一般方程和Numpy“最小二乘”、“求解”方法在回归中的差异？

Question

我在做多元线性回归。我试图通过使用正规方程方法（使用矩阵逆）、Numpy最小二乘numpy.linalg.lstsq工具和np.linalg.solve工具来获得θ（系数）。在我的数据中，我有n=143特征和m=13000训练示例。

---

对于正则化的正规方程方法，我使用以下公式：

Sources:

• Regularization (Andrew Ng, Stanford)
• Normal equations (Andrew Ng, Stanford)

正则化用于解决矩阵不可逆的潜在问题（矩阵可能变为奇异/不可逆）

---

数据准备代码：

import pandas as pd
import numpy as np

path = 'DB2.csv'  
data = pd.read_csv(path, header=None, delimiter=";")

data.insert(0, 'Ones', 1)
cols = data.shape[1]

X = data.iloc[:,0:cols-1]  
y = data.iloc[:,cols-1:cols] 

IdentitySize = X.shape[1]
IdentityMatrix= np.zeros((IdentitySize, IdentitySize))
np.fill_diagonal(IdentityMatrix, 1)

---

对于最小二乘法，我使用Numpy的Numpy.linalg.lstsq。下面是Python代码：

lamb = 1
th = np.linalg.lstsq(X.T.dot(X) + lamb * IdentityMatrix, X.T.dot(y))[0]            

我还使用了numpy的np.linalg.solve工具：

lamb = 1
XtX_lamb = X.T.dot(X) + lamb * IdentityMatrix
XtY = X.T.dot(y)
x = np.linalg.solve(XtX_lamb, XtY);

对于正态方程我使用：

lamb = 1
xTx = X.T.dot(X) + lamb * IdentityMatrix
XtX = np.linalg.inv(xTx)
XtX_xT = XtX.dot(X.T)
theta = XtX_xT.dot(y)

