# Short Time Series Forecasting: Recommended Methods and Techniques

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State of the Art of Crime Forecasting

## 3. Models and Methods

#### 3.1. Data Extraction

#### 3.2. Data Preparation

#### 3.3. Forecasting Methods

- [Jaganathan] combines numerous statistical and machine learning methods [22]: naive/snaive; ExponenTial Smoothing (ETS); dampened ETS; bagged ETS; exponential smoothing, complex exponential smoothing, general exponential smoothing; multi-aggregation prediction algorithm (MAPA); temporal hierarchical forecasting; autoregressive integrated moving average (ARIMA); ThetaH; hybrid Theta; forecast pro; seasonal and trend decomposition using loess forecast; trigonometric Box–Cox transform, ARMA errors, trend and seasonal components (TBATS); double seasonal Holt-Winters; and multilayer perceptron and extreme learning machines.
- [FFORMA] uses the following forecasting methods [23]: naive, random walk with drift, seasonal naive, theta method, automated ARIMA algorithm, ETS, TBATS, STLM-AR seasonal and trend decomposition, and neural network time series forecasts (NNETAR).
- [Hybrid] ensembles [24]: auto ARIMA, ETS, Theta, NNETAR, seasonal and trend decomposition using loess, TBATS, snaive.
- [LightGBM] [21] applies decision trees.

