Natural hazards (events that may cause actual disasters) are established in the literature as major causes of various massive and destructive problems worldwide. The occurrences of earthquakes, floods and heat waves affect millions of people through several impacts. These include cases of hospitalisation, loss of lives and economic challenges. The focus of this study was on the risk reduction of the disasters that occur because of extremely high temperatures and heat waves. Modelling average maximum daily temperature (AMDT) guards against the disaster risk and may also help countries towards preparing for extreme heat. This study discusses the use of the

Most of the classical statistical techniques that are frequently used in the energy sector and meteorological analysis are classified into regression analysis, time series, state space and Kalman filtering (Hahn, Meyer-Nieberg & Pickl

In this study, the use of the

The _{r}) gains preference over the ordinary GEV distribution for block maxima. Coles (

The rest of the article is organised as follows: A brief discussion of disaster risk management and reduction is given in the section ‘Disaster risk management and reduction’. The ‘Models’ section presents the extreme value analysis techniques that are used in this study. The empirical results are presented and discussed in the ‘Empirical results and discussion’ section. The ‘Conclusion and recommendation’ section concludes and recommends the possible areas for further research.

Twigg (

In the presence of hottest days, people switch on the cooling systems until a point at which all the cooling systems are on, resulting in an extreme increase in electricity demand (Munoz et al.

This study is aimed at fitting the GEV distribution for _{r}) on the AMDT. Our objectives include, among others, the use of the maximum likelihood estimation (MLE) method in estimating the target parameters, the choice of the best model out of several nested models using the likelihood ratio test that is based on the deviance statistic, the use of entropy difference test (EDT) in assessing the goodness of fit of the models as well as the estimation of the return levels using the quantile function.

The GEV distribution is discussed by Fisher and Tippett (_{1},…,X_{n} constitutes a random sample of size _{n} = max{X_{1},…,X_{n}}, then the extremal types theorem states that for the suitable normalising constants {_{n} > 0} and {_{n}}, the distribution function of the re-scaled block maxima

for a non-degenerate distribution function

defined for

where − ∞ <

The ordinary GEV distribution in _{γ}_{γ}

If _{γ}

The survival distribution of the GEV distribution is given by:

defined for

By letting

As

The quantile function given in

and then identify the limiting behaviour of this variable, for fixed

defined on

where:

with:

The GEV distribution for the

is the joint probability density function given in Coles (

valid for − ^{(r)} ≤ ^{(r–1)} ≤ …≤ ^{(1)}; and:

The likelihood function for the

defined for

However, the validity of the GEV distribution for

A plot of the AMDT in degrees Celsius using data for the non-winter season (September–April of each year) is given in _{i}), for _{i}) seem to be stable for _{r} distribution when

Plot of average maximum daily temperature (°C) for the period 2000−2010. Only data for the period September–April of each year are included.

Maximised log-likelihoods _{i}

λ_{i} |
95% confidence interval for γ | |||||||
---|---|---|---|---|---|---|---|---|

Parameter estimate | Standard error | Parameter estimate | Standard error | Parameter estimate | Standard error | |||

1 | −12.5255 | 30.8813 | 0.3141 | 0.9551 | 0.2643 | −0.6848 | 0.2376 | (−1.1505; −0.2191) |

2 | −15.9670 | 31.0993 | 0.2243 | 0.7540 | 0.1133 | −0.6206 | 0.1784 | (−0.9703; −0.2709) |

3 | −11.3280 | 31.1311 | 0.1660 | 0.6220 | 0.0688 | −0.4803 | 0.1173 | (−0.7102; −0.2504) |

4 | −8.5826 | 31.1190 | 0.1576 | 0.6107 | 0.0616 | −0.4515 | 0.1062 | (−0.6597; −0.2433) |

5 | −0.8328 | 31.1405 | 0.1427 | 0.5743 | 0.0551 | −0.4226 | 0.0887 | (−0.5965; −0.2487) |

6 | −11.1508 | 31.1421 | 0.1358 | 0.5571 | 0.0535 | −0.3963 | 0.0824 | (−0.5578; −0.2348) |

The sufficient evidence of the validity of the Weibull distribution family towards modelling of AMDT in South Africa is revealed by the estimates of the shape parameter for all values of _{4} = − 8.5826 seems to be the one that possesses a reasonably good fit. The graphical diagnostics which are the P–P, Q–Q, return level and the density plots are given for

Diagnostic plots illustrating the fit of the data (annual average maximum temperature) to the generalised extreme value distribution for

This leads us to conclude that at a 95% level of confidence, the value of the shape parameter

There are various tests that are considered in the literature as relevant to assess the goodness of fit of the models to the data. The Anderson–Darling test is suitable for assessing the goodness of fit of the heavy-tailed distributions. In this article, we use the EDT that is discussed in Bader et al. (_{r} for _{r} and GEV_{r-1} models. Considering a continuous random variable

which is the mean of the negative log-likelihood that is estimable by the sample average of the contribution to the log-likelihood from the observed data. Suppose that _{r-1} distribution. Then the difference in the log-likelihood between GEV_{r-1} and GEV_{r} provides a measure of deviation from the null hypothesis

is given by:

It is critical to test the goodness of fit of the GEV_{r} distribution with a sequence of null hypotheses (Bader et al. _{r} distribution fits the sample of the

The results of the test are summarised in _{r} model to the data. This is based on _{r=4} is the most appropriate model for AMDT in South Africa. Looking at the test statistics in column 4, the critical value _{α/2} = 1.645 also supports that we fail to reject

