To assess probability, I used a *point process* model for hot spells (see here). This blends several distributions to model:

- the frequency of hot spells (a Poisson fit to annual counts)
- the duration of individual spells (a geometric distribution)
- the maximum hot spell temperature (a generalized extreme value distribution)

There are a few assumptions that are violated here – specifically, hot spells are assumed to be independent and identically distributed, which is not necessarily true due to autocorrelation in temperature. Similarly, a geometric distribution assumes all trials are independent, which again is violated by autocorrelation (the chance of exceeding the threshold increases with the previous day’s temperature; our t isn’t fixed!). The first issue is partly addressed by ‘declustering’ the data; while individual days are definitely not independent, individual hot spells separated by sufficient time will be. Here, we consider a one day separation between spells enough to ensure independence – really, we should probably use something longer (2-3 days), but it doesn’t drastically change the results. As for the concerns around the geometric distribution: they aren’t addressed, but we’ll file it under ‘good enough’.

To fit the distributions, hot spells were identified in bias-corrected station data for St. John’s Airport (get it here); these were defined as periods with daily maximum temperatures above 24^{o}C. This threshold choice matches the WMO definition of a heat wave well, but is also justified by threshold tests commonly used in generalized Pareto (GP) distributions (specifically, GP parameters are stable for thresholds above 23-23.7^{o}C).

Once hot spells were identified, the starting date, duration, and peak temperature were recorded. Using data from 1950-2011, the following distribution parameters represented the best fit to our hot spell model:

**Frequency (Poisson Model):**annual counts of hot spells were used to fit the distribution; results gave lambda = 8.53 (the mean number per year). With an annual count variance of 11.37, the model is somewhat overdispersed but remains a reasonable fit to the data.**Duration (Geometric):**A geometric distribution is useful when estimating the length of unusual events – or number of trials before a success occurs. Here, the question is how many days typically pass before a heat wave ends. It uses a single parameter, theta, which here gives the probability a day below the threshold will occurring. Fitting to the hot spell durations gives theta = 0.3633.**Maximum Temperature (Generalized Pareto Distribution):**Fitting the generalized Pareto distribution to hot spell temperature maxima (recorded as the highest temperature recorded during each spell) gives a scale parameter of 2.407 and a shape parameter of 0.026. Based on 529 individual hot spells, the results are relative robust (low parameter uncertainty).

In order to estimate how unusual our record breaking string of four heat waves is, I created a huge number of fake ‘years’, in which i) the number of hot spells was determined by a random sample of the fitted Poisson distribution ii) and the duration of these spells was given by a random sample of the fitted geometric distribution. In 10^{6} ‘years’, the best-fit parameters given above give a 1.3% chance that 4 or more heat waves will occur. Taking into account uncertainty in the Poisson and geometric parameters, the 99% confidence interval on this is [0.4%, 3.15%].

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