Comparing Statistical and Machine Learning Methods for Time Series Forecasting in Data-Driven Logistics—A Simulation Study
Abstract
:1. Introduction
2. Methods
2.1. Time Series Methods
2.1.1. ARIMA
- AR (Autoregression): Represents the regression of the time series on its own past values, capturing dependencies through lagged observations. The number of lagged observations included in the models is given by p.
- I (Integrated): The differencing order (d) indicates the number of times the time series is differenced to achieve stationarity. This transformation involves subtracting the current observation from its d-th lag, which is crucial for stabilizing the mean and addressing trends.
- MA (Moving Average): Incorporates a moving average model to account for dependencies between observations and the residual errors of the lagged observations (q).
2.1.2. SARIMA
2.1.3. TBATS
- T (Trend): Captures the overall trend in the time series using an exponential smoothing mechanism.
- B (Box–Cox Transformation): Applies the Box–Cox transformation [50] to stabilize variance and ensure the homogeneity of variances.
- A (ARIMA Errors): Incorporates ARIMA errors to capture any remaining non-seasonal dependencies.
- S (Seasonal): Utilizes trigonometric functions to model multiple seasonal components, accommodating various seasonal patterns.
2.2. Machine Learning Methods
2.2.1. XGBoost
2.2.2. Random Forest
- B is the number of grown trees. Note that this parameter is usually not tuned since it is known that more trees are better.
- The cardinality of the sample of features at every node is .
- The minimum number of observations that each terminal node should contain (stopping criteria).
3. Simulation Set-Up
3.1. Data Generating Processes
3.2. Additional Complexities
3.3. Additional Queueing Models
3.4. Number of Different Settings
3.5. Data Preprocessing
3.6. Choice of Parameters
3.7. Evaluation Measure
4. Results
4.1. Predictive Power in Queueing Models
4.2. Predictive Power in the Different Time Series Settings
4.3. Influence of the Additional Complexities on the Predictive Power
4.4. Summarizing All Results
5. Real-World Data Example
6. Summary, Discussion, and Outlook
6.1. Summary with Highlights
- The out-of-the-box Random Forest emerged as the ML benchmark method.
- Training on differentiated time series can significantly improve the ML resilience.
- ML models are more robust with respect to additional (nonlinear) complexity, settings in which they outperformed statistical time series approaches.
- In all other settings, the time series approaches were at least competitive or even performed better.
6.2. Detailed Discussion and Outlook
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
References
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Model Type | Variant(s) | Data Generating Process |
---|---|---|
Autoregressive | AR | |
Bilinear | BL 1 | |
BL2 | ||
Nonlinear Autoregressive | NAR 1 | |
NAR2 | ||
Nonlinear Moving Average | NMA | |
Sign Autoregressive | SAR 1 | , |
SAR 2 | , | |
Smooth Transition Autoregressive | STAR 1 | , |
STAR 2 | , | |
Threshold Autoregressive | TAR 1 | |
TAR 2 |
DGP | RF | RF Diff | XGBoost | XGBoost Diff | ARIMA | SARIMA | TBATS | Naive | |
---|---|---|---|---|---|---|---|---|---|
Queueing Models | 7 | 1 | 7 | 5 | 2.5 | 3.5 | 3 | 6 | |
DGPS | no add. Compl. | 1 | 6 | 5 | 7 | 3 | 3 | 3 | 8 |
from | Jumps | 7 | 1 | 7 | 5 | 3 | 3 | 3 | 6 |
Table 1 | Random Walks | 5 | 1 | 7 | 6 | 3 | 3 | 3 | 8 |
with | Both | 7 | 1 | 7 | 6 | 3 | 3 | 3 | 5 |
MAPE | MSE | |||||
---|---|---|---|---|---|---|
Method | Prod. A | Prod. B | Prod. C | Prod. A | Prod. B | Prod. C |
Random Forest | 24.30 | 35.05 | 30.79 | 22.39 | 262.41 | 695.70 |
Random Forest Diff | 6.67 | 21.80 | 15.84 | 4.91 | 197.23 | 1.97 |
XGBoost | 25.06 | 41.62 | 19.51 | 22.34 | 376.62 | 147.20 |
XGBoost Diff | 10.70 | 37.98 | 27.15 | 13.10 | 841.56 | 41.00 |
(S)ARIMA | 28.57 | 49.30 | 33.56 | 29.48 | 1142.14 | 655.88 |
TBATS | 28.37 | 36.17 | 33.56 | 43.14 | 446.18 | 663.78 |
Naive | 33.18 | 30.71 | 30.59 | 25.10 | 194.21 | 82.03 |
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Schmid, L.; Roidl, M.; Kirchheim, A.; Pauly, M. Comparing Statistical and Machine Learning Methods for Time Series Forecasting in Data-Driven Logistics—A Simulation Study. Entropy 2025, 27, 25. https://doi.org/10.3390/e27010025
Schmid L, Roidl M, Kirchheim A, Pauly M. Comparing Statistical and Machine Learning Methods for Time Series Forecasting in Data-Driven Logistics—A Simulation Study. Entropy. 2025; 27(1):25. https://doi.org/10.3390/e27010025
Chicago/Turabian StyleSchmid, Lena, Moritz Roidl, Alice Kirchheim, and Markus Pauly. 2025. "Comparing Statistical and Machine Learning Methods for Time Series Forecasting in Data-Driven Logistics—A Simulation Study" Entropy 27, no. 1: 25. https://doi.org/10.3390/e27010025
APA StyleSchmid, L., Roidl, M., Kirchheim, A., & Pauly, M. (2025). Comparing Statistical and Machine Learning Methods for Time Series Forecasting in Data-Driven Logistics—A Simulation Study. Entropy, 27(1), 25. https://doi.org/10.3390/e27010025