#### 3.4. Proposed Forecasting Techniques

## 4. Experimental Result

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Arnoso Martínez, A.; Vozmediano Sanz, L.; Martínez de Taboada Kutz, C. Inseguridad subjetiva y representaciones sociales de la delincuencia. Univ. Psychol.
**2018**, 17, 1–14. [Google Scholar] [CrossRef] - Livier, M.; Martínez, G. Confianza, victimización y desorden en la percepción de inseguridad en una población mexicana Trust, victimization and disorder in a Mexican population’s perception of insecurity Resumen. Psicumex
**2019**, 9, 1–17. [Google Scholar] - Envipe, S.P. Encuesta Nacional De Victimización Y Percepción Sobre Seguridad Pública (Envipe) 2020. Inst. Nac. Estadística Geogr.
**2020**, 10, 1–58. [Google Scholar] - Santos, T.; Jiménez, M.A. El miedo de las víctimas: Diseccionando la Criminología del Control. Utopía Prax. Latinoam.
**2019**, 24, 133–153. [Google Scholar] - Armesto, A. Quality of government, crime victimization and particularistic political participation in Latin America. Perfiles Latinoam.
**2019**, 27, 1–27. [Google Scholar] [CrossRef] - Millán-Valenzuela, H.; Pérez-Archundia, E. Education, poverty and crime: Links of violence in Mexico? Convergencia
**2019**, 80, 1–26. [Google Scholar] [CrossRef] - Pavel, M.; Román, D.; Cecilia, M.; Minchel, J.; Lara, O.P. Reflexiones alternas en torno al tratamiento de las violencias y la delincuencia desde América Latina: La prevención del delito como estrategia. Med. Soc.
**2019**, 12, 110–117. [Google Scholar] - Guilmartin, C.E.K. No hay “delitos comunes” Un Planteamiento Alternativo Para Asegurar Puntos Calientes Globales y áreas Urbanas Densamente Pobladas. 2019. Available online: https://www.armyupress.army.mil/Journals/Edicion-Hispanoamericana/Archivos/Segundo-Trimestre-2019/No-hay-delitos-comunes/ (accessed on 28 May 2022).
- Ordóñez, H.; Cobos, C.; Bucheli, V. Machine learning model for predicting theft trends in Colombia | Modelo de machine learning para la predicción de las tendencias de hurto en Colombia. RISTI -Rev. Iber. Sist. Tecnol. Inf.
**2020**, 2020, 494–506. [Google Scholar] - Cichosz, P. Urban crime risk prediction using point of interest data. ISPRS Int. J. Geo-Inf.
**2020**, 9, 459. [Google Scholar] [CrossRef] - Chun, S.A.; Pathak, R.; Paturu, V.A.; Atluri, V.; Yuan, S.; Adam, N.R. Crime Prediction Model using Deep Neural Networks. In Proceedings of the 20th Annual International Conference on Digital Government, Dubai, United Arab Emirates, 18–20 June 2019; pp. 512–514. [Google Scholar] [CrossRef]
- Wang, K.; Li, W. Application of Electrical Automation Technology in Power System. J. Power Energy Eng.
**2019**, 7, 8–13. [Google Scholar] [CrossRef] [Green Version] - Liu, M.; Lu, T. A Hybrid Model of Crime Prediction. J. Phys. Conf. Ser.
**2019**, 1168, 032031. [Google Scholar] [CrossRef] - Jha, S.; Yang, E.; Almagrabi, A.O.; Bashir, A.K.; Joshi, G.P. Comparative analysis of time series model and machine testing systems for crime forecasting. Neural Comput. Appl.
**2021**, 33, 10621–10636. [Google Scholar] [CrossRef] - Yadav, R.; Kumari, S.; Savita. Autoregressive Model for Multivariate Crime Prediction; Springer: Singapore, 2020; pp. 301–307. [Google Scholar]
- Shi, L.; Lu, Y.; Pickett, J.T. The public salience of crime, 1960–2014: Age–period–cohort and time–series analyses. Criminology
**2020**, 58, 568–593. [Google Scholar] [CrossRef] - Melgarejo, M.; Rodriguez, C.; Mayorga, D.; Obregón, N. Time Series from Clustering: An Approach to Forecast Crime Patterns. In Recent Trends in Artificial Neural Networks: From Training to Prediction; IntechOpen: London, UK, 2020; pp. 1–20. [Google Scholar] [CrossRef] [Green Version]
- Izonin, I.; Tkachenko, R.; Shakhovska, N.; Lotoshynska, N. The additive input-doubling method based on the svr with nonlinear kernels: Small data approach. Symmetry
**2021**, 13, 612. [Google Scholar] [CrossRef] - Tkachenko, R.; Mishchuk, O.; Izonin, I.; Kryvinska, N.; Stoliarchuk, R. A non-iterative neural-like framework for missing data imputation. Procedia Comput. Sci.
**2019**, 155, 319–326. [Google Scholar] [CrossRef] - Makridakis, S.; Spiliotis, E.; Assimakopoulos, V. The M4 Competition: 100,000 time series and 61 forecasting methods. Int. J. Forecast.
**2020**, 36, 54–74. [Google Scholar] [CrossRef] - Spiliotis, E.; Assimakopoulos, V.; Makridakis, S.; Assimakopoulos, V. The M5 Accuracy competition: Results, findings and conclusions. Int. J. Forecast.
**2022**, in press. [Google Scholar] [CrossRef] - Jaganathan, S.; Prakash, P.K. A combination-based forecasting method for the M4-competition. Int. J. Forecast.
**2020**, 36, 98–104. [Google Scholar] [CrossRef] - Montero-Manso, P.; Athanasopoulos, G.; Hyndman, R.J.; Talagala, T.S. FFORMA: Feature-based forecast model averaging. Int. J. Forecast.
**2020**, 36, 86–92. [Google Scholar] [CrossRef] - Atıcı, R.; Pala, Z. Prediction of the Ionospheric foF2 Parameter Using R Language Forecasthybrid Model Library Convenient Time Series Functions. Wirel. Pers. Commun.
**2022**, 122, 3293–3312. [Google Scholar] [CrossRef] - Hyndman, R.J. Seasonal Decomposition of Short Time Series. 2018. Available online: https://robjhyndman.com/hyndsight/tslm-decomposition/ (accessed on 2 June 2022).
- Hyndman, R.J.; Athanasopoulos, G. Forecasting: Principles and Practice, 2nd ed.; OTexts: Melbourne, Australia, 2018. [Google Scholar]
- Hyndman, R.J.; Athanasopoulos, G. 12.7 Very long and very short time series. In Forecasting: Principles and Practice, 2nd ed.; OTexts: Melbourne, Australia, 2018; Chapter 12. [Google Scholar]
- Moffat, I.U.; Akpan, E.A. White Noise Analysis: A Measure of Time Series Model Adequacy. Appl. Math.
**2019**, 10, 989–1003. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Time series example of the four types of crime considered in this paper. (

