# Short Term Electrical Load Forecasting Using Mutual Information Based Feature Selection with Generalized Minimum-Redundancy and Maximum-Relevance Criteria

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Mutual Information-Based Generalized Minimal-Redundancy and Maximal-Relevance

_{m}with m features ($n\ll m$).

_{i}and target l is described as follows:

_{n}

_{−1}with n−1 features that has been selected. The aim is to select the nth feature from the rest of set {F

_{m}-J

_{n}

_{−1}} according to Equation (4). The incremental search method with respect to the condition is as follows:

_{n}

_{−1}| refers to the number of features in J

_{n}

_{−1}.

#### 2.2. Random Forest

#### 2.2.1. CART

_{i}is the number of ith class.

_{1}and D

_{2}subset, wherein the Gini index after the split is:

#### 2.2.2. Bagging

#### 2.2.3. RF

**x**is the input vector and $\left\{{\mathsf{\Theta}}_{k}\right\}$ represents the independent identically distributed random vectors. The modeling process of RF is:

- (1)
- k training sets are sampled with replacement from the dataset B by bootstrap.
- (2)
- Each training set grows up to a tree according to CART algorithm. Supposing dataset B has M features and mtry features are randomly selected from B for each non-leaf node. Afterward, the node is split by a feature selected from these mtry features.
- (3)
- Each tree grows completely without pruning.
- (4)
- The forecasting result is solved by calculating the mean value of the consequences of each tree predicted.

- (1)
- The same capacity of the training set sampled by bootstrap guarantees each sample in dataset B to be appraised equally. A situation that one sample may appear many times in the same training set and some may not causes low correlation among the trees.
- (2)
- The manner of selecting feature for node split applies randomness, and ensures the generalized performance of RF.

## 3. Data Analysis

_{t}

_{-168}, L

_{t}

_{-167}, …, L

_{t}

_{-2}, L

_{t-}

_{1}} are extracted as part of original feature set. When doing a day ahead load forecasting, assuming the current moment is t, the load values from the moment t-1 to t-24 are unknown. Therefore, the variables {L

_{t}

_{-24}, L

_{t}

_{-23}, …, L

_{t}

_{-1}} are eliminated from the original feature set. In addition, the features, such as hour of day, the day is within weekday or weekend, day of week and season, are considered for constructing the original feature set.

- F
_{Hour}means the moment of hour, which is tagged by the numbers from 1 to 24. - F
_{WW}is either weekday or weekend marked by binary numbers, wherein 0 means weekend and 1 means weekday. - F
_{DW}refers to the day of week, which is labeled by the numbers from 1 to 7. - F
_{S}uses the numbers from 1 to 4.

- F
_{L(t-25)}is the load 25 h before, F_{L(t-26)}means the load 26 h before, and so on.

## 4. The Proposed Feature Selection Method and STLF Model

_{i}is the actual value of load, ${\widehat{Z}}_{i}$ is the forecasting value, N is the number of sample.

#### 4.1. G-mRMR for Feature Selection

_{m}including m features and a selected feature set J. The detail of feature selection process is enumerated below:

- (1)
- Initialization Ø→J.
- (2)
- Compute the relevance between each feature and target variable l. Pick out the feature from F
_{m}which satisfies Equation (2) and add it into J. - (3)
- Find the feature in the rest of m−1 features in F
_{m}that satisfies Equation (4) and add it in to J. - (4)
- Repeat step (3) until F
_{m}becomes Ø. - (5)
- Rank the features in feature set J in descending order in accordance with the measured mRMR value.

#### 4.2. Wrapper for Feature Selection

_{m}, in the first step, the wrapper searched for the feature subset with only one feature, marked as S

_{1}, wherein the feature x

_{1}selected in S

_{1}leads to the largest prediction error reduction. In the second step, the wrapper selects the feature x

_{2}from {F

_{m}-S

_{1}} and combines with S

_{1}lead to the largest prediction error reduction. The search schedule is repeated until the prediction stops decreasing.

