Efficient probabilistic reconciliation of forecasts for real-valued and count time series Lorenzo ZambonDario AzzimontiGiorgio Corani

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Eﬃcient probabilistic reconciliation of forecasts

for real-valued and count time series

Lorenzo Zambon∗,†Dario Azzimonti∗Giorgio Corani∗

Abstract

Hierarchical time series are common in several applied ﬁelds. The fore-

casts for these time series are required to be coherent, that is, to satisfy

the constraints given by the hierarchy. The most popular technique to

enforce coherence is called reconciliation, which adjusts the base forecasts

computed for each time series. However, recent works on probabilistic

reconciliation present several limitations. In this paper, we propose a new

approach based on conditioning to reconcile any type of forecast distribu-

tion. We then introduce a new algorithm, called Bottom-Up Importance

Sampling, to eﬃciently sample from the reconciled distribution. It can

be used for any base forecast distribution: discrete, continuous, or in

the form of samples, providing a major speedup compared to the current

methods. Experiments on several temporal hierarchies show a signiﬁcant

improvement over base probabilistic forecasts.

Keywords: Forecast reconciliation, Probabilistic reconciliation, Tempo-

ral hierarchies, Importance sampling

1 Introduction

Often time series are organized into a hierarchy. For example, the total

visitors of a country can be divided into regions and the visitors of each

region can be further divided into sub-regions. Such data structures are

referred to as hierarchical time series; they are common in ﬁelds such as

retail sales [Makridakis et al., 2021] and energy modelling [Taieb et al.,

2021].

The forecasts for hierarchical time series should respect some summing

constraints, in which case they are referred to as coherent. For instance,

the sum of the forecasts for the sub-regions should match the forecast for

the entire region. However, the forecasts independently produced for each

time series (base forecasts) are generally incoherent.

∗IDSIA, Dalle Molle Institute for Artiﬁcial Intelligence, CH-6962, Lugano, Switzerland;

{lorenzo.zambon, dario.azzimonti, giorgio.corani}@idsia.ch

†Department of Mathematics, University of Pavia, 27100, Pavia, Italy

arXiv:2210.02286v3 [stat.ML] 12 Oct 2023

Reconciliation algorithms [Hyndman et al., 2011, Wickramasuriya et al.,

2019] adjust the incoherent base forecasts, making them coherent. Rec-

onciled forecasts are generally more accurate than base forecasts: indeed,

forecast reconciliation is a special case of forecast combination [Hollyman

et al., 2021]. An important application of reconciliation algorithms is con-

stituted by temporal hierarchies [Athanasopoulos et al., 2017, Kourentzes

and Athanasopoulos, 2021], which make coherent the forecasts produced

for the same time series at diﬀerent temporal scales.

Most reconciliation algorithms [Hyndman et al., 2011, Wickramasuriya

et al., 2019, 2020, Di Fonzo and Girolimetto, 2021, 2022] provide only rec-

onciled point forecasts. It is however clear [Kolassa, 2023] that reconciled

predictive distributions are needed for decision making.

Probabilistic reconciliation has been addressed only recently; earlier

attempts [Jeon et al., 2019, Taieb et al., 2021], though experimentally ef-

fective, lacked a strong formal justiﬁcation. For the case of Gaussian base

forecasts, Corani et al. [2020] obtains the reconciled distribution in ana-

lytical form introducing the approach of reconciliation via conditioning.

Panagiotelis et al. [2023] provides a framework for probabilistic reconcil-

iation via projection. However this approach cannot reconcile discrete

distributions. Corani et al. [2023] performs probabilistic reconciliation

via conditioning of count time series by adopting the concept of virtual

evidence [Pearl, 1988]. However its implementation in probabilistic pro-

gramming, based on Markov Chain Monte Carlo (MCMC), is too slow on

large hierarchies; moreover it requires the base forecast distribution to be

in parametric form.

The main contribution of this paper is the Bottom-Up Importance

Sampling (BUIS) algorithm, which samples from the reconciled distribu-

tion obtained via conditioning with a substantial speedup with respect to

Corani et al. [2023]. BUIS can be used even when the base forecast dis-

tribution is only available through samples. This is the case of forecasts

returned by models for time series of counts [Liboschik et al., 2017] or

based on deep learning [Salinas et al., 2020]. We prove the convergence

of BUIS to the actual reconciled distribution. An implementation of the

algorithm in the Rlanguage is available in the Rpackage bayesRecon

[Azzimonti et al., 2023].

