Estimation when the signal is small and the world keeps moving.

1. Non-stationary state estimation

We model a latent state $x_t$ with linear dynamics and observe it through additive noise:

$$x_t = F x_{t-1} + w_t, \qquad y_t = H x_t + v_t, \qquad w_t \sim \mathcal{N}(0, Q),\quad v_t \sim \mathcal{N}(0, R).$$

The filtered mean $\hat x_{t\mid t}$ is the conditional expectation $\mathbb{E}[x_t \mid y_{1:t}]$ under Gaussianity. Non-stationarity enters through $Q$: a larger process variance lets the estimator react faster, at the cost of passing more observation noise through to the estimate. Choosing $Q$ is therefore a bias-variance decision, not a cosmetic one.

We study adaptive estimators of $Q$ based on innovation statistics, and we check filter consistency with normalized innovation tests. See Kalman filtering for latent signals.

2. High-dimensional covariance estimation

When the number of variables $p$ is comparable to the number of observations $n$, the sample covariance matrix $S$ is a noisy and poorly conditioned estimate of $\Sigma$. Linear shrinkage replaces it with a convex combination of $S$ and a structured target $F$:

$$\hat\Sigma_\delta = \delta\, F + (1 - \delta)\, S, \qquad \delta \in [0, 1].$$

The intensity $\delta$ is chosen to minimize expected loss, and can be estimated from the data without tuning. See Shrinkage for high-dimensional covariance estimation.

3. Reproducible compute infrastructure

Estimation results are only as credible as the pipeline that produced them. We treat research code as an engineering system:

• every experiment is fully determined by a typed configuration and a seed;
• every artifact is content-addressed, so a result can be traced to the exact inputs that produced it;
• evaluation protocols are fixed before results are inspected.

See Deterministic research pipelines for implementation notes.

Methodological commitments

1. Uncertainty over point estimates. We report intervals and dispersion, not single summary numbers.
2. Temporal honesty. Validation respects time ordering, with purging and embargoes where labels overlap. See walk-forward validation.
3. No performance promises. Simulated and historical results describe methods. They do not forecast outcomes.