#### 4.3. The Proposed STLF Model

## 5. Case Study and Results Analysis

#### 5.1. Feature Selection Results Based on G-mRMR and RF

- (1)
- The training set and test set with optimal features are used for the experiment.
- (2)
- The initial number of tree nTree = 1.
- (3)
- Training RF and testing with different nTree value with increment of 1 until nTree = 500.

#### 5.2. Comparison Experiments for STLF

#### 5.2.1. Comparison of Different Feature Selection Methods

#### 5.2.2. Comparison of Different Intelligent Methods

^{2}= 2 [41].

_{neu}= 2p+1 [42], and the iteration is T = 2000 [43].

## 6. Conclusions

- (1)
- MI is adopted as the criterion to measure the relevance between features and time series of load and the dependency among features, which is the basis of quantitative analysis of feature selection by mRMR.
- (2)
- The correlation between features and load as well as the redundancy of these features are considered. As compared to the maximum relevance method, the G-mRMR method for feature selection reduces the number of optimal feature subset and avoids the association of STLF accuracy with the redundancy of features. For the time being, the relevance and redundancy are balanced by using a variable weighting factor. The features selected by G-mRMR make the accuracy of RF more precise than mRMR.
- (3)
- The optimal structure of RF is designed for reducing the complexity of the model and for improving the accuracy of STLF.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Yearly load curve analysis: (

**a**) Average daily load from 8 January 2005 to 31 December 2012; (

**b**) The population and GDP from 2005 to 2012; (

**c**) Hourly load autocorrelation of historical load data.

**Figure 8.**Prediction error curves: (

**a**) Prediction error curves corresponding to different weighting factor α; (

**b**) The enlarged figure of red box in (

**a**).

**Figure 10.**Prediction error curves: (

**a**) Prediction error curves corresponding to different feature selection methods; (

**b**) The enlarge figure of red box in (

**a**).

**Figure 11.**Load curves of forecasting results of four weeks in four seasons and the true values: (

**a**) Forecasting from 23 to 29 February 2012; (

**b**) Forecasting from 13 to 19 May 2012; (

**c**) Forecasting from 21 to 27 August 2012; (

**d**) Forecasting from 24 to 30 November 2012.

**Figure 12.**Forecasting error profiles of different predictors: (

**a**) Forecasting from 23 to 29 February 2012; (

**b**) Forecasting from 13 to 19 May 2012; (

**c**) Forecasting from 21 to 27 August 2012; (

**d**) Forecasting from 24 to 30 November 2012.

Feature Type | Original Feature |
---|---|

Exogenous features | 1.F_{Hour}, 2.F_{WW}, 3.F_{DW}, 4.F_{S} |

Endogenous features | 5.F_{L(t-25)}, 6.F_{L(t-26)}, 7.F_{L(t-27)}, 8.F_{L(t-28)}, …, 146.F_{L(t-166)}, 147.F_{L(t-167)}, 148.F_{L(t-168)} |

Data Set | Information | Purpose |
---|---|---|

Training Set | January, February, May, June, August, September, October, December | Train RF |

Validation Set | March, April, July, November | Use for obtain the best weighting factor |

Test Set | 23–29 February 2012 (Winter) 13–19 May 2012 (Spring) 21–27 August 2012 (Summer) 24–30 November 2012 (Fall) | Test performance of RF |

α | Min MAPE (%) | Number of Features | Feature Subset |
---|---|---|---|

0.1 | 2.5640 | 26 | F_{L(t-168)}, F_{L(t-25)}, F_{L(t-48)}, F_{L(t-144)}, F_{L(t-72)}, F_{Hour}, F_{L(t-47)}, F_{L(t-26)}, F_{L(t-120)}, F_{L(t-167)}, F_{WW}, F_{S}, F_{DW}, F_{L(t-34)}, F_{L(t-158)}, F_{L(t-103)}, F_{L(t-27)}, F_{L(t-96)}, F_{L(t-162)}, F_{L(t-132)}, F_{L(t-44)}, F_{L(t-88)}, F_{L(t-149)}, F_{L(t-153)}, F_{L(t-37)}, F_{L(t-107)} |

