# Multiple Novel Decomposition Techniques for Time Series Forecasting: Application to Monthly Forecasting of Electricity Consumption in Pakistan

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## Abstract

**:**

## 1. Introduction

## 2. The Proposed Forecasting Methodology

#### 2.1. The Proposed Decomposition Techniques

#### 2.1.1. Regression Spline Decomposition Method

#### 2.1.2. Smoothing Splines Decomposition Method

#### 2.1.3. Seasonal Trend Decomposition Method

#### 2.2. Modeling the Decomposed Subseries

#### 2.2.1. Linear Autoregressive Model

#### 2.2.2. Nonlinear Autoregressive Model

#### 2.2.3. Autoregressive Moving Average Model

#### 2.3. Accuracy Measures

## 3. Case Study Evaluation

**Table 6.**Pakistan’s electricity consumption (kWh): mean performance measures of the proposed versus the literature.

S.No | Models | MAPE | MAE | RMSE | CORR |
---|---|---|---|---|---|

1 | ${}^{\mathrm{c}}$DH${}_{\mathrm{c}}^{\mathrm{b}}$ | 1.9718 | 157.7533 | 199.5219 | 0.9957 |

2 | AR | 9.7316 | 841.3092 | 1116.3690 | 0.8618 |

3 | NPAR | 9.0549 | 817.5962 | 1156.6528 | 0.8598 |

4 | Proposed model [48] | 7.6291 | 665.7315 | 974.3326 | 0.9033 |

5 | Proposed model 1 [15] | 7.1039 | 607.8114 | 860.4425 | 0.9303 |

6 | Proposed model 2 [15] | 6.4823 | 569.1609 | 855.5536 | 0.9386 |

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Specific characteristics of the electricity consumption time series (kWh): (

**A**) electricity consumption time series for the period from January 1990 to June 2020 (original time series–gray; linear curve–black; nonlinear curve–red); (

**B**) annual periodicity for the period from January 2009 to December 2012; (

**C**) seasonal plot for the period from January 1990 to June 2020 (winter–blue; summer–dark green; spring–red; autumn–gray), and (

**D**) box-plots for yearly observation for the period from January 1990 to December 2020.

**Figure 2.**Electricity consumption (kWh) in Pakistan: The monthly electricity consumption series is decomposed by the three proposed decomposition methods: (

**a**) DSS, (

**b**) DRS, (

**c**) DH, and (

**d**) the benchmark decomposing method DSTL. In each sub-figure, the top panel shows the long-term trend (${\mathfrak{t}}_{\mathfrak{m}}$), the middle panel shows the seasonal (${\mathfrak{s}}_{\mathfrak{m}}$) component, and the bottom panel shows the stochastic component (${\mathfrak{r}}_{\mathfrak{m}}$).

**Figure 3.**Performance measures: the MAPE (

**top**), MAE (

**center**), and RMSE (

**bottom**) for all combination models using three proposed and the benchmark decomposition methods.

**Figure 4.**Performance measures: the accuracy measures for the best twelve models: MAPE (

**top**), MAE (

**center**), and RMSE (

**bottom**).

**Figure 5.**Scatter plot for the electricity consumption forecasting models along with the correlation coefficient (CORR), ${}^{c}$DH${}_{c}^{b}$ (

**1st**), ${}^{c}$DH${}_{c}^{a}$ (

**2nd**), ${}^{a}$DH${}_{c}^{a}$ (

**3rd**), and ${}^{a}$DH${}_{c}^{b}$ (

**4th**).

**Figure 6.**Original and forecasted electricity consumption for four of the best models over five years.

**Figure 7.**Performance measures: the proposed versus the literature. (

**A**) MAE; (

**B**) RMSE; and (

**C**) MAPE.

**Table 1.**Pakistan’s electricity consumption (kWh): out-of-sample one-month-ahead average forecast error for all combined models with the DSS method.

