In many real-world applications, time-series data are organized in structured forms—for example, products within categories, stores within regions, or states within a country. When these series are related through aggregation, they are referred to as Hierarchical or Grouped time series.
These structures arise naturally in:
Retail forecasting (products, stores, regions)
Tourism demand (states, purposes, demographics)
Supply chain management (warehouses, distribution centers)
Economic data (national, regional, sectoral aggregates)
Understanding the difference between hierarchical and grouped structures is crucial for forecast reconciliation and maintaining consistency across aggregation levels.
2 Hierarchical Time Series (HTS)
2.1 Definition
A hierarchical time series is organized in a tree-like structure, where observations at higher levels are obtained by summing the series at lower levels.
Key characteristics:
Each parent node in the hierarchy is the aggregate of its child nodes
Aggregation flows along a single path, typically from the bottom (most detailed level) to the top (overall total)
Forms a strict tree structure with no cross-classification
2.2 Example: Retail Sales Hierarchy
Consider a retail company tracking sales across a geographic hierarchy:
Total (Country)
├── State A
│ ├── Region A1
│ │ ├── Store A1a
│ │ └── Store A1b
│ └── Region A2
│ ├── Store A2a
│ └── Store A2b
└── State B
├── Region B1
│ └── Store B1a
└── Region B2
└── Store B2a
Here, store-level sales sum to region-level totals, region-level totals sum to state totals, and state totals sum to the national level.
2.3 Mathematical Representation
Let \(y_t\) denote the total series at time \(t\), and let \(\mathbf{b}_t\) denote the vector of bottom-level series. The hierarchical structure can be expressed using a summing matrix\(\mathbf{S}\):
\[
\mathbf{y}_t = \mathbf{S} \mathbf{b}_t
\]
where \(\mathbf{y}_t\) contains all series in the hierarchy (from top to bottom), and \(\mathbf{S}\) defines the aggregation structure.
2.3.1 Example with 3 bottom-level series
Suppose we have three stores (A, B, C) in two regions:
A grouped time series does not follow a single hierarchical tree. Instead, the same observations can be aggregated along multiple independent dimensions—for example, region, product type, or customer segment.
Key characteristics:
Each dimension provides a different aggregation view of the same base data
The structure allows cross-classification, meaning a single observation may belong to several groups simultaneously
Multiple valid aggregation paths exist
3.2 Example: Tourism Data
A tourism company tracks visitor numbers by both state and purpose of travel:
State
Purpose
Visitors
NSW
Business
1,200
NSW
Leisure
3,400
VIC
Business
800
VIC
Leisure
2,100
This dataset has two aggregation dimensions:
By State: NSW, VIC (geographical aggregation)
By Purpose: Business, Leisure (functional aggregation)
Because these dimensions intersect, the same dataset can be summarized in multiple ways:
State totals: NSW total, VIC total
Purpose totals: Business total, Leisure total
Grand total: All visitors
3.3 Grouped Structure Visualization
Total
/ \
/ \
By State By Purpose
/ \ / \
NSW VIC Business Leisure
/ \ / \
NSW×Bus NSW×Leis VIC×Bus VIC×Leis
Note that the bottom level (NSW×Business, NSW×Leisure, etc.) is the same in both aggregation paths, but the intermediate levels differ.
3.4 Mathematical Representation
Similar to HTS, we can express a grouped time series using a summing matrix:
\[
\mathbf{y}_t = \mathbf{S} \mathbf{b}_t
\]
However, \(\mathbf{S}\) now contains multiple aggregation paths corresponding to different grouping variables.
3.4.1 Example: Two-way Grouping
For the tourism example with 2 states × 2 purposes = 4 bottom-level series:
More complex, must reconcile across multiple views
Use Case
Forecasts that roll up cleanly through levels
Forecasts that can be sliced by different dimensions
Flexibility
Single view of aggregation
Multiple simultaneous views
Important
You are not asking a different research question when you model as GTS vs HTS. You are asking the same economic question (e.g., forecasting tourism demand), but you are telling the software how the data are organized and should be reconciled.
