# A Novel Hybrid Algorithm to Forecast Functional Time Series Based on Pattern Sequence Similarity with Application to Electricity Demand

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Methodology

- Initialization ${\mathit{\theta}}^{\left(0\right)}$;
- Repeat until convergence:
- (a)
- E step: compute ${t}_{ik}^{\left(q\right)}=\mathbb{E}[{Z}_{i}|{\mathbf{C}}_{i}^{\left(q\right)},{\mathit{\theta}}^{\left(q\right)}]$;
- (b)
- M step: update the model parameters as follows:
- ${\pi}_{k}^{(q+1)}=\frac{{\eta}_{k}^{\left(q\right)}}{n}$ with ${\eta}_{k}^{\left(q\right)}={\sum}_{i=1}^{n}{t}_{ik}^{\left(q\right)}$;
- ${\mathit{\mu}}_{k}^{(q+1)}=\frac{1}{{\eta}_{k}^{\left(q\right)}}{\sum}_{i=1}^{n}{t}_{ik}^{\left(q\right)}{\mathbf{C}}_{i}$;
- the elements ${\mathsf{\Sigma}}_{k}^{(q+1)}$ are obtained from the diagonalization of ${W}^{1/2}{D}_{k}^{\left(q\right)}{W}^{1/2}$ where ${D}_{k}^{\left(q\right)}=\frac{1}{{\eta}_{k}^{\left(q\right)}}{\sum}_{i=1}^{n}{t}_{ik}^{\left(q\right)}({\mathbf{C}}_{i}-{\mu}_{k}^{\left(q\right)}){({\mathbf{C}}_{i}-{\mu}_{k}^{\left(q\right)})}^{t}$, and W contains the inner products between the basis function ${({\varphi}_{r}^{j})}_{rj}$.

## 4. Results

#### 4.1. Data Description

#### 4.2. Quality Measures

#### 4.3. Hardware and Software Resources

#### 4.4. Performance Assessment

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Algorithm | |

PSF | Pattern Sequence Forecasting |

funPSF | Functional Pattern Sequence Forecasting |

ANN | Artificial Neural Networks |

ARIMA | Autoregressive Integrated Moving Average |

ARCH | Autoregressive Conditional Heteroskedasticity |

ARMAX | Autoregressive Moving Average with Exogenous inputs model |

EM | Expectation-Maximization |

ANFIS | Adaptive Neuro-Fuzzy Inference System |

Variables | |

K | Number of clusters |

W | Length of the window |

D | Dataset |

L | Labeled dataset |

T | Test set |

${S}_{W}$ | Pattern sequence |

M | Number of matches |

h | Prediction horizon |

Metrics | |

MAE | Mean Absolute Error |

MAPE | Mean Absolute Error |

RMSE | Root Mean Squared Error |

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Hardware | |
---|---|

Processors | Intel(R) Core(TM) i7-5820K CPU |

Cores | 6 (12 threads) |

Clock speed | 2.13 GHz |

Cache | 16 MB |

Network | Gigabit (1 Gb/s) |

RAM | 48 GB |

Software | |
---|---|

Framework | RStudio 1.1 |

OS | Ubuntu 16.04.1 LTS |

K | Silhouette | Dunn | Davies–Bouldin | BIC |
---|---|---|---|---|

Best | 3 | 4 | 3 | 4 |

Second best | 4 | 3 | 4 | 6 |

**Table 4.**Summary of the achieved results in terms of mean absolute error (MAE) (MW), mean absolute percentage error (MAPE) (%), and Root Mean Squared Error (RMSE).