S.No | Models | MAPE | MAE | RMSE | CORR |
---|---|---|---|---|---|

1 | ${}^{\mathrm{a}}$DSS${}_{\mathrm{a}}^{\mathrm{a}}$ | 3.8513 | 303.1823 | 395.5292 | 0.9833 |

2 | ${}^{\mathrm{a}}$DSS${}_{\mathrm{b}}^{\mathrm{a}}$ | 3.7886 | 299.5319 | 396.5169 | 0.9833 |

3 | ${}^{\mathrm{a}}$DSS${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.3521 | 191.2760 | 256.6520 | 0.9930 |

4 | ${}^{\mathrm{a}}$DSS${}_{\mathrm{a}}^{\mathrm{b}}$ | 3.6722 | 289.3194 | 394.0852 | 0.9834 |

5 | ${}^{\mathrm{a}}$DSS${}_{\mathrm{b}}^{\mathrm{b}}$ | 3.6107 | 285.5530 | 395.6597 | 0.9832 |

6 | ${}^{\mathrm{a}}$DSS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.3908 | 195.8973 | 259.4971 | 0.9929 |

7 | ${}^{\mathrm{a}}$DSS${}_{\mathrm{a}}^{\mathrm{c}}$ | 3.8175 | 300.6711 | 397.5553 | 0.9832 |

8 | ${}^{\mathrm{a}}$DSS${}_{\mathrm{b}}^{\mathrm{c}}$ | 3.7450 | 296.2238 | 398.2182 | 0.9831 |

9 | ${}^{\mathrm{a}}$DSS${}_{\mathrm{c}}^{\mathrm{c}}$ | 2.2939 | 185.8645 | 250.0828 | 0.9934 |

10 | ${}^{\mathrm{b}}$DSS${}_{\mathrm{a}}^{\mathrm{a}}$ | 3.8355 | 303.7990 | 400.3169 | 0.9833 |

11 | ${}^{\mathrm{b}}$DSS${}_{\mathrm{b}}^{\mathrm{a}}$ | 3.7651 | 299.5174 | 401.5727 | 0.9832 |

12 | ${}^{\mathrm{b}}$DSS${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.3829 | 193.9457 | 263.4854 | 0.9928 |

13 | ${}^{\mathrm{b}}$DSS${}_{\mathrm{a}}^{\mathrm{b}}$ | 3.6738 | 291.3070 | 399.1517 | 0.9833 |

14 | ${}^{\mathrm{b}}$DSS${}_{\mathrm{b}}^{\mathrm{b}}$ | 3.6180 | 288.4040 | 400.9866 | 0.9831 |

15 | ${}^{\mathrm{b}}$DSS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.4173 | 198.4902 | 266.6493 | 0.9926 |

16 | ${}^{\mathrm{b}}$DSS${}_{\mathrm{a}}^{\mathrm{c}}$ | 3.8088 | 301.8532 | 402.9203 | 0.9830 |

17 | ${}^{\mathrm{b}}$DSS${}_{\mathrm{b}}^{\mathrm{c}}$ | 3.7361 | 297.4499 | 403.8527 | 0.9830 |

18 | ${}^{\mathrm{b}}$DSS${}_{\mathrm{c}}^{\mathrm{c}}$ | 2.3342 | 189.5097 | 258.0311 | 0.9931 |

19 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{a}}^{\mathrm{a}}$ | 3.8141 | 300.0557 | 396.0669 | 0.9832 |

20 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{b}}^{\mathrm{a}}$ | 3.7811 | 298.4973 | 396.7102 | 0.9831 |

21 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.3239 | 188.2632 | 248.0112 | 0.9935 |

22 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{a}}^{\mathrm{b}}$ | 3.7218 | 292.1390 | 395.1023 | 0.9832 |

23 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{b}}^{\mathrm{b}}$ | 3.6642 | 288.2662 | 396.3294 | 0.9831 |

24 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.2911 | 188.6474 | 251.7045 | 0.9933 |

25 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{a}}^{\mathrm{c}}$ | 3.7967 | 298.9096 | 398.5101 | 0.9830 |

26 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{b}}^{\mathrm{c}}$ | 3.7508 | 296.1589 | 398.8303 | 0.9829 |

27 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{c}}^{\mathrm{c}}$ | 2.2382 | 181.4303 | 241.8992 | 0.9938 |

**Table 2.**Pakistan’s electricity consumption (kWh): out-of-sample one-month-ahead average forecast error for all combined models with the DRS method.

