Methods

The returns of extreme events are analyzed by applying the extreme value theory statistical methods named as Block Maxima (BM), and Peaks over Threshold (POT). POT is applied to investigate the temperature and precipitation extremes in Sindh. POT is preferred here due to short duration of data (33 years). While the seas surface temperature extremes are estimated by BM approach, due to long duration of data (125 years).

Extreme Value Theory (EVT) is used for describing the distribution of rare events, especially in financial, insurance, hydrology , meteorology or environmental applications, where the risk of extreme events is of interest (Reiss et al., 2001). Where inference about extremes can be challenging due to the scarcity of data, extreme value models are used to study the behaviour of the tail of the distribution. The purpose of extreme value analysis is to model the risk of extreme, rare events, by finding reliable estimates of the frequency of these events. In statistical terms, the frequency is usually expressed as a return period and intensity corresponds to return level.

i. **Block Maxima (BM).** determines the statistical properties of extremes in a period (Block), typically daily or annual intervals are used. The maxima are distributed according to the generalized extreme value distribution (GEV). The GEV is a single representation of three types of distributions: Gumbel, Fréchet and Weibull. The GEV distribution has three parameters the location μ, the scale parameter σ, and the shape parameter ξ. Coles, (2001) gives the generalized extreme value distribution as

The return values are then calculated by solving G(zp) = 1/p, where the return values are z_p, and p is the return period so that z_p is the value that is expected to be exceeded, on average, once every 1/p years (Coles, 2001). The equation can be solved analytically, and is given by

ii. **Peaks Over Threshold (POT).** determines the distribution of the exceedances over a threshold, which has to be carefully selected. The exceedances are asymptotically distributed according to the generalized Pareto distribution (GPD), which is characterized by two parameters, the shape ξ and the scale σ. GPD is universal in terms of asymptotic behavior, so when convergence is achieved can vary geographically The GPD for exceedances x-u of a random variable x reads as

Where u is the threshold. The choice of the threshold u is done in order to ensure that the model in (3) provides a reasonable fit to exceedances of this threshold. The result for the two parameters shape ξ and scale σ depend on the threshold u (Coles, 2001).

The return values are then calculated by the equation given as

Where N represents the return period, ny is the number of observations per year, ζ_u is the probability of an individual observation exceeding the thresholdu, the shape parameter is ξ and the scale parameter is σ.

Coles, S.: An Introduction to Statistical Modeling of Extreme Values, Springer London, London., 2001.

Lucarini, V., Faranda, D., .Freitas, A.C.M., Freitas, J.M., Holland, M., Kuna, T., Nicol, M., Todd, M., Vaienti, S.: Extremes and Recurrence in Dynamical Systems, John Wiley & Sons Inc,2016.

Reiss, M. Thomas, and R. Reiss. Statistical analysis of extreme values, volume 2. Springer, 2001.