examining the effect of temporal aggregation on forecasting … · 2017. 4. 7. · spiliotis...

National Technical University of Athens- Forecasting & Strategy Unit

34th International Symposium on Forecasting Rotterdam, Netherlands – Hierarchical Time Series

Spiliotis Evangelos – Forecasting & Time Series Prediction stream

Examining the effect of temporal aggregation on forecasting accuracy for hierarchical energy consumption time series

Evangelos Spiliotis

Co-Authors: Dr. Fotios Petropoulos Dr. Nikolaos Kourentzes Prof. Vassilios Assimakopoulos

National Technical University of Athens Forecasting & Strategy Unit Lancaster University Lancaster Centre for Forecasting

©1-July-2014, Evangelos Spiliotis

A challenging problem

Hourly data (energy consumption in kWh)

Duration of ~9.5 weeks (1612 hours, from 5-Jan-12 to 12-March-12)

Data source: The monitoring systems installed in the bank branches

Bank

Branch1

HVAC1

UPS1

Lighting1

Branch2

HVAC2

UPS2

Lighting2

Branch3

HVAC3

UPS3

Lighting3

Branch4

HVAC4

UPS4

Lighting4

Branch5

HVAC5

UPS5

Lighting5

We need to forecast the electrical consumption of a Greek bank across three hierarchical levels: The bank (Lvl 0) The five branches composing it (Lvl 1) The end uses of each branch (Lvl 2): HVAC,

devices connected to UPS and Lighting

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34th International Symposium on Forecasting Rotterdam, Netherlands - Hierarchical Time Series

What about hierarchical aggregation? In commercial building sector, cross-sectional aggregation refers to the aggregation of multiple energy loads for the formation of

families of energy demand

Why using it? Reconciliation of forecasts Improvements on forecasting performance (maybe)

How? Hierarchical time series are commonly forecasted using:

Bottom-Up Top-Down or Optimal method (Hyndman, Ahmed, Athanasopoulos, Shang, 2011)

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Why not combine with temporal aggregation?

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Emphasizes different time-series characteristics by transforming the original data to alternative time frequencies

Possible gains:

Improvements on forecasting accuracy (A study on ARIMA models-Abraham, 1982)

Reduction of bias (Mohammadipour & Boylan, 2012)

MAPA (Kourentzes, Petropoulos & Trapero, 2014) is a generalized methodology for applying temporal aggregation Aggregates time series into different levels (typically up to annual data) Fits an appropriate model to each aggregation level Combines information from all models to a reconciled so that both short and

long term forecasts are performing well

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Our Motivation Empirical studies have shown that temporal aggregation, and

especially MAPA, can boost the performance of forecasting in time series with characteristics similar to those of our case-study.

Proper reconciliation of forecasts can complement and reinforce our results

The combination of sub-aggregation and cross-sectional aggregation leads to better forecasts than B-U and T-D approaches individually (Tabar, Babai, Ducq & Syntetos, OR55 – 2013)

Research questions Does a combination of temporal and hierarchical aggregation lead to better forecasts? If yes, which is the contribution of each aggregation method?

Main issues in our case-study

• Capture effectively the intense seasonal pattern of the time series

• Choose an appropriate forecasting model

• Deal with the noise which becomes dominant on the low levels of the pyramid

• Remain accurate for long-term forecasts

• Manage special events & bank holidays

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Capturing seasonality

We decided to use classical mean multiplicative decomposition of seasonal frequency 168 Seasonal plots (Hyndman & Athanasopoulos -2012) were provided by R: Forecast Package

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Model selection for MAPA process Simple exponential smoothing: Fast moving data - no trend

Consequently, MAPA gets a simplified form

SES leads to zero trend components

A priori decomposition leads to zero seasonal components

f_MAPA = mapasimple(des_insample, ppy, fh = fh, minimumAL = 1, maximumAL = ppy, comb = "median", output = "forecast", paral = 0, display = 0, outplot = 2, hybrid = FALSE, model = "ANN")


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Methodological Approach

Classical multiplicative

decomposition Frequency 168

Bank

Branches

End Uses

SES level 1

SES

SES

level 2

level k

.

.

.

Temporal Aggregated time series

level 2

Temporal Aggregated time series

level k

Average level

Seasonalization of data

Bottom-Up Optimal Top-Down

Hierarchical levels h=0,1,2 h=2 h=0 h=1

Bank Branches

End Uses

Final forecasts per hierarchical level

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Experimental design 10

Accuracy evaluation through symmetric Mean Absolute Percentage error

𝑠𝑀𝐴𝑃𝐸 =1

𝑛 𝑌𝑖 − 𝐹𝑖𝑌𝑖 + 𝐹𝑖2

100(%)

𝑛

𝑖=1


21 time series referring to 5 bank branches and 3 end uses Forecasting horizons tested: 1 to 7 days ahead Out-of sample evaluation Rolling origin procedure was used for better evaluating the forecasts. The

procedure stops when the forecasting horizon corresponds to the last of the available sample (Tashman - 2000)