S.No | Models | MAPE | MAE | RMSE | CORR |
---|---|---|---|---|---|

1 | ${}^{\mathrm{a}}$DRS${}_{\mathrm{a}}^{\mathrm{a}}$ | 3.6219 | 286.1814 | 385.6078 | 0.9838 |

2 | ${}^{\mathrm{a}}$DRS${}_{\mathrm{b}}^{\mathrm{a}}$ | 3.5769 | 283.3066 | 380.3018 | 0.9843 |

3 | ${}^{\mathrm{a}}$DRS${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.2938 | 179.0583 | 233.9822 | 0.9941 |

4 | ${}^{\mathrm{a}}$DRS${}_{\mathrm{a}}^{\mathrm{b}}$ | 3.5534 | 280.5790 | 379.7724 | 0.9843 |

5 | ${}^{\mathrm{a}}$DRS${}_{\mathrm{b}}^{\mathrm{b}}$ | 3.4699 | 275.1287 | 375.1131 | 0.9848 |

6 | ${}^{\mathrm{a}}$DRS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.2535 | 176.4146 | 236.3622 | 0.9939 |

7 | ${}^{\mathrm{a}}$DRS${}_{\mathrm{a}}^{\mathrm{c}}$ | 3.6576 | 288.3619 | 389.5358 | 0.9834 |

8 | ${}^{\mathrm{a}}$DRS${}_{\mathrm{b}}^{\mathrm{c}}$ | 3.6066 | 284.6170 | 383.8062 | 0.9840 |

9 | ${}^{\mathrm{a}}$DRS${}_{\mathrm{c}}^{\mathrm{c}}$ | 2.3016 | 179.1916 | 234.8334 | 0.9940 |

10 | ${}^{\mathrm{b}}$DRS${}_{\mathrm{a}}^{\mathrm{a}}$ | 3.6197 | 288.3346 | 388.2273 | 0.9839 |

11 | ${}^{\mathrm{b}}$DRS${}_{\mathrm{b}}^{\mathrm{a}}$ | 3.5768 | 286.0483 | 382.8333 | 0.9845 |

12 | ${}^{\mathrm{b}}$DRS${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.4216 | 190.4513 | 242.9584 | 0.9937 |

13 | ${}^{\mathrm{b}}$DRS${}_{\mathrm{a}}^{\mathrm{b}}$ | 3.5288 | 281.4996 | 383.0212 | 0.9843 |

14 | ${}^{\mathrm{b}}$DRS${}_{\mathrm{b}}^{\mathrm{b}}$ | 3.4665 | 278.1669 | 378.2763 | 0.9849 |

15 | ${}^{\mathrm{b}}$DRS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.4007 | 189.9531 | 246.1694 | 0.9935 |

16 | ${}^{\mathrm{b}}$DRS${}_{\mathrm{a}}^{\mathrm{c}}$ | 3.6585 | 290.9357 | 392.1562 | 0.9835 |

17 | ${}^{\mathrm{b}}$DRS${}_{\mathrm{b}}^{\mathrm{c}}$ | 3.6215 | 288.7784 | 386.3424 | 0.9841 |

18 | ${}^{\mathrm{b}}$DRS${}_{\mathrm{c}}^{\mathrm{c}}$ | 2.4383 | 191.8828 | 243.8220 | 0.9937 |

19 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{a}}^{\mathrm{a}}$ | 3.7136 | 293.0724 | 391.8557 | 0.9832 |

20 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{b}}^{\mathrm{a}}$ | 3.6323 | 287.2605 | 386.0430 | 0.9838 |

21 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.2916 | 179.9492 | 232.3815 | 0.9941 |

22 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{a}}^{\mathrm{b}}$ | 3.6498 | 287.2037 | 386.8070 | 0.9837 |