K | funPSF | 7-funPSF | ANN | ARIMA |
---|---|---|---|---|

MAE | 95.85 | 68.33 | 106.44 | 146.56 |

MAPE | 7.94 | 5.63 | 9.78 | 13.40 |

RMSE | 123.92 | 93.18 | 141.20 | 142.37 |

Hour | funPSF | 7-funPSF | ANN | ARIMA |
---|---|---|---|---|

1 | 61.17 | 59.89 | 148.53 | 123.04 |

2 | 53.81 | 53.75 | 88.75 | 116.14 |

3 | 48.52 | 48.05 | 76.05 | 103.54 |

4 | 46.84 | 45.62 | 91.62 | 103.48 |

5 | 50.38 | 45.85 | 71.85 | 99.74 |

6 | 61.46 | 49.94 | 96.94 | 120.84 |

7 | 85.27 | 52.78 | 80.78 | 135.63 |

8 | 119.46 | 59.30 | 92.30 | 156.46 |

9 | 128.83 | 64.36 | 93.36 | 159.11 |

10 | 122.46 | 62.71 | 109.71 | 157.90 |

11 | 114.57 | 61.20 | 92.20 | 148.67 |

12 | 112.18 | 61.14 | 86.14 | 153.19 |

13 | 107.36 | 64.56 | 111.56 | 148.76 |

14 | 108.44 | 71.09 | 101.09 | 161.72 |

15 | 112.54 | 76.18 | 101.18 | 161.23 |

16 | 113.93 | 79.46 | 127.46 | 161.64 |

17 | 110.22 | 78.31 | 125.31 | 155.12 |

18 | 105.02 | 76.23 | 122.23 | 150.75 |

19 | 121.00 | 102.10 | 135.10 | 179.46 |

20 | 131.11 | 115.82 | 159.82 | 202.16 |

21 | 119.38 | 98.31 | 124.31 | 183.41 |

22 | 99.10 | 75.52 | 107.52 | 158.77 |

23 | 86.88 | 68.33 | 98.33 | 139.92 |

24 | 80.42 | 69.48 | 112.48 | 136.74 |

Average | 95.85 | 68.33 | 106.44 | 146.56 |

Hour | funPSF | 7-funPSF | ANN | ARIMA |
---|---|---|---|---|

January | 121.97 | 103.93 | 128.93 | 181.14 |

February | 97.18 | 75.86 | 105.86 | 154.23 |

March | 71.02 | 43.84 | 73.84 | 114.51 |

April | 88.90 | 57.98 | 100.98 | 135.73 |

May | 106.77 | 68.27 | 104.27 | 153.82 |

June | 115.70 | 84.01 | 127.01 | 169.03 |

July | 87.35 | 63.43 | 98.43 | 142.87 |

August | 72.66 | 48.38 | 98.38 | 121.83 |

September | 79.79 | 52.16 | 95.16 | 128.87 |

October | 73.19 | 43.15 | 85.15 | 115.40 |

November | 106.23 | 76.19 | 116.19 | 161.05 |

December | 128.36 | 102.02 | 143.02 | 180.23 |

Average | 95.85 | 68.33 | 106.44 | 146.56 |

Regular Days | Festivities | |||||||
---|---|---|---|---|---|---|---|---|

Day | funPSF | 7-funPSF | ANN | ARIMA | funPSF | 7-funPSF | ANN | ARIMA |

Monday | 156.22 | 65.20 | 94.53 | 143.87 | 134.66 | 64.41 | 145.07 | 142.60 |

Tuesday | 84.08 | 65.64 | 84.18 | 141.42 | 257.26 | 359.83 | 312.26 | 253.67 |

Wednesday | 80.12 | 67.08 | 86.34 | 152.47 | 153.18 | 202.39 | 256.73 | 258.32 |

Thursday | 89.00 | 73.24 | 101.83 | 133.09 | 129.00 | 133.31 | 142.87 | 201.54 |

Friday | 87.61 | 76.15 | 111.06 | 140.29 | 79.29 | 106.58 | 121.72 | 163.51 |

Saturday | 72.11 | 69.24 | 117.02 | 145.00 | 42.41 | 40.15 | 99.03 | 241.80 |

Sunday | 100.00 | 54.80 | 120.34 | 153.32 | 87.33 | 42.70 | 117.77 | 145.61 |

Average | 95.59 | 67.34 | 102.19 | 144.21 | 126.16 | 135.62 | 170.78 | 201.01 |

Hour | funPSF | 7-funPSF | ANN | ARIMA |
---|---|---|---|---|

1 | 5.33 | 5.21 | 5.38 | 10.82 |

2 | 5.12 | 5.10 | 8.48 | 13.00 |

