# 1 9 th International Conference Zaragoza-Pau on Applied Mathematics and Statistics On heat wave definition Abaurrea J., Cebrián A.C., Asín J., Centelles.

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1 9 th International Conference Zaragoza-Pau on Applied Mathematics and Statistics On heat wave definition Abaurrea J., Cebrián A.C., Asín J., Centelles A. 19-21 September 2005

2 Introduction (1) Heat waves have not a standard operational definition Usual approach to define them: An excess of daily maximum temperature, Tx, over a fixed threshold (POT) Other conditions required: A minimum duration of the event The daily minimum temperature exceeds another threshold Problems and evidences not considered: Greater effects on morbidity and mortality of heat waves occurring in the early summer Possibility of longer heat waves including intermediate “cool” days

3 Kysely (2000): A group of consecutive days is considered a heat wave if: a) Tx ≥ T1 for, at least, three days b) Tx ≥ T2 for every day c) Mean (Tx) ≥ T1 Tx: Daily maximum temperature T1: Threshold for hot days T2: Threshold for warm days For Central Europe T1 = 30ºC and T2 = 25ºC Introduction (2)

4 a)and b) as Kysely c)Mean(Tx)  T1 for the whole period and for each partial sequence, HC, HCHC, etc., where H stands for a hot spell and C a cool spell d)The length and the area (the accumulated sum of differences to T1) for each cool spell included in the wave, must be lower than the corresponding 90 th percentile in the reference period T1 and T2 are, respectively, the 95 th and 50 th percentiles of Tx values observed in June, July and August, in the reference period 1961-1990 A time period is considered a heat wave if: Introduction (3) Abaurrea et al. (2004):

5 Heat wave: A period of arbitrary length where Tx exceeds a “shot temperature” Shot temperature = extraordinary increase of mortality –Madrid (36.5ºC) –Barcelona (30.3ºC) Introduction (4) Díaz et al. (2003): –Sevilla (41ºC) –Lisboa (33.5ºC) These thresholds are the 95 th percentiles of the corresponding Tx value distributions in JJAS, 1991-2002 In this way, they obtain the heat wave thresholds for main Spanish towns: Zaragoza (37.3ºC), Huesca (36.1ºC)

6 Data and preliminary analysis Zaragoza and Huesca Daily Tx and Tn data for 1951-2004 Daily mortality data for 1975-2002 (people aged 65 or more years)

7 Temperature Tx and Tn evolution during the studied period 1951-75 stability 1976-90 increase 1991-96 stability 1997-2004 increase Data and preliminary analysis

8 Tx evolution in different summer periods Temperature Data and preliminary analysis Lowess (40) of Tx data in 7 overlapping 5-week intervals

9 Lowess (30) of Mr corresponding to seven overlapping 10 year long intervals  Mr decreases between 1975 and 2002  Change in the seasonal profile Mortality Data and preliminary analysis Mr: Daily mortality rate per 1000 inhabitants

10 The effect of hot temperatures on mortality occurs in the short term (1-3 days) (Díaz et al. 2005) For daily variables, the correlation between Tx and Mr is maximum with a 24 hour delay The greatest correlation between 3 days averaged values is also obtained for a 24 hour delay Temperature-Mortality relationship Data and preliminary analysis

11 Data and preliminary analysis The impact on Mr of a fixed high temperature changes in time and along the summer To show this property we select a temperature value, 33.3ºC, the 90 th percentile of daily Tx values in June, 1975- 81 1975-02 is divided into four 7-year periods and we consider observations from June, July and August Temperature-Mortality relationship

12 Data and preliminary analysis Decrease of Mr 90 th percentile and mean values in time and along the summer 33.3ºC is a critical value, regarding the Mr response, for the first period and it is not for the last one Temperature-Mortality relationship

13 Data and preliminary analysis Decrease of Mr 90 th percentile and mean values in time and along the summer 33.3ºC is a critical value, regarding the Mr response, for the first period and it is not for the last one Temperature-Mortality relationship

14 Mortality Excess We define the mortality excess, Mex(t), in day t as the difference between the number of deaths, Mf(t), and its expected value, Me(t) Mex (t) = Mf(t) –Me(t) The expected mortality is obtained by fitting a regression model including: a)Time terms until the second order (long term evolution) b)Harmonic terms until the fourth order (seasonality) c)Dummy variables for indicating the periods 75-86, 87-96 and 97-02, in order to fit different seasonal patterns

15 Expected and observed mortality lowess Mortality Excesss

16 We try to identify the ‘shot temperature’ for each 7-year period and summer interval, looking for the change point in the lowess smoother of Mex vs. Tx PROBLEMS  Smoothed curves are frequently erratic because of small sample sizes  A proper shot temperature doesn’t appear in many graphs Threshold selection

17 Threshold selection

18 Threshold selection

19 High excess crossing temperatures increase in time but remain constant when they are transformed to a percentile scale Threshold selection

20 Threshold selection process The process to define T1 threshold needs several steps a)Analysed interval: 14-May to 16-September in 1975-2002 b)Four 7-year periods (1975-81, 1982-88, 1989-95, 1996-2002) c)Several divisions of the 14/5-16/9 interval using different length cells: 3-weeks, 4-weeks, month,...

21 d)Identifying 1.25-excess crossing temperature percentiles Threshold selection process

22 Threshold selection process e)Percentile-Threshold allocation to 11 selected dates along the summer 1.25-excess crossing temperature percentile values

23 f)Transformation of the percentile-threshold into its equivalent temperature-threshold (T1) using an adequate probabilistic distribution Threshold selection process Probabilistic distributions: N: Normal LN: Lognormal W: Weibull EV: Extreme Value L: Logistic LL: Log-Logistic

24 g)Estimation of the daily T1 threshold for each 7-year period Threshold selection process

25 4 th June: 32.3-34ºC 27 th August: 37-39.8ºC The increase of T1 along the period 1975-2002 is about 2ºC The bigger slope of the 3 rd period is due to temperature warming in August and July, whereas the smaller slope of the 4 th period is due to strong temperature increase in June Threshold selection process T1 thresholds for the period 1975-2002

26 in comparison with the T1-Díaz performance Results and conclusions Evaluation of T1- threshold results

27 References Abaurrea, J., et al. (2004). Modelling hot extreme temperature events using a non homogeneous Poisson model. 6th World Congress of Bernoulli Society for Mathematical Statistics and Probability, Barcelona. Díaz, J., et al. (2002). Effects of extremely hot days on people older than 65 years in Seville (Spain) from 1986 to 1997. Int. J. Biometeorology, 46, 145-149. Díaz, J., Linares, C., García-Herrera, R. (2005). Impacto de las temperaturas extremas en la salud pública. Futuras actuaciones. Rev. Esp. Salud Pública, 79, 145-157. Kysely, J. (2002). Temporal fluctuations in heat waves at Prague, the Czech republic, from 1901-97 and their relationship to atmospheric circulation. Int. J. Climatol., 22, 33-50. Robinson, P. J. (2001). On the definition of a heat wave. J. of Applied Meteorology, 40, 762-75.

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