#
Energy Efficiency Forecast as an Inverse Stochastic Problem: A Cross-Entropy Econometrics Approach^{ †}

^{1}

^{2}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Energy Efficiency and Its Measurement

- the energy efficiency level,
- the energy efficiency trends,
- the energy efficiency policies, and
- the overall energy efficiency.

- Scoring is done separately for the four considered sectors (households, transport, industry, and services) and for all sectors together.
- The score by sector is based on scores computed for statistically selected indicators of end uses in buildings or modes in transport. For the industry sector, an aggregate score is obtained from various industrial branch scores that account for the energy efficiency characteristics of each of them.
- The score by sector is calculated as a weighted score of each indicator. The weights correspond to the average shares over the last 3 years of each end use or transport mode in the sector consumption.

## 3. Mathematical Problem Setting

- (a)
- Inverse problem and the maximum entropy principal

- -
- G is the amounts observed in rows or columns;
- -
- f is the unknown regional cross-sectoral energy efficiency coefficient matrix;
- -
- D defines the model Hilbert support space;
- -
- $h$ is the transformation kernel associating measures G and f; and
- -
- $b$ explains the random components.

- (b)
- Non-extensive cross-entropy energy model and confidence interval area

- -
- sample statistics are linear or collinear for various reasons;
- -
- non-stationary or non-cointegrating variables result from poor model specification;
- -
- data from the sampling plan are insufficient and/or incomplete due to technical or financial constraints—official statistics on small areas could illustrate this situation;
- -
- The Gaussian properties of random disturbances are questioned, among others, due to systematic errors resulting from the research process;
- -
- the model is not linear, and the last option is an approximate linearization; and
- -
- observations of aggregated data (in time or space) may hide a very complex system represented, for example, by a PL distribution, and there may be multifractal properties of the system.

## 4. Outputs and Comment

#### Limitations of the Study and Prospective Research Area

- -
- The power law distribution generalizes most of the known statistical laws and has proven to be designed for analytically resolving non-stationary functions. Its application leads to the analytical closed form outputs of the model.
- -
- The Kullback-Leibler relative entropy, being a combination between the properties of entropy and Bayesian formalism, stands for a strong information metric, particularly in the case of inverse problem modelling. Thanks to this formalism, several related hypotheses using the least squares method become obsolete.
- -
- While these two scientific sub-disciplines are based on solid hypotheses, joining traditional econometrics to them leads to the model proposed in this paper.

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Polish Province | Industry | Transport | Households | Services | Average Energy Intensity Coefficients (GWh\mln zl Value Added) |
---|---|---|---|---|---|

${Y}_{\xb71}{P}_{11}$ | ${Y}_{\xb72}{P}_{12}$ | ${Y}_{\xb7l}{P}_{1l}$ | ${Y}_{\xb7L}{P}_{1L}$ | 0.08 | |

Mazowieckie | ${Y}_{\xb71}{P}_{21}$ | .. | ${Y}_{\xb7l}{P}_{2l}$ | ${Y}_{\xb7l}{P}_{2L}$ | 0.05 |

Malopolskie | .. | .. | .. | .. | 0.06 |

Silesian | .. | .. | .. | .. | 0.09 |

Lublin | .. | .. | .. | .. | 0.07 |

Podkarpackie | .. | .. | .. | .. | 0.06 |

Podlaskie | .. | .. | .. | .. | 0.06 |

Swietokrzyskie | .. | .. | .. | .. | 0.09 |

Lubuskie | ${Y}_{\xb71}{P}_{k1}$ | .. | ${Y}_{\xb7l}{P}_{kl}$ | ${Y}_{\xb7L}{P}_{kL}$ | 0.07 |

Wielkopolska | .. | .. | .. | .. | 0.05 |

Zachodnia pomors | .. | .. | .. | .. | 0.07 |

Dolnoslaskie | .. | .. | .. | .. | 0.07 |

Opole | .. | .. | .. | .. | 0.11 |

Kujawska-pomorska | .. | .. | .. | .. | 0.07 |

Pomorska | .. | .. | .. | .. | 0.06 |

Warminsko-mazurskie | ${Y}_{\xb71}{P}_{K1}$ | .. | ${Y}_{\xb7l}{P}_{Kl}$ | ${Y}_{\xb7L}{P}_{KL}$ | 0.06 |

