Drift and noise are not identifiable objects without assumptions. The Doob–Meyer decomposition explains exactly what can—and cannot—be separated in stochastic systems, and why martingales are the correct lens for trend extraction.
In kernel DMD and Koopman analysis, many extracted modes are not features of the dynamics but artifacts of finite-rank projection. Increasing model capacity often amplifies spectral pollution rather than revealing structure.
A conceptual map of generation: subword tokenization → learned embeddings → attention-based next-token prediction. Concrete diagrams show what each stage buys you and what it assumes.
Normalizing flows turn density estimation into calculus: invertible transforms + Jacobians give exact log-likelihoods. This post builds the change-of-variables view and why architectures like RealNVP make it tractable.
In non-normal problems, eigenvalues are not stable summaries: tiny perturbations can move them drastically. Pseudospectra replace point eigenvalues with uncertainty-aware regions—crucial for DMD/Koopman-style analysis.
GARCH-style volatility can look calibrated—until the data regime changes. This post shows why coverage breaks under structural shifts and how Adaptive Conformal Inference re-calibrates online without distributional assumptions.