0.2 | 2.5857 | 25 | F_{L(t-168)}, F_{L(t-25)}, F_{L(t-48)}, F_{L(t-144)}, F_{Hour}, F_{S}, F_{WW}, F_{L(t-71)}, F_{DW}, F_{L(t-27)}, F_{L(t-106)}, F_{L(t-162)}, F_{L(t-38)}, F_{L(t-127)}, F_{L(t-93)}, F_{L(t-156)}, F_{L(t-88)}, F_{L(t-32)}, F_{L(t-29)}, F_{L(t-96)}, F_{L(t-44)}, F_{L(t-134)}, F_{L(t-26)}, F_{L(t-166)}, F_{L(t-59)} |

0.3 | 2.5858 | 27 | F_{L(t-168)}, F_{L(t-25)}, F_{L(t-48)}, F_{Hour}, F_{WW}, F_{S}, F_{L(t-144)}, F_{DW}, F_{L(t-103)}, F_{L(t-37)}, F_{L(t-162)}, F_{L(t-70)}, F_{L(t-131)}, F_{L(t-28)}, F_{L(t-88)}, F_{L(t-153)}, F_{L(t-106)}, F_{L(t-75)}, F_{L(t-159)}, F_{L(t-34)}, F_{L(t-125)}, F_{L(t-96)}, F_{L(t-43)}, F_{L(t-165)}, F_{L(t-109)}, F_{L(t-31)}, F_{L(t-26)} |

0.4 | 2.5597 | 15 | F_{L(t-168)}, F_{L(t-25)}, F_{L(t-48)}, F_{WW}, F_{S}, F_{L(t-127)}, F_{L(t-85)}, F_{L(t-139)}, F_{DW}, F_{L(t-34)}, F_{L(t-160)}, F_{L(t-70)}, F_{L(t-28)}, F_{L(t-120)}, F_{L(t-141)} |

0.5 | 2.5897 | 80 | F_{L(t-168)}, F_{L(t-25)}, F_{L(t-47)}, F_{WW}, F_{S}, F_{L(t-127)}, F_{L(t-86)}, F_{DW}, F_{L(t-139)}, F_{L(t-35)}, F_{L(t-99)}, F_{L(t-160)}, F_{L(t-69)}, F_{L(t-29}), F_{L(t-154)}, F_{L(t-120)}, F_{L(t-41)}, F_{L(t-81)}, F_{L(t-133)}, F_{L(t-148)}, F_{L(t-166)}, F_{L(t-32)}, F_{L(t-63)}, F_{L(t-92)}, F_{L(t-26)}, F_{L(t-108)}, F_{L(t-162)}, F_{L(t-78)}, … |

0.6 | 2.5868 | 46 | F_{L(t-168)}, F_{L(t-25)}, F_{Hour}, F_{L(t-47)}, F_{S}, F_{L(t-127)}, F_{L(t-88)}, F_{DW}, F_{L(t-156)}, F_{L(t-139)}, F_{L(t-76)}, F_{L(t-34)}, F_{L(t-110)}, F_{L(t-69)}, F_{L(t-149)}, F_{L(t-120)}, F_{L(t-41)}, F_{L(t-81)}, F_{L(t-27)}, F_{L(t-165)}, F_{L(t-37)}, F_{L(t-162)}, F_{L(t-98)}, F_{L(t-30)}, F_{L(t-131)}, F_{L(t-159)}, F_{L(t-104)}, F_{L(t-44)}, … |