23 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{b}}^{\mathrm{b}}$ | 3.5598 | 280.8756 | 381.6342 | 0.9842 |

24 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.2163 | 175.0277 | 235.9146 | 0.9940 |

25 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{a}}^{\mathrm{c}}$ | 3.7834 | 297.8234 | 396.4955 | 0.9828 |

26 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{b}}^{\mathrm{c}}$ | 3.6797 | 289.7838 | 390.2820 | 0.9834 |

27 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{c}}^{\mathrm{c}}$ | 2.2904 | 179.2011 | 234.5493 | 0.9940 |

**Table 3.**Pakistan’s electricity consumption (kWh): out-of-sample one-month-ahead average forecast error for all models combined with the DH method.

S.No | Models | MAPE | MAE | RMSE | CORR |
---|---|---|---|---|---|

1 | ${}^{\mathrm{a}}$DH${}_{\mathrm{a}}^{\mathrm{a}}$ | 3.6829 | 290.3322 | 386.3345 | 0.9837 |

2 | ${}^{\mathrm{a}}$DH${}_{\mathrm{b}}^{\mathrm{a}}$ | 3.6204 | 286.9580 | 384.0921 | 0.9840 |

3 | ${}^{\mathrm{a}}$DH${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.0068 | 158.7059 | 199.4186 | 0.9957 |

4 | ${}^{\mathrm{a}}$DH${}_{\mathrm{a}}^{\mathrm{b}}$ | 3.6247 | 286.6047 | 385.6102 | 0.9838 |

5 | ${}^{\mathrm{a}}$DH${}_{\mathrm{b}}^{\mathrm{b}}$ | 3.5200 | 280.4262 | 384.4030 | 0.9840 |

6 | ${}^{\mathrm{a}}$DH${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.0393 | 162.0884 | 204.0827 | 0.9955 |

7 | ${}^{\mathrm{a}}$DH${}_{\mathrm{a}}^{\mathrm{c}}$ | 3.7206 | 292.9335 | 390.9844 | 0.9833 |

8 | ${}^{\mathrm{a}}$DH${}_{\mathrm{b}}^{\mathrm{c}}$ | 3.6233 | 286.8009 | 388.9185 | 0.9836 |

9 | ${}^{\mathrm{a}}$DH${}_{\mathrm{c}}^{\mathrm{c}}$ | 2.0839 | 164.6258 | 206.6300 | 0.9954 |

10 | ${}^{\mathrm{b}}$DH${}_{\mathrm{a}}^{\mathrm{a}}$ | 3.6624 | 291.3618 | 389.4783 | 0.9838 |

11 | ${}^{\mathrm{b}}$DH${}_{\mathrm{b}}^{\mathrm{a}}$ | 3.6307 | 290.7418 | 387.5633 | 0.9841 |

12 | ${}^{\mathrm{b}}$DH${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.0744 | 164.3205 | 211.3162 | 0.9953 |

13 | ${}^{\mathrm{b}}$DH${}_{\mathrm{a}}^{\mathrm{b}}$ | 3.6016 | 287.6479 | 389.0995 | 0.9838 |

14 | ${}^{\mathrm{b}}$DH${}_{\mathrm{b}}^{\mathrm{b}}$ | 3.5387 | 284.6500 | 388.2118 | 0.9841 |

15 | ${}^{\mathrm{b}}$DH${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.1090 | 168.4910 | 216.3346 | 0.9951 |

16 | ${}^{\mathrm{b}}$DH${}_{\mathrm{a}}^{\mathrm{c}}$ | 3.6959 | 293.8540 | 393.9049 | 0.9834 |

17 | ${}^{\mathrm{b}}$DH${}_{\mathrm{b}}^{\mathrm{c}}$ | 3.6419 | 291.0633 | 392.1600 | 0.9837 |

18 | ${}^{\mathrm{b}}$DH${}_{\mathrm{c}}^{\mathrm{c}}$ | 2.1345 | 169.5054 | 217.7980 | 0.9950 |

