# Nonparametric Conditional Heteroscedastic Hourly Probabilistic Forecasting of Solar Radiation

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## Abstract

**:**

## 1. Introduction

## 2. Data and Preliminaries

## 3. Point Forecast

#### Point Forecast Performance Evaluation

## 4. Probabilistic Forecasting

Algorithm 1: Algorithm for generating (100-$\alpha $) prediction intervals using the conditional method. |

## 5. Probabilistic Forecast Performance Evaluation

#### 5.1. Prediction Interval Coverage Probability

#### 5.2. Prediction Interval Normalized Averaged Width

#### 5.3. Winkler Score

#### 5.4. A Closer Look at Darwin

#### 5.5. Results in the Literature

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Glossary

${I}_{t}$ | hourly solar irradiation |

${F}_{t}$ | seasonal component of the solar irradiation |

${A}_{t}$ | autoregressive component of the solar irradiation |

${Z}_{t}$ | noise term for the solar forecast model |

${B}_{i,j}$ | two-dimensional array for binning the noise, according to the sun position |

i corresponds to the sun elevation and j to the sun hour angle | |

${\psi}_{t}^{2}$ | variance of the noise term at time t |

$\beta $ | exponential smoothing parameter |

${\widehat{I}}_{t}$ | forecast of ${I}_{t}$ at time $t-1$ |

${\gamma}_{t}$ | transformation of ${Z}_{t}$ to the corresponding value in probability in $N(0,1)$ |

${\widehat{\tau}}_{t}$ | transformation of ${\widehat{I}}_{t}$ to the corresponding value in probability in $N(0,1)$ |

$G({Z}_{t},i,j)$ | empirical cumulative distribution function of the noise term ${Z}_{t}$ |

$H({Z}_{t},i,j)$ | empirical cumulative distribution function of the solar forecast ${\widehat{I}}_{t}$ |

${L}_{t}^{100-\alpha}$ | lower bound of the prediction interval |

${U}_{t}^{100-\alpha}$ | upper bound of the prediction interval |

$\alpha $ | probability level for determining the prediction interval. For example, for a 95% prediction interval, |

$\alpha =2.5\%$ | |

${\delta}_{t}$ | the width of the prediction interval at time t |

## References

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**Figure 1.**Map of Australia showing the three locations used in this paper: Adelaide, Darwin and Mildura.

**Figure 2.**GHI prediction intervals using the unconditional and conditional probabilistic forecasting methods for Adelaide.

**Figure 3.**GHI prediction intervals using the unconditional and conditional probabilistic forecasting methods for Darwin.

**Figure 4.**GHI prediction intervals using the unconditional and conditional probabilistic forecasting methods for Mildura.

**Figure 5.**Reliability of prediction intervals in the form of quantile–quantile plots over the out-of-sample period with 95% consistency bars using the unconditional and conditional probabilistic forecasting methods, for Adelaide, Darwin, and Mildura. The black line represents is the 45-degree reference line.

**Figure 6.**Prediction interval normalized averaged width (PINAW) over the out-of-sample period using the unconditional and conditional probabilistic forecasting methods, for Adelaide, Darwin and Mildura.

**Figure 7.**Normalized Winkler score over the out-of-sample period using the unconditional and conditional probabilistic forecasting methods, for Adelaide, Darwin, and Mildura.

**Figure 10.**Reliability of prediction intervals in the form of quantile–quantile plots over the out-of-sample period with 95% consistency bars for the unconditional and conditional probabilistic forecasting methods, for Darwin, for summer and winter seasons. The black line represents is the 45-degree reference line.

**Figure 11.**Normalized Winkler score over the out-of-sample period for unconditional and conditional probabilistic forecasting methods for Darwin, for summer and winter seasons. The black line represents is the 45-degree reference line.

Location | Data Period | Köppen-Geiger |
---|---|---|

Climate Classification | ||

Adelaide | 2005 to 2014 | Hot Mediterranean |

Darwin | 1995 to 2004 | Tropical |

Mildura | 1995 to 2004 | Semi-arid |

**Table 2.**Out-of-sample normalized root means square error (NRMSE), mean bias error (MBE) and mean absolute error (MAE) for the point forecasting method for Adelaide, Darwin, and Mildura.

Point Forecast | NRMSE (%) | MBE (%) | MAE (%) |
---|---|---|---|

Adelaide | 19.14 | 0.72 | 13.25 |

Darwin | 22.74 | 0.81 | 15.83 |

Mildura | 15.29 | 1.32 | 10.83 |

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**MDPI and ACS Style**

Boland, J.; Grantham, A. Nonparametric Conditional Heteroscedastic Hourly Probabilistic Forecasting of Solar Radiation. *J* **2018**, *1*, 174-191.
https://doi.org/10.3390/j1010016

**AMA Style**

Boland J, Grantham A. Nonparametric Conditional Heteroscedastic Hourly Probabilistic Forecasting of Solar Radiation. *J*. 2018; 1(1):174-191.
https://doi.org/10.3390/j1010016

**Chicago/Turabian Style**

Boland, John, and Adrian Grantham. 2018. "Nonparametric Conditional Heteroscedastic Hourly Probabilistic Forecasting of Solar Radiation" *J* 1, no. 1: 174-191.
https://doi.org/10.3390/j1010016