Optimal forecasts for hierarchical and grouped time series
Creators: name: Wickramasuriya, Shanika L
Publisher: name: Figshare
Year: 2017
Subjects: subject: Statistics
subject: FOS: Mathematics

description: Large collections of time series often have aggregation constraints due
 to product or geographical groupings. When the time series 
disaggregation is unique we refer to it as a hierarchical structure 
whereas when it involves different paths of disaggregation we refer to 
it as a grouped structure. In both of these structures, forecasts for 
the most disaggregated series are usually required to add-up exactly to 
the forecasts of the aggregated series, a constraint known as 
aggregation consistency. Common approaches to obtaining a set of 
aggregate-consistent forecasts include the "bottom-up'', "top-down'' and
 "middle-out'' approaches. The bottom-up approach forecasts all series 
at the most disaggregated level and sums the forecasts to form the 
forecasts for higher levels. In contrast, the top-down approach 
forecasts the most aggregated series and distributes these forecasts 
based on some proportions to form the forecasts for the disaggregated 
series. The middle-out approach is a combination of both bottom-up and 
top-down approaches. 
  <br>    
  <br>   Considering the drawbacks of these approaches, Hyndman, Ahmed, 
Athanasopoulos, and Shang (2011) proposed forecasting all disaggregated 
and aggregated series independently, and then reconciling the forecasts 
so they become aggregate consistent. Their approach is based on a 
generalized least squares estimator which required an estimate of the 
covariance matrix of the reconciliation errors that arise due to 
aggregate inconsistency. In this thesis I show that this covariance 
matrix is impossible to estimate in practice due to identifiability 
conditions. 
  <br>    
  <br>   To overcome this issue, I propose a new forecast reconciliation
 approach that minimizes the mean squared error of aggregate-consistent 
forecasts across the entire collection of time series under the 
assumption of unbiasedness. The minimization problem has a closed form 
solution similar to that of Hyndman et al. (2011), except that the 
required covariance matrix is estimable leading to a feasible solution. I
 make this solution scalable by providing a computationally less 
demanding alternative representation. I refer to this approach as MinT. 
  <br>    
  <br>   I evaluate the performance of MinT compared to alternative 
approaches in the literature using a series of simulation designs that 
take into account various features of the collection of time series. 
This is followed by an empirical application using Australian domestic 
tourism data. The results indicate that MinT performs well with both 
simulated and real data. 
  <br>    
  <br>   Many of the applications encountered in practice have fewer 
observations per series than the number of series in the structure. In 
such situations, the sample variance covariance matrix is known to 
perform poorly which makes the estimation of the covariance matrix 
challenging and requires the use of special strategies. I mainly focused
 on estimators that belong to the Stein-type shrinkage, sparse, and low 
rank + sparse classes. For applications with a hierarchical structure, 
it is reasonable to assume that the covariance matrix is a 
representation of some lower dimension; for example, a block diagonal 
structure where the series with the same parental node tend to behave 
more similarly than the series that are less closely related. Hence, I 
consider modifications for some of these estimators to incorporate prior
 knowledge about important relationships resulting from the aggregation 
constraints. In addition, I explore several possibilities of reducing 
the computational burden of low rank + sparse type of estimators. 
  <br>    
  <br>   I evaluate the performances of these competing estimators using
 a series of simulation designs and several empirical applications using
 Australian domestic tourism data. The results indicate that most of the
 covariance estimators contribute to substantially improving the 
forecast accuracy of MinT. In addition, I observe that prior knowledge 
of the correlation structure does not play a key role in improving the 
forecast accuracy. One of the estimators that belongs to low rank + 
sparse category tends to behave unexpectedly in both simulation and 
empirical evaluations. Through a case study I identify the causes and 
propose remedies for obtaining an estimator that produces consistently 
better performance. 
  <br>    
  <br>   MinT and its variants can result in negative values for the 
reconciled forecasts even when the original set of forecasts are 
non-negative. This should be avoided in certain applications that are 
inherently non-negative in nature, especially when the forecasts are 
used for decisions or policy implementation processes. Hence, MinT is 
reformulated as a least squares minimization problem with non-negativity
 constraints. Considering the dimension and sparsity of the matrices 
involved and the alternative representation of MinT, this problem is 
solved using three algorithms: block principal pivoting, projected 
conjugate gradient and scaled gradient projection. I compare the 
performances of these algorithms using a series of simulations and an 
empirical application. The results reveal that the block principal 
pivoting algorithm outperforms the rest where a structure with nearly 
six hundred thousand series can be reconciled in less than one minute 
using a standard desktop computer. In addition, it is observed that the 
gains or losses in forecast accuracy due to non-negativity constraints 
are negligible