【机器学习】回归算法公式推导

短命女 2023-06-09 14:21 21阅读 0赞

### 文章目录 ###

*  回归算法
 *   *  1、least-squares (LS)
     *  2、regularized LS (RLS)
     *  3、L1-regularized LS (LASSO)

# 回归算法 #

Define the following quantities:  
 y = \[ y 1 ⋮ y n \] = \[ y 1 , ⋯   , y n \] T ( n × 1 ) y = \\left\[ \\begin\{array\} \{c\} y\_1 \\\\ \\vdots \\\\ y\_n \\end\{array\}\\right\] =\\left\[y\_\{1\}, \\cdots, y\_\{n\}\\right\]^\{T\} \\quad (n×1) y=⎣⎢⎡y1⋮yn⎦⎥⎤=\[y1,⋯,yn\]T(n×1)

ϕ ( x ) = \[ 1 , x , , x 2 , ⋯   , x K \] T ( k + 1 × 1 ) \\phi(x)= \[1,x,,x^2,\\cdots,x^K\]^T \\quad (k+1×1) ϕ(x)=\[1,x,,x2,⋯,xK\]T(k\+1×1)

Φ = \[ ϕ ( x 1 ) , ⋯   , ϕ ( x n ) \] ( 1 × n ) \\Phi = \\left\[\\phi(x\_1), \\cdots, \\phi(x\_n)\\right\] \\quad (1×n) Φ=\[ϕ(x1),⋯,ϕ(xn)\](1×n)

X = \[ x 1 , ⋯   , x n \] ( 1 × n ) X=\\left\[x\_1, \\cdots,x\_n\\right\] \\quad (1×n) X=\[x1,⋯,xn\](1×n)

θ = \[ θ 0 , ⋯   , θ K \] T ( K × 1 ) \\theta = \[\\theta\_0, \\cdots ,\\theta\_K\]^T \\quad (K×1) θ=\[θ0,⋯,θK\]T(K×1)

## 1、least-squares (LS) ##

objective function  ∥ y − Φ T θ ∥ 2 \\left\\|y-\\Phi^\{T\} \\theta\\right\\|^\{2\} ∥∥y−ΦTθ∥∥2  
 a r g m i n θ ∥ y − Φ T θ ∥ 2 = a r g m i n θ ( y − Φ T θ ) T ( y − Φ T θ ) = a r g m i n θ ( y T − θ T Φ ) ( y − Φ T θ ) = a r g m i n θ \[ y T y − y T Φ T θ − θ T Φ y + θ T Φ Φ T θ \] ( y T Φ T θ 是 实 数 ， 所 以 类 似 A T = A ) = a r g m i n θ \[ y T y − 2 y T Φ T θ + θ T Φ Φ T θ \] \\begin\{aligned\} \\underset\{\\theta\}\{argmin\} \\left\\|y-\\Phi^\{T\} \\theta\\right\\|^\{2\} &= \\underset\{\\theta\}\{argmin\} (y-\\Phi^T\\theta)^T(y-\\Phi^T\\theta) \\\\ &= \\underset\{\\theta\}\{argmin\} (y^T-\\theta^T\\Phi)(y-\\Phi^T\\theta) \\\\ &= \\underset\{\\theta\}\{argmin\} \[y^Ty-y^T\\Phi^T\\theta-\\theta^T\\Phi y+\\theta^T\\Phi\\Phi^T\\theta\] \\\\ (y^T\\Phi^T\\theta是实数，所以类似A^T=A)&= \\underset\{\\theta\}\{argmin\} \[y^Ty-2y^T\\Phi^T\\theta+\\theta^T\\Phi\\Phi^T\\theta\] \\end\{aligned\} θargmin∥∥y−ΦTθ∥∥2(yTΦTθ是实数，所以类似AT=A)=θargmin(y−ΦTθ)T(y−ΦTθ)=θargmin(yT−θTΦ)(y−ΦTθ)=θargmin\[yTy−yTΦTθ−θTΦy\+θTΦΦTθ\]=θargmin\[yTy−2yTΦTθ\+θTΦΦTθ\]  
derivatives  
 ∂ ∂ θ \[ y T y − 2 y T Φ T θ + θ T Φ Φ T θ \] = − 2 Φ y + Φ Φ T θ + θ T Φ Φ T ( 还 是 因 为 实 数 ) = − 2 Φ y + 2 Φ Φ T θ \\begin\{aligned\} \\frac\{\\partial\}\{\\partial\\theta\}\[y^Ty-2y^T\\Phi^T\\theta+\\theta^T\\Phi\\Phi^T\\theta\] & = -2\\Phi y + \\Phi\\Phi^T\\theta + \\theta^T\\Phi\\Phi^T \\\\ (还是因为实数)& = -2\\Phi y +2\\Phi \\Phi^T \\theta \\end\{aligned\} ∂θ∂\[yTy−2yTΦTθ\+θTΦΦTθ\](还是因为实数)=−2Φy\+ΦΦTθ\+θTΦΦT=−2Φy\+2ΦΦTθ  
because  ∂ x T a ∂ x = ∂ a T x ∂ x = a \\frac\{\\partial x^T a\}\{\\partial\{x\}\}=\\frac\{\\partial a^T x\}\{\\partial\{x\}\} = a ∂x∂xTa=∂x∂aTx=a and  ∂ x T = ( ∂ x ) T \\partial x^T= (\\partial x)^T ∂xT=(∂x)T

