# Comparison between Time- and Observation-Based Gaussian Process Regression Models for Global Horizontal Irradiance Forecasting

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## Abstract

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## 1. Introduction

- What input should be chosen, time or past GHI observations?
- For each of those inputs, what is most appropriate kernel?
- What are the advantages and disadvantages of multi-horizon GPR models and horizon-specific GPR models?

## 2. Description of the Forecasting Models Included in This Study

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^{®}CPU E7-4890 v2 @ 2.80 GHz. The Gaussian Processes for Machine Learning (GPML) Matlab toolbox [28] was used for the GPR models’ implementation.

#### 2.1. Scaled Persistence Model

- t is the time;
- ${\theta}_{z}$ is the solar zenith angle;
- ${I}_{0}$ is the extraterrestrial solar irradiance: the solar irradiance incident on the outer limit of the Earth’s atmosphere, considering the whole solar spectrum;
- m is the relative optical air mass, defined as the ratio of the optical path length of the solar beam through the atmosphere to the optical path through a standard atmosphere at sea level with the Sun at the zenith; it is calculated here by the Kasten and Young’s formula [37], widely used by the scientific community:$$\begin{array}{c}\hfill m\left(t\right)={\left(cos\left({\theta}_{z}\left(t\right)\right)+\frac{{a}_{1}}{{\left({a}_{2}-{\theta}_{z}\left(t\right)\right)}^{{a}_{3}}}\right)}^{-1}\end{array}$$
- ${T}_{L}$ is the Linke turbidity coefficient [38], defined as the number of atmospheres without aerosols or water vapor necessary to produce the observed attenuation of the solar extraterrestrial irradiance;
- ${f}_{h1}$, ${f}_{h2}$, ${c}_{1}$, and ${c}_{2}$ are parameters given by:$$\begin{array}{cc}\hfill {f}_{h1}& =exp\left(-\frac{h}{8000}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {f}_{h2}& =exp\left(-\frac{h}{1250}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {c}_{1}& =5.09\times {10}^{-5}h+0.868\hfill \end{array}$$$$\begin{array}{cc}\hfill {c}_{2}& =3.92\times {10}^{-5}h+0.0387\hfill \end{array}$$

#### 2.2. Gaussian Processes

#### 2.2.1. Definition

#### 2.2.2. Gaussian Process Regression

#### 2.2.3. Training a GPR Model

#### 2.2.4. Kernels Used with Time-Based GPR Models

- The hyperparameter ${\sigma}^{2}$ was initialized to the variance of the training data;
- The period P was set to a day, indicating the daily periodicity of GHI;
- The remaining hyperparameters were randomly drawn from an uniform distribution $\mathrm{Uniform}(0,1)$.

- The hyperparameters ${\sigma}_{1}^{2}$ and ${\sigma}_{2}^{2}$ were initialized to the variance of the training data;
- The periods ${P}_{1}$ and ${P}_{2}$ were, respectively, set to one day and 365 days, indicating the daily and annual periodicity of GHI, respectively;
- The remaining hyperparameters were randomly drawn from a uniform distribution $\mathrm{Uniform}(0,1)$.

