**4. Analysis**

This section presents the method we use to estimate the Tsallis mean value of the positive daily variations of calls during an extreme weather period. We describe the method giving as an example

the 2010 cold wave period. Figure 4 shows the daily total number of 999 emergency ambulance calls in London from 1 July 2010 to 31 May 2011. Note that the minimum number of calls for this period is chosen to be 3500. We need to set a minimum threshold (baseline) for the number of calls for each extreme weather episode to avoid analyzing very low numbers of calls that are not caused by temperature variations. They could be due to other causes such as an ambulance operations problem. Overall, despite the fact that our study gives special attention to the occurrence of high numbers of calls, we try to avoid analyzing extremely low numbers of calls unrelated to temperature changes. In other words, we do not calculate positive daily differences for calls below this baseline. For example, we can clearly see in Figure 1 that a plausible baseline for the number of ambulance calls during the 2010 cold wave period is 3500. However, there are two days within this episode which had 3000 calls. If we keep those two days in the dataset the positive daily difference of calls would be erroneously high and unrelated to temperature. In addition, to focus only on peaks that are caused solely by extreme temperature, we substituted the peak observed on the 1 January 2011 (7455 calls) by the average number of calls (4201).

**Figure 4.** The daily total number of 999 emergency ambulance calls (blue dots) in London from 1 July 2010 to 31 May 2011.

For each extreme weather episode, we specify three periods. We define the extreme weather period as Period B. This is a period of one month in duration centered at the peak of the positive daily difference of calls (Figure 5). Other durations were chosen for sensitivity analysis purposes (see below). Although usually extreme temperature episodes last for a few days (few events) we choose a period of one month in order to have a sufficient number of events (31 events) for calculating the associated probability distribution. Using the actual number of days that concern the extreme temperature episode would lead us to a poor estimation of the probability distribution due to the small number of events selected. We define further two associated periods, one before (Period A) and one straight after (Period C) the extreme weather period (Period B). Periods A and C are two-month periods where we expect "normal" numbers of calls (i.e., corresponding to "normal weather conditions") to be recorded.

As we have explained, we use the positive daily difference of ambulance 999 calls (x) as the measure for analysis and we estimate the parameter q and the Tsallis CCDF for each one of the extreme temperature periods. We estimate the Tsallis probability distribution of the time series and the associated Tsallis expected value (referred as generalized q-expectation value in NESM's theoretical framework, see Tsallis 2009) for each of the datasets. Tsallis probability distribution and the associated expected value are calculated for each period of the four selected extreme weather episodes (i.e., before (Period A), during (Period B) and right after (Period C) the episode). We focus on the Tsallis expected value of the extreme temperature in Period B because it measures the increase of the mean value of the positive daily difference of calls relative to periods A and C. Moreover, as we show in the following sections, by analyzing Period B we obtain an accurate measure of the true mean of the positive daily difference of calls.

**Figure 5.** The positive daily difference of calls of 999 emergency ambulance incidents (blue dots) in London from 1 July 2010 to 31 May 2011. The dataset is divided into three periods. Period A corresponds to the period before the extreme temperature, Period B is the extreme temperature period and Period C is after the extreme temperature.

Parameter q is calculated by fitting Tsallis CCDF to the dataset using the Levenberg–Marquardt (LM) algorithm [23] (Figure 6). The LM algorithm is an iterative numerical procedure that is suited to solving nonlinear least squares problems [24–26]. Figure 7 shows the exceedance probability distribution function corresponding to each period for the 2010 cold wave. When the time series exhibits extremes (i.e., during the extreme weather period) the q value acquires its highest value and the distribution becomes "fatter" at the tail end, which means that the probability of getting high values of calls (x) is not small.

**Figure 6.** Log–log plot of the exceedance probability distribution function. The dataset (blue dots) and the Tsallis fitting curve (Equation (1), red line) for period C of the 2010 cold wave. The q value is calculated equal to 1.02.

**Figure 7.** The exceedance probability distribution function corresponding to each period. It should be noted that the axis in this plot is not in logarithmic scale as in Figure 6.

Based on the PMRM method, we partition the exceedance probability distribution function of the positive daily difference of ambulance calls-outs (D) into a number of ranges; three ranges are selected here: low D/high exceedance probability (Range 1), medium D/medium exceedance probability (Range 2) and high D/low exceedance probability (Range 3) (Figure 8). In terms of extreme events, Range 3 is the most important, and the PMRM provides a robust method for its interpretation and analysis. Range 4 includes all the three ranges 1 to 3 (Figure 8).

**Figure 8.** The partition of the positive daily difference of calls into three ranges of the exceedance probability (red line), i.e., high, medium and low exceedance probability, for the whole dataset, in London from 1 July 2010 to 31 May 2011. The x-axis is in logarithmic scale.

Figure 8 shows the partition of the exceedance probability function of the positive daily difference of calls into three ranges of the exceedance probability, i.e., high, medium and low exceedance probability, for the whole dataset, in London from 1 July 2010 to 31 May 2011. Without loss of generality, the partitioning probabilities a1 and a2 are selected to be 0.35 and 0.68 respectively (y-axis). The associated positive daily difference of calls (x-axis) is b1 = 220 and b2 = 80. This selection of the partition probabilities defines Range 2 (medium exceedance probability) as the linear part of the curve of the exceedance probability (red line) shown in Figure 8. The same partitioning probabilities are used throughout this paper to calculate the unconditional and conditional expected values of the positive daily difference of the ambulance 999 calls in the four ranges.

In the analysis, the conditional and unconditional Tsallis expected values are calculated for each of the three periods (A, B, and C) and four ranges by using NESM and the partitioned multi-objective risk method. The results are presented in the following section.