---

在所有的方法中我都使用正则化。下面是结果（θ系数），可以看出这三种方法之间的差异：

Normal equation:        np.linalg.lstsq         np.linalg.solve
[-27551.99918303]       [-27551.95276154]       [-27551.9991855]
[-940.27518383]         [-940.27520138]         [-940.27518383]
[-9332.54653964]        [-9332.55448263]        [-9332.54654461]
[-3149.02902071]        [-3149.03496582]        [-3149.02900965]
[-1863.25125909]        [-1863.2631435]         [-1863.25126344]
[-2779.91105618]        [-2779.92175308]        [-2779.91105347]
[-1226.60014026]        [-1226.61033117]        [-1226.60014192]
[-920.73334259]         [-920.74331432]         [-920.73334194]
[-6278.44238081]        [-6278.45496955]        [-6278.44237847]
[-2001.48544938]        [-2001.49566981]        [-2001.48545349]
[-715.79204971]         [-715.79664124]         [-715.79204921]
[ 4039.38847472]        [ 4039.38302499]        [ 4039.38847515]
[-2362.54853195]        [-2362.55280478]        [-2362.54853139]
[-12730.8039209]        [-12730.80866036]       [-12730.80392076]
[-24872.79868125]       [-24872.80203459]       [-24872.79867954]
[-3402.50791863]        [-3402.5140501]         [-3402.50793382]
[ 253.47894001]         [ 253.47177732]         [ 253.47892472]
[-5998.2045186]         [-5998.20513905]        [-5998.2045184]
[ 198.40560401]         [ 198.4049081]          [ 198.4056042]
[ 4368.97581411]        [ 4368.97175688]        [ 4368.97581426]
[-2885.68026222]        [-2885.68154407]        [-2885.68026205]
[ 1218.76602731]        [ 1218.76562838]        [ 1218.7660275]
[-1423.73583813]        [-1423.7369068]         [-1423.73583793]
[ 173.19125007]         [ 173.19086525]         [ 173.19125024]
[-3560.81709538]        [-3560.81650156]        [-3560.8170952]
[-142.68135768]         [-142.68162508]         [-142.6813575]
[-2010.89489111]        [-2010.89601322]        [-2010.89489092]
[-4463.64701238]        [-4463.64742877]        [-4463.64701219]
[ 17074.62997704]       [ 17074.62974609]       [ 17074.62997723]
[ 7917.75662561]        [ 7917.75682048]        [ 7917.75662578]
[-4234.16758492]        [-4234.16847544]        [-4234.16758474]
[-5500.10566329]        [-5500.106558]          [-5500.10566309]
[-5997.79002683]        [-5997.7904842]         [-5997.79002634]
[ 1376.42726683]        [ 1376.42629704]        [ 1376.42726705]
[ 6056.87496151]        [ 6056.87452659]        [ 6056.87496175]
[ 8149.0123667]         [ 8149.01209157]        [ 8149.01236827]
[-7273.3450484]         [-7273.34480382]        [-7273.34504827]
[-2010.61773247]        [-2010.61839251]        [-2010.61773225]
[-7917.81185096]        [-7917.81223606]        [-7917.81185084]
[ 8247.92773739]        [ 8247.92774315]        [ 8247.92773722]
[ 1267.25067823]        [ 1267.24677734]        [ 1267.25067832]
[ 2557.6208133]         [ 2557.62126916]        [ 2557.62081337]
[-5678.53744654]        [-5678.53820798]        [-5678.53744647]
[ 3406.41697822]        [ 3406.42040997]        [ 3406.41697836]
[-8371.23657044]        [-8371.2361594]         [-8371.23657035]
[ 15010.61728285]       [ 15010.61598236]       [ 15010.61728304]
[ 11006.21920273]       [ 11006.21711213]       [ 11006.21920284]
[-5930.93274062]        [-5930.93237071]        [-5930.93274048]
[-5232.84459862]        [-5232.84557665]        [-5232.84459848]
[ 3196.89304277]        [ 3196.89414431]        [ 3196.8930428]
[ 15298.53309912]       [ 15298.53496877]       [ 15298.53309919]
[ 4742.68631183]        [ 4742.6862601]         [ 4742.68631172]
[ 4423.14798495]        [ 4423.14765013]        [ 4423.14798546]
[-16153.50854089]       [-16153.51038489]       [-16153.50854123]
[-22071.50792741]       [-22071.49808389]       [-22071.50792408]
[-688.22903323]         [-688.2310229]          [-688.22904006]
[-1060.88119863]        [-1060.8829114]         [-1060.88120546]
[-101.75750066]         [-101.75776411]         [-101.75750831]
[ 4106.77311898]        [ 4106.77128502]        [ 4106.77311218]
[ 3482.99764601]        [ 3482.99518758]        [ 3482.99763924]
[-1100.42290509]        [-1100.42166312]        [-1100.4229119]
[ 20892.42685103]       [ 20892.42487476]       [ 20892.42684422]
[-5007.54075789]        [-5007.54265501]        [-5007.54076473]
[ 11111.83929421]       [ 11111.83734144]       [ 11111.83928704]
[ 9488.57342568]        [ 9488.57158677]        [ 9488.57341883]
[-2992.3070786]         [-2992.29295891]        [-2992.30708529]
[ 17810.57005982]       [ 17810.56651223]       [ 17810.57005457]
[-2154.47389712]        [-2154.47504319]        [-2154.47390285]
[-5324.34206726]        [-5324.33913623]        [-5324.34207293]
[-14981.89224345]       [-14981.8965674]        [-14981.89224973]
[-29440.90545197]       [-29440.90465897]       [-29440.90545704]
