Editing Openai/693eaf5a-1494-8007-b1e3-00abefb11523 (section)

=== I use these notations: ===
* ptp_tpt = price at day ttt (price_max when using the most recent)
* dtd_tdt = numeric date at day ttt (e.g., days since start or day-of-year)
* DtD_tDt = day-of-year (1..365)
* hth_tht = tetrahedral apex height (a function we will parametrize)
* Lt=∥ODt∥L_t = \| O D_t \|Lt=∥ODt∥ length from origin to apex
* p^t+1\hat p_{t+1}p^t+1 predicted next max price

==== Start from earlier result: ====

Lt=2(dt2+pt2)L_t = \sqrt{2(d_t^2 + p_t^2)}Lt=2(dt2+pt2)
project onto price axis:

p^t+1=α⋅ptdt2+pt2⋅Lt=α2 pt\hat p_{t+1} = \alpha \cdot \frac{p_t}{\sqrt{d_t^2 + p_t^2}} \cdot L_t
= \alpha \sqrt{2}\, p_tp^t+1=α⋅dt2+pt2pt⋅Lt=α2pt
Here α\alphaα is a single learned scale parameter (start with α=1\alpha = 1α=1). Fit α\alphaα to minimize historical error. This captures the geometry but lets the data scale the amplitude.

==== Introduce exponents to control influence of time vs price: ====

ht=k⋅(dtγ+ptδ)1/2h_t = k \cdot (d_t^\gamma + p_t^\delta)^{1/2}ht=k⋅(dtγ+ptδ)1/2
and

Lt=dt2+pt2+ht2L_t = \sqrt{d_t^2 + p_t^2 + h_t^2}Lt=dt2+pt2+ht2
project:

p^t+1=β⋅Lt⋅ptμ(dt2+pt2)ν/2\hat p_{t+1} = \beta \cdot L_t \cdot \frac{p_t^\mu}{(d_t^2 + p_t^2)^{\nu/2}}p^t+1=β⋅Lt⋅(dt2+pt2)ν/2ptμ
Parameters to fit: k,γ,δ,β,μ,νk,\gamma,\delta,\beta,\mu,\nuk,γ,δ,β,μ,ν. Use regularization to avoid overfitting. This family is flexible and can learn seasonal/higher-order interactions.

==== Make the tetrahedron from three different past times (e.g., ttt, t−7t-7t−7, t−30t-30t−30) as the three spatial vertices (time axis, price axis, and a secondary axis). Use those to compute a 3D simplex volume or the longest edge, then project back to price. ====

Construct points:
* A=(dt,0,0)A=(d_t,0,0)A=(dt,0,0)
* B=(0,pt,0)B=(0,p_t,0)B=(0,pt,0)
* C=(dt−τ,pt−τ,h)C=(d_{t-\tau}, p_{t-\tau}, h)C=(dt−τ,pt−τ,h) for some lag τ\tauτ

Use either:
* Longest edge length LmaxL_{\text{max}}Lmax or
* Tetrahedral volume V=16∣det⁡([A−O,B−O,C−O])∣V = \tfrac{1}{6} | \det([A-O, B-O, C-O]) |V=61∣det([A−O,B−O,C−O])∣

Then set:

p^t+1=λ1pt+λ2⋅projprice(Lmax⁡)+λ3⋅V\hat p_{t+1} = \lambda_1 p_t + \lambda_2 \cdot \text{proj}_{price}(L_{\max}) + \lambda_3 \cdot Vp^t+1=λ1pt+λ2⋅projprice(Lmax)+λ3⋅V
Fit λ\lambdaλ’s; this mixes geometric metrics and direct persistence.

==== Make hth_tht proportional to short-term volatility (std of returns over last n days): ====

ht=k⋅σtη,σt=std(Δpt−n+1:t)h_t = k \cdot \sigma_{t}^{\eta},\qquad \sigma_t = \text{std}(\Delta p_{t-n+1:t})ht=k⋅σtη,σt=std(Δpt−n+1:t)
Then use LtL_tLt projection as in family (2). This captures the idea that "taller" tetrahedra (larger h) should occur during high-volatility periods and produce larger predictions.