Spline Regression V2

Spline Regression V2

Function i

\[ \begin{aligned} f_i(x) &= a_i(x-x_i)^3 + b_i(x-x_i)^2 + c_i(x-x_i) + y_i + d_i \\ f^{'}_i(x) &= 3a_i(x-x_i)^2 + 2b_i(x-x_i) + c_i \\ f^{''}_i(x) &= 6a_i(x-x_i) + 2b_i \end{aligned} \]

Constrains

$\forall i \in {2 \ldots n-1} $:

\[ \begin{aligned} f_{i}(x_{i}) - f_{i-1}(x_{i}) &= 0 \\ f^{'}_{i}(x_{i}) - f^{'}_{i-1}(x_{i}) &= 0 \\ f^{''}_{i}(x_{i}) - f^{''}_{i-1}(x_{i}) &= 0 \end{aligned} \]

\(additonally:\)

\[ \begin{aligned} f^{''}_{1}(x_{1}) &= 0 \\ f^{''}_{n-1}(x_{n}) &= 0 \end{aligned} \]

Cost Function

\[\begin{aligned} F_{i} &= \begin{bmatrix}f_i & f_i^{'} & f_i^{''}\end{bmatrix}^{T} \\ Y_{i} &= \begin{bmatrix}y_i & 0 & 0\end{bmatrix}^{T} \\ E_i^j &= F_i(x_j) - Y_j \\ G_i &= F_{i-1}(x_i)-F_i(x_i) \\ G_1 &= \begin{bmatrix} g_1 \end{bmatrix} = \begin{bmatrix} f_1^{''}(x_1) \end{bmatrix} \\ G_n &= \begin{bmatrix} g_n \end{bmatrix}= \begin{bmatrix} f_{n-1}^{''}(x_n) \end{bmatrix} \\ \phi_{i} &= \begin{bmatrix}\phi_i^1 & \phi_i^2 & \phi_i^3\end{bmatrix}^{T} \\ \phi_1 &= [ \phi_1^1 ] \\ \phi_n &= [ \phi_n^1 ] \\ \end{aligned} \]

\[\begin{aligned} O = &\sum_{i=1}^{n-1}({E_i^i}^TWE_i^i + {E_i^{i+1}}^TWE_i^{i+1}) +\sum_{i=1}^{n}\phi_{i}^TG_i \end{aligned} \]

\[ \begin{aligned} \beta_i &= \begin{bmatrix}a_i & b_i & c_i & d_i\end{bmatrix}^{T} \\ X &= \begin{bmatrix}\phi_1^T & \beta_1^T & \phi_2^T & \beta_2^T &\ldots & \phi_{n-1}^T & \beta_{n-1}^T & \phi_n^T\end{bmatrix}^{T} \end{aligned} \]

\[ \begin{aligned} \frac{\partial O}{\partial X} = \begin{bmatrix} \frac{\partial O}{\partial X_1} & \ldots & \frac{\partial O}{\partial X_m} \end{bmatrix}^{T} \end{aligned} \]

\[ \begin{aligned} H = \frac{\partial^2 O}{\partial X^2} = \frac{\partial }{\partial X}(\frac{\partial O}{\partial X}) = \begin{bmatrix} \frac{\partial^2 O}{\partial X_1^2} & \frac{\partial^2 O}{\partial X_1\partial X_2} & \ldots & \frac{\partial O}{\partial X_1\partial X_m} \\ \frac{\partial^2 O}{\partial X_2\partial X_1} & \frac{\partial^2 O}{\partial X_2^2} & \ldots & \frac{\partial O}{\partial X_2\partial X_m} \\ \vdots & \vdots & \ddots &\vdots \\ \frac{\partial^2 O}{\partial X_m\partial X_1} & \frac{\partial^2 O}{\partial X_m\partial X_2} & \ldots & \frac{\partial^2 O}{\partial X_m^2} \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} B &= -\left.\frac{\partial O}{\partial X}\right| _{X=0} \\ B &= \begin{bmatrix} B_1^{\phi} & B_1^{\beta} & B_2^{\phi} & B_2^{\beta} & \ldots & B_{n-1}^{\phi} & B_{n-1}^{\beta} & B_n^{\phi} \end{bmatrix}^T \\ B_i^{\phi} &= -\left.\frac{\partial O}{\partial \phi_i}\right| _{\phi_i=0} = \begin{bmatrix} y_i-y_{i-1} && 0 && 0 \end{bmatrix}^T\\ B_i^{\beta} &= -\left.\frac{\partial O}{\partial \beta_i}\right| _{\beta_i=0} = \begin{bmatrix} 2w_1(y_{i+1}-y_i)(x_{i+1}-x_i)^3 + 6w_2y_{i+1}^{'}(x_{i+1}-x_i)^2 + 12w_3y_{i+1}^{''}(x_{i+1}-x_i) \\ 2w_1(y_{i+1}-y_i)(x_{i+1}-x_i)^2 + 4w_2y_{i+1}^{'}(x_{i+1}-x_i) + 4w_3(y_i^{''} + y_{i+1}^{''}) \\ 2w_1(y_{i+1} - y_i)(x_{i+1}-x_i) + 2w_2(y_i^{'} + y_{i+1}^{'}) \\ 2w_1(y_{i+1} - y_i) \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} HX = B \\ X = H^{-1}B \end{aligned} \]

