Using Trigonometry to Estimate Influenza Deaths



In a nutshell, the sin() and cos() terms are periodic curves, and the weighting of the various coefficients is what allows proper regression fits.  Let’s take a closer look at the formula, and try to make sense of it.

As t (weeks) increases to 52, \frac{t}{52} goes from 0 to 1  (\frac{0}{52}\frac{1}{52}\frac{2}{52}\frac{3}{52}, …, \frac{52}{52})  Once it goes past 52, it just cycles around again.  Recall 2\pi radians = 360 degrees.  Since  \frac{t}{52} is multiplied by 2\pi, it is multiplying 360 by some number.   So, it seems the sin() and cos() terms simply use t weeks to scale across multiples of 360 degrees .

For example, as t goes from 0 to 52, \frac{t}{52} goes from 0 to 1, 2\pi * \frac{t}{52} goes from 0 and 360.  (and then it repeats since sin repeats in multiples of 2\pi and therefore sin(2\pi * \frac{t}{52}) goes from sin(0) to sin(360) which is a full periodic cycle of this function.  Note the same logic applies to the cos() term in the formula.

The picture says it all.

Further Reading:  Automated Detection of Influenza Epidemics with Hidden Markov Models

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