Entropy difference test for diagnosing generalised extreme value distribution for

ForwardStop | StrongStop | Statistic | |||||
---|---|---|---|---|---|---|---|

2 | 0.061593823 | 0.4240636 | 0.49817771 | 1.8692084 | 31.05364 | 0.7615547 | −0.5923202 |

3 | 0.001607416 | 0.5442273 | 0.07780948 | 3.1545576 | 31.13994 | 0.6271697 | −0.4868675 |

4 | 0.783431762 | 0.8155366 | 0.10330571 | 0.2748496 | 31.10788 | 0.6074107 | −0.4538394 |

5 | 0.096269066 | 0.1012236 | 0.01989029 | 1.6632167 | 31.12734 | 0.5824586 | −0.4292671 |

Estimates of extreme quantiles and probabilities of exceedance levels of the annual maximum distribution are obtained using the quantile function given in _{4} which is established as the best fitting model to the data. The quantity _{p} in _{p} is expected to be exceeded on average once every 1/

where _{i}) and _{j}) are the maximum likelihood functions of _{i} and _{j}, respectively. A test of the validity of model based on _{i} relative to _{j} at the _{i} if _{(i,j)} >

The point estimates _{0.04} = 32.15 is expected to be exceeded on average once every 25 years. For the greater accuracy in the estimation of the return level and parameters in the model for _{1}) and _{2}) is calculated as _{(1,2)} = 2 (−15.9670 – (−12.5255)) = −6.88. Similar to Soares and Scotto (_{(2,3)}, _{(3,4)} and _{(4,5)}. The comparison is invalid for _{(1,2)} and _{(5,6)}. The test for selecting the best model is conducted at 1% level of significance for which _{(2,3)} and _{(4,5)}, _{3}) and _{5}) are rejected as _{(2,3)} > 6.64 and _{(4,5)} > 6.64. It is therefore clear that _{(3,4)} = 5.49 < 6.64 reveals failure to reject _{4}), which implies the validity of

Return values for 5, 10, 25, 50 and 100 years.

5 years | 10 years | 25 years | 50 years | 100 years |
---|---|---|---|---|

31.78 | 31.98 | 32.15 | 32.24 | 32.30 |

Deviance statistics and

D(1,2) | D(2,3) | D(3,4) | D(4,5) | D(5,6) |
---|---|---|---|---|

−6.88 | 9.23 | 5.49 | 15.50 | −20.64 |

0.009999 | 0.009999 | 0.009999 | 0.009999 | 0.009999 |

Tail and quantile estimation for the generalised extreme value distribution for annual maxima with

Quantiles | Temperatures (°C) (_{p}) |
Observed number of exceedances | GEV distribution no. of exceedances |
---|---|---|---|

90th | 32.0 | 4 | 2 |

95th | 32.1 | 2 | 1 |

97.5th | 32.2 | 1 | 0 |

99th | 32.3 | 0 | 0 |

GEV, generalised extreme value.

The number of observations that are above the estimated tail quantile _{0.1} = 32.0 are then counted and found to be 2. For the observed number of exceedances, we get 0.1 × 44 = 4.4 ≈ 4, where 44 is the number of observations available in 11 years by four order statistics.

_{p}

Natural hazards are established in the literature as major causes of various massive and destructive problems worldwide. The occurrence of extremely high temperatures, heat waves and hot spells affects millions of people through several impacts. These include cases of hospitalisation, loss of lives and economic challenges. It is important that countries should always be prepared for natural hazards that cause actual disasters as uncertain events. One of the steps towards the preparation is to manage the risk of the occurrence of heat waves, which is performed in this study through modelling the frequency of the occurrence of extremely high temperatures. A brief discussion of actual disasters together with their corresponding impacts is given in this study. The main focus is on the reduction of disaster risk that could occur as a result of extreme heat. This study has demonstrated the modelling approach that is relevant for guarding against the disaster risk that could occur as a result of extreme high temperatures. The GEV_{r} distribution for _{r} distribution to the data. The choice of

The methodology and results of this study are important to disaster risk managers as the modelling framework demonstrated in this article is valid for modelling occurrence of extreme heat. This may be useful for guarding against risk of disasters that may occur as a result of hot spells and heat waves. The use of statistical approaches that concentrate on the mean instead of the tails of the distributions is not recommendable for modelling the occurrence of extreme temperatures. The Weibull family of distributions is recommended for modelling AMDTs in South Africa. This study is also important for power utility companies such as Eskom, South Africa’s power utility company, as demand for electricity significantly increases during a heat wave period. This will therefore help system operators to determine the amount of electricity that is consumed during a heat wave period, which will then help them to schedule and dispatch the increased demand in electricity. Extreme high temperatures cause transmission lines to sag. Combination of extreme heat and the added demand for electricity to run air conditioning causes transmission line temperatures to rise.

The use of extreme value analysis techniques such as the GEV distribution for

The authors are grateful to the National Research Foundation of South Africa for funding this research, Eskom for providing the data, the University of Limpopo for its resources and the numerous people for their helpful comments on this article.

The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.

Both authors contributed equally to the writing of this article.

Diagnostic plots illustrating the fit of the data (annual average maximum temperature) to the generalised extreme value distribution for

Diagnostic plots illustrating the fit of the data (annual average maximum temperature) to the generalised extreme value distribution for

Diagnostic plots illustrating the fit of the data (annual average maximum temperature) to the generalised extreme value distribution for

Profile log-likelihood for the location parameter μ using estimates of the model with

Profile log-likelihood for the scale parameter

Profile log-likelihood for the shape parameter