To aggregate or not to aggregate: Forecasting of finiteautocorrelated series

.center[.title[.small[To aggregate or not to aggregate: Forecasting of finiteautocorrelated series]]]
.sticker-float[![fable](resources/carbs-white-background.jpg)]
.sticker-left[![isf](resources/DLSG.png)]

.center[JORS Research in Focus Webinar - 25 July 2025]

Bahman Rostami-Tabar, Professor of Analytics 
Director of Data Lab for Social Good Reserach Group, Cardiff University 
[Read paper:https://doi.org/10.1080/01605682.2022.2118631](https://www.tandfonline.com/doi/full/10.1080/01605682.2022.2118631) 
Slides: https://bahmanrt.netlify.app/talk/

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## Outline

- Introduction & Framing

- Literature Background & Contributions

- Model Setup

- Results: Analytical & Empirical

- Conclusion, Limitations & Future Work

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## Outline

- .remember[Introduction & Framing]

- Literature Background & Contributions

- Model Setup

- Results: Analytical & Empirical

- Conclusion, Limitations & Future Work

]

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## Should we forecast at hourly or weekly frequency?

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### Motivating example: hospital admissions

- Hourly data on patient admissions is available.

- The planning task requires weekly forecasts to allocate resources.

- Key decision: .remember[Forecast using hourly data directly and then aggregate forecast or aggregate to weekly and then forecast?]
]

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### Common across domains and time granulrity

Data is often collected at higher frequency (e.g., Minutes, hours) than the forecasting target frequency (e.g., day, week, month, quarter).

- Retail
- Energy
- Transportation
- Finance
- Manufacturing
- Agriculture
- and more

]

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## Temporal aggregation
### Transforming higher-frequency data into lower-frequency data

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## Temporal aggregation - a tardeoff

.pull-left[
<img src="figure/jors2025/TAexample.jpg" width ="450px" >
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- Signal-to-noise

- Model complexity
]

.footnote[Kourentzes, Nikolaos, Bahman Rostami-Tabar, and Devon K. Barrow. "series forecasting by temporal aggregation: Using optimal or multiple aggregation levels?." Journal of Business Research 78 (2017): 1-9.]

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## How TA affects time series features
### .small[Rostami-Tabar & Mircetic. Neurocomputing 548 (2023): 126376.]

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<img src="figure/jors2025/tas1.jpg" width ="400px" >
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<img src="figure/jors2025/tas2.jpg" width ="400px" >
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## Outline

- .graylight[Introduction & Framing]

- .remember[Literature Background & Contributions]

- Model Setup

- Results: Analytical & Empirical

- Conclusion, Limitations & Future Work

]

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## Temporal aggregation - a very brief history

.pull-left[

- The term .rememebr[temporal aggregation (TA)] emerged in the context of econometrics and time series analysis, dating back to the 1970s.
- "TA affects the specification of models, estimation of parameters an efficiency of forecasting" (Brewer (1973), Wei (1979).
]

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The term became particularly important in studies on:

- Macroeconomic modeling
- Autoregressive Integrated Moving Average (ARMA) processes.
- Forecasting

]

.footnote[Brewer, K.R.W. (1973). Some consequences of temporal aggregation and systematic sampling for ARIMA and ARMAX models. J. Econometrics 1, 133-154.
 
 Wei, W.W.S. (1979). Some consequences of temporal aggregation in seasonal time series models. In Seasonal Analysis of Economic Time Series, Ed. A. Zellner, pp. 433-444. Washington, D.C.; U.S. Department of
 Commerce, Bureau of the Census]
 
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## Two distinct approaches to forecasting with temporal aggregation

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### Approach 1: understanding and optimizing

- Investigates how and when TA improves forecast accuracy.

- Focuses on finding the optimal aggregation level for a given forecasting task.

- Evaluates trade-offs between noise reduction and information loss.
]

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### Approach 2: Combining information across temporal levels

- Leverages data from multiple levels of aggregation simultaneously (e.g., hourly + daily + weekly).

- Aims to improve forecast performance through multi-scale modeling or reconciliation.

- Reflects the hierarchical nature of many real-world decision processes.
]

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## How and when TA is useful? Finding optmial aggregation level

- Nikolopoulos, Konstantinos, et al. JORS 62.3 (2011): 544-554.
  - Empirical evaluation on intermittent series
  - TA can improve accuracy of forecasts
  - There might be an optimal aggregation level

- Rostami‐Tabar, Bahman, et al. Naval Research Logistics (NRL) 60.6 (2013): 479-498.
  - Assuming autocorrelated series, AR processes, and SES
  - Analytical MSE expressions for non-aggregated and non-overlapping aggregated series
  - We also provided an analytical proof showing when the non-overlapping TA approach outperforms the non-aggregated alternative.

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## Combining information from different levels

.pull-left[
- Kourentzes, Nikolaos, et al. International Journal of Forecasting 30.2 (2014): 291-302.
 - Multiple temporal aggregation levels
 
 <img src="figure/jors2025/mapa.png" width ="500px" >
]

.pull-right[
- Athanasopoulos, George, et al. EJOR 262.1 (2017): 60-74.
 - Temporal Hierarchies
 
 <img src="figure/jors2025/ta-hierarchy.png" width ="500px" >
]

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### Motivation for this paper

- Previous research focused solely on non-overlapping temporal aggregation 
- Previous research assumed infinite history length
- This paper considers both overlapping and non-overlapping temporal aggregation and compare with non-aggregation approach.

]

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### Objectives

[1.] We derive analytical MSE expressions under the three approaches when a finite history length is used.

[2.] We evaluate the performance of the three approaches by analysing
the impact of the length of the series, the aggregation level and the process parameters on the forecast performance.