3 | 4.88 | 4.83 | 7.10 | 9.68 |

4 | 4.86 | 4.71 | 5.80 | 13.44 |

5 | 5.27 | 4.77 | 5.66 | 8.29 |

6 | 6.34 | 5.09 | 7.23 | 10.60 |

7 | 8.49 | 5.17 | 10.72 | 11.91 |

8 | 11.33 | 5.46 | 5.86 | 15.70 |

9 | 11.48 | 5.63 | 15.23 | 20.94 |

10 | 10.29 | 5.21 | 7.28 | 14.34 |

11 | 9.29 | 4.92 | 8.43 | 11.30 |

12 | 8.86 | 4.78 | 10.43 | 11.08 |

13 | 8.39 | 4.98 | 13.35 | 17.46 |

14 | 8.48 | 5.46 | 6.83 | 18.41 |

15 | 8.92 | 5.91 | 14.97 | 13.89 |

16 | 9.11 | 6.21 | 6.40 | 11.11 |

17 | 8.86 | 6.18 | 11.69 | 14.28 |

18 | 8.39 | 6.03 | 8.56 | 13.70 |

19 | 9.32 | 7.94 | 8.39 | 15.56 |

20 | 9.76 | 8.75 | 17.16 | 16.40 |

21 | 8.37 | 6.99 | 16.82 | 17.77 |

22 | 6.82 | 5.24 | 8.43 | 8.02 |

23 | 6.29 | 4.97 | 9.97 | 14.23 |

24 | 6.34 | 5.49 | 14.55 | 9.54 |

Average | 7.94 | 5.63 | 9.78 | 13.40 |

Month | funPSF | 7-funPSF | ANN | ARIMA |
---|---|---|---|---|

January | 10.17 | 8.75 | 14.10 | 16.24 |

February | 8.30 | 6.40 | 8.70 | 10.82 |

March | 6.67 | 4.08 | 9.08 | 9.67 |

April | 6.58 | 3.87 | 10.34 | 11.14 |

May | 7.49 | 4.88 | 7.95 | 12.83 |

June | 8.03 | 5.05 | 9.73 | 13.19 |

July | 8.52 | 6.19 | 11.08 | 13.18 |

August | 8.14 | 5.82 | 8.45 | 18.06 |

September | 7.40 | 5.28 | 10.92 | 12.17 |

October | 6.64 | 4.35 | 7.09 | 20.64 |

November | 7.11 | 4.67 | 8.56 | 13.04 |

December | 10.20 | 8.13 | 11.43 | 9.80 |

Average | 7.94 | 5.62 | 9.78 | 13.40 |

Regular Days | Festivities | |||||||
---|---|---|---|---|---|---|---|---|

Day | funPSF | 7-funPSF | ANN | ARIMA | funPSF | 7-funPSF | ANN | ARIMA |

Monday | 12.25 | 5.28 | 19.30 | 25.61 | 10.51 | 5.60 | 20.18 | 24.33 |

Tuesday | 6.53 | 5.24 | 13.10 | 7.09 | 32.55 | 44.95 | 46.51 | 44.83 |

Wednesday | 6.28 | 5.27 | 8.00 | 20.27 | 15.32 | 20.47 | 23.24 | 34.97 |

Thursday | 6.93 | 5.74 | 13.55 | 18.16 | 13.65 | 14.08 | 28.26 | 20.71 |

Friday | 6.91 | 6.01 | 11.02 | 14.32 | 6.98 | 9.44 | 11.88 | 24.93 |

Saturday | 6.27 | 5.82 | 9.34 | 15.72 | 4.00 | 3.56 | 18.84 | 13.31 |

Sunday | 9.97 | 5.06 | 14.26 | 10.75 | 8.64 | 3.98 | 20.87 | 17.29 |

Average | 7.88 | 5.49 | 12.65 | 15.99 | 13.09 | 14.58 | 24.25 | 25.77 |

Hour | funPSF | 7-funPSF | ANN | ARIMA |
---|---|---|---|---|

1 | 79.11 | 79.21 | 118.63 | 119.41 |

2 | 73.00 | 74.32 | 108.83 | 107.84 |

3 | 68.20 | 69.02 | 98.81 | 103.20 |

4 | 66.16 | 65.77 | 96.57 | 101.87 |

5 | 68.27 | 64.40 | 95.65 | 103.25 |

6 | 77.45 | 66.53 | 102.11 | 104.87 |

7 | 100.52 | 67.02 | 116.15 | 114.47 |

8 | 139.49 | 79.47 | 149.05 | 143.03 |

9 | 152.67 | 86.58 | 156.51 | 159.09 |

10 | 148.60 | 86.28 | 165.66 | 164.56 |

11 | 140.82 | 84.56 | 152.15 | 157.58 |

12 | 138.15 | 84.81 | 134.84 | 138.15 |

13 | 134.03 | 89.64 | 164.42 | 167.23 |

14 | 136.79 | 98.12 | 143.62 | 143.91 |

15 | 142.12 | 105.26 | 158.49 | 158.74 |

16 | 144.23 | 109.68 | 155.01 | 160.63 |

