Transformation of the Mean, Standard Deviation and Skewness

Transformation of the mean, standard deviation and skewness

For each grid-point and model, let and be the local detrended time series from the daily 30-yr control and (B2 or A2) scenario 2-meter temperature dataset, respectively. The mean (), standard deviation () and skewness () of are computed, and used to transform into , so that

,(1)

,(2)

.(3)

We first apply the transformation of the skewness coefficient (if not, , the time series is not changed in this first step: ). If the skewness is increased (), we transform into

.(4)

Note that for each time slice of the 30-yr period, and therefore the transformation is a well-defined function. Here, represents a unique parameter so that and have the same skewness coefficient. This simple correspondence dilates (contracts) percentiles within the warm (cold) PDF tail, so that the larger , the larger the increase of the skewness coefficient. Similarly, when the skewness is decreased (), is transformed into

,(5)

with . Again, for each time slice, and therefore the transformation is a well-defined function. In this latter case, the skewness is decreased because , and therefore the larger , the larger the decrease of the skewness coefficient. Note that many other transformations could be defined to readjust the skewness coefficient, and that changes in the PDF tail are especially sensitive to this choice. Here we adopt and test the simplest possible choice.

Afterwards, the resulting time series is standardized, i.e.,

,(6)

and the mean and/or the standard deviation are then adjusted by using a simple linear rescaling transformation:

.(7)

The skewness is invariant for standardizing and rescaling transformations, therefore and have the same skewness coefficient.

The transformation of skewness ( into , see equations 4 or 5) slightly changes the structure of the data (i.e., the temporal Pearson correlation between and , , is almost always greater than 0.9), while the transformation of the mean and standard deviation ( into , see equations 6 and 7) does not introduce any change in the structure of the data (i.e., ).

Regional anomalies in the structure of the PDF

Supplementary Figure 2 depicts A2 scenario changes in the structure of the PDF for several continental subregions: Western Mediterranean (WMED, [-11,10]E x [30,46]N), Eastern Mediterranean (EMED, [10,31]E x [30,46]N), Central Europe (CEUR, [-5,30]E x [44,55]N), Northwestern Europe (NWEUR, [-11,20]E x [50,70]N) and Northeastern Europe (NEEUR, [20,50]E x [53,70]N). Note that similar results and conclusions can be inferred for B2 (not shown). These regions were specifically defined in order to isolate areas with homogenous changes in the mean, standard deviation or skewness (see Figure 1). For example, the minimum annual mean temperature rise is found in NWEUR, whereas the standard deviation strongly increases (decreases) in WMED (NEEUR), and skewness increase is especially small in WMED.

The skill of the MSW approach is further stressed in this subcontinental analysis, because it is the only scheme that successfully reproduces basic changes in both tails for the 5 subregions. Particularly, it correctly simulates the relatively small (large) warming of the extreme warm tail in NEEUR (in WMED, EMED, CEUR and NWEUR) and the relatively small (large) warming of the extreme cold tail in WMED (in EMED, CEUR, NWEUR and NEEUR). On the other hand, the MS procedure only reproduces changes in the extreme warm tail, and only for some regions (WMED, EMED and CEUR). Finally, the M approach fails in almost all cases and regions.