, points)))
# 返回更新后的權(quán)重和偏置頂
return [w_new, b_new]
def runner(w_start, b_start, points, learning_rate, num_iterations):
"""
迭代訓(xùn)練
:param w_start: 初始weight
:param b_start: 初始bias
:param points: 數(shù)據(jù)
:param learning_rate: 學(xué)習(xí)率
:param num_iterations: 迭代次數(shù)
:return: 訓(xùn)練好的權(quán)重和偏執(zhí)頂
"""
# 定義w_end, b_end, 存放返回權(quán)重
w_end = w_start
b_end = b_start
# 更新權(quán)重
for i in range(1, num_iterations + 1):
w_end, b_end = step_gradient(i, w_end, b_end, points, learning_rate)
# 返回訓(xùn)練好的b, w
return [w_end, b_end]
if __name__ == "__main__": # 判斷是否為直接運(yùn)行
# 執(zhí)行主函數(shù)
run()
輸出結(jié)果:
Starting gradient descent at w = 0, b = 0, error = 5611.166153823905
Running...
This is the 10th iterations w = 0.5954939346814911, b = 0.011748797759247776, error = 2077.4540105037636
This is the 20th iterations w = 0.9515563561471605, b = 0.018802975867006404, error = 814.0851271130122
This is the 30th iterations w = 1.1644557718428263, b = 0.023050105300353223, error = 362.4068500146176
This is the 40th iterations w = 1.291753898278705, b = 0.02561881917471017, error = 200.92329896151622
This is the 50th iterations w = 1.3678685455519075, b = 0.027183959773995233, error = 143.18984477036037
This is the 60th iterations w = 1.4133791147591803, b = 0.02814903475888354, error = 122.54901023376003
This is the 70th iterations w = 1.4405906232245687, b = 0.028755312994862656, error = 115.16948797045545
This is the 80th iterations w = 1.4568605956220553, b = 0.029147056093611835, error = 112.53113537539161
This is the 90th iterations w = 1.4665883081088924, b = 0.029410522232548166, error = 111.58784050644537
This is the 100th iterations w = 1.4724042147529013, b = 0.029597287663210802, error = 111.25056079777497
This is the 110th iterations w = 1.475881139890538, b = 0.029738191313600983, error = 111.12994295811941
This is the 120th iterations w = 1.477959520545057, b = 0.02985167266801462, error = 111.08678583026905
This is the 130th iterations w = 1.479201671130221, b = 0.029948757225817496, error = 111.07132237076124
This is the 140th iterations w = 1.4799438156483897, b = 0.03003603745100295, error = 111.06575992136905
This is the 150th iterations w = 1.480386992125614, b = 0.030117455167888288, error = 111.06373727064113
This is the 160th iterations w = 1.4806514069946144, b = 0.030195367306897165, error = 111.0629801653088
This is the 170th iterations w = 1.4808089351476725, b = 0.030271183144693698, error = 111.06267551686379
This is the 180th iterations w = 1.4809025526554018, b = 0.030345745328433527, error = 111.0625326308038
This is the 190th iterations w = 1.4809579561496398, b = 0.030419557701150367, error = 111.0624475783524
This is the 200th iterations w = 1.480990510387525, b = 0.030492921525124016, error = 111.06238320300855
This is the 210th iterations w = 1.4810094024003952, b = 0.030566016933760057, error = 111.06232622062124
This is the 220th iterations w = 1.4810201253791957, b = 0.030638951634017437, error = 111.0622718818556
This is the 230th iterations w = 1.4810259638611891, b = 0.030711790026994222, error = 111.06221848873447
This is the 240th iterations w = 1.481028881765914, b = 0.030784570619965538, error = 111.06216543419914
This is the 250th iterations w = 1.4810300533774932, b = 0.030857316437543122, error = 111.06211250121454
This is the 260th iterations w = 1.4810301808342632, b = 0.03093004124680784, error = 111.06205961218657
This is the 270th iterations w = 1.4810296839649824, b = 0.031002753279495907, error = 111.06200673937376
This is the 280th iterations w = 1.4810288137973704, b = 0.031075457457601333, error = 111.06195387285815
This is the 290th iterations w = 1.48102772042814, b = 0.031148156724127858, error = 111.06190100909376
This is the 300th iterations w = 1.4810264936044433, b = 0.03122085283878386, error = 111.06184814681296
This is the 310th iterations w = 1.4810251869886903, b = 0.0312935468537513, error = 111.06179528556238
This is the 320th iterations w = 1.4810238326671836, b = 0.031366239398161695, error = 111.0617424251801
This is the 330th iterations w = 1.4810224498252484, b = 0.031438930848192506, error = 111.06168956560795
This is the 340th iterations w = 1.481021049934344, b = 0.03151162142877266, error = 111.06163670682551
This is the 350th iterations w = 1.4810196398535866, b = 0.03158431127439525, error = 111.06158384882504
This is the 360th iterations w = 1.4810182236842395, b = 0.03165700046547913, error = 111.0615309916041
This is the 370th iterations w = 1.4810168038785667, b = 0.031729689050110664, error = 111.06147813516172
This is the 380th iterations w = 1.4810153819028469, b = 0.03180237705704362, error = 111.06142527949757
This is the 390th iterations w = 1.48101395863381, b = 0.03187506450347233, error = 111.06137242461139
This is the 400th iterations w = 1.48101253459568, b = 0.03194775139967933, error = 111.06131957050317
This is the 410th iterations w = 1.4810111101019028, b = 0.03202043775181446, error = 111.06126671717288
This is the 420th iterations w = 1.4810096853398989, b = 0.032093123563556446, error = 111.06121386462064
This is the 430th iterations w = 1.4810082604217312, b = 0.032165808837106485, error = 111.06116101284626
This is the 440th iterations w = 1.481006835414406, b = 0.03223849357378233, error = 111.06110816184975
This is the 450th iterations w = 1.4810054103579875, b = 0.03231117777437349, error = 111.06105531163115
This is the 460th iterations w = 1.4810039852764323, b = 0.0323838614393536, error = 111.06100246219052
This is the 470th iterations w = 1.4810025601840635, b = 0.032456544569007456, error = 111.0609496135277
This is the 480th iterations w = 1.4810011350894463, b = 0.03252922716350693, error = 111.06089676564281
This is the 490th iterations w = 1.4809997099977015, b = 0.032601909222956374, error = 111.06084391853577
This is the 500th iterations w = 1.4809982849118903, b = 0.032674590747419754, error = 111.0607910722065
After 500 iterations w = 1.4809982849118903, b = 0.032674590747419754, error = 111.0607910722065
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