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
\(T_t = \begin{cases} \begin{pmatrix}0 &0 \\0&\rho \end{pmatrix},& {t+1} \mod {c} = 0\\ \begin{pmatrix} 0 & \gamma_t\\ 0 & \rho\end{pmatrix},& {t} \mod {c} = 0\\\begin{pmatrix} 1 & \gamma_t\\ 0 & \rho\end{pmatrix},& otherwise\end{cases}\)
Measurement
\(Z_t = \begin{cases} \begin{pmatrix} 0 & \gamma_t\end{pmatrix},& {t} \mod {c} = 0 \\ \begin{pmatrix} 1 & \gamma_t \end{pmatrix},& {t} \mod {c} \neq 0\end{cases}\)
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 $\hat x_t=z_t+\gamma_t\hat\mu_t$