[3.] Using monthly time series from the M4 competition, we empirically evaluate  the performance of the three approaches.

]

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## Outline

- .graylight[Introduction & Framing]

- .graylight[Literature Background & Contributions]

- .remember[Model Setup]

- Results: Analytical & Empirical

- Conclusion, Limitations & Future Work

]

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## Assumption about data

We assume that the non-aggregated series `$d_{t}$`, follows an ARMA(1,1) process:

`$$d_t = C + \epsilon_t + \phi d_{t-1} - \theta \epsilon_{t-1} \quad \text{where } |\theta| \leq 1,\ |\phi| \leq 1$$`

with a constant `$C$`, autoregressive coefficient `$\phi$`, and moving average coefficient `$\theta$`, and `$\epsilon_t \sim \mathcal{N}(0, \sigma^2)$`

`$\gamma_k = \text{Cov}(d_t, d_{t-k})$` =

`\begin{cases}
\displaystyle \frac{(1 - 2\phi\theta + \theta^2)}{1 - \phi^2} \sigma^2 & k = 0, \\
\displaystyle \frac{(\phi - \theta)(1 - \phi\theta)}{1 - \phi^2} \sigma^2 & |k| = 1, \\
\phi^{|k|-1} \gamma_1 & |k| > 1.
\end{cases}`

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## Autocorrelation associated with an ARMA(1,1) process

.center[<img src="figure/jors2025/acf.png" width ="600px" >]

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## Forecast

- Simple Exponential Smoothing (SES) is used
- We aim to forecast the cumulative (aggregated) series, written as follows:

`$$D_T = d_t + d_{t+1} + \cdots + d_{t+m-1}.$$`

`$$f_t = \sum_{k=1}^{N} \alpha (1 - \alpha)^{k-1} d_{t-k} + (1 - \alpha)^N f_0$$`

`$$F_{T,\text{NOA}} = \sum_{k=1}^{\left\lceil \frac{N}{m} \right\rceil} \beta_N (1 - \beta_N)^{k-1} D_{T-k,\text{NOA}} + \left(1 - \beta_N\right)^{\left\lceil \frac{N}{m} \right\rceil} F_{0,\text{NOA}}$$`

`$$F^{1}_{T,\text{OA}} = \sum_{k=1}^{N - m + 1} \beta_0 (1 - \beta_0)^{k - 1} D_{T - k,\text{OA}} + (1 - \beta_0)^{N - m + 1} F_{0,\text{OA}}$$`

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## Compare three approaches

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## MSE for three approaches
.pull-left[
 
`$$\text{MSE}_{\text{NA}} = \mathrm{var}(D_T - f_t^m) ,$$`

`$$\text{MSE}_{\text{NOA}} = \mathrm{var}\left(D_T - F_{T,\text{NOA}}^1\right),$$`

`$$\text{MSE}_{\text{OA}} = \mathrm{var}\left(D_T - F_{T,\text{OA}}^1\right).$$`
]

.pull-right[
- NA: forecasting method applied to non-aggregated data, then summed over horizon `$m$`

- NOA: forecasting method applied to non-overlapping aggregated data but a direct model for the cumulative target,

- OA: forecasting method applied to overlapping aggregated data but a direct model for the cumulative target,
]

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## MSE of non-aggrgate approach
.center[<img src="figure/jors2025/msena.png" width ="1000px" >]

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## MSE of non-overlapping aggregation approach
.center[<img src="figure/jors2025/msenoa.png" width ="1000px" >]

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## MSE of overlapping aggregation approach
.center[<img src="figure/jors2025/mseoa.png" width ="1000px" >]

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## Outline

- .graylight[Introduction & Framing]

- .graylight[Literature Background & Contributions]

- .graylight[Model Setup]

- .remember[Results: Analytical & Empirical]

- Conclusion, Limitations & Future Work

]

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## Results

.center[<img src="figure/jors2025/ratio.jpg" width ="650px" >]

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## Results - empirical data

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## Outline

- .graylight[Introduction & Framing]

- .graylight[Literature Background & Contributions]

- .graylight[Model Setup]

- .graylight[Results: Analytical & Empirical]

- .remember[Conclusion, Limitations & Future Work]

]

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## Conclusion
The question we address is fundamental— .remember[persisting over time] and .remember[remaining relevant] across diverse domains and temporal granularities.

- .remember[High positive autocorrelation:] Non-aggregated data yields lower MSEs.
- .remember[Negative autocorrelation]: TA outperforms non-aggregated forecasts.
- .remember[Alternating autocorrelation signs]: TA performs better than non-aggregated approaches.
- .remember[Longer forecast horizons:] Both overlapping and non-overlapping TA show improved accuracy.
- .remember[Short time series:] Overlapping TA is superior; differences diminish as history length increases.
- .remember[Diminishing returns:] The improvement in forecast accuracy decreases slowly beyond a certain series length for TA approaches.

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## Limitations

- .remember[Data generation process]: We assume that the disaggregated (non-aggregated) time series follows a stationary ARMA(1,1) process.

- .remember[Forecasting model]: We rely exclusively on Simple Exponential Smoothing (SES) as the forecasting method.

- .remember[Forecasting horizon]: The current framework focuses on generating a cumulative forecast over a fixed horizon `$M$`, effectively making it a one-step-ahead forecast in the aggregated setting.

- .remember[Empirical data]: The empirical evaluation is limited to the M4 competition data, which primarily consists of positive autocorrelated series.

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## Future Work
### Despite the contributions of this work, a key .remember[open question] remains

>There is still no .remember[general analytical framework] that explains when and how temporal aggregation affects forecast accuracy.

>While specific cases (e.g., ARMA processes with SES) can be studied in isolation, a global understanding—one that applies across models, aggregation schemes, and forecasting horizons—remains elusive.

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