19 | ${}^{\mathrm{c}}$DH${}_{\mathrm{a}}^{\mathrm{a}}$ | 3.7636 | 296.7260 | 393.3650 | 0.9831 |

20 | ${}^{\mathrm{c}}$DH${}_{\mathrm{b}}^{\mathrm{a}}$ | 3.6728 | 290.6076 | 390.3498 | 0.9834 |

21 | ${}^{\mathrm{c}}$DH${}_{\mathrm{c}}^{\mathrm{a}}$ | 1.9815 | 157.0250 | 194.1687 | 0.9959 |

22 | ${}^{\mathrm{c}}$DH${}_{\mathrm{a}}^{\mathrm{b}}$ | 3.7257 | 293.5816 | 392.9407 | 0.9831 |

23 | ${}^{\mathrm{c}}$DH${}_{\mathrm{b}}^{\mathrm{b}}$ | 3.6367 | 287.9581 | 390.9442 | 0.9834 |

24 | ${}^{\mathrm{c}}$DH${}_{\mathrm{c}}^{\mathrm{b}}$ | 1.9718 | 157.7533 | 199.5219 | 0.9957 |

25 | ${}^{\mathrm{c}}$DH${}_{\mathrm{a}}^{\mathrm{c}}$ | 3.8181 | 300.0332 | 398.4989 | 0.9826 |

26 | ${}^{\mathrm{c}}$DH${}_{\mathrm{b}}^{\mathrm{c}}$ | 3.7237 | 293.8677 | 395.6701 | 0.9829 |

27 | ${}^{\mathrm{c}}$DH${}_{\mathrm{c}}^{\mathrm{c}}$ | 2.0413 | 161.5169 | 202.6835 | 0.9956 |

**Table 4.**Pakistan’s electricity consumption (kWh): out-of-sample one-month-ahead average forecast error for all models combined with the DSTL method.

S.No | Models | MAPE | MAE | RMSE | CORR |
---|---|---|---|---|---|

1 | ${}^{\mathrm{a}}$DSTL${}_{\mathrm{a}}^{\mathrm{a}}$ | 11.4390 | 936.2145 | 1042.3530 | 0.8795 |

2 | ${}^{\mathrm{a}}$DSTL${}_{\mathrm{b}}^{\mathrm{a}}$ | 11.4826 | 940.0003 | 1048.3239 | 0.8778 |

3 | ${}^{\mathrm{a}}$DSTL${}_{\mathrm{c}}^{\mathrm{a}}$ | 10.4287 | 840.1818 | 954.3405 | 0.8965 |

4 | ${}^{\mathrm{a}}$DSTL${}_{\mathrm{a}}^{\mathrm{b}}$ | 11.5359 | 943.3328 | 1042.6842 | 0.8794 |

5 | ${}^{\mathrm{a}}$DSTL${}_{\mathrm{b}}^{\mathrm{b}}$ | 11.5795 | 947.1186 | 1048.4946 | 0.8778 |

6 | ${}^{\mathrm{a}}$DSTL${}_{\mathrm{c}}^{\mathrm{b}}$ | 10.5400 | 847.9374 | 954.6524 | 0.8964 |

7 | ${}^{\mathrm{a}}$DSTL${}_{\mathrm{a}}^{\mathrm{c}}$ | 11.4614 | 936.9204 | 1040.3136 | 0.8800 |

8 | ${}^{\mathrm{a}}$DSTL${}_{\mathrm{b}}^{\mathrm{c}}$ | 11.5049 | 940.7062 | 1046.2201 | 0.8783 |

9 | ${}^{\mathrm{a}}$DSTL${}_{\mathrm{c}}^{\mathrm{c}}$ | 10.4823 | 842.8418 | 953.5488 | 0.8966 |

10 | ${}^{\mathrm{b}}$DSTL${}_{\mathrm{a}}^{\mathrm{a}}$ | 11.3617 | 925.8832 | 1033.5390 | 0.8800 |

