# Ensemble FARIMA Prediction with Stable Infinite Variance Innovations for Supermarket Energy Consumption

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## Abstract

**:**

## 1. Introduction

## 2. The FARIMA Model

**Assumption**

**1.**

**Assumption**

**2.**

**Remark**

**1.**

**Theorem**

**1**

**Theorem**

**2**

**Theorem**

**3**

- when $t=0$, ${Y}_{0}=0$, ${W}_{0}=0$;
- when $t=1$, ${W}_{1}=f\left(0\right){Y}_{1}+f\left(1\right){Y}_{0}=f\left(0\right){Y}_{1}$;
- when $t=2$, ${W}_{2}=f\left(0\right){Y}_{2}+f\left(1\right){Y}_{1}$;
- when $t=3$, ${W}_{3}=f\left(0\right){Y}_{3}+f\left(1\right){Y}_{2}+f\left(2\right){Y}_{1}$;
- $\cdots \cdots $
- when $t=N$, ${W}_{N}=f\left(0\right){Y}_{N}+f\left(1\right){Y}_{N-1}+f\left(2\right){Y}_{N-2}+\cdots +f\left(N\right){Y}_{1}$.

## 3. Fractional Analytics for Industrial Data

#### 3.1. General Framework

#### 3.2. Fractional Feature Extraction

**$\alpha $-stable distribution**Traditional data processing assumes that the data fit as a Gaussian distribution due to its ease of analysis. Many methods can transform non-Gaussian data into a Gaussian distribution. However, the problem is that the process information carried by the raw data may be lost in the transformation. Therefore, it is important to analyze raw non-Gaussian information in data analysis. For this reason, an $\alpha $-stable distribution can be employed to describe non-Gaussian signals. The characteristic function of an $\alpha $-stable distribution is given as follows:

**LRD and Hurst exponent**An LRD is a typical non-Gaussian behavior always accompanying the industrial processes. It denotes the autocorrelation of time series and is important for trend forecasts. The Hurst exponent is a measurement of how the range of fluctuations in a time series varies with the time span. It is a common tool for analyzing LRD features. The definition of the Hurst exponent are interpreted in the time domain and the frequency domain. Here, we use its definition in the time domain:

**Multifractal analysis**Self-similarity is a property maintained when scaling in time or space. Due to the homogeneous nature of continuous process products, self-similarity is widely found in the industrial process. Fractal theory usually describes the self-similarity of time series, and the fractal dimension is a measurement of fractal complexity to evaluate the validity of space occupied and the irregularity. The quantitative index for fractal dimension is given as follows:

## 4. Study Case

#### 4.1. Fractional Feature Extraction

#### 4.2. Energy Prediction Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Variable | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathit{\delta}$ | Hurst |
---|---|---|---|---|---|

Global dew point temp. | 1.6845 | 1 | 2.1572 | 56.0584 | 0.9199 |

Indoor temperature | 1.7927 | −1 | 0.7185 | 72.5350 | 0.9342 |

Suction capacity | 1.4950 | 0.1196 | 1.0821 | 50.7036 | 0.8126 |

Compressor load | 1.9798 | 1 | 1.7060 | 16.3246 | 0.9798 |

Variable | $\mathit{q}=1$ | $\mathit{q}=3$ | $\mathit{q}=5$ | $\mathit{q}=7$ | $\Delta \mathit{H}$ |
---|---|---|---|---|---|

Global dew point temp. | 0.6896 | 0.5021 | 0.3271 | 0.2599 | 0.4297 |

Indoor temperature | 0.7265 | 0.5582 | 0.5044 | 0.4741 | 0.2524 |

Suction capacity | 0.2791 | 0.1913 | 0.2298 | 0.2697 | 0.0878 |

Compressor load | 0.5902 | 0.4783 | 0.3970 | 0.3208 | 0.2694 |

Training | Testing | ||||
---|---|---|---|---|---|

Model | RMSE | MAE | RMSE | Max Error | Predict Mean |

ARMA | 1.1035 | 1.5613 | 2.0244 | 7.6000 | 14.6497 |

ARIMA | 2.0942 | 13.7962 | 17.3131 | 24.6000 | 9.9240 |

FARIMA | 0.3591 | 1.5497 | 2.0372 | 8.0000 | 14.6479 |

ARMA + FARIMA | − | 1.5301 | 1.9980 | 7.8000 | 14.6488 |

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**MDPI and ACS Style**

Wang, J.; Liu, Y.; Wu, H.; Lu, S.; Zhou, M.
Ensemble FARIMA Prediction with Stable Infinite Variance Innovations for Supermarket Energy Consumption. *Fractal Fract.* **2022**, *6*, 276.
https://doi.org/10.3390/fractalfract6050276

**AMA Style**

Wang J, Liu Y, Wu H, Lu S, Zhou M.
Ensemble FARIMA Prediction with Stable Infinite Variance Innovations for Supermarket Energy Consumption. *Fractal and Fractional*. 2022; 6(5):276.
https://doi.org/10.3390/fractalfract6050276

**Chicago/Turabian Style**

Wang, Jing, Yi Liu, Haiyan Wu, Shan Lu, and Meng Zhou.
2022. "Ensemble FARIMA Prediction with Stable Infinite Variance Innovations for Supermarket Energy Consumption" *Fractal and Fractional* 6, no. 5: 276.
https://doi.org/10.3390/fractalfract6050276