[-6925.31991443]        [-6925.32123144]        [-6925.31992383]
[ 104.98071593]         [ 104.97886085]         [ 104.98071152]
[-5184.94477582]        [-5184.9447972]         [-5184.94477792]
[ 1555.54536625]        [ 1555.54254362]        [ 1555.5453638]
[-402.62443474]         [-402.62539068]         [-402.62443718]
[ 17746.15769322]       [ 17746.15458093]       [ 17746.15769074]
[-5512.94925026]        [-5512.94980649]        [-5512.94925267]
[-2202.8589276]         [-2202.86226244]        [-2202.85893056]
[-5549.05250407]        [-5549.05416936]        [-5549.05250669]
[-1675.87329493]        [-1675.87995809]        [-1675.87329255]
[-5274.27756529]        [-5274.28093377]        [-5274.2775701]
[-5424.10246845]        [-5424.10658526]        [-5424.10247326]
[-1014.70864363]        [-1014.71145066]        [-1014.70864845]
[ 12936.59360437]       [ 12936.59168749]       [ 12936.59359954]
[ 2912.71566077]        [ 2912.71282628]        [ 2912.71565599]
[ 6489.36648506]        [ 6489.36538259]        [ 6489.36648021]
[ 12025.06991281]       [ 12025.07040848]       [ 12025.06990358]
[ 17026.57841531]       [ 17026.56827742]       [ 17026.57841044]
[ 2220.1852193]         [ 2220.18531961]        [ 2220.18521579]
[-2886.39219026]        [-2886.39015388]        [-2886.39219394]
[-18393.24573629]       [-18393.25888463]       [-18393.24573872]
[-17591.33051471]       [-17591.32838012]       [-17591.33051834]
[-3947.18545848]        [-3947.17487999]        [-3947.18546459]
[ 7707.05472816]        [ 7707.05577227]        [ 7707.0547217]
[ 4280.72039079]        [ 4280.72338194]        [ 4280.72038435]
[-3137.48835901]        [-3137.48480197]        [-3137.48836531]
[ 6693.47303443]        [ 6693.46528167]        [ 6693.47302811]
[-13936.14265517]       [-13936.14329336]       [-13936.14267094]
[ 2684.29594641]        [ 2684.29859601]        [ 2684.29594183]
[-2193.61036078]        [-2193.63086307]        [-2193.610366]
[-10139.10424848]       [-10139.11905454]       [-10139.10426049]
[ 4475.11569903]        [ 4475.12288711]        [ 4475.11569421]
[-3037.71857269]        [-3037.72118246]        [-3037.71857265]
[-5538.71349798]        [-5538.71654224]        [-5538.71349794]
[ 8008.38521357]        [ 8008.39092739]        [ 8008.38521361]
[-1433.43859633]        [-1433.44181824]        [-1433.43859629]
[ 4212.47144667]        [ 4212.47368097]        [ 4212.47144686]
[ 19688.24263706]       [ 19688.2451694]        [ 19688.2426368]
[ 104.13434091]         [ 104.13434349]         [ 104.13434091]
[-654.02451175]         [-654.02493111]         [-654.02451174]
[-2522.8642551]         [-2522.88694451]        [-2522.86424254]
[-5011.20385919]        [-5011.22742915]        [-5011.20384655]
[-13285.64644021]       [-13285.66951459]       [-13285.64642763]
[-4254.86406891]        [-4254.88695873]        [-4254.86405637]
[-2477.42063206]        [-2477.43501057]        [-2477.42061727]
[ 0.]                   [  1.23691279e-10]      [ 0.]
[-92.79470071]          [-92.79467095]          [-92.79470071]
[ 2383.66211583]        [ 2383.66209637]        [ 2383.66211583]
[-10725.22892185]       [-10725.22889937]       [-10725.22892185]
[ 234.77560283]         [ 234.77560254]         [ 234.77560283]
[ 4739.22119578]        [ 4739.22121432]        [ 4739.22119578]
[ 43640.05854156]       [ 43640.05848841]       [ 43640.05854157]
[ 2592.3866707]         [ 2592.38671547]        [ 2592.3866707]
[-25130.02819215]       [-25130.05501178]       [-25130.02819515]
[ 4966.82173096]        [ 4966.7946407]         [ 4966.82172795]
[ 14232.97930665]       [ 14232.9529959]        [ 14232.97930363]
[-21621.77202422]       [-21621.79840459]       [-21621.7720272]
[ 9917.80960029]        [ 9917.80960571]        [ 9917.80960029]
[ 1355.79191536]        [ 1355.79198092]        [ 1355.79191536]
[-27218.44185748]       [-27218.46880642]       [-27218.44185719]
[-27218.04184348]       [-27218.06875423]       [-27218.04184318]
[ 23482.80743869]       [ 23482.78043029]       [ 23482.80743898]
[ 3401.67707434]        [ 3401.65134677]        [ 3401.67707463]
[ 3030.36383274]        [ 3030.36384909]        [ 3030.36383274]
[-30590.61847724]       [-30590.63933424]       [-30590.61847706]
[-28818.3942685]        [-28818.41520495]       [-28818.39426833]
[-25115.73726772]       [-25115.7580278]        [-25115.73726753]
[ 77174.61695995]       [ 77174.59548773]       [ 77174.61696016]
[-20201.86613672]       [-20201.88871113]       [-20201.86613657]
[ 51908.53292209]       [ 51908.53446495]       [ 51908.53292207]
[ 7710.71327865]        [ 7710.71324194]        [ 7710.71327865]
[-16206.9785119]        [-16206.97851993]       [-16206.9785119]

