Plotter for seasonal variation

Month Seasonal effect added to log(mortality) © www.longevitas.co.uk

Seasonal variations in mortality are pronounced, with elevated mortality in winter and reduced mortality in summer. The seasonal adjustment to log(mortality) at calendar time y is:

exp(SeasonalExcess)*cos(2π(y-SeasonalPeak)).

Only two parameters are required for this simple model of seasonal variation:

  1. SeasonalPeak, the point in the year when mortality peaks, and
  2. SeasonalExcess, the log(amplitude) of the mortality peak.

Note that this parameterisation ensures that we always identify the winter mortality peak addition of exp(SeasonalExcess). This will usually occur around at a fraction 0-0.1 of the year after January 1st in the northern hemisphere, and 0.5-0.6 in the southern hemisphere.

An optional third parameter controls the sharpness of the winter peak:

  1. SeasonalShape, the shape parameter according to Richards, Ramonat, Vesper and Kleinow (2020).

where the following equation is used when SeasonalShape is non-zero:

exp(SeasonalExcess)*2[(exp(SeasonalShape/2*(1+cos(2π(y-SeasonalPeak))))-1) / (exp(SeasonalShape)-1)]-1.

Parameters:
SeasonalPeakFraction of year after January 1st when mortality peaks.
SeasonalExcessSeasonal excess winter mortality (log scale).
SeasonalShapeSeasonal shape parameter.

A future development might be an optional parameter to permit SeasonalExcess to increase with age.

The cyclic seasonal effect shown above is added to the basic age pattern for log(mortality). One can also add selection effects and an age-related time trend.

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