# Generalised Regression Hypothesis Induction for Energy Consumption Forecasting

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Symbolic Regression

#### 2.2. Multi-Objective Optimisation Paradigm

## 3. Methods

#### 3.1. Straight Line Programs for Time Series Prediction

#### 3.2. Single-Objective Problem Formulation

#### 3.3. Multi-Objective Problem Formulation

#### 3.4. Algorithm Description

- Crossover operator. Two parents ${P}_{1}$ and ${P}_{2}$ are used in both single- and multi-objective approaches in order to generate two new children ${C}_{1}$ and ${C}_{2}$. The operator starts out selecting a random rule ${U}_{i}\in \{{U}_{1},{U}_{2},\dots ,{U}_{N-1}\}$ from ${P}_{1}$. After that, an ordered set of rules R is calculated as the set of rules $U=\{{U}_{1},{U}_{2},\dots ,{U}_{i-1}\}$ that can be reached from the selected rule ${U}_{i}$. Then, a random rule ${U}_{k}\in \{{U}_{1},{U}_{2},\dots ,{U}_{N-\left|R\right|+1}\}$ from ${P}_{2}$ is selected, where $\left|R\right|$ is the number of rules included in R. The offspring ${C}_{1}$ is created as a copy of the parent ${P}_{2}$ and the rules in R are copied into ${C}_{1}$ and renamed from ${U}_{k-\left|R\right|+1}$ to ${U}_{k}$. Finally, the offspring ${C}_{2}$ is generated with the same procedure, but exchanging the roles of both parents ${P}_{1}$ and ${P}_{2}$. An example of this operator is shown in Figure 1, where rule ${r}_{1}=5$ was selected randomly from parent ${P}_{1}$. After that, the ruleset U is created as the set of rules that can be reached from ${U}_{5}$. In this case, $U=\{{U}_{5},{U}_{2}{,}_{1}\}$, and is renamed as $R=\{{r}_{1},{r}_{2},{r}_{3}\}$. Then, a random position ${r}_{2}=4$ is selected in ${P}_{2}$, and the offspring ${C}_{1}$ is created as a copy of ${P}_{2}$ with the replacement of rules in R, starting from ${r}_{2}$.
- Mutation operator. Given a SLP table of an individual of the population, a random element of the consequent of a random rule is exchanged for another random symbol. If the selected element is an operator, it is exchanged by another valid operator and if the selected element is an operand, it is exchanged by a terminal symbol or a reference to other rule, as shown in Figure 2. On the other hand, if the mutation operator exchanges a binary operator by an unary operator, the second operand of the rule is left to the value ∅. Nevertheless, if the operator mutes an unary operator to a binary operator, then the second operand is randomly selected from the set of valid operands of the production rule (independent variables, parameters or references to other rules of the SLP table).

## 4. Experimentation

- A first study attempted to empirically validate whether the described problem can be solved with the proposed formulations. Thus, we built different scenarios with synthetic data, with the aim of validating the performance of each approach under a controlled experimental environment that eases the analysis of performance of the approaches. To that end, Section 4.1 describes a set of benchmark algebraic expressions to be used in the experiments, and the results obtained with each approach. This section ends with a discussion of the results obtained.
- In the second experiment (Section 4.2), we tackled a real problem about energy consumption prediction. In it, we were provided with the energy consumption time series of a set of buildings for which we assumed there is a medium or high correlation and the goal was to find a single parameterised algebraic expression $F({X}_{i},{W}_{i})$ that can explain the global/common behaviour of all related energy consumption data series.

#### 4.1. Experimentation with Synthetic Data

#### 4.1.1. Data Acquisition

#### 4.1.2. Experimental Settings

#### 4.1.3. Results and Discussion

#### 4.2. Experimentation with Real Data

#### 4.2.1. Data Acquisition

#### 4.2.2. Experimental Settings

#### 4.2.3. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Energy consumption data series of cluster of buildings ${E}_{1}$ and ${E}_{2}$ during 500 days.

**Figure 8.**Box plots of accuracy for both single- and multi-objective approach and cluster of buildings.

**Figure 9.**Real and predicted energy consumption from both single- and multi-objective approaches in cluster of buildings ${E}_{1}$.