We provide two further formal contributions. The ﬁrst is a deﬁnition

of coherence for probabilistic forecasts that applies to both discrete and

continuous distributions. The second is a novel interpretation of the rec-

onciliation via conditioning, in which the base forecast distribution is con-

ditioned on the hierarchy constraints. This allows for a uniﬁed treatment

of the reconciliation of discrete and continuous forecast distributions. We

test our method exhaustively on temporal hierarchies reporting positive

results both for the accuracy and the eﬃciency of our method.

The paper is organized as follows. In Sec. 2, we introduce the notation

and the reconciliation of point forecasts. In Sec. 3, we introduce our ap-

proach to reconciliation via conditioning and we compare it to the existing

literature. In Sec. 4, we introduce the Bottom-Up Importance Sampling

algorithm. We empirically verify its correctness in Sec. 5, while in Sec. 6

we test it on diﬀerent data sets. We present the conclusions in Sec. 7.

B1B2

B3B4

Figure 1: A hierarchy with 4bottom and 3upper variables.

2 Notation

Consider the hierarchy of Fig. 1. We denote by b= [b1,...,bnb]Tthe

vector of bottom variables, and by u= [u1,...,unu]Tthe vector of upper

variables. We then denote by

y="u

b#∈Rn

the vector of all the variables. The hierarchy can be expressed as a set of

linear constraints:

y=Sb,where S="A

I#.(1)

We refer to I∈Rnb×nbas the identity matrix, to S∈Rn×nbas the

summing matrix and to A∈Rnu×nbas the aggregating matrix. We can

thus write the constraints as u=Ab. For example, the aggregating

matrix of the hierarchy in Fig. 1 is:

A=





1111

1100

0011





.

A point y∈Rnis coherent if it satisﬁes the constraints given by the

hierarchy. We denote by Sthe set of coherent points, which is a linear

subspace of Rn:

S:= {y∈Rn:y=Sb}.(2)

2.1 Temporal hierarchies

In temporal hierarchies [Athanasopoulos et al., 2017, Kourentzes and

Athanasopoulos, 2021], forecasts are generated for the same time series

at diﬀerent temporal scales. For instance, a quarterly time series can be

aggregated to the semi-annual and the annual scale. If we are interested

in predictions up to one year ahead, we compute four quarterly fore-

casts ˆq1,ˆq2,ˆq3,ˆq4, two semi-annual forecasts ˆs1,ˆs2, and an annual forecast

ˆa1. We then obtain the hierarchy in Fig. 1. The base point forecasts,

independently computed at each frequency, are ˆ

b= [ˆq1,ˆq2,ˆq3,ˆq4]Tand

u= [ˆa1,ˆs1,ˆs2]T.

2.2 Point forecasts reconciliation

Let us denote by ˆ

y=ˆ

uT|ˆ

bTTthe vector of the base (incoherent) fore-

casts. Note that, for ease of notation, we drop the time subscript. Point

reconciliation is generally performed in two steps [Hyndman et al., 2011,

Wickramasuriya et al., 2019]. First, the reconciled bottom forecasts are

computed by linearly combining the base forecasts of the entire hierarchy:

b=Gˆ

for some matrix G∈Rm×n. Then, the reconciled forecasts for the whole

hierarchy are given by:

y=S˜

The state-of-the-art reconciliation method is MinT [Wickramasuriya et al.,

2019], which deﬁnes Gas:

G= (STW−1S)−1STW−1,

where Wis the covariance matrix of the errors of the base forecasts. This

method minimizes the expected sum of the squared errors of the reconciled

forecasts, under the assumption of unbiased base forecasts.

2.3 Probabilistic reconciliation

Probabilistic reconciliation requires a probabilistic framework, in which

forecasts are in the form of probability distributions. We denote by

ˆν∈ P(Rn) the forecast distribution for y, where P(Rn) is the space of

probability measures on (Rn,B(Rn)), and B(Rn) is the Borel σ-algebra

on Rn. Moreover, we denote by ˆνuand ˆνbthe marginal distributions of,

respectively, the forecasts for the upper and the bottom components of y.

