Forward-Backward Linear Predictor (FBLP)

Improving the efficiency of our prediction.

Forward Predictor

Suppose we have series $x[t]$ and we want to predict the next values using the previous $m$ observations. This problem is written as

\begin{align*} x[t] = -\sum_{i=1}^m a_i x[t-i] + f[t] \end{align*}

where $f[t]$ is the forward error predictor. Note that $a_i$ corresponds to the coefficient on the observation that is $i$ time-steps before $x[t]$ .

This premise leads to the following matrix formulation:

\begin{align*} \pmb x &= A\pmb h + \pmb f \\ \begin{bmatrix}x[m+1] \\ x[m+2] \\ \vdots \\ x[N]\end{bmatrix} &= \begin{bmatrix} x[m] & x[m-1] & \dots & x[1] \\ x[m+1] & x[m] & \dots & x[2] \\ \vdots & \vdots & \ddots & \vdots \\ x[N-1] & x[N] & \dots & x[N-m] \end{bmatrix} \begin{bmatrix}-a_1 \\ -a_2 \\ \vdots \\ -a_m\end{bmatrix} + \begin{bmatrix}f[m+1] \\ f[m+2] \\ \vdots \\ f[N-1] \end{bmatrix} \end{align*}

Backward Predictor

Using the same data as before, we're now interested in predicting the preceeding values using $m$ future values. Due flipping the order, we write this problem using the complex conjugate.

\begin{align*} \overline x[t] = -\sum_{i=1}^m a_i \overline x[t+i] + \overline b[t] \end{align*}

where $b[t]$ is the backward error predictor. Note that $a_i$ corresponds to the coefficients on the observation that is $i$ time-steps after $x[t]$ .

Sampling a various values of $t$ we obtain a system of equations that can be written in matrix form as

\begin{align*} \overline{\pmb x} &= \overline A \pmb h + \overline{\pmb b} \\ \begin{bmatrix}\overline x[1] \\ \overline x[2] \\ \vdots \\ x[N-m]\end{bmatrix} &= \begin{bmatrix}\overline x[2] & \overline x[3] & \dots & \overline x[m+1] \\ \overline x[3] & \overline x[4] & \dots & \overline x[m+2] \\ \vdots & \vdots & \ddots & \vdots \\ \overline x[N-m+1] & \overline x[N-m+2] & \dots & \overline x[N]\end{bmatrix} \begin{bmatrix}-a_1 \\ -a_2 \\ \vdots \\ -a_m\end{bmatrix} + \begin{bmatrix} \overline b[1] \\ \overline b[2] \\ \vdots \\ \overline b[N-m] \end{bmatrix} \end{align*}

Forward-Backward Linear Predictor

The insight for the FBLP predictor is seeing that we can stack these $A$ matrices to find a lower variance estimate of $\pmb h$ . The assumption being that the relationship is symmetrical.

By this, I mean that the coefficient $a$ in the expression $a x[t] + f[t] = x[t+1]$ is the same as the coefficient $a$ found in the expression $a x[t+1] + b[t+1] = x[t]$ .

Now we simply stack the the two $A$ matrices and $\pmb x$ matrices to get

\begin{align*} \begin{bmatrix}x[m+1] \\ x[m+2] \\ \vdots \\ x[N] \\ \overline x[1] \\ \overline x[2] \\ \vdots \\ x[N-m]\end{bmatrix} = \begin{bmatrix} x[m] & x[m-1] & \dots & x[1] \\ x[m+1] & x[m] & \dots & x[2] \\ \vdots & \vdots & \ddots & \vdots \\ x[N-1] & x[N] & \dots & x[N-m] \\ \overline x[2] & \overline x[3] & \dots & \overline x[m+1] \\ \overline x[3] & \overline x[4] & \dots & \overline x[m+2] \\ \vdots & \vdots & \ddots & \vdots \\ \overline x[N-m+1] & \overline x[N-m+2] & \dots & \overline x[N] \end{bmatrix} \begin{bmatrix}-a_1 \\ -a_2 \\ \vdots \\ -a_m\end{bmatrix} + \begin{bmatrix} f[m+1] \\ f[m+2] \\ \vdots \\ f[N-1] \\ \overline b[1] \\ \overline b[2] \\ \vdots \\ \overline b[N-m] \end{bmatrix} \end{align*}