Re: Derivation of the Cost Function

The maximum value used in the simulation was 1000. I'll generate a new
version of the graph which has 1000 as the maximum.

On Wed, Oct 7, 2020 at 8:02 AM Chris Lilley <chris@w3.org> wrote:

> Thanks! That makes it much clearer for me.
>
> The graph at
> https://docs.google.com/document/d/1kx62tpy5hGIbHh6tHMAryon9Sgye--W_IsHTeCMlmEo/edit#
>
> has M= 100 but the Python has M = 1000. Which is correct, for the final
> version of the cost function?
>
> Also, I have a screenshot of the sigmoid graph for now but would love to
> see the graph as a separate graphic, if that is convenient?
> On 2020-10-07 02:30, Garret Rieger wrote:
>
> Here's how the cost function as presented here
> <https://docs.google.com/document/d/1kx62tpy5hGIbHh6tHMAryon9Sgye--W_IsHTeCMlmEo/edit#heading=h.4fz1x8661i63> was
> derived:
>
>
>    - Start with the logistic function (a sigmoid): M/ ( 1 + e^(-k(x -
>    x_0)))
>    - M is the maximum height of the function.
>    - k scales the width of the function
>    - and x_0 shifts the function left/right.
>
> We want a function that starts rising at x = T_z and hits it's maximum at
> T_m:
>
>    - Width of the period it rises over is W = T_m - T_z
>    - Scale k = K / W. Where K is a hand selected constant which controls
>    the width. By dividing by W we normalize the scale against the width. A
>    value of K = 11.5 was found to give a near maximum and minimum value at T_z
>    and T_m.
>    - x_0 = W/2 + T_z = T_m/2 - T_z/2 + T_z = T_m/2 + T_z/2
>       -  This moves the function W/2 units to the right (the logistic
>       function starts centered on x = 0)
>       - And then an additional T_z to the right to add the initial period
>       of zero cost.
>
> If you plug that all into M / (1 + e^(-k(x - x_0))) you get:
>
> M / ( 1 + e^(-11.5/(T_m-T_z) * ( x - T_m/2 - T_z/2)))
>
> There's a similar explanation in the actual implementation in code:
> https://github.com/w3c/PFE-analysis/blob/master/analysis/cost.py
>
> --
> Chris Lilley
> @svgeesus
> Technical Director @ W3C
> W3C Strategy Team, Core Web Design
> W3C Architecture & Technology Team, Core Web & Media
>
>