Reducing mean deltas in numbers adding up to a certain value

Question

This is likely quite common, I have 12 numbers that add up to 1, in fact they are seasonality numbers, % of occurences per month. I want to reduce the deltas (closer but not to the mean), while still adding up to 1, this would achieve the effect of smoothing the graph and reducing the spikes / uneveness without making it a flat line. Thanks

Answer 1

Instead of $x_1,x_2,\ldots, x_{12}$

you could use $y_1,y_2,\ldots, y_{12}$

where $y_i=kx_i +\frac{1-k}{12}$ for some constant $k$ of your choice with $0 < k < 1$.

$k$ closer to $0$ will make the pattern almost flat, while $k$ closer to $1$ will be close to the original pattern