0.7 | 2.5891 | 88 | F_{L(t-168)}, F_{L(t-25)}, F_{WW}, F_{S}, F_{L(t-103)}, F_{L(t-61)}, F_{L(t-139)}, F_{DW}, F_{L(t-47)}, F_{L(t-160)}, F_{L(t-82)}, F_{L(t-124)}, F_{L(t-30)}, F_{L(t-93)}, F_{L(t-156)}, F_{L(t-41)}, F_{L(t-146)}, F_{L(t-33)}, F_{L(t-110)}, F_{L(t-72)}, F_{L(t-152)}, F_{L(t-164)}, F_{L(t-27)}, F_{L(t-90)}, F_{L(t-131)}, F_{L(t-39)}, F_{L(t-118)}, F_{L(t-77)}, … |

0.8 | 2.6046 | 93 | F_{L(t-168)}, F_{L(t-25)}, F_{WW}, F_{S}, F_{L(t-103)}, F_{L(t-61)}, F_{L(t-139)}, F_{DW}, F_{L(t-47)}, F_{L(t-160)}, F_{L(t-82)}, F_{L(t-124)}, F_{L(t-30)}, F_{L(t-93)}, F_{L(t-156)}, F_{L(t-41)}, F_{L(t-146)}, F_{L(t-33)}, F_{L(t-110)}, F_{L(t-166)}, F_{L(t-75)}, F_{L(t-152)}, F_{L(t-90)}, F_{L(t-72)}, F_{L(t-44)}, F_{L(t-131)}, F_{L(t-28)}, F_{L(t-39)}, … |

0.9 | 2.5918 | 35 | F_{L(t-168)}, F_{L(t-25)}, F_{WW}, F_{S}, F_{L(t-103)}, F_{L(t-67)}, F_{L(t-133)}, F_{DW}, F_{L(t-34)}, F_{L(t-160)}, F_{L(t-46)}, F_{L(t-148)}, F_{L(t-96)}, F_{L(t-29)}, F_{L(t-84)}, F_{L(t-140)}, F_{L(t-39)}, F_{L(t-153)}, F_{L(t-75)}, F_{L(t-114)}, F_{L(t-165)}, F_{L(t-56)}, F_{L(t-122)}, F_{L(t-62)}, F_{L(t-155)}, F_{L(t-126)}, F_{L(t-41)}, F_{L(t-119)}, … |

Day | G-mRMR-RF (α = 0.4) | MI-RF | PCC-RF | SFS-RF | RF with Full Features | |||||
---|---|---|---|---|---|---|---|---|---|---|

MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | |

Day 1 | 1.93 | 75.24 | 1.79 | 69.42 | 10.28 | 401.01 | 2.07 | 79.58 | 1.91 | 70.74 |

Day 2 | 1.77 | 66.63 | 1.78 | 67.90 | 9.78 | 388.26 | 2.22 | 79.46 | 1.80 | 69.13 |

Day 3 | 1.58 | 53.24 | 1.63 | 51.63 | 7.59 | 285.51 | 1.47 | 49.33 | 1.50 | 50.49 |

Day 4 | 1.69 | 79.28 | 1.59 | 70.02 | 5.35 | 189.65 | 2.52 | 105.32 | 1.98 | 76.33 |

Day 5 | 2.26 | 90.72 | 2.66 | 104.16 | 11.14 | 440.91 | 2.04 | 83.68 | 2.91 | 113.32 |

Day 6 | 1.58 | 57.73 | 2.37 | 83.87 | 9.78 | 396.44 | 1.61 | 57.41 | 2.54 | 87.59 |

Day 7 | 1.28 | 51.92 | 0.97 | 36.35 | 9.26 | 362.46 | 1.87 | 73.03 | 1.29 | 44.60 |

Average | 1.72 | 67.82 | 1.82 | 69.05 | 9.02 | 352.03 | 1.97 | 75.40 | 1.99 | 73.17 |