11 | ${}^{\mathrm{b}}$DSTL${}_{\mathrm{b}}^{\mathrm{a}}$ | 11.4053 | 929.6691 | 1039.4770 | 0.8784 |

12 | ${}^{\mathrm{b}}$DSTL${}_{\mathrm{c}}^{\mathrm{a}}$ | 10.3991 | 839.0876 | 954.7580 | 0.8961 |

13 | ${}^{\mathrm{b}}$DSTL${}_{\mathrm{a}}^{\mathrm{b}}$ | 11.4586 | 933.0015 | 1033.5924 | 0.8801 |

14 | ${}^{\mathrm{b}}$DSTL${}_{\mathrm{b}}^{\mathrm{b}}$ | 11.5021 | 936.7874 | 1039.3701 | 0.8785 |

15 | ${}^{\mathrm{b}}$DSTL${}_{\mathrm{c}}^{\mathrm{b}}$ | 10.5044 | 845.3495 | 954.7660 | 0.8961 |

16 | ${}^{\mathrm{b}}$DSTL${}_{\mathrm{a}}^{\mathrm{c}}$ | 11.3840 | 926.5891 | 1031.5245 | 0.8805 |

17 | ${}^{\mathrm{b}}$DSTL${}_{\mathrm{b}}^{\mathrm{c}}$ | 11.4276 | 930.3750 | 1037.3973 | 0.8790 |

18 | ${}^{\mathrm{b}}$DSTL${}_{\mathrm{c}}^{\mathrm{c}}$ | 10.4620 | 842.4702 | 954.0124 | 0.8963 |

19 | ${}^{\mathrm{c}}$DSTL${}_{\mathrm{a}}^{\mathrm{a}}$ | 11.4204 | 937.2199 | 1044.5191 | 0.8798 |

20 | ${}^{\mathrm{c}}$DSTL${}_{\mathrm{b}}^{\mathrm{a}}$ | 11.4640 | 941.0057 | 1050.8107 | 0.8781 |

21 | ${}^{\mathrm{c}}$DSTL${}_{\mathrm{c}}^{\mathrm{a}}$ | 10.3579 | 838.3250 | 951.3494 | 0.8972 |

22 | ${}^{\mathrm{c}}$DSTL${}_{\mathrm{a}}^{\mathrm{b}}$ | 11.5173 | 944.3382 | 1044.6312 | 0.8799 |

23 | ${}^{\mathrm{c}}$DSTL${}_{\mathrm{b}}^{\mathrm{b}}$ | 11.5609 | 948.1240 | 1050.7638 | 0.8782 |

24 | ${}^{\mathrm{c}}$DSTL${}_{\mathrm{c}}^{\mathrm{b}}$ | 10.4727 | 846.3646 | 951.4223 | 0.8972 |

25 | ${}^{\mathrm{c}}$DSTL${}_{\mathrm{a}}^{\mathrm{c}}$ | 11.4428 | 937.9258 | 1042.2494 | 0.8804 |

26 | ${}^{\mathrm{c}}$DSTL${}_{\mathrm{b}}^{\mathrm{c}}$ | 11.4864 | 941.7116 | 1048.4788 | 0.8787 |

27 | ${}^{\mathrm{c}}$DSTL${}_{\mathrm{c}}^{\mathrm{c}}$ | 10.4041 | 840.5131 | 950.2979 | 0.8974 |

**Table 5.**Pakistan’s electricity consumption (kWh): mean forecast error of one-month-ahead post-sample for the best four models with DSS, DRS and DH decompositions.

