Notation for forecast reconciliation
Recently I spent a few weeks visiting Professor Tommaso Di Fonzo at the University of Padova (Italy), and one of the things we discussed was finding a notation we were both happy with so we could be more consistent in our future papers.
This is what we came up with. Hopefully others will agree and use it too!
For readers new to forecast reconciliation, Chapter 11 of FPP3 provides an introduction.
We observe n time series at time t, written as \bm{y}_t. The base forecasts of \bm{y}_{T+h} given data \bm{y}_1,\dots,\bm{y}_T are denoted by \hat{\bm{y}}_h.
Structural representation
This was the original formulation of the problem due to Hyndman et al. (2011), but presented here in our new notation.
Let \bm{b}_t be a vector of n_b “bottom-level” time series at time t, and let \bm{a}_t be a corresponding vector of n_a = n-n_b aggregated time series, where \bm{a}_t = \bm{A}\bm{b}_t and \bm{A} is the n_a\times n_b “aggregation” matrix specifying how the bottom-level series \bm{b}_t are to be aggregated to form \bm{a}_t. The full vector of time series is given by \bm{y}_t = \begin{bmatrix}\bm{a}_t \\\bm{b}_t\end{bmatrix}. This leads to the n\times n_b “summing” or “structural” matrix given by \bm{S} = \begin{bmatrix}\bm{A} \\ \bm{I}_{n_b}\end{bmatrix} such that \bm{y}_t = \bm{S}\bm{b}_t.
All bottom-up, middle-out, top-down and linear reconciliation methods can be written as \tilde{\bm{y}}_h = \bm{S}\bm{G}\hat{\bm{y}}_h, for different matrices \bm{G}.
Optimal reconciled forecasts are obtained with \bm{G}=(\bm{S}'\bm{W}^{-1}\bm{S})^{-1}\bm{S}'\bm{W}^{-1}, or \tilde{\bm{y}}_h = \bm{M}\hat{\bm{y}}_h, where the n\times n “mapping” matrix is given by \bm{M} = \bm{S}(\bm{S}'\bm{W}^{-1}\bm{S})^{-1}\bm{S}'\bm{W}^{-1}, \tag{1} \hat{\bm{y}}_h are the h-step forecasts of y_{T+h} given data to time T, and \bm{W} is an n \times n positive definite matrix. Different choices for \bm{W} lead to different solutions such as OLS, WLS and MinT (Wickramasuriya, Athanasopoulos, and Hyndman 2019).
Linear combinations
There is actually no reason for \bm{a}_t to be restricted to aggregates of \bm{b}_t. They can include any linear combination of the bottom-level series \bm{b}_t, so the corresponding \bm{A} and \bm{S} matrices may contain any real values, not just 0s and 1s. Nevertheless, we will use the same notation for this more general setting.
Zero-constrained representation
This representation is more efficient and was used by Di Fonzo and Girolimetto (2021). It was also discussed in Wickramasuriya, Athanasopoulos, and Hyndman (2019). Here it is in the new notation.
We can express the structural representation using the n_a \times n constraint matrix \bm{C} = \begin{bmatrix}\bm{I}_{n_a} & -\bm{A}\end{bmatrix}, so that \bm{C}\bm{y}_t = \bm{0}_{n_a}. Then we can write the mapping matrix as \bm{M} = \bm{I}_n - \bm{W}\bm{C}'(\bm{C}\bm{W}\bm{C}')^{-1}\bm{C}. \tag{2} Note that Equation 2 involves inverting an n_a \times n_a matrix, rather than the n_b \times n_b matrix in Equation 1. For most practical problems, n_a < n_b, so Equation 2 is more efficient.
This form of the mapping matrix also allows us to interpret the reconciliation as an additive adjustment to the base forecasts. If the base forecasts are already reconciled, then \bm{C}\hat{\bm{y}}_h = \bm{0}_{n_a} and so \bm{M}=\bm{I}_{n}.
General zero-constrained representation
The most general way to express the problem is not to denote individual series as bottom-level or aggregated, but to define the linear constraints \bm{\Gamma}\bm{y}_t = \bm{0} where \bm{\Gamma} is an r\times n matrix, not necessarily full rank, which may contain any real values.
