Linear observations

LinearObservation fits Gaussian data that observe sums or other linear combinations of a finer prediction grid. This covers temporal disaggregation, geographical reconciliation and benchmarked nowcasting without inventing a pseudo-response at the fine level.

If the model's predictor on the supplied grid is

\[ \eta = o + Zx, \]

a linear observation declares

\[ y_j = (C\eta)_j + \varepsilon_j, \qquad \varepsilon_j \sim N(0,\sigma_j^2). \]

from pylgm import Gaussian, LGM, LinearConstraint, LinearObservation

result = model.fit(
    grid,
    observations=[
        LinearObservation(national_quarterly, C_quarter, sigma=1.0),
        LinearObservation(regional_annual, C_region_year, sigma=2.0),
    ],
    constraints=[
        LinearConstraint(C_accounting, national_annual),
    ],
)

Each operator has one row per aggregate value and one column per row of grid, in the order in which the caller supplied those rows. Sparse SciPy matrices are accepted and are preferable for large aggregation systems. pyLGM realigns the columns if canonical panel sorting changes the internal row order.

The response column named by LGM.response may be absent when linear observations or constraints are supplied. If it is present, its non-null rows are combined with the linear observations, using the model's Gaussian sigma; each LinearObservation uses its own scalar or row-specific sigma.

Soft observations versus exact constraints

Use LinearObservation for published estimates, preliminary releases and other measurements with uncertainty. Its sigma is the standard deviation of the aggregate measurement, not of each fine-grid cell.

Use LinearConstraint only for an identity that must hold exactly. pyLGM translates C @ eta = e into a constraint on the latent field, C @ Z @ x = e - C @ o, rejects incompatible systems and removes redundant rows before inference.

The returned predictive_mean and predictive_variance remain aligned with the original fine grid, rather than with the shorter aggregate-observation vector. The likelihood is standardized internally, including the Jacobian normalization, so posterior inference and log marginal likelihood retain the declared heterogeneous observation variances.

Scope

Linear observations currently support Gaussian LGM models with pandas input. They work with fixed fits, empirical-Bayes optimization and INLA integration. Exact cross-block constraints use the dense Gaussian path; models large enough to route to the sparse solver still inherit its existing restriction against cross-block constraint rows.