S.No | Models | MAPE | MAE | RMSE | CORR |
---|---|---|---|---|---|

1 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.2382 | 181.4303 | 241.8992 | 0.9938 |

2 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.2911 | 188.6474 | 251.7045 | 0.9933 |

3 | ${}^{\mathrm{c}}$DSS${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.3239 | 188.2632 | 248.0112 | 0.9935 |

4 | ${}^{\mathrm{a}}$DSS${}_{\mathrm{c}}^{\mathrm{c}}$ | 2.2939 | 185.8645 | 250.0828 | 0.9934 |

5 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.2163 | 175.0277 | 235.9146 | 0.9940 |

6 | ${}^{\mathrm{a}}$DRS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.2535 | 176.4146 | 236.3622 | 0.9939 |

7 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.2904 | 179.2011 | 234.5493 | 0.9940 |

8 | ${}^{\mathrm{c}}$DRS${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.2916 | 179.9492 | 232.3815 | 0.9941 |

9 | ${}^{\mathrm{c}}$DH${}_{\mathrm{c}}^{\mathrm{b}}$ | 1.9718 | 157.7533 | 199.5219 | 0.9957 |

10 | ${}^{\mathrm{c}}$DH${}_{\mathrm{c}}^{\mathrm{a}}$ | 1.9815 | 157.0250 | 194.1687 | 0.9959 |

11 | ${}^{\mathrm{a}}$DH${}_{\mathrm{c}}^{\mathrm{a}}$ | 2.0068 | 158.7059 | 199.4186 | 0.9957 |

12 | ${}^{\mathrm{a}}$DH${}_{\mathrm{c}}^{\mathrm{b}}$ | 2.0393 | 162.0884 | 204.0827 | 0.9955 |

**Table 7.**Pakistan’s electricity consumption (kWh): results (p-value) of the DM test for the best twelve models given in Table 5.

Models | ${}^{\mathbf{c}}\mathbf{DSS}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{c}}\mathbf{DSS}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{c}}\mathbf{DSS}_{\mathbf{c}}^{\mathbf{a}}$ | ${}^{\mathbf{a}}\mathbf{DSS}_{\mathbf{c}}^{\mathbf{c}}$ | ${}^{\mathbf{c}}\mathbf{DRS}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{a}}\mathbf{DRS}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{c}}\mathbf{DRS}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{c}}\mathbf{DRS}_{\mathbf{c}}^{\mathbf{a}}$ | ${}^{\mathbf{c}}\mathbf{DH}_{\mathbf{c}}^{\mathbf{a}}$ | ${}^{\mathbf{c}}\mathbf{DH}_{\mathbf{c}}^{\mathbf{a}}$ | ${}^{\mathbf{a}}\mathbf{DH}_{\mathbf{c}}^{\mathbf{a}}$ | ${}^{\mathbf{a}}\mathbf{DH}_{\mathbf{c}}^{\mathbf{b}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${}^{\mathrm{c}}$DSS${}_{\mathrm{c}}^{\mathrm{b}}$ | 0.000 | 0.966 | 0.977 | 0.947 | 0.356 | 0.345 | 0.320 | 0.283 | 0.002 | 0.002 | 0.002 | 0.002 |

${}^{\mathrm{c}}$DSS${}_{\mathrm{c}}^{\mathrm{b}}$ | 0.034 | 0.000 | 0.289 | 0.412 | 0.173 | 0.148 | 0.160 | 0.147 | 0.001 | 0.002 | 0.002 | 0.001 |

${}^{\mathrm{c}}$DSS${}_{\mathrm{c}}^{\mathrm{a}}$ | 0.023 | 0.711 | 0.000 | 0.658 | 0.248 | 0.225 | 0.215 | 0.188 | 0.002 | 0.001 | 0.001 | 0.001 |

${}^{\mathrm{a}}$DSS${}_{\mathrm{c}}^{\mathrm{c}}$ | 0.053 | 0.588 | 0.343 | 0.000 | 0.227 | 0.187 | 0.198 | 0.175 | 0.003 | 0.002 | 0.001 | 0.001 |

${}^{\mathrm{c}}$DRS${}_{\mathrm{c}}^{\mathrm{b}}$ | 0.644 | 0.827 | 0.752 | 0.773 | 0.000 | 0.527 | 0.392 | 0.306 | 0.004 | 0.005 | 0.021 | 0.019 |

${}^{\mathrm{a}}$DRS${}_{\mathrm{c}}^{\mathrm{b}}$ | 0.655 | 0.852 | 0.775 | 0.813 | 0.473 | 0.000 | 0.402 | 0.314 | 0.004 | 0.005 | 0.008 | 0.006 |