If \bm{\Gamma} is full rank, then Equation 2 holds with \bm{C} = \bm{\Gamma}.
Temporal reconciliation
Temporal reconciliation was proposed by Athanasopoulos et al. (2017). Here it is in our new notation.
For simplicity we will assume the original (scalar) time series is observed with a single seasonality of period m (e.g., m=12 for monthly data), and the total length of the series T is an integer multiple of m. We will denote the original series by y, and the various temporally aggregated series by x.
Let \{k_1,\dots,k_p\} denote the p factors of m in ascending order, where k_1=1 and k_p=m. For each factor k of m, we can construct a temporally aggregated series x_j^{[k]} = \sum_{t = (j-1)k+1}^{jk} y_t for j=1,\dots,T/k. Of course, x_j^{[1]} = y_t.
Since the observation index j varies with each aggregation level, we define \tau as the observation index of the most aggregated level (e.g., annual), so that j=\tau at that level.
For each aggregation level, we stack the observations in the column vectors \bm{x}_\tau^{[k]} = \begin{bmatrix*}[l] x_{M_k(\tau-1)+1}^{[k]}\\ x_{M_k(\tau-1)+2}^{[k]}\\ \quad\vdots\\ x_{M_k\tau}^{[k]} \end{bmatrix*}, where k=\{k_1,\dots,k_p\}, \tau=1,\dots,N, and N=T/m. Collecting these in one column vector, we obtain \bm{x}_\tau = \begin{bmatrix*}[l] {x_\tau^{[k_p]}}\\ {\bm{x}_\tau^{[k_{p-1}]}}\\ \quad\vdots\\ {\bm{x}_\tau^{[k_1]}}\\ \end{bmatrix*}.
The structural representation of this formulation is \bm{x}_\tau = \bm{S} \bm{x}_\tau^{[1]} where \bm{S} = \begin{bmatrix}\bm{A}\\\bm{I}\end{bmatrix} and \bm{A} = \begin{bmatrix} \bm{1}'_m \\ \bm{I}_{m/k_{p-1}} \otimes \bm{1}'_{k_{p-1}} \\ \vdots\\ \bm{I}_{m/k_{2}} \otimes \bm{1}'_{k_{2}} \\ \end{bmatrix}
The zero-constrained representation is \bm{C} = \begin{bmatrix} \bm{I} & -\bm{A}\end{bmatrix}.
If there are multiple seasonalities that are not integer multiples of each other, the resulting additional temporal aggregations can simply be stacked in \bm{x}_t, and \bm{A} can be extended accordingly.
Cross-temporal reconcilation
Now consider the case where we have both cross-sectional and temporal aggregations, as discussed in Di Fonzo and Girolimetto (2021).
Suppose we have observed \bm{y}_t at the most temporally disaggregated level, including all the cross-sectionally disaggregated and aggregated (or constrained) series. Let y_{i,t} be the ith element of the vector \bm{y}_t, i=1,\dots,n. For each i, we can expand y_{i,t} to include all the temporally aggregated variants, giving a vector of length p: \bm{x}_{i,\tau} = \begin{bmatrix} {x_{i,\tau}^{[k_p]}}\\ \vdots\\ {\bm{x}_{i,\tau}^{[k_1]}} \end{bmatrix}. These can then be stacked into a long vector: \bm{x}_\tau = \begin{bmatrix} \bm{x}_{1,\tau}\\ \vdots\\ \bm{x}_{n,\tau} \end{bmatrix}.
If \bm{S}_{cs} denotes the structural matrix for the cross-sectional reconciliation, and \bm{S}_{te} denotes the structural matrix for the temporal reconciliation, then the cross-temporal structural matrix is \bm{S}_{ct} = \bm{S}_{cs} \otimes \bm{S}_{te}, so that \bm{x}_\tau = \bm{S}_{ct} \bm{b}_\tau, where the bottom-level series \bm{b}_\tau = \begin{bmatrix} {\bm{x}_{1,\tau}^{[1]}} \\ \vdots\\ {\bm{x}_{n,\tau}^{[1]}} \end{bmatrix}.