Linear Prediction for Time Series

36-467/36-667

20 September 2018

\[ \newcommand{\Expect}[1]{\mathbb{E}\left[ #1 \right]} \newcommand{\Var}[1]{\mathrm{Var}\left[ #1 \right]} \newcommand{\SampleVar}[1]{\widehat{\mathrm{Var}}\left[ #1 \right]} \newcommand{\Cov}[1]{\mathrm{Cov}\left[ #1 \right]} \newcommand{\TrueRegFunc}{\mu} \newcommand{\EstRegFunc}{\widehat{\TrueRegFunc}} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator*{\argmin}{argmin} \DeclareMathOperator{\det}{det} \newcommand{\TrueNoise}{\epsilon} \newcommand{\EstNoise}{\widehat{\TrueNoise}} \]

In our last episode

Optimal linear prediction for time series

\[\begin{eqnarray} \EstRegFunc(t_0) & = & \alpha + \vec{\beta} \cdot \left[\begin{array}{c} X(t_1) \\ X(t_2) \\ \vdots \\ X(t_n) \end{array}\right]\\ \alpha & = & \Expect{X(t_0)} - \vec{\beta} \cdot \left[\begin{array}{c} \Expect{X(t_1)}\\ \Expect{X(t_2)} \\ \vdots \\ \Expect{X(t_n)}\end{array}\right] ~ \text{(goes away if everything's centered)}\\ \vec{\beta} & = & {\left[\begin{array}{cccc} \Var{X(t_1)} & \Cov{X(t_1), X(t_2)} & \ldots & \Cov{X(t_1), X(t_n)}\\ \Cov{X(t_1), X(t_2)} & \Var{X(t_2)} & \ldots & \Cov{X(t_2), X(t_n)}\\ \vdots & \vdots & \ldots & \vdots\\ \Cov{X(t_1), X(t_n)} & \Cov{X(t_2), X(t_n)} & \ldots & \Var{X(t_n)}\end{array}\right]}^{-1} \left[\begin{array}{c} \Cov{X(t_0), X(t_1)}\\ \Cov{X(t_0), X(t_2)}\\ \vdots \\ \Cov{X(t_0), X(t_n)}\end{array}\right] \end{eqnarray}\]

Interpolation

Back to Kyoto

What we didn’t tell R to make up

When did the cherries flower in 1015?

Similarly for extrapolation

Getting the expectations and covariances

Repeating the experiment