基於最小二乘法的一般多元線性迴歸的實戰

一、說明

我是在jupyter完成的，然後導出成markdown格式，ipynb文件導出爲markdown的命令如下：

jupyter nbconvert --to markdown 

源代碼和數據文件，點擊這裏獲取

二、數據項說明

	Name		Data Type	Meas.	Description
	----		---------	-----	-----------
	Sex		nominal			M, F, and I (infant)
	Length		continuous	mm	Longest shell measurement
	Diameter	continuous	mm	perpendicular to length
	Height		continuous	mm	with meat in shell
	Whole weight	continuous	grams	whole abalone
	Shucked weight	continuous	grams	weight of meat
	Viscera weight	continuous	grams	gut weight (after bleeding)
	Shell weight	continuous	grams	after being dried
	Rings		integer			+1.5 gives the age in years

現在有8個數據字段，前面7個是特徵值，最最後一個Rings爲預測，具體請查閱文件內容

三、實戰部分

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

dataframe01 = pd.read_excel('abalone.xlsx', sheet_name='data')

dataframe01.head(10)

	Sex	Length	Diameter	Height	Whole weight	Shucked weight	Viscera weight	Shell weight	Rings
0	M	0.455	0.365	0.095	0.5140	0.2245	0.1010	0.150	15
1	M	0.350	0.265	0.090	0.2255	0.0995	0.0485	0.070	7
2	F	0.530	0.420	0.135	0.6770	0.2565	0.1415	0.210	9
3	M	0.440	0.365	0.125	0.5160	0.2155	0.1140	0.155	10
4	I	0.330	0.255	0.080	0.2050	0.0895	0.0395	0.055	7
5	I	0.425	0.300	0.095	0.3515	0.1410	0.0775	0.120	8
6	F	0.530	0.415	0.150	0.7775	0.2370	0.1415	0.330	20
7	F	0.545	0.425	0.125	0.7680	0.2940	0.1495	0.260	16
8	M	0.475	0.370	0.125	0.5095	0.2165	0.1125	0.165	9
9	F	0.550	0.440	0.150	0.8945	0.3145	0.1510	0.320	19

# 查看數據容量 
dataframe01.shape

(4177, 9)

dataframe01.columns # 特徵名字

Index(['Sex', 'Length', 'Diameter', 'Height', 'Whole weight', 'Shucked weight',
       'Viscera weight', 'Shell weight', 'Rings'],
      dtype='object')

# 清洗數據
# 替換特徵值，將性別中的字符類型轉化爲整數
dataframe02 = dataframe01.copy()

dataframe02.Sex[dataframe01['Sex']=='I']=0
dataframe02.Sex[dataframe01['Sex']=='F']=1
dataframe02.Sex[dataframe01['Sex']=='M']=2

dataframe02.head(10)

	Sex	Length	Diameter	Height	Whole weight	Shucked weight	Viscera weight	Shell weight	Rings
0	2	0.455	0.365	0.095	0.5140	0.2245	0.1010	0.150	15
1	2	0.350	0.265	0.090	0.2255	0.0995	0.0485	0.070	7
2	1	0.530	0.420	0.135	0.6770	0.2565	0.1415	0.210	9
3	2	0.440	0.365	0.125	0.5160	0.2155	0.1140	0.155	10
4	0	0.330	0.255	0.080	0.2050	0.0895	0.0395	0.055	7
5	0	0.425	0.300	0.095	0.3515	0.1410	0.0775	0.120	8
6	1	0.530	0.415	0.150	0.7775	0.2370	0.1415	0.330	20
7	1	0.545	0.425	0.125	0.7680	0.2940	0.1495	0.260	16
8	2	0.475	0.370	0.125	0.5095	0.2165	0.1125	0.165	9
9	1	0.550	0.440	0.150	0.8945	0.3145	0.1510	0.320	19

# 導入線性迴歸的庫
from sklearn.linear_model import LinearRegression as LR
from sklearn.model_selection import train_test_split
from sklearn.model_selection import cross_val_score

data_index = list(dataframe01.columns)

data_index

['Sex',
 'Length',
 'Diameter',
 'Height',
 'Whole weight',
 'Shucked weight',
 'Viscera weight',
 'Shell weight',
 'Rings']

# 獲取特徵矩陣X 的index
X_index = data_index[0:-1]
Y_index = data_index[-1]

X_index, Y_index

(['Sex',
  'Length',
  'Diameter',
  'Height',
  'Whole weight',
  'Shucked weight',
  'Viscera weight',
  'Shell weight'],
 'Rings')

X = dataframe02[X_index]

X.head()

	Sex	Length	Diameter	Height	Whole weight	Shucked weight	Viscera weight	Shell weight
0	2	0.455	0.365	0.095	0.5140	0.2245	0.1010	0.150
1	2	0.350	0.265	0.090	0.2255	0.0995	0.0485	0.070
2	1	0.530	0.420	0.135	0.6770	0.2565	0.1415	0.210
3	2	0.440	0.365	0.125	0.5160	0.2155	0.1140	0.155
4	0	0.330	0.255	0.080	0.2050	0.0895	0.0395	0.055

Y = dataframe02[Y_index]

Y.head()

0    15
1     7
2     9
3    10
4     7
Name: Rings, dtype: int64

# 劃分訓練集和測試集
Xtrain, Xtest, Ytrain, Ytest = train_test_split(X,Y,test_size=0.2,random_state=420)

Xtrain.head()