− 2 Φ y + 2 Φ Φ T θ = 0 ⟹ θ = ( Φ Φ T ) − 1 Φ y -2\\Phi y +2\\Phi \\Phi^T \\theta = 0 \\Longrightarrow \\theta = (\\Phi\\Phi^T)^\{-1\}\\Phi y −2Φy\+2ΦΦTθ=0⟹θ=(ΦΦT)−1Φy  
parameter estimate:  
 θ ^ = ( Φ Φ T ) − 1 Φ y \\hat\{\\theta\} = (\\Phi\\Phi^T)^\{-1\}\\Phi y θ^=(ΦΦT)−1Φy

## 2、regularized LS (RLS) ##

objective function  ∥ y − Φ T θ ∥ 2 + λ ∥ θ ∥ 2 \\left\\|y-\\Phi^\{T\} \\theta\\right\\|^\{2\} + \\lambda \\|\\theta\\|^2 ∥∥y−ΦTθ∥∥2\+λ∥θ∥2  
 a r g m i n θ ∥ y − Φ T θ ∥ 2 + λ ∥ θ ∥ 2 = a r g m i n θ ( y − Φ T θ ) T ( y − Φ T θ ) + λ θ T θ = a r g m i n θ ( y T − θ T Φ ) ( y − Φ T θ ) + λ θ T θ = a r g m i n θ \[ y T y − y T Φ T θ − θ T Φ y + θ T Φ Φ T θ + λ θ T θ \] ( y T Φ T θ 是 实 数 ， 所 以 类 似 A T = A ) = a r g m i n θ \[ y T y − 2 y T Φ T θ + θ T Φ Φ T θ + λ θ T θ \] \\begin\{aligned\} \\underset\{\\theta\}\{argmin\} \\left\\|y-\\Phi^\{T\} \\theta\\right\\|^\{2\} + \\lambda \\|\\theta\\|^2 &= \\underset\{\\theta\}\{argmin\} (y-\\Phi^T\\theta)^T(y-\\Phi^T\\theta) + \\lambda \\theta^T \\theta \\\\ &= \\underset\{\\theta\}\{argmin\} (y^T-\\theta^T\\Phi)(y-\\Phi^T\\theta) + \\lambda \\theta^T \\theta\\\\ &= \\underset\{\\theta\}\{argmin\} \[y^Ty-y^T\\Phi^T\\theta-\\theta^T\\Phi y+\\theta^T\\Phi\\Phi^T\\theta + \\lambda \\theta^T \\theta \] \\\\ (y^T\\Phi^T\\theta是实数，所以类似A^T=A)&= \\underset\{\\theta\}\{argmin\} \[y^Ty-2y^T\\Phi^T\\theta+\\theta^T\\Phi\\Phi^T\\theta + \\lambda \\theta^T \\theta\] \\end\{aligned\} θargmin∥∥y−ΦTθ∥∥2\+λ∥θ∥2(yTΦTθ是实数，所以类似AT=A)=θargmin(y−ΦTθ)T(y−ΦTθ)\+λθTθ=θargmin(yT−θTΦ)(y−ΦTθ)\+λθTθ=θargmin\[yTy−yTΦTθ−θTΦy\+θTΦΦTθ\+λθTθ\]=θargmin\[yTy−2yTΦTθ\+θTΦΦTθ\+λθTθ\]  
derivatives  
 ∂ ∂ θ \[ y T y − 2 y T Φ T θ + θ T Φ Φ T θ + λ θ T θ \] = 2 Φ y + Φ Φ T θ + θ T Φ Φ T + 2 λ θ ( 还 是 因 为 实 数 ) = − 2 Φ y + 2 Φ Φ T θ + 2 λ θ \\begin\{aligned\} \\frac\{\\partial\}\{\\partial\\theta\}\[y^Ty-2y^T\\Phi^T\\theta+\\theta^T\\Phi\\Phi^T\\theta+ \\lambda \\theta^T \\theta\] & = 2\\Phi y + \\Phi\\Phi^T\\theta + \\theta^T\\Phi\\Phi^T + 2 \\lambda \\theta \\\\ (还是因为实数)& = -2\\Phi y +2\\Phi \\Phi^T \\theta + 2 \\lambda \\theta \\end\{aligned\} ∂θ∂\[yTy−2yTΦTθ\+θTΦΦTθ\+λθTθ\](还是因为实数)=2Φy\+ΦΦTθ\+θTΦΦT\+2λθ=−2Φy\+2ΦΦTθ\+2λθ  
because  ∂ x T x ∂ x = 2 x \\frac\{\\partial x^Tx\}\{\\partial\{x\}\} = 2x ∂x∂xTx=2x  
 − 2 Φ y + 2 Φ Φ T θ + 2 λ θ = 0 ⟹ θ = ( Φ Φ T + λ I ) − 1 Φ y -2\\Phi y +2\\Phi \\Phi^T \\theta + 2 \\lambda \\theta = 0 \\Longrightarrow \\theta =(\\Phi\\Phi^T+\\lambda I)^\{-1\}\\Phi y −2Φy\+2ΦΦTθ\+2λθ=0⟹θ=(ΦΦT\+λI)−1Φy  
parameter estimate:  
 θ ^ = ( Φ Φ T + λ I ) − 1 Φ y \\hat\{\\theta\} = (\\Phi\\Phi^T+\\lambda I)^\{-1\}\\Phi y θ^=(ΦΦT\+λI)−1Φy

## 3、L1-regularized LS (LASSO) ##

objective function  ∥ y − Φ T θ ∥ 2 + λ ∥ θ ∥ 1 \\left\\|y-\\Phi^\{T\} \\theta\\right\\|^\{2\} + \\lambda \\|\\theta\\|\_1 ∥∥y−ΦTθ∥∥2\+λ∥θ∥1