Time anchor (reminder)
Use the midpoint anchor from training.t0 = midpoint(train_start, train_stop) ; t = days_since(date, t0)
Unwrapped OLS targets
Fit on the continuous series (unwrap first; wrap only for display).
- fixed-n:
y(t) = L_actual_unwrapped(t) − n*t - free-n :
y(t) = L_actual_unwrapped(t)
Regressor design for a candidate Ω = {w_k}
- fixed-n:
X(t) = [ 1 , sin(w1*t), cos(w1*t), ... , sin(wm*t), cos(wm*t) ] - free-n :
X(t) = [ 1 , t , sin(w1*t), cos(w1*t), ... , sin(wm*t), cos(wm*t) ]
Estimation (OLS + BIC with clamp)beta_hat = argmin_beta || y − X*beta ||_2^2RSS = || y − X*beta_hat ||_2^2 ; k = ncols(X) ; N = nrows(X)BIC = k*log(N) + N*log( max(RSS/N, 1e-16) ) (lower is better)
Event-aware training loss (degree still dominant)
Let L_hat_unwrapped be the model on the unwrapped series.
- Speed (central differences):
v_hat = | d/dt L_hat_unwrapped |;v_act = | d/dt L_actual_unwrapped | - Speed MAE:
mae_v = mean( | v_hat − v_act | ) - Degree MAE on train:
MAE_deg_train - Cusp MAE on train (ASCII): define
cusp_dist_deg(x) = min( (x % 30) , 30 − (x % 30) ), then average|cusp(model) − cusp(actual)|.
Combined loss:loss = 1.0*MAE_deg_train + 0.3*cusp_MAE_train + 0.4*mae_v
Admissibility gate (strict parsimony)ADMISSIBLE iff (BIC_EXTRA ≤ BIC_BASE − 6.0) and (loss_EXTRA < loss_BASE).
Selection rule
Pick the admissible model with the smallest (loss, then BIC); otherwise fall back to BASE. Keep t0 and the time convention fixed across comparisons.
Notes & tips
- Use the same train/test windows and sampling mode as declared in §2.5–2.6; no leakage.
- Compute derivatives via central differences (
Δt=1on daily grids; one-sided at ends). - Treat
ΔBIC ≥ 6as a decisive improvement; otherwise reject extras. - Maintain the midpoint anchor to reduce phase coupling and edge bias.
Navigation
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