Cholette (SSF)

State space form for the Cholette algorithm

Introduction

Given an initial time series $z_t,\; 0 \le t < n$ we have to find the corresponding $x_t$ that respects the (observed) aggregation constraints $\mathbf{x_T} =\sum_{t \in T}x_t$ and that follows the short-term movements of $z_t$.

In the Cholette’s method, this is achieved by minimizing the following quadratic penalty function:

$\sum_t{\left( \frac{x_t-z_t}{\vert z_t \vert^\lambda}-\rho \frac{x_{t-1}-z_{t-1}}{\vert z_{t-1} \vert^\lambda} \right)^2}$

It is easy to see that the quadratic function of Cholette corresponds, from a formal point of view, to the sum of the square residuals generated by the auto-regressive process:

$\frac{x_t-z_t}{|z_t|^\lambda}=\frac{\delta_t}{\vert z_t \vert ^\lambda}=\mu_t$ $\mu_t = \rho \mu_{t-1} + \epsilon_t$

To simplify the notations, we will use hereafter $\vert z_t \vert^\lambda=\gamma_t$ .

Starting from that observation, we can give the Cholette’s algorithm a state space representation.

State vector

We define the state vector as

$\alpha_t= \begin{pmatrix} \left( \gamma_t \mu_t \right)^{\underline C} \\ \mu_t \end{pmatrix}$

Initialization

$\alpha_{-1} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$

Not diffuse: $rho<1$

$P_{*} = \begin{pmatrix} 0 & 0 \\ 0 & \frac{1}{1-\rho^2} \end{pmatrix}$

Diffuse (Denton variant): $\rho=1$

$B = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$

$P_{\infty} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$

Dynamics

$T_t = \begin{cases} \begin{pmatrix}0 &0 \\0&\rho \end{pmatrix},& {t+1} \mod {c} = 0\\ \begin{pmatrix} 0 & \gamma_t\\ 0 & \rho\end{pmatrix},& {t} \mod {c} = 0\\\begin{pmatrix} 1 & \gamma_t\\ 0 & \rho\end{pmatrix},& otherwise\end{cases}$

$V_t = \begin{pmatrix}0 &0 \\0&1 \end{pmatrix} ,\quad S_t = \begin{pmatrix}0 \\ 1 \end{pmatrix}$

Measurement

$Z_t = \begin{cases} \begin{pmatrix} 0 & \gamma_t\end{pmatrix},& {t} \mod {c} = 0 \\ \begin{pmatrix} 1 & \gamma_t \end{pmatrix},& {t} \mod {c} \neq 0\end{cases}$

$h_t = 0$

The “observations” are defined by $Z_t \alpha_t=\left(\gamma_t \mu_t \right)^C$. They are available for the periods $t$ such that $t=q c-1$ and missing for the other periods. For the observed periods, they correspond to $\mathbf{x_T}-\mathbf{z_T}$.

Once $\hat\mu_t$ have been estimated, we retrieve $\hat x_t$ by $\hat x_t=z_t+\gamma_t\hat\mu_t$