That structure determines:
how forecasts aggregate or cross-classify,
which reconciliation methods are valid, and
how is_aggregated() will recognize levels for plotting or filtering.
You are defining what belongs to what, not yet what to forecast.
The model itself (ETS, ARIMA, etc.) doesn’t “know” whether it’s HTS or GTS — it’s the reconciliation step that uses the structure.
5 Forecast Reconciliation
5.1 The Challenge
When forecasting hierarchical or grouped time series, forecasts made at different levels may be incoherent—that is, the forecasts at upper levels may not equal the sum of forecasts at lower levels.
Example of incoherence:
Forecast for Total = 1,000 units
Forecast for State A = 600 units
Forecast for State B = 500 units
Problem: 600 + 500 = 1,100 ≠ 1,000
5.2 Reconciliation Methods
Reconciliation adjusts forecasts to ensure internal consistency across all levels of the hierarchy or grouping structure.
5.2.1 Common Approaches
Bottom-up: Forecast only at the bottom level, then aggregate
Pros: Simple, respects detailed patterns
Cons: Bottom-level forecasts may be noisy
Top-down: Forecast at the top level, then disaggregate using proportions
Pros: Stable aggregate forecast
Cons: May lose detailed information
Middle-out: Forecast at an intermediate level, then aggregate up and disaggregate down
Optimal reconciliation (MinT): Use all forecasts and find optimal combination that minimizes forecast error variance
These four methods — bottom-up, top-down, middle-out, and MinT — are reconciliation methods, not forecasting models.
Forecasting models, on the other hand, are statistical or machine learning models used to produce base forecasts for each series before reconciliation like ETS, ARIMA, TSLM, Neural network, etc. Each of these models produces unreconciled (independent) forecasts.
where \(\mathbf{G}\) is a matrix chosen to minimize forecast error variance subject to the constraint that \(\mathbf{S}\mathbf{G}\mathbf{S} = \mathbf{S}\) (aggregating reconciled forecasts at the bottom level equals the reconciled forecasts at higher levels).
S enforces aggregation structure.
G defines how adjustments propagate to ensure coherence.
Note
All reconciliation methods (BU, TD, MO, MinT) apply to both HTS and GTS.
- The underlying math (via the summing matrix (S)) is the same.
- HTS: simpler tree structure.
- GTS: multiple crossing dimensions → larger (S), more complex reconciliation.
6 Practical Considerations
6.1 When to Use HTS vs GTS
Use Hierarchical Time Series when:
Data has a natural tree structure (geography, products)
Single aggregation dimension is sufficient
Simpler reconciliation is desired
Use Grouped Time Series when:
Multiple independent categorizations exist
Cross-tabulations are meaningful
Flexibility in aggregation views is needed
6.2 Software Implementation
Both HTS and GTS can be handled in R using the fable and fabletools packages:
A dataset containing the quarterly overnight trips from 1998 Q1 to 2016 Q4 across Australia.
A tsibble with 23,408 rows and 5 variables:
Quarter: Year quarter (index)
Region: The tourism regions are formed through the aggregation of Statistical Local Areas (SLAs) which are defined by the various State and Territory tourism authorities according to their research and marketing needs
State: States and territories of Australia
Purpose: Stopover purpose of visit:
“Holiday”
“Visiting friends and relatives”
“Business”
“Other reason”
Trips: Overnight trips in thousands
In a tsibble (tidy time series), each observation is uniquely identified by:
A time index → e.g. Quarter
A key → one or more variables that define individual series
We can see that there are 5 total columns, with 4 character vectors and only 1 numeric variable. We can examine the variable names in multiple ways -
We can see from attr(*, "key")=tsibble[304x4] that there are 304 unique time series in the tsibble, each defined by the key that is a combination of Region + State + Purpose.
