Multivariate Cholette
Introduction
Given a set of initial time series $\left\lbrace z_{i,t}\right\rbrace_{i \in I}$ we have to find the corresponding $\left\lbrace x_{i,t}\right\rbrace_{i \in I}$ that respect temporal aggregation constraints, represented by $\mathbf{x_{i,T}}=\sum_{t \in T}x_{i,t}$ and contemporaneous constraints given by $q_{k,t}=\sum_{j \in J_k}{w_{k,j}x_{j,t}}$ or, in matrix form $q_{k,t} = w_k x_t$
The multi-variate Cholette’s method consists in minimizing the following quadratic penalty function:
\[\sum_{i,t}{\left( \left( \frac{x_{i,t}-z_{i,t}}{|z_{i,t}|^\lambda} \right)-\rho \left( \frac{x_{i,t-1}-z_{i,t-1}}{|z_{i,t-1}|^\lambda} \right) \right)^2}\]JD+ uses state space models for representing the Cholette method.
State space form
The state space representation of the multi-variate benchmarking model is obtained by juxtaposing the different matrices of the univariate models(see univariate Cholette) (one for each series involved in the model) and by adding, for each linear constraint, the corresponding “measurement” equation.
The model is defined more precisely below
State vector
\[\alpha_t = \begin{pmatrix} \left( \gamma_{0,t} \mu_{0,t}\right)^{\underline C} \\ \mu_{0,t} \\ \vdots \\ \left( \gamma_{m,t} \mu_{m,t}\right)^{\underline C} \\ \mu_{m,t} \end{pmatrix}\]Initial conditions and dynamics
Simple juxtaposition of the initial conditions and of the dynamics of the univariate models
Measurement
A first set of “observations” corresponds to the univariate temporal aggregation constraints, as they are defined in the univariate case:
\[\mathbf{\delta_{iT}}=\mathbf{x_{iT}}-\mathbf{z_{iT}}\]A second set of “observations” corresponds to the contemporaneous constraints, modified to fit the definition of the state vector. More exactly, we define $\tilde q_{k,t} = q_{k,t} - w_k z_t = w_k \left(x_t - z_t \right)$
They ared defined for all the time periods. However, for the periods that correspond to the temporal constraints, they are redundant and removed from the model.
So the matrix of the observations is:
\[\begin{pmatrix} \vdots & \vdots\\ - & \tilde q_{k,t}\\ \vdots & \vdots &\\ \delta_{i,T} & - \\ \vdots & \vdots\\ \end{pmatrix}\]The corresponding measurement matrix is defined by
\[\begin{pmatrix} \cdots & 0 \,or\, 1 & \gamma_{i,t} & \cdots & \cdots \\ \cdots & \cdots & \cdots & w_{k, t}\gamma_{k,t} &\cdots \\ \end{pmatrix}\]By construction, the smoothed states contain MMSE estimates of the $\mu_{i,t}$ that respect all the constraints of the model.
Once $\hat\mu_{i,t}$ have been estimated, we retrieve $\hat x_{i,t}$ by $\hat x_{i,t}=z_{i,t}+\gamma_{i,t}\hat\mu_{i,t}$