Optimal Linear Estimation of Functionals

• The paper derives explicit formulas for the optimal linear estimator of a functional from noisy, incomplete observations using Hilbert space projections and spectral analysis.
• It presents a mean-square error minimization framework that extends classical Wiener theory to accommodate missing data and spectral uncertainty.
• The robust minimax approach identifies least favorable spectral densities, ensuring reliable estimation under varying noise and signal conditions.

the mean-square optimal linear estimation of the functional $A\xi = \int_{R^s} a(t) \xi(-t) dt$ for a stationary process $\xi(t)$ observed with noise and missing data. The summary highlights both the spectral certainty and minimax (robust) approaches, including explicit formulas and the implications for the general problem.

1. Problem Setting

• Signal: $\xi(t)$, a real stationary process with known or partially known spectral density $f(\lambda)$.
• Noise: $\eta(t)$, an uncorrelated stationary process with spectral density $g(\lambda)$.
• Observations: $y(t) = \xi(t) + \eta(t)$ are available for $t \in \mathbb{R} \setminus S$, i.e., with missing values on a set $S$.
• Functional to estimate: $A\xi = \int_{R^s} a(t) \xi(-t) dt$ where $a(t)$ is a given function.
• Goal: Find the linear estimator of $A\xi$ from the observations, minimizing mean-square error. This generalizes classical Wiener filtering to functionals and missing data.

2. Spectral Certainty: Explicit Estimator and Error Formulas

Key Method: Hilbert Space Projection

The problem is solved via Hilbert space projection methods, leveraging the spectral (Fourier) representation of stationary processes.

• Spectral representation:

$$\xi(t) = \int_{-\infty}^\infty e^{i t \lambda} Z_\xi(d\lambda),\quad \eta(t) = \int_{-\infty}^\infty e^{i t \lambda} Z_\eta(d\lambda)$$

• Fourier Transform of the functional:

$$A(e^{i\lambda}) = \int_{R^s} a(t) e^{-i t \lambda} dt$$

Spectral Characteristic and Estimator

• Optimal linear estimator $\widehat{A\xi}$:

$$\widehat{A\xi} = \int_{-\infty}^{\infty} h(e^{i \lambda}) (Z_\xi(d\lambda) + Z_\eta(d\lambda))$$

where the filter $h(e^{i\lambda})$ (spectral characteristic) is given by:

$$h(e^{i \lambda}) = \frac{A(e^{i \lambda}) f(\lambda) - C(e^{i \lambda})}{f(\lambda) + g(\lambda)}$$

• Correction term accounting for missing data:

$$C(e^{i \lambda}) = \sum_{l=1}^{s} \int_{-M_l-N_l}^{-M_l} (B^{-1} R a)(t) e^{i t \lambda} dt + \int_0^\infty (B^{-1} R a)(t) e^{i t \lambda} dt$$

Mean-Square Error:

$$\boxed{ \Delta(h;f,g) = \frac{1}{2\pi}\int_{-\infty}^\infty \frac{|A(e^{i\lambda})g(\lambda) + C(e^{i\lambda})|^2}{|f(\lambda)+g(\lambda)|^2} f(\lambda)d\lambda + \frac{1}{2\pi}\int_{-\infty}^\infty \frac{|A(e^{i\lambda})f(\lambda) - C(e^{i\lambda})|^2}{|f(\lambda)+g(\lambda)|^2} g(\lambda)d\lambda }$$

3. Minimax (Robust) Filtering under Spectral Uncertainty

Motivation and Approach

In practice, spectral densities $f(\lambda)$ and $g(\lambda)$ are often not known exactly. When they belong to certain admissible sets ($D_f, D_g$), a minimax approach is used.

• Objective: Find the estimator minimizing the worst-case mean-square error over all allowed spectral densities.
• Least Favorable Spectral Densities $(f_0, g_0)$:

$$\Delta(f_0, g_0) = \max_{(f,g)\in D_f \times D_g} \Delta(h(f_0, g_0); f, g)$$

• Minimax Spectral Characteristic:

$h^0(e^{i\lambda}) = h(f_0, g_0)$

Formulas and Conditions for Finding Least Favorable Densities

The optimal densities satisfy: $$\Delta(h(f_0, g_0); f_0, g_0) = \max_{(f,g) \in D_f \times D_g} \Delta(h(f_0, g_0); f, g)$$ Using constraints (e.g., $L_1$ or $L_2$ bounds on deviations from nominal spectral densities).

4. Implications & Generalizations

• Extends classical Wiener-Kolmogorov theory to include missing data and observation noise.
• Detailed treatment of spectral uncertainty helps tailor filters that are robust to model errors.
• The approach allows computation in broad settings, including block-missing observations and robust filtering against imprecise knowledge of spectra.

5. Summary of Key Formulas

Spectral Characteristic: $$h(e^{i \lambda}) = \frac{A(e^{i \lambda}) f(\lambda) - C(e^{i \lambda})}{f(\lambda) + g(\lambda)}$$

Mean-Square Error: $$\Delta(h;f,g) = \langle Ra, B^{-1}Ra \rangle + \langle Qa, a \rangle$$

Minimax Filtering:

Find $(f_0, g_0)$ minimizing worst-case MSE; use $h^0(e^{i\lambda})$ built from these.

Conclusion

The paper provides a robust estimation framework for the linear functional of stationary processes with missing data, extending classical approaches by integrating spectral uncertainty in the filter design. This minimizes error in practical applications where precise spectral information may be unattainable.