**a**) shoplifting, (

**b**) vehicular, (

**c**) theft, (

**d**) burglary.

**Figure 4.**The regular forecasting of the time series minus the seasonal component (ReMS). (

**a**) — Train, (

**b**) — Seasonal, (

**c**) — (Train-Seasonal) — Forecasting ReMS.

**Figure 5.**The ReMS plus the average of the seasonal component (CAS). (

**a**) — Forecasting ReMS, (

**b**) — Seasonal - - - Average Seasonal, (

**c**) — Forecasting CAS.

**Figure 6.**CAHS uses the same structure as CAS; however, it only uses the last h values instead of the whole seasonal component. (

**a**) — Forecasting ReMS, (

**b**) — Seasonal - - - - Average of last h = 0.1458, (

**c**) — Forecasting CAHS.

**Figure 7.**The ReMS plus the independent forecasting of the seasonal component (CReMSPlusFS). (

**a**) — Forecasting ReMS, (

**b**) — Seasonal — Seasonal Forecasting, (

**c**) — Forecasting (CReMSPlusFS).

F. Method | F. Configuration | Average sMAPE |
---|---|---|

ARIMA | Re | 37.5052 |

Jaganathan | Re | 37.6426 |

SMA | Re | 38.535 |

Hybrid | CAS | 39.4308 |

Hybrid | ReMS | 39.7137 |

SMA | CReMSPlusFS | 39.7689 |

SMA | ReMS | 40.0366 |

Hybrid | Re | 40.2257 |

SMA | CAS | 40.2651 |

Hybrid | CReMSPlusFS | 40.6069 |

Hybrid | CAHS | 40.8827 |

Jaganathan | CReMSPlusFS | 41.1746 |

ARIMA | CAHS | 41.793 |

SMA | CAHS | 41.7961 |

FFORMA | ReMS | 42.8928 |

FFORMA | CReMSPlusFS | 42.934 |

FFORMA | CAS | 43.0287 |

Jaganathan | CAS | 43.0652 |

Jaganathan | ReMS | 43.3135 |

FFORMA | Re | 43.8414 |

FFORMA | CAHS | 44.1245 |

Jaganathan | CAHS | 45.7345 |

ARIMA | CReMSPlusFS | 45.9772 |

ARIMA | CAS | 46.1315 |

ARIMA | ReMS | 46.6199 |

HW | Re | 47.3814 |

HW | CAS | 52.8533 |

HW | ReMS | 52.9905 |

HW | CAHS | 55.4958 |

LigthGBM | ReMS | 57.7173 |

LigthGBM | CAS | 58.2243 |

LigthGBM | CAHS | 59.0801 |

LigthGBM | Re | 64.1037 |

ANN | ReMS | 68.258 |

ANN | CAS | 68.289 |

ANN | CAHS | 69.6555 |

HW | CReMSPlusFS | 69.7481 |

LigthGBM | CReMSPlusFS | 72.307 |

ANN | Re | 72.5841 |

ANN | CReMSPlusFS | 74.0957 |

**Table 2.**Experimentation configurations ordered by Mean Rank. Here, ↑$\times r$ or ↓$\times k$ means that the experimentation configuration went up or down r or k rows, respectively, regarding Table 1.

