Cubic spline

Introduction

We try to approximate a time series $y_t$ by a smooth function $\mu_t$, chosen by minimizing the criterion:

$\sum_{t=0}^{t<n}\left[y_t-\mu_t\right]^2 + \lambda \sum_{t=0}^{t<n}\left[\Delta^2 \mu_t\right]^2$

We consider the following model:

$y_t = \mu_t + \epsilon_t$ $\Delta^2 \mu_t = \eta_t$ $var(\epsilon_t)= \sigma^2, \quad var(\eta_t)= \sigma^2/\lambda$

Initialization

Dynamics

$T_t = \begin{pmatrix}1 & 1 \\ 0 & 1 \end{pmatrix}$ $V_t = \begin{pmatrix} 1/3 & 1/2 \\ 1/2 & 1 \end{pmatrix}$ $T_t = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$

Cubic spline

Cubic spline

Introduction

Initialization

Dynamics

Measurement

Implementation