remove(list =ls())library(fable)
Loading required package: fabletools
Registered S3 method overwritten by 'tsibble':
method from
as_tibble.grouped_df dplyr
library(fabletools)library(tsibble)
Attaching package: 'tsibble'
The following objects are masked from 'package:base':
intersect, setdiff, union
library(skimr)library(dplyr)
Attaching package: 'dplyr'
The following objects are masked from 'package:stats':
filter, lag
The following objects are masked from 'package:base':
intersect, setdiff, setequal, union
# A tsibble: 6 x 5 [1Q]
# Key: Region, State, Purpose [1]
Quarter Region State Purpose Trips
<qtr> <chr> <chr> <chr> <dbl>
1 1998 Q1 Adelaide South Australia Business 135.
2 1998 Q2 Adelaide South Australia Business 110.
3 1998 Q3 Adelaide South Australia Business 166.
4 1998 Q4 Adelaide South Australia Business 127.
5 1999 Q1 Adelaide South Australia Business 137.
6 1999 Q2 Adelaide South Australia Business 200.
6.3 Understanding the key
Confirm the key.
key_vars(tourism)
[1] "Region" "State" "Purpose"
index_var(tourism)
[1] "Quarter"
# more brute forcetsibble::tourism %>%distinct(Region, State, Purpose) %>%count()
# A tibble: 1 × 1
n
<int>
1 304
table(tourism$State)
ACT New South Wales Northern Territory Queensland
320 4160 2240 3840
South Australia Tasmania Victoria Western Australia
3840 1600 6720 1600
table(tourism$Region)
Adelaide Adelaide Hills
320 320
Alice Springs Australia's Coral Coast
320 320
Australia's Golden Outback Australia's North West
320 320
Australia's South West Ballarat
320 320
Barkly Barossa
320 320
Bendigo Loddon Blue Mountains
320 320
Brisbane Bundaberg
320 320
Canberra Capital Country
320 320
Central Coast Central Highlands
320 320
Central Murray Central NSW
320 320
Central Queensland Clare Valley
320 320
Darling Downs Darwin
320 320
East Coast Experience Perth
320 320
Eyre Peninsula Fleurieu Peninsula
320 320
Flinders Ranges and Outback Fraser Coast
320 320
Geelong and the Bellarine Gippsland
320 320
Gold Coast Goulburn
320 320
Great Ocean Road High Country
320 320
Hobart and the South Hunter
320 320
Kakadu Arnhem Kangaroo Island
320 320
Katherine Daly Lakes
320 320
Lasseter Launceston, Tamar and the North
320 320
Limestone Coast MacDonnell
320 320
Macedon Mackay
320 320
Mallee Melbourne
320 320
Melbourne East Murray East
320 320
Murraylands New England North West
320 320
North Coast NSW North West
320 320
Northern Outback
320 320
Outback NSW Peninsula
320 320
Phillip Island Riverina
320 320
Riverland Snowy Mountains
320 320
South Coast Spa Country
320 320
Sunshine Coast Sydney
320 320
The Murray Tropical North Queensland
320 320
Upper Yarra Western Grampians
320 320
Whitsundays Wilderness West
320 320
Wimmera Yorke Peninsula
320 320
table(tourism$Region, tourism$State)
ACT New South Wales Northern Territory
Adelaide 0 0 0
Adelaide Hills 0 0 0
Alice Springs 0 0 320
Australia's Coral Coast 0 0 0
Australia's Golden Outback 0 0 0
Australia's North West 0 0 0
Australia's South West 0 0 0