Day | G-mRMR-RF (α = 0.4) | MI-RF | PCC-RF | SFS-RF | RF with Full Features | |||||
---|---|---|---|---|---|---|---|---|---|---|

MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | |

Day 1 | 1.20 | 42.36 | 1.22 | 39.15 | 3.76 | 110.39 | 1.43 | 50.90 | 1.57 | 47.92 |

Day 2 | 1.64 | 60.32 | 1.33 | 50.26 | 8.98 | 273.78 | 1.28 | 46.10 | 1.37 | 53.34 |

Day 3 | 2.04 | 66.88 | 2.04 | 67.09 | 6.56 | 246.64 | 2.03 | 69.43 | 2.00 | 66.78 |

Day 4 | 0.94 | 34.38 | 0.96 | 34.48 | 7.04 | 263.29 | 0.89 | 34.98 | 1.11 | 41.54 |

Day 5 | 1.55 | 53.26 | 1.40 | 46.62 | 7.17 | 261.54 | 1.40 | 50.04 | 1.50 | 52.38 |

Day 6 | 1.28 | 41.45 | 1.34 | 44.68 | 6.66 | 237.55 | 1.28 | 40.22 | 1.45 | 40.03 |

Day 7 | 0.84 | 26.82 | 0.99 | 36.97 | 5.51 | 178.83 | 0.92 | 50.61 | 1.01 | 49.05 |

Average | 1.35 | 46.49 | 1.33 | 48.03 | 6.53 | 224.57 | 1.32 | 48.90 | 1.40 | 50.15 |

Day | G-mRMR-RF (α = 0.4) | MI-RF | PCC-RF | SFS-RF | RF with Full Features | |||||
---|---|---|---|---|---|---|---|---|---|---|

MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | |

Day 1 | 2.88 | 92.71 | 2.61 | 83.05 | 6.69 | 258.31 | 3.32 | 104.41 | 2.89 | 90.68 |

Day 2 | 1.48 | 55.30 | 1.55 | 56.81 | 8.22 | 319.74 | 1.77 | 62.53 | 1.59 | 57.62 |

Day 3 | 0.91 | 31.93 | 0.82 | 29.02 | 7.04 | 263.33 | 1.00 | 38.68 | 1.07 | 36.28 |

Day 4 | 1.88 | 76.95 | 2.27 | 90.86 | 8.97 | 344.82 | 1.99 | 84.88 | 2.17 | 87.44 |

Day 5 | 1.77 | 54.77 | 1.87 | 56.56 | 6.42 | 227.25 | 2.16 | 70.95 | 1.91 | 58.15 |

Day 6 | 2.08 | 73.60 | 1.78 | 71.44 | 5.91 | 181.33 | 1.71 | 65.13 | 1.86 | 74.78 |

Day 7 | 6.12 | 208.00 | 6.77 | 237.00 | 11.26 | 458.19 | 6.98 | 247.66 | 6.57 | 227.17 |

Average | 2.45 | 72.83 | 2.52 | 89.25 | 7.79 | 293.28 | 2.70 | 96.32 | 2.58 | 90.30 |

Day | G-mRMR-RF (α = 0.4) | MI-RF | PCC-RF | SFS-RF | RF with Full Features | |||||
---|---|---|---|---|---|---|---|---|---|---|

MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | |

Day 1 | 1.61 | 58.60 | 1.64 | 58.26 | 6.80 | 263.40 | 1.96 | 68.90 | 1.78 | 63.62 |

Day 2 | 1.05 | 48.24 | 1.12 | 43.26 | 6.97 | 242.74 | 2.11 | 78.43 | 1.09 | 40.30 |

Day 3 | 1.98 | 74.62 | 2.06 | 74.67 | 10.78 | 427.39 | 1.98 | 73.90 | 2.12 | 76.64 |

Day 4 | 1.47 | 57.14 | 1.33 | 48.09 | 9.21 | 387.30 | 1.80 | 67.13 | 1.50 | 57.63 |