**Figure 10.**Real and predicted energy consumption from both single- and multi-objective approaches in cluster of buildings ${E}_{2}$.

Items | 3 Datasets | 5 Datasets | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | ${\mathit{f}}_{3}$ | ${\mathit{f}}_{4}$ | ${\mathit{f}}_{5}$ | ${\mathit{f}}_{6}$ | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | ${\mathit{f}}_{3}$ | ${\mathit{f}}_{4}$ | ${\mathit{f}}_{5}$ | ${\mathit{f}}_{6}$ | |

TRAIN | ||||||||||||

Single objective | ||||||||||||

Median | $3.12\times {10}^{-16}$ | $1.53\times {10}^{-2}$ | $1.89\times {10}^{8}$ | $1.45\times {10}^{-7}$ | $4.92$ | $4.46\times {10}^{-5}$ | $7.81\times {10}^{-10}$ | $5.5\times {10}^{-2}$ | $3.99\times {10}^{7}$ | $6.27\times {10}^{-6}$ | $3.75$ | $5.55\times {10}^{-5}$ |

Best | $6.45\times {10}^{-18}$ | $6.2\times {10}^{-4}$ | $2.18\times {10}^{6}$ | $1.37\times {10}^{-9}$ | $1.2\times {10}^{-1}$ | $2.62\times {10}^{-11}$ | $2.97\times {10}^{-11}$ | $2.1\times {10}^{-3}$ | $8.77\times {10}^{6}$ | $2.03\times {10}^{-9}$ | $2.1\times {10}^{-1}$ | $4.91\times {10}^{-11}$ |

Worst | $1.52\times {10}^{-14}$ | $4.5\times {10}^{-2}$ | $4.86\times {10}^{8}$ | $6.61\times {10}^{-3}$ | $1.76\times {10}^{1}$ | $1.77\times {10}^{-3}$ | $5.67\times {10}^{-8}$ | $8.9\times {10}^{-2}$ | $9.41\times {10}^{7}$ | $6.33\times {10}^{-3}$ | $1.83\times {10}^{1}$ | $3.18\times {10}^{-3}$ |

Time (s.) | $1.03\times {10}^{3}$ | $1.3\times {10}^{3}$ | $1.81\times {10}^{3}$ | $1.21\times {10}^{3}$ | $1.35\times {10}^{3}$ | $1.17\times {10}^{3}$ | $2.03\times {10}^{3}$ | $2.14\times {10}^{3}$ | $2.14\times {10}^{3}$ | $2.06\times {10}^{3}$ | $2.16\times {10}^{3}$ | $1.96\times {10}^{3}$ |

Size | $8.93$ | $10.6$ | $11.76$ | $9.96$ | $11.16$ | $11.23$ | $9.63$ | $10.5$ | $10.76$ | $10.33$ | $11.56$ | $10.9$ |

Parameters | $2.8$ | $3.33$ | $2.83$ | $3.2$ | $3.16$ | $3.33$ | $3.13$ | $3.3$ | $2.66$ | $2.83$ | $3.36$ | $3.2$ |

Multi objective | ||||||||||||

Median | $4.6\times {10}^{-15}$ | $3.1\times {10}^{-3}$ | $8.49\times {10}^{6}$ | $2.87\times {10}^{-7}$ | $9.4\times {10}^{-1}$ | $2.12\times {10}^{-6}$ | $1.13\times {10}^{-8}$ | $8\times {10}^{-2}$ | $4.43\times {10}^{7}$ | $5.62\times {10}^{-7}$ | $1.75\times {10}^{1}$ | $3.95\times {10}^{-6}$ |

Best | $3.54\times {10}^{-18}$ | $3.83\times {10}^{-5}$ | $1.21\times {10}^{5}$ | $1.36\times {10}^{-9}$ | $1.09\times {10}^{-1}$ | $2.58\times {10}^{-11}$ | $2.92\times {10}^{-11}$ | $1.2\times {10}^{-3}$ | $1.11\times {10}^{6}$ | $2\times {10}^{-9}$ | $7.2\times {10}^{-1}$ | $4.86\times {10}^{-11}$ |

