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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