#### 2.2.5. Kernels Used with Observation-Based GPR Models

## 3. Description of Training and Test Datasets

## 4. Results and Discussion

#### 4.1. Performance Criteria

#### 4.1.1. Normalized Root-Mean-Square Error

#### 4.1.2. Dynamic Mean Absolute Error

#### 4.1.3. Skill Score

#### 4.1.4. Interval Score

#### 4.2. Forecasting Results

- The scaled persistence model;
- The time-based multi-horizon GPR models, with kernels ${k}_{\mathrm{Per}\times \mathrm{RQ}}$ and ${k}_{\mathrm{Per}\times \mathrm{RQ}+\mathrm{Per}}$;
- The observation-based multi-horizon GPR models (obtained by iterating one-step-ahead forecasting models until the desired forecast horizon is reached), with kernels ${k}_{\mathrm{SE}-\mathrm{ARD}}$, ${k}_{\mathrm{RQ}-\mathrm{ARD}}$, ${k}_{{\mathrm{M}}_{1/2}-\mathrm{ARD}}$, ${k}_{{\mathrm{M}}_{3/2}-\mathrm{ARD}}$, and ${k}_{{\mathrm{M}}_{5/2}-\mathrm{ARD}}$;
- The observation-based horizon-specific GPR models, with kernels ${k}_{\mathrm{SE}-\mathrm{ARD}}$, ${k}_{\mathrm{RQ}-\mathrm{ARD}}$, ${k}_{{\mathrm{M}}_{1/2}-\mathrm{ARD}}$, ${k}_{{\mathrm{M}}_{3/2}-\mathrm{ARD}}$, and ${k}_{{\mathrm{M}}_{5/2}-\mathrm{ARD}}$.

#### 4.2.1. Multi-Horizon GPR Models: Time or Past Observations as Input?

^{®}Xeon

^{®}E7-4890 v2 @ 2.80 GHz processor. As a result, time-based multi-horizon GPR models outperform observation-based multi-horizon GPR models while also having a lower computational cost.

#### 4.2.2. Multi-Horizon or Horizon-Specific GPR Models?

#### 4.2.3. Contribution of the Annual Periodic Kernel

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ANN | Artificial neural network |

ARD | Automatic relevance determination |

CPU | Central processing unit |

DMAE | Dynamic mean absolute error |

DHI | Diffuse horizontal irradiance |

DNI | Direct normal irradiance |

GHI | Global horizontal irradiance |

${GHI}_{\mathrm{cs}}$ | Clear-sky global horizontal irradiance |

GP | Gaussian process |

GPML | Gaussian Processes for Machine Learning (Matlab toolbox) |

GPR | Gaussian process regression |

IS | Interval score |

kNN | k-nearest neighbours |

(n)MAE | (Normalised) mean absolute error |

${\mathrm{M}}_{\nu}$ | Matérn (kernel) of degree $\nu $ |

NWP | Numerical weather prediction |

PV | Photovoltaic |

Per | Periodic (kernel) |

(n)RMSE | (Normalised) root mean square error |

RQ | Rational quadratic (kernel) |

SE | Squared exponential (kernel) |

SS | Skill score |

SVR | Support vector regression |

UVA | Ultraviolet A |

UVB | Ultraviolet B |

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**Figure 1.**GHI decomposition during one day: the periodic pattern is modeled by the periodic kernel and the stochastic pattern is modeled by the rational quadratic kernel.

**Figure 3.**GHI data covering years 2016 and 2017. A seasonal pattern appears when considering data over several years. Measurements taken at PROMES-CNRS laboratory in Perpignan.

**Figure 5.**Observation-based one-step-ahead GPR model with kernel ${k}_{\mathrm{SE}-\mathrm{ARD}}$, from which a multi-horizon model is obtained by iteration. $GHI(t-\Delta t),\phantom{\rule{4pt}{0ex}}GHI(t-2\Delta t),\phantom{\rule{4pt}{0ex}}\dots ,\phantom{\rule{4pt}{0ex}}GHI(t-D\Delta t)$ is a vector of observations, and $\Delta t=10$ min is the time step.

**Figure 6.**Observation-based horizon-specific GPR model with kernel ${k}_{\mathrm{SE}-\mathrm{ARD}}$. $GHI(t-\Delta t)$, $GHI(t-2\Delta t),\phantom{\rule{4pt}{0ex}}\dots ,\phantom{\rule{4pt}{0ex}}GHI(t-D\Delta t)$ is a vector of observations, and $\Delta t=10$ min is the time step.

**Figure 7.**GHI data covering years 2016 and 2017. Measurements are taken with a rotating shadowband irradiometer at PROMES-CNRS laboratory, located in Perpignan (southern France).

**Figure 8.**Comparison between multi-horizon GPR models: performance criteria values for time-based (with triangular marks) and observation-based (with square marks) models. Note that the latter are obtained by iterating one-step-ahead models until the desired forecast horizon is reached. Values are calculated using test data (year 2017).