The forecast distribution ˆνmay be either discrete or absolutely con-

tinuous. In the following, if there is no ambiguity, we will use ˆπto denote

either its probability mass function, in the former case, or its density, in

the latter. Therefore, if ˆνis discrete, we have

ˆν(F) = X

x∈F

ˆπ(x),

for any F∈ B(Rn). Note that the sum is well-deﬁned as ˆπ(x)>0 for at

most countably many x’s. On the contrary, if ˆνis absolutely continuous,

for any F∈ B(Rn) we have

ˆν(F) = ZF

ˆπ(x)dx.

3 Probabilistic Reconciliation

We now discuss coherence in the probabilistic framework and our approach

to probabilistic reconciliation.

Recall that a point forecast is incoherent if it does not belong to the

set S, deﬁned as in (2). Let ˆν∈ P(Rn) be a forecast distribution. Thus,

ˆνis incoherent if there exists a set Tof incoherent points, i.e. T∩ S =∅,

such that ˆν(T)>0. Or, equivalently, if supp(ˆν)⊈S. We now deﬁne the

summing map s:Rnb→Rnas

s(b) = Sb.(3)

The image of sis given by S. Moreover, from (3) and (1), sis injective.

Hence, sis a bijective map between Rnband S, with inverse given by

s−1(y) = b, where y= (u,b)∈ S. As explained in Panagiotelis et al.

[2023], for any ν∈ P(Rnb) we may obtain a distribution ˜ν∈ P(S) as

˜ν=s#ν, namely the pushforward of νusing s:

˜ν(F) = ν(s−1(F)),∀F∈ B(S),

where s−1(F) := {b∈Rnb:s(b)∈F}is the preimage of F. In other

words, s#builds a probability distribution for ysupported on the coherent

subspace Sfrom a distribution on the bottom variables b. Since sis a

measurable bijective map, s#is a bijection between P(Rnb) and P(S),

with inverse given by (s−1)#(Appendix A). We thus propose the following

deﬁnition.

Deﬁnition 1. We call coherent distribution any distribution ν∈ P(Rnb).

This deﬁnition works with any type of distribution. Moreover, it can

be used even if the constraints are not linear, as it does not require sto

be a linear map.

3.1 Probabilistic reconciliation

The aim of probabilistic reconciliation is to obtain a coherent reconciled

distribution ˜ν∈ P(Rnb) from the base forecast distribution ˆν∈ P(Rn).

The probabilistic bottom-up approach, which simply ignores any prob-

abilistic information about the upper series, is obtained by setting ˜ν= ˆνb.

Panagiotelis et al. [2023] proposes a reconciliation method based on

projection. Given a continuous map ψ:Rn→ S, the reconciled dis-

tribution ˜ν∈ P(S) is deﬁned as the push-forward of the base forecast

distribution ˆνusing ψ:

˜ν=ψ#ˆν,

i.e. ˜ν(F) = ˆν(ψ−1(F)), for any F∈ B(Rn). Hence, if y1,...,yNare inde-

pendent samples from ˆν, then ψ(y1),...,ψ(yN) are independent samples

from ˜ν. The map ψis expressed as ψ=s◦g, where g:Rn→Rnbcom-

bines information from all the levels by projecting on the bottom level.

gis assumed to be in the form g(y) = d+Gy, and the parameters

γ:= (d,vec(G)) ∈Rnb+nb×nare optimized through stochastic gradient

descent (SGD) to minimize a chosen scoring rule. This approach therefore

can only be used with continuous distributions.

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Efficientprobabilisticreconciliationofforecastsforreal-valuedandcounttimeseriesLorenzoZambon∗,†DarioAzzimonti∗GiorgioCorani∗AbstractHierarchicaltimeseriesarecommoninseveralappliedfields.Thefore-castsforthesetimeseriesarerequiredtobecoherent,thatis,tosatisfytheconstraintsgivenbythehierarchy.Themostpo...

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Efficient probabilistic reconciliation of forecasts for real-valued and count time series Lorenzo ZambonDario AzzimontiGiorgio Corani

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