Average energy efficiency ratio | 0.372 | 0.612 | 0.621 | 0.689 |

**Table 2.**Post-entropic 2021 energy efficiency ratio forecasts and efficiency progress in 2020/2021(%).

Industry | Transport | Households | Services | Average Efficiency Ratio per Province | Efficiency Ratio Change_2021/2020 | |
---|---|---|---|---|---|---|

Lodz | 0.4 | 0.667 | 0.677 | 0.754 | 0.625 | −0.12 |

Slaskie | 0.429 | 0.729 | 0.741 | 0.83 | 0.682 | 0.81 |

Mazowieckie | 0.302 | 0.471 | 0.477 | 0.521 | 0.443 | −3.16 |

Wielkopolska | 0.333 | 0.526 | 0.533 | 0.583 | 0.494 | 7.49 |

Dolnoslaskie | 0.369 | 0.602 | 0.61 | 0.676 | 0.564 | −1.29 |

Opole | 0.482 | 0.847 | 0.861 | 0.974 | 0.791 | 2.43 |

Malopolskie | 0.336 | 0.536 | 0.543 | 0.596 | 0.503 | −2.44 |

Swietokrzyskie | 0.445 | 0.764 | 0.777 | 0.873 | 0.715 | 5.32 |

Zachodnia pomors | 0.373 | 0.61 | 0.618 | 0.685 | 0.572 | 0.04 |

Kujawska-pomorska | 0.37 | 0.604 | 0.613 | 0.678 | 0.566 | −0.88 |

Pomorska | 0.336 | 0.536 | 0.543 | 0.597 | 0.503 | −2.44 |

Lublin | 0.369 | 0.602 | 0.61 | 0.675 | 0.564 | −1.29 |

Podkarpackie | 0.336 | 0.536 | 0.543 | 0.597 | 0.503 | −2.33 |

Lubuskie | 0.386 | 0.636 | 0.645 | 0.716 | 0.596 | 4.07 |

Podlaskie | 0.336 | 0.536 | 0.543 | 0.597 | 0.503 | −2.39 |

Warminsko-mazurskie | 0.339 | 0.541 | 0.548 | 0.602 | 0.507 | −1.53 |

Sector average ratio | 0.371 | 0.609 | 0.618 | 0.685 | ||

Progress_2021/2020(in %) ratio | 0.067 | 0.369 | 0.385 | 0.548 |

Industry | Transport | Households | Services | |
---|---|---|---|---|

Lodz | −4.901 | −4.569 | −4.566 | −4.558 |

Slaskie | −9.481 | −8.137 | −8.098 | −7.812 |

Mazowieckie | 14.89 | 11.26 | 11.13 | 10.104 |

Wielkopolska | 14.89 | 11.26 | 11.13 | 10.104 |

Dolnoslaskie | 0.405 | −0.418 | −0.455 | −0.76 |

Opole | −17.153 | −14.151 | −14.051 | −13.297 |

Malopolskie | 6.77 | 4.634 | 4.552 | 3.9 |

Swietokrzyskie | −9.481 | −8.137 | −8.098 | −7.812 |

Zachodnia pomors | 0.405 | −0.418 | −0.455 | −0.76 |

Kujawska-pomorska | 0.405 | −0.418 | −0.455 | −0.76 |

Pomorska | 6.77 | 4.634 | 4.552 | 3.9 |

Lublin | 0.405 | −0.418 | −0.455 | −0.76 |

Podkarpackie | 6.77 | 4.634 | 4.552 | 3.9 |

Lubuskie | 0.405 | −0.418 | −0.455 | −0.76 |

Podlaskie | 6.77 | 4.634 | 4.552 | 3.9 |

Warminsko-mazurskie | 6.77 | 4.634 | 4.552 | 3.9 |

AEE ratio | −0.289 | −0.908 | −0.937 | −1.178 |

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**MDPI and ACS Style**

Bwanakare, S.
Energy Efficiency Forecast as an Inverse Stochastic Problem: A Cross-Entropy Econometrics Approach. *Energies* **2023**, *16*, 7715.
https://doi.org/10.3390/en16237715

**AMA Style**

Bwanakare S.
Energy Efficiency Forecast as an Inverse Stochastic Problem: A Cross-Entropy Econometrics Approach. *Energies*. 2023; 16(23):7715.
https://doi.org/10.3390/en16237715

**Chicago/Turabian Style**

Bwanakare, Second.
2023. "Energy Efficiency Forecast as an Inverse Stochastic Problem: A Cross-Entropy Econometrics Approach" *Energies* 16, no. 23: 7715.
https://doi.org/10.3390/en16237715