Re: Derivation of the Cost Function

Added updated cost function graph to the repo:
https://github.com/w3c/PFE-analysis/blob/master/results/07-08-2020/cost.png

On Wed, Oct 7, 2020 at 10:58 AM Garret Rieger <grieger@google.com> wrote:

> 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
>>
>>