17 | 139.65 | 106.87 | 159.44 | 159.69 |

18 | 131.36 | 99.80 | 164.19 | 165.10 |

19 | 148.13 | 125.33 | 160.33 | 161.06 |

20 | 159.52 | 141.37 | 187.04 | 188.27 |

21 | 149.88 | 124.69 | 159.83 | 156.41 |

22 | 128.41 | 98.02 | 163.12 | 162.75 |

23 | 111.13 | 88.72 | 133.68 | 133.42 |

24 | 101.56 | 89.14 | 144.72 | 142.53 |

Average | 123.92 | 93.18 | 142.37 | 141.20 |

Month | funPSF | 7-funPSF | ARIMA | ANN |
---|---|---|---|---|

January | 121.57 | 99.60 | 128.53 | 131.57 |

February | 88.62 | 56.62 | 90.13 | 76.31 |

March | 112.70 | 78.87 | 131.02 | 130.85 |

April | 134.19 | 89.88 | 199.27 | 207.99 |

May | 147.41 | 107.42 | 212.22 | 234.47 |

June | 112.48 | 80.89 | 118.75 | 111.29 |

July | 93.93 | 64.13 | 109.20 | 108.78 |

August | 100.78 | 67.52 | 109.69 | 108.06 |

September | 90.38 | 56.48 | 98.64 | 94.92 |

October | 136.24 | 99.03 | 166.41 | 156.41 |

November | 166.16 | 136.69 | 174.10 | 168.03 |

December | 150.96 | 134.33 | 170.44 | 166.45 |

Average | 123.92 | 93.18 | 142.37 | 141.20 |

Regular Days | Festivities | |||||||
---|---|---|---|---|---|---|---|---|

Day | funPSF | 7-funPSF | ANN | ARIMA | funPSF | 7-funPSF | ANN | ARIMA |

Monday | 185.06 | 86.96 | 197.24 | 201.66 | 160.26 | 89.33 | 182.31 | 190.01 |

Tuesday | 109.87 | 88.84 | 112.19 | 117.92 | 297.31 | 396.56 | 388.40 | 376.32 |

Wednesday | 103.08 | 90.92 | 111.63 | 118.21 | 181.39 | 231.48 | 247.91 | 262.93 |

Thursday | 116.50 | 98.37 | 124.97 | 124.99 | 151.17 | 161.60 | 164.88 | 170.75 |

Friday | 110.64 | 100.38 | 115.01 | 118.37 | 102.80 | 141.73 | 153.04 | 160.24 |

Saturday | 93.99 | 92.06 | 100.87 | 102.55 | 55.12 | 56.84 | 70.32 | 88.32 |

Sunday | 123.27 | 76.29 | 131.36 | 132.60 | 105.21 | 58.61 | 110.67 | 120.68 |

Average | 120.34 | 90.54 | 127.61 | 130.90 | 150.47 | 162.30 | 188.22 | 195.61 |

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## Share and Cite

**MDPI and ACS Style**

Martínez-Álvarez, F.; Schmutz, A.; Asencio-Cortés, G.; Jacques, J.
A Novel Hybrid Algorithm to Forecast Functional Time Series Based on Pattern Sequence Similarity with Application to Electricity Demand. *Energies* **2019**, *12*, 94.
https://doi.org/10.3390/en12010094

**AMA Style**

Martínez-Álvarez F, Schmutz A, Asencio-Cortés G, Jacques J.
A Novel Hybrid Algorithm to Forecast Functional Time Series Based on Pattern Sequence Similarity with Application to Electricity Demand. *Energies*. 2019; 12(1):94.
https://doi.org/10.3390/en12010094

**Chicago/Turabian Style**

Martínez-Álvarez, Francisco, Amandine Schmutz, Gualberto Asencio-Cortés, and Julien Jacques.
2019. "A Novel Hybrid Algorithm to Forecast Functional Time Series Based on Pattern Sequence Similarity with Application to Electricity Demand" *Energies* 12, no. 1: 94.
https://doi.org/10.3390/en12010094