Ballarat 0 0 0
Barkly 0 0 320
Barossa 0 0 0
Bendigo Loddon 0 0 0
Blue Mountains 0 320 0
Brisbane 0 0 0
Bundaberg 0 0 0
Canberra 320 0 0
Capital Country 0 320 0
Central Coast 0 320 0
Central Highlands 0 0 0
Central Murray 0 0 0
Central NSW 0 320 0
Central Queensland 0 0 0
Clare Valley 0 0 0
Darling Downs 0 0 0
Darwin 0 0 320
East Coast 0 0 0
Experience Perth 0 0 0
Eyre Peninsula 0 0 0
Fleurieu Peninsula 0 0 0
Flinders Ranges and Outback 0 0 0
Fraser Coast 0 0 0
Geelong and the Bellarine 0 0 0
Gippsland 0 0 0
Gold Coast 0 0 0
Goulburn 0 0 0
Great Ocean Road 0 0 0
High Country 0 0 0
Hobart and the South 0 0 0
Hunter 0 320 0
Kakadu Arnhem 0 0 320
Kangaroo Island 0 0 0
Katherine Daly 0 0 320
Lakes 0 0 0
Lasseter 0 0 320
Launceston, Tamar and the North 0 0 0
Limestone Coast 0 0 0
MacDonnell 0 0 320
Macedon 0 0 0
Mackay 0 0 0
Mallee 0 0 0
Melbourne 0 0 0
Melbourne East 0 0 0
Murray East 0 0 0
Murraylands 0 0 0
New England North West 0 320 0
North Coast NSW 0 320 0
North West 0 0 0
Northern 0 0 0
Outback 0 0 0
Outback NSW 0 320 0
Peninsula 0 0 0
Phillip Island 0 0 0
Riverina 0 320 0
Riverland 0 0 0
Snowy Mountains 0 320 0
South Coast 0 320 0
Spa Country 0 0 0
Sunshine Coast 0 0 0
Sydney 0 320 0
The Murray 0 320 0
Tropical North Queensland 0 0 0
Upper Yarra 0 0 0
Western Grampians 0 0 0
Whitsundays 0 0 0
Wilderness West 0 0 0
Wimmera 0 0 0
Yorke Peninsula 0 0 0
Queensland South Australia Tasmania Victoria
Adelaide 0 320 0 0
Adelaide Hills 0 320 0 0
Alice Springs 0 0 0 0
Australia's Coral Coast 0 0 0 0
Australia's Golden Outback 0 0 0 0
Australia's North West 0 0 0 0
Australia's South West 0 0 0 0
Ballarat 0 0 0 320
Barkly 0 0 0 0
Barossa 0 320 0 0
Bendigo Loddon 0 0 0 320
Blue Mountains 0 0 0 0
Brisbane 320 0 0 0
Bundaberg 320 0 0 0
Canberra 0 0 0 0
Capital Country 0 0 0 0
Central Coast 0 0 0 0
Central Highlands 0 0 0 320
Central Murray 0 0 0 320
Central NSW 0 0 0 0
Central Queensland 320 0 0 0
Clare Valley 0 320 0 0
Darling Downs 320 0 0 0
Darwin 0 0 0 0
East Coast 0 0 320 0
Experience Perth 0 0 0 0
Eyre Peninsula 0 320 0 0
Fleurieu Peninsula 0 320 0 0
Flinders Ranges and Outback 0 320 0 0
Fraser Coast 320 0 0 0
Geelong and the Bellarine 0 0 0 320
Gippsland 0 0 0 320
Gold Coast 320 0 0 0
Goulburn 0 0 0 320
Great Ocean Road 0 0 0 320
High Country 0 0 0 320
Hobart and the South 0 0 320 0
Hunter 0 0 0 0
Kakadu Arnhem 0 0 0 0
Kangaroo Island 0 320 0 0
Katherine Daly 0 0 0 0
Lakes 0 0 0 320
Lasseter 0 0 0 0
Launceston, Tamar and the North 0 0 320 0
Limestone Coast 0 320 0 0
MacDonnell 0 0 0 0
Macedon 0 0 0 320
Mackay 320 0 0 0
Mallee 0 0 0 320
Melbourne 0 0 0 320
Melbourne East 0 0 0 320
Murray East 0 0 0 320
Murraylands 0 320 0 0
New England North West 0 0 0 0
North Coast NSW 0 0 0 0
North West 0 0 320 0
Northern 320 0 0 0
Outback 320 0 0 0
Outback NSW 0 0 0 0
Peninsula 0 0 0 320
Phillip Island 0 0 0 320
Riverina 0 0 0 0
Riverland 0 320 0 0
Snowy Mountains 0 0 0 0
South Coast 0 0 0 0
Spa Country 0 0 0 320
Sunshine Coast 320 0 0 0