Form of H Matrix

\[ \begin{aligned} H = \begin{bmatrix} 0 & T_1^T \\ T_1 & A_1 & R_1 \\ & R_1^T & 0 & T_2^T \\ & & T_2 & A_2 & R_2 \\ & & & R_2^T & 0 & T_3^T \\ & & & & \vdots & \vdots & \vdots \\ & & & & & R_{n-2}^T & 0 & T_{n-1}^T \\ & & & & & & T_{n-1} & A_{n-1} & R_{n-1} \\ & & & & & & & R_{n-1}^T & 0 \end{bmatrix} \\ \end{aligned} \]

\[\begin{aligned} A_i^{11} &= 2w_1(x_{i+1}-x_i)^6+18w_2(x_{i+1}-x_i)^4+72w_3(x_{i+1}-x_i)^2 \\ A_i^{12} &= 2w_1(x_{i+1}-x_i)^5+12w_2(x_{i+1}-x_i)^3+24w_3(x_{i+1}-x_i) \\ A_i^{13} &= 2w_1(x_{i+1}-x_i)^4+6w_2(x_{i+1}-x_i)^2 \\ A_i^{14} &= 2w_1(x_{i+1}-x_i)^3 \\ \\ A_i^{21} &= 2w_1(x_{i+1}-x_i)^5+12w_2(x_{i+1}-x_i)^3+24w_3(x_{i+1}-x_i) \\ A_i^{22} &= 2w_1(x_{i+1}-x_i)^4+8w_2(x_{i+1}-x_i)^2+16w_3 \\ A_i^{23} &= 2w_1(x_{i+1}-x_i)^3+4w_2(x_{i+1}-x_i) \\ A_i^{24} &= 2w_1(x_{i+1}-x_i)^2 \\ \\ A_i^{31} &= 2w_1(x_{i+1}-x_i)^4+6w_2(x_{i+1}-x_i)^2 \\ A_i^{32} &= 2w_1(x_{i+1}-x_i)^3+4w_2(x_{i+1}-x_i) \\ A_i^{33} &= 2w_1(x_{i+1}-x_i)^2+4w_2 \\ A_i^{34} &= 2w_1(x_{i+1}-x_i) \\ \\ A_i^{41} &= 2w_1(x_{i+1}-x_i)^3 \\ A_i^{42} &= 2w_1(x_{i+1}-x_i)^2 \\ A_i^{43} &= 2w_1(x_{i+1}-x_i) \\ A_i^{44} &= 4w_1 \\ \\ T_i &= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -2 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix} \\ R_i &= \begin{bmatrix} (x_{i+1}-x_i)^3 & 3(x_{i+1}-x_i)^2 & 6(x_{i+1}-x_i) \\ (x_{i+1}-x_i)^2 & 2(x_{i+1}-x_i) & 2 \\ (x_{i+1}-x_i) & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} \\ T_1 &= \begin{bmatrix} 0 & 2 &0 & 0 \end{bmatrix}^T \\ R_{n-1} &= \begin{bmatrix} 6(x_n-x_{n-1}) & 2 &0 & 0 \end{bmatrix}^T \end{aligned} \]