**Figure 9.**Comparison between multi-horizon and horizon-specific models: performance criteria values for time-based GPR models (with triangular marks) and for observation-based horizon-specific GPR models (with square marks). Values are calculated using test data (year 2017).

**Figure 10.**Performance criteria values for time-based GPR models. Values are calculated considering test data (year 2017).

**Figure 11.**Examples of 10 min GHI forecasts over one week, given by the scaled persistence model and the GPR models included in this study.

**Figure 12.**Examples of 1 h GHI forecasts over one week, given by the scaled persistence model and the GPR models included in this study.

**Figure 13.**Examples of 3 h GHI forecasts over one week, given by the scaled persistence model and the GPR models included in this study.

**Figure 14.**Examples of 5 h GHI forecasts over one week, given by the scaled persistence model and the GPR models included in this study.

**Figure 15.**Examples of 24 h GHI forecasts over one week, given by the scaled persistence model and the GPR models included in this study.

**Table 1.**Main recent studies including Gaussian process regression models. For the meaning of abbreviations, see the end of the paper.

Authors | Input Variables When Using GPR | Output Variables | Data Sampling Time | Forecast Horizons | Database |
---|---|---|---|---|---|

Gbémou et al. (2021) [11] | GHI | GHI | 10 min | 10 min to 4 h | Two-year dataset: one year for training and one year for testing |

Lubbe et al. (2020) [25] | GHI, DNI, DHI, UVA, UVB, air temperature, barometric pressure, relative humidity, wind speed, wind direction, standard deviation in wind direction | GHI | 1 h | 1 h | Fourteen-day dataset: nine days for training and five days for testing |

Tolba et al. (2020) [24] | Time | GHI | 30 min | 30 min to 48 h | Two 45-day datasets: 30 days for training and 15 days for testing |

Gbémou et al. (2019) [29] | Time | GHI | 30 min | 30 min to 24 h | Two-year dataset: one year for training and one year for testing |

Sharifzadeh et al. (2019) [16] | PV power generation, temperature, DHI, DNI | PV power generation | 1 h | 1 h to 6 h | From 1985 to 2014 |

Guermoui et al. (2018) [30] | Air temperature, relative humidity, sunshine duration | Clearness index | 24 h | 24 h | From 2005 to 2008 |

Rohani et al. (2018) [23] | Temperature, relative humidity, sea level pressure, sunshine duration, extraterrestrial irradiation | GHI | 24 h | 24 h | From 2009 to 2014 |

Huang et al. (2016) [31] | Clearness index | Clearness index | 24 h | 24 h | Three-year dataset: two years for training and one year for testing |

Lauret et al. (2015) [10] | Clear-sky index | Clear-sky index | 1 h | 1 h to 6 h | Two-year dataset: one year for training and one year for testing, in each of three sites |

Salcedo-Sanz et al. (2014) [32] | Aerosol optical depth, ozone concentration, total precipitable water, NWP data, time | GHI | 24 h | 24 h | Two-year dataset: one year for training and one year for testing |

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**MDPI and ACS Style**

Gbémou, S.; Eynard, J.; Thil, S.; Grieu, S.
Comparison between Time- and Observation-Based Gaussian Process Regression Models for Global Horizontal Irradiance Forecasting. *Solar* **2022**, *2*, 445-468.
https://doi.org/10.3390/solar2040027

**AMA Style**

Gbémou S, Eynard J, Thil S, Grieu S.
Comparison between Time- and Observation-Based Gaussian Process Regression Models for Global Horizontal Irradiance Forecasting. *Solar*. 2022; 2(4):445-468.
https://doi.org/10.3390/solar2040027

**Chicago/Turabian Style**

Gbémou, Shab, Julien Eynard, Stéphane Thil, and Stéphane Grieu.
2022. "Comparison between Time- and Observation-Based Gaussian Process Regression Models for Global Horizontal Irradiance Forecasting" *Solar* 2, no. 4: 445-468.
https://doi.org/10.3390/solar2040027