${}^{\mathrm{c}}$DRS${}_{\mathrm{c}}^{\mathrm{b}}$ | 0.680 | 0.840 | 0.785 | 0.802 | 0.608 | 0.598 | 0.000 | 0.263 | 0.004 | 0.003 | 0.013 | 0.018 |

${}^{\mathrm{c}}$DRS${}_{\mathrm{c}}^{\mathrm{a}}$ | 0.717 | 0.853 | 0.812 | 0.825 | 0.694 | 0.686 | 0.737 | 0.000 | 0.010 | 0.004 | 0.018 | 0.029 |

${}^{\mathrm{c}}$DH${}_{\mathrm{c}}^{\mathrm{b}}$ | 0.998 | 0.999 | 0.999 | 0.997 | 0.996 | 0.996 | 0.996 | 0.990 | 0.000 | 0.185 | 0.496 | 0.759 |

${}^{\mathrm{c}}$DH${}_{\mathrm{c}}^{\mathrm{a}}$ | 0.999 | 0.998 | 0.999 | 0.999 | 0.995 | 0.996 | 0.998 | 0.996 | 0.815 | 0.000 | 0.783 | 0.891 |

${}^{\mathrm{a}}$DH${}_{\mathrm{c}}^{\mathrm{a}}$ | 0.998 | 0.998 | 0.999 | 0.999 | 0.979 | 0.992 | 0.987 | 0.982 | 0.504 | 0.218 | 0.000 | 0.773 |

${}^{\mathrm{a}}$DH${}_{\mathrm{c}}^{\mathrm{b}}$ | 0.998 | 0.999 | 0.999 | 0.999 | 0.982 | 0.994 | 0.982 | 0.971 | 0.241 | 0.109 | 0.227 | 0.000 |

**Table 8.**Pakistan’s electricity consumption (kWh): results (p-value) of the DM test for the bestproposed models versus the literature and the benchmark models given in Table 6.

Models | ${}^{\mathbf{c}}\mathbf{DH}_{\mathbf{c}}^{\mathbf{b}}$ | AR | NPAR | ARMA | P-ARMA | NP-ARMA |
---|---|---|---|---|---|---|

${}^{\mathrm{c}}\mathrm{DH}_{\mathrm{c}}^{\mathrm{b}}$ | - | <0.99 | <0.99 | <0.99 | <0.99 | <0.99 |

AR | >0.01 | - | 0.14 | 0.99 | <0.99 | 0.94 |

NPAR | >0.01 | 0.86 | - | <0.99 | <0.99 | 0.98 |

ARMA | >0.01 | 0.01 | >0.01 | - | 0.76 | 0.01 |

P-ARMA | >0.01 | >0.01 | >0.01 | 0.24 | - | >0.01 |

NP-ARMA | >0.01 | 0.06 | 0.02 | 0.99 | <0.99 | - |

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**MDPI and ACS Style**

Iftikhar, H.; Bibi, N.; Canas Rodrigues, P.; López-Gonzales, J.L. Multiple Novel Decomposition Techniques for Time Series Forecasting: Application to Monthly Forecasting of Electricity Consumption in Pakistan. *Energies* **2023**, *16*, 2579.
https://doi.org/10.3390/en16062579

**AMA Style**

Iftikhar H, Bibi N, Canas Rodrigues P, López-Gonzales JL. Multiple Novel Decomposition Techniques for Time Series Forecasting: Application to Monthly Forecasting of Electricity Consumption in Pakistan. *Energies*. 2023; 16(6):2579.
https://doi.org/10.3390/en16062579

**Chicago/Turabian Style**

Iftikhar, Hasnain, Nadeela Bibi, Paulo Canas Rodrigues, and Javier Linkolk López-Gonzales. 2023. "Multiple Novel Decomposition Techniques for Time Series Forecasting: Application to Monthly Forecasting of Electricity Consumption in Pakistan" *Energies* 16, no. 6: 2579.
https://doi.org/10.3390/en16062579