Multidimensional extremes: joint study of precipitation and temperature time series

To study climate change, it is essential to analyze extremes as well. The study of extremes can be done on the one hand by examining the time series of extreme climatic events and on the other hand by examining the extremes of climatic time series. In the latter case, if we analyze a single element, the extreme is the maximum or minimum of the given time series. In the present study, we determine the extreme values of climatic time series by examining several meteorological elements together and thus determining the extremes. In general, the main difficulties are connected with the different probability distribution of the variables and the handling of the stochastic connection between them. The first issue can be solved by the standardization procedures, i.e. to transform the variables into standard normal ones. For example, the Standardized Precipitation Index (SPI) uses precipitation sums assuming gamma distribution, or the standardization of temperature series assumes normal distribution. In case of more variables, the problem of stochastic connection can be solved on the basis of the vector norm of the variables defined by their covariance matrix. According to this methodology we have developed a new index in order to examine the precipitation and temperature variables jointly. We present the new index with the mathematical background, furthermore some examples for spatio-temporal examination of these indices using our software MASH (Multiple Analysis of Series for Homogenization; Szentimrey) and MISH (Meteorological Interpolation based on Surface Homogenized Data Basis; Szentimrey, Bihari). For our study, we used the daily average temperature and precipitation time series in Hungary for the period 1870-2020. First of all, our analyses indicate that even though some years may not be considered extreme if only either precipitation or average temperature is taken in to account, but examining the two elements together these years were extreme years indeed. Based on these, therefore, the study of the extremes of multidimensional climate time series complements and thus makes the study of climate change more efficient compared to examining only one-dimensional time series.