Spatio-Temporal Covariance & Lohr Survey De-Biasing

In This Component

Naive sample covariance estimation over high-growth marketplace logs suffers from secular trend confounding: platform onboarding growth inflates the first principal component (${\text{PC}}_1$) and induces spurious $+0.90$ correlations between unrelated product categories.

Secular Detrending & Kronecker-Separable DSTM

To isolate true cross-category co-movement across the Top-$P$ ($P=10$) categories over $T=87$ operational weeks, Aegis decomposes weekly demand into secular platform trajectory, calendar seasonality, and stationary residuals:

$D_{t,p}={\mu }_{t,p}^{\text{trend}}+s_{t,p}^{\text{season}}+{ϵ}_{t,p}$

Spectral decomposition ${\Sigma }_{ϵ}={\sum }_{j=1}^{10}{\lambda }_j{e}_j{e}_j^T$ reconstructs correlated Monte Carlo scenarios with only 2.16% Frobenius relative error.

To prevent dimensional rank deficiency when modeling joint hub-by-category dynamics across $K=5$ hubs and $P=10$ categories (${\mathbb{R}}^{50}$ state space with $T=87$ weekly observations), the Wikle Dynamic Spatio-Temporal Model (DSTM) enforces Kronecker separability on the innovation covariance:

$W_{t+1}=M{W}_t+{\eta }_{t+1},{\eta }_{t+1}\sim \mathcal{N}\left(0,{\Sigma }_{\text{space}}^{(K\times K)}\otimes {\Sigma }_{\text{cat}}^{(P\times P)}\right)$

Continuous weekly innovations modulate discrete order-level SKU arrivals via a Hierarchical Log-Gaussian Cox Process (LGCP).

Sharon Lohr Finite Survey Sampling & De-Biasing

Because Southeast Brazil (São Paulo) generates 42–47% of historical platform traffic, Aegis anchors stratum inclusion probabilities ${\pi }_h=n_h/N_h$ against external IBGE demographic frames, computing unbiased Horvitz-Thompson regional totals and Neyman optimal allocations:

${\hat{t}}_{\text{HT}}={\sum }_{i\in \mathcal{S}}\frac{y_i}{{\pi }_i},n_h=n\frac{N_h{S}_h}{{\sum }_{k=1}^H{N}_k{S}_k}$