	Sex	Length	Diameter	Height	Whole weight	Shucked weight	Viscera weight	Shell weight
2763	0	0.550	0.425	0.135	0.6560	0.2570	0.1700	0.203
439	2	0.500	0.415	0.165	0.6885	0.2490	0.1380	0.250
1735	2	0.670	0.520	0.165	1.3900	0.7110	0.2865	0.300
751	2	0.485	0.355	0.120	0.5470	0.2150	0.1615	0.140
1626	1	0.570	0.450	0.135	0.7805	0.3345	0.1850	0.210

Ytrain.head()

2763    10
439     13
1735    11
751     10
1626     8
Name: Rings, dtype: int64

#恢復索引
for i in [Xtrain, Xtest]:
    i.index = range(i.shape[0])

#恢復索引
for i in [Ytrain, Ytest]:
    i.index = range(i.shape[0])

Xtrain.head()   # 查看X訓練集頭部

	Sex	Length	Diameter	Height	Whole weight	Shucked weight	Viscera weight	Shell weight
0	0	0.550	0.425	0.135	0.6560	0.2570	0.1700	0.203
1	2	0.500	0.415	0.165	0.6885	0.2490	0.1380	0.250
2	2	0.670	0.520	0.165	1.3900	0.7110	0.2865	0.300
3	2	0.485	0.355	0.120	0.5470	0.2150	0.1615	0.140
4	1	0.570	0.450	0.135	0.7805	0.3345	0.1850	0.210

Ytrain.head()

0    10
1    13
2    11
3    10
4     8
Name: Rings, dtype: int64

# 先用訓練集訓練(fit)標準化的類，然後用訓練好的類分別轉化(transform)訓練集和測試集

# 開始建模
reg = LR().fit(Xtrain, Ytrain)

yhat = reg.predict(Xtest) #預測我們的yhat

yhat.min()

4.22923686878166

yhat.max()

22.656846035572762

reg.coef_ # w,係數向量

array([  0.40527178,  -0.88791132,  13.01662939,  10.39250886,
         9.64127293, -20.87747601, -10.50683081,   7.70632772])

Xtrain.columns

Index(['Sex', 'Length', 'Diameter', 'Height', 'Whole weight', 'Shucked weight',
       'Viscera weight', 'Shell weight'],
      dtype='object')

[*zip(Xtrain.columns,reg.coef_)]

[('Sex', 0.4052717783379893),
 ('Length', -0.8879113179582045),
 ('Diameter', 13.016629389061475),
 ('Height', 10.39250886428478),
 ('Whole weight', 9.64127293101552),
 ('Shucked weight', -20.87747600529615),
 ('Viscera weight', -10.506830809919672),
 ('Shell weight', 7.706327719866024)]

# 特徵說明

Name Data Type Meas. Description

---

​ Sex nominal M, F, and I (infant)
​ Length continuous mm Longest shell measurement
​ Diameter continuous mm perpendicular to length
​ Height continuous mm with meat in shell
​ Whole weight continuous grams whole abalone
​ Shucked weight continuous grams weight of meat
​ Viscera weight continuous grams gut weight (after bleeding)
​ Shell weight continuous grams after being dried
​ Rings integer +1.5 gives the age in years

# 截距
reg.intercept_

2.7888240054011835

# 自定義最小二乘法嘗試
def my_least_squares(x_array, y_array):
    '''
    :param x: 列表，表示m*n矩陣
    :param y: 列表，表示m*1矩陣
    :return: coef：list 迴歸係數(1*n矩陣)   intercept: float 截距
    '''
    # 矩陣對象化
    arr_x_01 = np.array(x_array)
    arr_y_01 = np.array(y_array)

    # x_array由 m*n矩陣轉化爲 m*(n+1)矩陣，其中第n+1列係數全爲1
    # 獲取行數
    row_num = arr_x_01.shape[0]

    # 生成常量係數矩陣  m*1矩陣
    arr_b = np.array([[1 for i in range(0, row_num)]])

    # 合併成m*(n+1)矩陣
    arr_x_02 = np.insert(arr_x_01, 0, values=arr_b, axis=1)

    # 矩陣運算
    w = np.linalg.inv(np.matmul(arr_x_02.T, arr_x_02))
    w = np.matmul(w, arr_x_02.T)
    w = np.matmul(w, arr_y_01)
    
    # w爲1*(n+1)矩陣
    # print(w)
    result = list(w)
    coef = result.pop(-1)
    intercept = result
    
    return coef, intercept

# debug中
my_least_squares(Xtrain,list(Ytrain))

# 梯度下降法嘗試
def costFunc(X,Y,theta):
    '''
    代價函數
    '''
    inner = np.power((X*theta.T)-Y,2)
    return np.sum(inner)/(2*len(X))

def gradientDescent(X,Y,theta,alpha,iters):
    '''
    梯度下降
    '''
    temp = np.mat(np.zeros(theta.shape))
    cost = np.zeros(iters)
    thetaNums = int(theta.shape[1])
    print(thetaNums)
    for i in range(iters):
        error = (X*theta.T-Y)
        for j in range(thetaNums):
            derivativeInner = np.multiply(error,X[:,j])
            temp[0,j] = theta[0,j] - (alpha*np.sum(derivativeInner)/len(X))

        theta = temp
        cost[i] = costFunc(X,Y,theta)

    return theta,cost