Day 5 | 1.12 | 42.84 | 0.90 | 33.83 | 10.29 | 413.17 | 1.15 | 46.87 | 1.26 | 45.01 |

Day 6 | 1.31 | 53.79 | 1.33 | 52.33 | 9.03 | 389.52 | 1.32 | 54.74 | 1.23 | 47.40 |

Day 7 | 1.10 | 42.86 | 1.22 | 45.31 | 9.53 | 387.69 | 1.06 | 44.21 | 1.18 | 43.42 |

Average | 1.38 | 54.01 | 1.37 | 50.82 | 8.93 | 358.74 | 1.63 | 62.02 | 1.45 | 53.43 |

Predictor | Min MAPE (%) | Number of Features | Feature Subset |
---|---|---|---|

G-mRMR-RF (α = 0.4) | 2.5389% | 15 | F_{L(t-168)}, F_{L(t-25)}, F_{L(t-48)}, F_{WW}, F_{S}, F_{L(t-127)}, F_{L(t-85)}, F_{L(t-139)}, F_{DW}, F_{L(t-34)}, F_{L(t-160)}, F_{L(t-70)}, F_{L(t-28)}, F_{L(t-120)}, F_{L(t-141)} |

G-mRMR-SVR (α = 0.3) | 3.3293% | 5 | F_{L(t-168)}, F_{L(t-25)}, F_{L(t-48)}, F_{Hour}, F_{WW} |

G-mRMR-BPNN (α = 0.1) | 2.7186% | 11 | F_{L(t-168)}, F_{L(t-25)}, F_{L(t-48)}, F_{L(t-144)}, F_{L(t-72)}, F_{Hour}, F_{L(t-47)}, F_{L(t-26)}, F_{L(t-120)}, F_{L(t-167)}, F_{WW} |

Test Set | MAPE (%) | G-mRMR-RF | G-mRMR-ANN | G-mRMR-SVR |
---|---|---|---|---|

23–29 February 2012 (Winter) | Max | 2.26 | 3.72 | 4.24 |

Min | 1.28 | 1.23 | 1.37 | |

Ave | 1.72 | 2.18 | 2.61 | |

13–19 May 2012 (Spring) | Max | 2.04 | 3.14 | 4.42 |

Min | 0.84 | 1.24 | 1.87 | |

Ave | 1.35 | 2.05 | 2.78 | |

21–27 August 2012 (Summer) | Max | 6.12 | 6.32 | 7.68 |

Min | 0.91 | 1.78 | 1.69 | |

Ave | 2.45 | 2.87 | 3.50 | |

24–30 November 2012 (Fall) | Max | 1.98 | 2.84 | 4.01 |

Min | 1.05 | 1.38 | 1.80 | |

Ave | 1.38 | 2.10 | 2.86 |

© 2016 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Huang, N.; Hu, Z.; Cai, G.; Yang, D.
Short Term Electrical Load Forecasting Using Mutual Information Based Feature Selection with Generalized Minimum-Redundancy and Maximum-Relevance Criteria. *Entropy* **2016**, *18*, 330.
https://doi.org/10.3390/e18090330

**AMA Style**

Huang N, Hu Z, Cai G, Yang D.
Short Term Electrical Load Forecasting Using Mutual Information Based Feature Selection with Generalized Minimum-Redundancy and Maximum-Relevance Criteria. *Entropy*. 2016; 18(9):330.
https://doi.org/10.3390/e18090330

**Chicago/Turabian Style**

Huang, Nantian, Zhiqiang Hu, Guowei Cai, and Dongfeng Yang.
2016. "Short Term Electrical Load Forecasting Using Mutual Information Based Feature Selection with Generalized Minimum-Redundancy and Maximum-Relevance Criteria" *Entropy* 18, no. 9: 330.
https://doi.org/10.3390/e18090330