F. Method | F. Technique | Mean Rank | Average sMAPE | Position Change |
---|---|---|---|---|

SMA | CAHS | 14.31 | 41.8 | ↑× 13 |

SMA | ReMS | 14.49 | 40.04 | ↑× 5 |

ARIMA | Re | 14.54 | 37.51 | ↓× 2 |

SMA | CAS | 14.63 | 40.27 | ↑× 5 |

Hybrid | CAS | 14.71 | 39.43 | ↓× 1 |

Jaganathan | Re | 14.77 | 37.64 | ↓× 4 |

SMA | Re | 15.23 | 38.54 | ↓× 4 |

Hybrid | ReMS | 15.66 | 39.71 | ↓× 3 |

SMA | CReMSPlusFS | 15.91 | 39.77 | ↓× 3 |

Hybrid | CAHS | 16.00 | 40.88 | ↑× 1 |

Hybrid | Re | 17.00 | 40.23 | ↓× 3 |

Jaganathan | CReMSPlusFS | 17.14 | 41.17 | - × - |

Hybrid | CReMSPlusFS | 17.46 | 40.61 | ↓× 3 |

FFORMA | ReMS | 17.60 | 42.89 | ↑× 1 |

FFORMA | CAS | 17.97 | 43.03 | ↑× 2 |

FFORMA | CAHS | 18.21 | 44.12 | ↑× 5 |

FFORMA | CReMSPlusFS | 18.24 | 42.93 | ↓× 1 |

ARIMA | CReMSPlusFS | 18.49 | 45.98 | ↑× 5 |

FFORMA | Re | 18.51 | 43.84 | ↑× 1 |

ARIMA | CAHS | 19.00 | 41.79 | ↓× 7 |

Jaganathan | CAS | 19.00 | 43.07 | ↓× 3 |

HW | ReMS | 19.07 | 52.99 | ↑× 6 |

Jaganathan | CAHS | 19.16 | 45.73 | ↓× 1 |

Jaganathan | ReMS | 19.41 | 43.31 | ↓× 5 |

HW | CAS | 19.46 | 52.85 | ↑× 2 |

ARIMA | CAS | 19.66 | 46.13 | ↓× 2 |

HW | Re | 20.34 | 47.38 | ↓× 1 |

HW | CAHS | 20.53 | 55.5 | ↑× 1 |

ARIMA | ReMS | 20.54 | 46.62 | ↓× 4 |

HW | CReMSPlusFS | 25.51 | 69.75 | ↑× 7 |

ANN | ReMS | 27.16 | 68.26 | ↑× 3 |

LigthGBM | ReMS | 27.67 | 57.72 | ↓× 2 |

LigthGBM | CAHS | 27.73 | 59.08 | ↓× 1 |

ANN | CAS | 27.83 | 68.29 | ↑× 1 |

LigthGBM | CAS | 27.89 | 58.22 | ↓× 4 |

ANN | Re | 28.20 | 72.58 | ↑× 3 |

ANN | CAHS | 28.24 | 69.66 | ↓× 1 |

LigthGBM | Re | 29.46 | 64.1 | ↓× 5 |

ANN | CReMSPlusFS | 30.74 | 74.1 | ↑× 1 |

LigthGBM | CReMSPlusFS | 32.51 | 72.31 | ↓× 2 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cruz-Nájera, M.A.; Treviño-Berrones, M.G.; Ponce-Flores, M.P.; Terán-Villanueva, J.D.; Castán-Rocha, J.A.; Ibarra-Martínez, S.; Santiago, A.; Laria-Menchaca, J.
Short Time Series Forecasting: Recommended Methods and Techniques. *Symmetry* **2022**, *14*, 1231.
https://doi.org/10.3390/sym14061231

**AMA Style**

Cruz-Nájera MA, Treviño-Berrones MG, Ponce-Flores MP, Terán-Villanueva JD, Castán-Rocha JA, Ibarra-Martínez S, Santiago A, Laria-Menchaca J.
Short Time Series Forecasting: Recommended Methods and Techniques. *Symmetry*. 2022; 14(6):1231.
https://doi.org/10.3390/sym14061231

**Chicago/Turabian Style**

Cruz-Nájera, Mariel Abigail, Mayra Guadalupe Treviño-Berrones, Mirna Patricia Ponce-Flores, Jesús David Terán-Villanueva, José Antonio Castán-Rocha, Salvador Ibarra-Martínez, Alejandro Santiago, and Julio Laria-Menchaca.
2022. "Short Time Series Forecasting: Recommended Methods and Techniques" *Symmetry* 14, no. 6: 1231.
https://doi.org/10.3390/sym14061231