Sydney 0 0 0 0
The Murray 0 0 0 0
Tropical North Queensland 320 0 0 0
Upper Yarra 0 0 0 320
Western Grampians 0 0 0 320
Whitsundays 320 0 0 0
Wilderness West 0 0 320 0
Wimmera 0 0 0 320
Yorke Peninsula 0 320 0 0
Western Australia
Adelaide 0
Adelaide Hills 0
Alice Springs 0
Australia's Coral Coast 320
Australia's Golden Outback 320
Australia's North West 320
Australia's South West 320
Ballarat 0
Barkly 0
Barossa 0
Bendigo Loddon 0
Blue Mountains 0
Brisbane 0
Bundaberg 0
Canberra 0
Capital Country 0
Central Coast 0
Central Highlands 0
Central Murray 0
Central NSW 0
Central Queensland 0
Clare Valley 0
Darling Downs 0
Darwin 0
East Coast 0
Experience Perth 320
Eyre Peninsula 0
Fleurieu Peninsula 0
Flinders Ranges and Outback 0
Fraser Coast 0
Geelong and the Bellarine 0
Gippsland 0
Gold Coast 0
Goulburn 0
Great Ocean Road 0
High Country 0
Hobart and the South 0
Hunter 0
Kakadu Arnhem 0
Kangaroo Island 0
Katherine Daly 0
Lakes 0
Lasseter 0
Launceston, Tamar and the North 0
Limestone Coast 0
MacDonnell 0
Macedon 0
Mackay 0
Mallee 0
Melbourne 0
Melbourne East 0
Murray East 0
Murraylands 0
New England North West 0
North Coast NSW 0
North West 0
Northern 0
Outback 0
Outback NSW 0
Peninsula 0
Phillip Island 0
Riverina 0
Riverland 0
Snowy Mountains 0
South Coast 0
Spa Country 0
Sunshine Coast 0
Sydney 0
The Murray 0
Tropical North Queensland 0
Upper Yarra 0
Western Grampians 0
Whitsundays 0
Wilderness West 0
Wimmera 0
Yorke Peninsula 0
6.3.1 Structure is confirmed as (State / Region) * Purpose.
Read data dictionary, eyeball/View(), use commands to confirm the data structure is HTS or GTS. Then model accordingly.
# 1. Create hierarchical and grouped structurestourism_hts <- tourism %>%aggregate_key(State / Region, Trips =sum(Trips))tourism_gts <- tourism %>%aggregate_key(State * Purpose, Trips =sum(Trips))# 2. Fit models (creates a mable)fit_gts <- tourism_gts %>%model(ets =ETS(Trips))# 3. Reconcile (applied to the model table, not forecasts)fit_reconciled <- fit_gts %>%reconcile(bu =bottom_up(ets), # Bottom-Up: sum from bottom-level series# td_fp = top_down(ets, method = "forecast_proportions"), # Top-Down: split using forecast proportions# td_ap = top_down(ets, method = "average_proportions"), # Top-Down: split using average proportionsmint_shrink =min_trace(ets, method ="mint_shrink"), # MinT Shrinkagemint_ols =min_trace(ets, method ="ols") # MinT OLS )# 4. Forecast from the reconciled modelfc <- fit_reconciled %>%forecast(h =8) # 8 quarters = 2 yearshead(fc)
# A fable: 6 x 6 [1Q]
# Key: State, Purpose, .model [1]
Loading required namespace: crayon
State Purpose .model Quarter
<chr*> <chr*> <chr> <qtr>
1 ACT Business ets 2018 Q1
2 ACT Business ets 2018 Q2
3 ACT Business ets 2018 Q3
4 ACT Business ets 2018 Q4
5 ACT Business ets 2019 Q1
6 ACT Business ets 2019 Q2
# ℹ 2 more variables: Trips <dist>, .mean <dbl>
6.4 Reconciliation Method Nuances
6.4.1 Bottom-Up (BU)
Use when: bottom-level data are rich and reliable.