正如你所看到的正规方程，最小二乘法和np.linalg.solve工具方法给出了不同程度的结果。问题是，为什么这三种方法给出的结果明显不同，而哪种方法给出的结果更有效和更准确？

假设： 正规方程法的结果与np.linalg.solve的结果非常接近。而np.linalg.lstsq的结果与两者不同。由于正态方程使用逆，所以我们不期望它的结果非常精确，因此np.linalg.求解工具的结果也是如此。似乎np.linalg.lstsq给出了更好的结果。

---

升级版：
正如Dave Hensley所提到的：
在np.fill_diagonal(IdentityMatrix, 1)行之后 应该添加此代码IdentityMatrix[0,0] = 0。

---

DB2.csv可在DropBox:DB2.csv上找到

完整的Python代码可以在DropBox上找到：Full code

Answer 1

正如Matthew Gunn所提到的，计算系数矩阵的显式逆作为解线性方程组的一种方法是不好的。直接获得解（see here）更快更准确。

你之所以看到np.linalg.solve和np.linalg.lstsq之间的差异，是因为这些函数对你试图求解的系统做出不同的假设，并使用不同的数值方法。

• 在引擎盖下，solve调用DGESV LAPACK routine，它使用LU分解，然后向前和向后替换来找到Ax = b的精确解。它要求系统是精确确定的，即A是平方的，并且是满秩的。

• lstsq改为调用DGELSD，它使用A的奇异值分解来找到一个最小二乘解。这也适用于超定和欠定的情况。

如果您的系统已完全确定，那么您应该使用solve，因为它需要更少的浮点操作，因此将更快、更精确。在您的情况下，由于正则化步骤，XtX_lamb被保证是满秩的。

Answer 2

不计算矩阵逆解线性系统

专业的算法不能解决矩阵的逆问题。它很慢，并且引入了不必要的错误。对于小型系统来说，这不是灾难，但为什么要做一些次优的事情呢？

基本上，当你看到数学写为：

x = A^-1 * b

你反而想要：

x = np.linalg.solve(A, b)

在你的情况下，你需要像：

XtX_lamb = X.T.dot(X) + lamb * IdentityMatrix
XtY = X.T.dot(Y)
x = np.linalg.solve(XtX_lamb, XtY);

Answer 3

另一个答案解释了为什么理论上一种计算方法比另一种好。然而，他们没有给出一种方法来测试哪种解决方案实际显示出更好的结果。这里是：

def test(a, x, b):
    res = a.dot(x).as_matrix() - b.as_matrix()
    print(np.linalg.norm(res))

test(XtX_lamb, x, XtY)
test(XtX_lamb, th, XtY)
test(XtX_lamb, theta, XtY)

计算了线性系统误差向量的范数2。 结果是：

np.linalg.solve - 0.000488340357871
np.linalg.lstsq - 1.75520748498
normal equation - 16.1628614202

因此，linalg.solve确实显示了最精确的结果。