Worst | $2.59\times {10}^{-13}$ | $4.3\times {10}^{-1}$ | $1.32\times {10}^{9}$ | $3.92\times {10}^{-2}$ | $3.78\times {10}^{2}$ | $1.19\times {10}^{-3}$ | $8.14\times {10}^{-7}$ | $2.22$ | $2\times {10}^{9}$ | $2.54\times {10}^{-1}$ | $3.07\times {10}^{2}$ | $6.88\times {10}^{-3}$ |

Time (s.) | $1.38\times {10}^{3}$ | $1.91\times {10}^{3}$ | $2.58\times {10}^{3}$ | $1.72\times {10}^{3}$ | $3.12\times {10}^{3}$ | $1.67\times {10}^{3}$ | $2.43\times {10}^{3}$ | $2.89\times {10}^{3}$ | $4.47\times {10}^{3}$ | $2.96\times {10}^{3}$ | $5.49\times {10}^{3}$ | $2.9\times {10}^{3}$ |

Size | $11.76$ | $12.1$ | $12.76$ | $12.7$ | 12 | $13.23$ | $7.56$ | $9.96$ | $11.76$ | $12.36$ | $10.33$ | $13.1$ |

Parameters | 3 | $3.53$ | $2.7$ | $3.3$ | $3.1$ | $3.86$ | $2.43$ | $3.03$ | $2.23$ | $3.53$ | $3.06$ | $3.8$ |

TEST | ||||||||||||

Single-Objective | ||||||||||||

Median | $2.01\times {10}^{-15}$ | $2.42\times {10}^{-2}$ | $8.21\times {10}^{4}$ | $8.94\times {10}^{-8}$ | $5.52$ | $9.63\times {10}^{-5}$ | $1.79\times {10}^{-9}$ | $4.9\times {10}^{-2}$ | $1.09\times {10}^{5}$ | $4.38\times {10}^{-5}$ | $3.91$ | $3.95\times {10}^{-4}$ |

Best | $7.94\times {10}^{-18}$ | $1.3\times {10}^{-3}$ | $2.26\times {10}^{4}$ | $7.19\times {10}^{-10}$ | $1.4\times {10}^{-1}$ | $2.44\times {10}^{-11}$ | $4.21\times {10}^{-11}$ | $3.2\times {10}^{-3}$ | $3.67\times {10}^{4}$ | $1.19\times {10}^{-9}$ | $1.5\times {10}^{-1}$ | $4.36\times {10}^{-11}$ |

Worst | $3.6\times {10}^{-1}$ | $6.18$ | $1.65\times {10}^{5}$ | $4.7\times {10}^{-2}$ | $9.38\times {10}^{2}$ | $1.97$ | $1.1\times {10}^{-5}$ | $2.4\times {10}^{-1}$ | $2.28\times {10}^{5}$ | $1.67\times {10}^{-2}$ | $1.23\times {10}^{3}$ | $2.72\times {10}^{-1}$ |

Multi-Objective | ||||||||||||

Median | $1.36\times {10}^{-12}$ | $1.05\times {10}^{1}$ | $1.65\times {10}^{5}$ | $5.3$ | $9.7\times {10}^{2}$ | $2.6\times {10}^{-1}$ | $1.59\times {10}^{-6}$ | $1.83\times {10}^{1}$ | $2.28\times {10}^{5}$ | $3.05$ | $9.33\times {10}^{2}$ | $6.6\times {10}^{-1}$ |

Best | $1.36\times {10}^{-12}$ | $8.57$ | $1.45\times {10}^{5}$ | $1.08$ | $4.67\times {10}^{2}$ | $6\times {10}^{-2}$ | $1.32\times {10}^{-6}$ | $1.23\times {10}^{1}$ | $1.97\times {10}^{5}$ | $1.65$ | $8.1\times {10}^{2}$ | $7\times {10}^{-2}$ |

Worst | 3 | $4.22\times {10}^{1}$ | $1.66\times {10}^{5}$ | $5.3$ | $1.08\times {10}^{3}$ | $1.52\times {10}^{1}$ | $1.58\times {10}^{-6}$ | $8.6\times {10}^{1}$ | $2.29\times {10}^{5}$ | $7.51$ | $1.28\times {10}^{3}$ | $6.7\times {10}^{-1}$ |