method = "mint_shrink" → shrinkage estimator for covariance (more stable)
Nuances:
Works for any structure — hierarchical, grouped, or mixed
Statistically optimal: uses correlations between series (“borrows strength”)
In 2-level hierarchies, acts like a weighted hybrid between top-down and bottom-up
Preferred in modern forecasting applications
6.4.5 Comparison Summary
Method
Works For
Needs Strict Hierarchy?
Behavior (2 Levels)
Strength
Limitation
Top-Down
Hierarchical only
Yes
Valid (splits total)
Uses stable totals
Fails for grouped structures
Bottom-Up
Hierarchical / Grouped
No
Simple summation
Coherent, interpretable
Propagates bottom-level noise
Middle-Out
Hierarchical (3+ levels)
Yes
Defaults to BU
Balances top & bottom
Not meaningful if only 2 levels
MinT-OLS / Shrink
All structures
No
Weighted optimum
Statistically efficient
Needs covariance estimation
7 Visualizing Reconciled Forecasts
After generating reconciled forecasts (fc), we visualize them to confirm:
Coherence – forecasts add up correctly across hierarchy levels.
Behavior – different reconciliation methods (Bottom-Up, Middle-Out, MinT) produce realistic, consistent trends.
Below we use autoplot() from fpp3 to examine forecasts at multiple levels.
# --- 1. Compare all reconciliation methods ------------------------------autoplot(fc, tourism_gts) +labs(title ="Forecast Reconciliation Methods (8-Quarter Horizon)",subtitle ="Comparing Bottom-Up, MinT (OLS), and MinT (Shrinkage)",y ="Trips",x ="Year" ) +facet_wrap(~ .model, scales ="free_y") +theme_minimal()
# --- 2. Focus on one geographic unit (Victoria) -------------------------fc %>%filter(State =="Victoria") %>%autoplot(tourism_gts) +labs(title ="Reconciled Forecasts for Victoria",subtitle ="Comparison across Bottom-Up and MinT methods",y ="Trips",x ="Year" ) +facet_wrap(~ .model) +theme_minimal()
# --- 3. Top-Level Aggregation: Total Tourism Trips ----------------------fc %>%filter(is_aggregated(State), is_aggregated(Purpose)) %>%autoplot(tourism_gts) +labs(title ="Total Tourism Trips – Coherent Forecasts",subtitle ="Top-Level Series after Reconciliation (All States × All Purposes)",y ="Trips",x ="Year" ) +facet_wrap(~ .model) +theme_minimal()
# --- 4. State-Level Totals (Aggregated over Purpose) --------------------fc %>%filter(!is_aggregated(State), is_aggregated(Purpose)) %>%autoplot(tourism_gts) +labs(title ="State-Level Total Tourism Trips (Aggregated over Purpose)",subtitle ="Useful for comparing geographic performance",y ="Trips",x ="Year" ) +facet_wrap(~ State) +theme_minimal()
# --- 5. Purpose-Level Totals (Aggregated over State) --------------------fc %>%filter(is_aggregated(State), !is_aggregated(Purpose)) %>%autoplot(tourism_gts) +labs(title ="Purpose-Level Total Tourism Trips (Aggregated over State)",subtitle ="Useful for analyzing trends by travel purpose",y ="Trips",x ="Year" ) +facet_wrap(~ Purpose) +theme_minimal()
Plot
Aggregation Level
Key Insight
1
All series
Compare reconciliation methods visually
2
Victoria only
Assess local forecast coherence
3
Total
Check top-level consistency
4
State totals
Compare states geographically
5
Purpose totals
Understand motivation trends
Forecasting Level (Your Choice)
Once the structure is declared and reconciled, you can forecast at any level you want — bottom, middle, or top.