Benchmarks: Naïve 2 and SES

Bias evaluation through Mean Scaled error

𝑠𝐸𝑖,ℎ =𝑦𝑁+ℎ − 𝐹ℎ1𝑁 𝑦𝑡𝑁𝑡=1

• An accuracy scaled error metric was also used providing similar conclusions with sMAPE (naïve was set as a benchmark)

• The performance differences between MAPA and the benchmark methods were consistent across the forecasting horizons. Average performance will be presented

Accuracy Evaluation 11


MAPA leads to more accurate forecasts regardless the hierarchical method used Bottom-up dominates the rest of the methods whether temporal aggregation is applied

or not Lvl2 is the level mostly benefited from temporal aggregation (29%), followed by lvl1

(19%) and lvl0 (7%) This is translated to a 19%, 3% and 18% gain in B.U., T.D. and Optimal method,

respectively: The higher the gain from MAPA the better the performance after reconciling the forecasts

0% 10% 20% 30% 40% 50%

Bottom-Up

Top-Down

Optimal

Average sMAPE gains of MAPA over SES per Hierarchical Method

Level 2

Level 1

Level 0

0.00 10.00 20.00 30.00 40.00

MAPA

SES

Naïve 2

Average sMAPE of hierarchical methods per forecasting method (equal weights per level)

Average Optimal

Average T.D.

Average B.U.

Bias Evaluation

MAPA leads to less biased forecasts regardless the hierarchical method used

Bottom-up dominates the rest of the methods when temporal aggregation is applied

The bottom-up approach is highly benefited mainly due to lvl2 performance. Similarly Top-down is mostly benefited by lvl0 performance, while in Optimal approach all levels are similarly affected

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-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

MAPA

SES

Naïve 2

Average sME of hierarchical methods per forecasting method (equal weights per level)

Average Optimal

Average T.D.

Average B.U.

0% 50% 100% 150% 200%

Bottom-Up

Top-Down

Optimal

Average sME gains of MAPA over SES per Hierarchical Method

Level 2

Level 1

Level 0

Summarizing our results

• The present work is a prototype of combining cross-sectional with temporal aggregation

• When MAPA forecasts are used for cross-sectional aggregation, hierarchical forecasting performance is boosted both in terms of bias and accuracy regardless the hierarchical method used

• Benefits are larger when long term forecasts are needed, or noisy data are available. This reinforces the existing findings in the literature (Kourentzes, Petropoulos & Trapero - 2014)

• If noise does not dominate in any hierarchical level, Optimal method seems to be the best choice for reorganizing forecasts across the hierarchy


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Further work to be made Specify the performance gains of hybrid aggregation over simple

cross-sectional aggregation

Examine the effect of different temporal and cross-sectional combinations on forecasting accuracy and bias: Cross-sectional aggregation first and then temporal aggregation

Cross-sectional and temporal aggregation at the same time

Evaluate MAPA’s performance on very short (hour level) forecasting horizons

Modify MAPA’s algorithm is order to weight temporal components averaged based on the forecasting horizon set

34th International Symposium on Forecasting

Rotterdam, Netherlands - Hierarchical Time Series

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References

• N. Kourentzes, F. Petropoulos, J. R. Trapero (2014). Improving forecasting by estimating time series structural components across multiple frequencies, International Journal of Forecasting, Vol. 30, pp. 291-302

• F. Petropoulos, N. Kourentzes (2014). Improving forecasting via multiple temporal aggregation, Foresight (forthcoming)

• F. Petropoulos, N. Kourentzes (2014). Forecast Combinations for Intermittent Demand, Journal of Operational Research Society (forthcoming)

• R.J. Hyndman, R.A. Ahmed, G. Athanasopoulos, H.L. Shang (2011). Optimal combination forecasts for hierarchical time series, Computational Statistics and Data Analysis, Vol. 55, pp. 2579-2589

• B. Abraham (1982). Temporal aggregation and time series, International Statistical Review, Vol. 50, pp. 285-291

• M. Mohammadipour & J.E. Boylan (2012). Forecast horizon aggregation in integer autoregressive moving average (INARMA) models, Omega, Vol. 40, issue 6, pp. 703-712

• L.J. Tashman (2000). Out-of-sample tests of forecasting accuracy: an analysis and review, International Journal of Forecasting, Vol. 16, pp. 437-450

• B.R. Tabar, M.-Z. Babai, Y. Ducq, A. Syntetos (2013). Demand forecasting by hybrid temporal and cross-sectional aggregation, OR55 conference

• G. Athanasopoulos, S. Razbash, D. Schmidt, Z. Zhou, Y. Khan, C. Bergmeir & E. Wang (2014). Forecast package for R v 5.4: Forecasting functions for time series and linear models

• N. Kourentzes & F. Petropoulos (2014). MAPA package for R v1.5: Multiple Aggregation Prediction Algorithm

34th International Symposium on Forecasting Rotterdam, Netherlands - Energy Forecasting Spiliotis

Evangelos – Forecasting & Time Series Prediction stream

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Thank you for your attention Any questions?

If you would like more information about our work contact me at:

[email protected]

Or visit forecasting & strategy unit’s website

http://www.fsu.gr

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mailto:[email protected]

http://www.fsu.gr/

examining the effect of temporal aggregation on forecasting … · 2017. 4. 7. · spiliotis...

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