**Table 2.**Statistical tests to compare algorithms in the results of all benchmark algebraic expressions.

Train | Test | |||
---|---|---|---|---|

3 Datasets | 5 Datasets | 3 Datasets | 5 Datasets | |

${f}_{1}$ | $1.7\times {10}^{-1}$ (x) | $4.9\times {10}^{-3}$ (+) | $2.87\times {10}^{-9}$ (+) | $1.39\times {10}^{-10}$ (+) |

${f}_{2}$ | $1.3\times {10}^{-1}$ (x) | $2.8\times {10}^{-2}$ (+) | $2.83\times {10}^{-11}$ (+) | $2.86\times {10}^{-11}$ (+) |

${f}_{3}$ | $1.4\times {10}^{-2}$ (-) | $7.9\times {10}^{-1}$ (x) | $3.95\times {10}^{-11}$ (+) | $1.22\times {10}^{-10}$ (+) |

${f}_{4}$ | $8.01\times {10}^{-1}$ (x) | $8.01\times {10}^{-1}$ (x) | $1.67\times {10}^{-11}$ (+) | $2.62\times {10}^{-11}$ (+) |

${f}_{5}$ | $5.8\times {10}^{-1}$ (x) | $2.8\times {10}^{-4}$ (+) | $2.79\times {10}^{-10}$ (+) | $2.41\times {10}^{-11}$ (+) |

${f}_{6}$ | $7\times {10}^{-3}$ (-) | $1.9\times {10}^{-1}$ (x) | $7.55\times {10}^{-11}$ (+) | $1.81\times {10}^{-11}$ (+) |

Train | Test | |||
---|---|---|---|---|

${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | ${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | |

Single-Objective | ||||

Median | $2.3\times {10}^{-2}$ | $6.7\times {10}^{-2}$ | $3.1\times {10}^{-2}$ | $7.2\times {10}^{-2}$ |

Best | $1.6\times {10}^{-2}$ | $5.5\times {10}^{-2}$ | $2.3\times {10}^{-2}$ | $5.3\times {10}^{-2}$ |

Worst | $3\times {10}^{-2}$ | $8.1\times {10}^{-2}$ | $8.4\times {10}^{-2}$ | $2.4\times {10}^{-1}$ |

Time | $1.02\times {10}^{3}$ | $2.53\times {10}^{3}$ | - | - |

Size | 10.83 | 11.56 | - | - |

Parameters | 2.43 | 3 | - | - |

Multi-Objective | ||||

Median | $1.9\times {10}^{-2}$ | $6.8\times {10}^{-2}$ | $3.1\times {10}^{-2}$ | $7.2\times {10}^{-2}$ |

Best | $1.4\times {10}^{-2}$ | $5.1\times {10}^{-2}$ | $2.3\times {10}^{-2}$ | $5.3\times {10}^{-2}$ |

Worst | $3.7\times {10}^{-2}$ | $1.1\times {10}^{-1}$ | $8.4\times {10}^{-2}$ | $2.4\times {10}^{-1}$ |

Time | $1.61\times {10}^{3}$ | $4.12\times {10}^{3}$ | - | - |

Size | 11.03 | 4.73 | - | - |

Parameters | 2.9 | 1.46 | - | - |

KW Test | $4.49\times {10}^{-5}$ (-) | $0.35$ (x) | $2.9\times {10}^{-11}$ (+) | $1.21\times {10}^{-10}$ (+) |

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**MDPI and ACS Style**

Rueda, R.; Cuéllar, M.P.; Molina-Solana, M.; Guo, Y.; Pegalajar, M.C. Generalised Regression Hypothesis Induction for Energy Consumption Forecasting. *Energies* **2019**, *12*, 1069.
https://doi.org/10.3390/en12061069

**AMA Style**

Rueda R, Cuéllar MP, Molina-Solana M, Guo Y, Pegalajar MC. Generalised Regression Hypothesis Induction for Energy Consumption Forecasting. *Energies*. 2019; 12(6):1069.
https://doi.org/10.3390/en12061069

**Chicago/Turabian Style**

Rueda, R., M. P. Cuéllar, M. Molina-Solana, Y. Guo, and M. C. Pegalajar. 2019. "Generalised Regression Hypothesis Induction for Energy Consumption Forecasting" *Energies* 12, no. 6: 1069.
https://doi.org/10.3390/en12061069