For example, in HTS:
forecast at Region → automatically sums to State and Total.
forecast at State → middle-out.
forecast at Total → top-down.
In GTS:
forecast at State × Purpose → bottom-up aggregation gives totals.
forecast at State (aggregated over Purpose) or at Purpose (aggregated over State).
You can flexibly choose the forecast level at the end — because the structure ensures all levels remain coherent.
8 Summary
Note
Hierarchical Time Series (HTS): Data arranged in a single hierarchy; each level aggregates the ones below it
Single aggregation path
Tree structure
Simpler reconciliation
Grouped Time Series (GTS): Data structured across multiple, possibly intersecting hierarchies; provides multiple valid aggregation paths
Cross-classification
Multiple aggregation dimensions
More complex reconciliation required
Feature
Hierarchical (HTS)
Grouped (GTS)
Structure
Pure tree (one parent per series)
Cross-classification (multiple valid groupings)
Aggregation Path
Unique (e.g., Country → State → Region)
Multiple (e.g., State and Purpose both aggregate to Total)
Number of constraints
Simpler, fewer summation rules
More complex, overlapping constraints
Computational cost
Lower
Higher
Interpretation
Forecasts roll up neatly through one tree
Forecasts must satisfy several simultaneous totals
Forecast Reconciliation: Essential for maintaining internal consistency across all aggregation views
Ensures forecasts add up correctly
Optimal methods can improve forecast accuracy at all levels
Grouped structures increase flexibility but require more sophisticated forecast reconciliation to maintain internal consistency across all aggregation views.
Reconciliation Method Selection Guide
Top-Down → disaggregate from stable totals (strict hierarchies only)
Bottom-Up → aggregate from detailed data (always safe)
Middle-Out → start in the middle (redundant for 2 levels)
MinT → optimal hybrid, handles all structures efficiently
Use Bottom-Up for simplicity, Top-Down for stable aggregates, Middle-Out for deep hierarchies, and MinT for statistical optimality.
Reconciliation methods come after modeling, and adjust forecasts so that they are coherent — meaning they add up correctly across the hierarchy or grouping.
Reconciliation Method
What It Does
When It’s Used
Bottom-Up
Aggregates forecasts from lowest level upward
When bottom-level data are reliable
Top-Down
Disaggregates total forecasts down to components
When aggregate data are more stable
Middle-Out
Combines both: model at an intermediate level
When middle level is best trade-off
MinT (Minimum Trace)
Uses all levels and error covariances for optimal reconciliation
When accuracy and coherence are both important
9 Further Reading
Hyndman, R.J., & Athanasopoulos, G. (2021). Forecasting: Principles and Practice (3rd ed.), Chapter 11: Forecasting hierarchical and grouped time series. OTexts.
Wickramasuriya, S.L., Athanasopoulos, G., & Hyndman, R.J. (2019). Optimal forecast reconciliation for hierarchical and grouped time series through trace minimization. Journal of the American Statistical Association, 114(526), 804-819.
Athanasopoulos, G., Hyndman, R.J., Kourentzes, N., & Petropoulos, F. (2017). Forecasting with temporal hierarchies. European Journal of Operational Research, 262(1), 60-74.
9.1 Step-by-step workflow
When you begin working with hierarchical or grouped time series (like the tourism data), your process looks like this:
