Didier Dacunha-Castelle

Non-stationary future return levels for extreme rainfall over Extremadura (southwestern Iberian Peninsula)

ABSTRACT Based on a previous study for temperature, a new method for the calculation of non-stationary return levels for extreme rainfall is described and applied to Extremadura, a region of southwestern Spain, using the peaks-over-threshold approach. Both all-days and rainy-days-only datasets were considered and the 20-year return levels expected in 2020 were estimated taking different trends into account: first, for all days, considering a time-dependent threshold and the trend in the scale parameter of the generalized Pareto distribution; and second, for rainy days only, considering how the mean, variance, and number of rainy days evolve. Generally, the changes in mean, variance and number of rainy days can explain the observed trends in extremes, and their extrapolation gives more robust estimations. The results point to a decrease of future return levels in 2020 for spring and winter, but an increase for autumn.

Changes in heat wave characteristics over Extremadura (SW Spain)

Heat wave (HW) events are becoming more frequent, and they have important consequences because of the negative effects they can have not only on the human population in health terms but also on biodiversity and agriculture. This motivated a study of the trends in HW events over Extremadura, a region in the southwest of Spain, with much of its area in summer devoted to the production of irrigated crops such as maize and tomatoes. Heat waves were defined for the study as two consecutive days with temperatures above the 95th percentile of the summer (June–August) maximum temperature (Tmax) time series. Two datasets were used: One consisted of 13 daily temperature records uniformly distributed over the Region, and the other was the SPAIN02 gridded observational dataset, extracting just the points corresponding to Extremadura. The trends studied were in the duration, intensity and frequency of HW events, and in other parameters such as the mean, low (25th percentile) and high (75th percentile) values. In general terms, the results showed significant positive trends in those parameters over the east, the northwest and a small area in the south of the region. In order to study changes in HW characteristics (duration, frequency and intensity) considering different subperiods, a stochastic model was used to generate 1000 time series equivalent to the observed ones. The results showed that there were no significant changes in HW duration in the last 10-year subperiod in comparison with the first. But, the results were different for warm events (WE), defined with a lower threshold (the 75th percentile), which are also important for agriculture. For several sites, there were significant changes in WE duration, frequency and intensity.

Martingales in Discrete Time

With martingales in discrete time, we tackle all the original ideas of the theory of processes. The following chapters will therefore be either applications of this, or extensions to continuous time. They will be technically more difficult, but will be based on the same ideas.

Random Fields and Stochastic Integrals

We shall see in Chapter 6 that every stationary process is the Fourier transform of a random measure carried by ∏= [−π,+ π[. We describe here the uncorrelated random fields which formalize the intuitive notion of random measure. The theory extends easily to more general spaces ∏.

Nonstationary ARMA Processes and Forecasting

To modelize a nonstationary random phenomenon, it is tempting to write Yn = f(n) + Xn where X is stationary and the trend f(n) is a deterministic function to be adequately selected; f(n) is often assumed to be the sum of a polynomial in n and of linear combinations of cos nλj and sin nλj,= 1,2, ..., K. A first approach is to estimate separately f (for instance by least squares methods (cf. [15]) and the covariance structure of X.

Discrete Time Random Processes

The experimental description of any random phenomenon involves a family of numbers Xt, t ɛ T. Since Kolmogorov, it has been mathematically convenient to summarize the impact of randomness through the stochastic choice of a point in an adequate set Ω (space of trials) and to consider the random variables Xt as well determined functions on Ω with values in ℝ.

Spectral Representation of Stationary Processes

Since we may identify the one dimensional torus with TT = [-π,π,[to each bounded measure v TT on we associate its Fourier transform

Asymptotic Maximum Likelihood

\hat \nu (n) = \int_{TT} {{e^{in\lambda }}d\nu \left( \lambda \right),\;n \in ZZ}