Generalized Multivariate Difference Estimator (GMDe) for data assimilation

Environmetrika primarily supports sample surveys that use repeated measurements of population elements over time, where those elements are selected with probability sampling. These supply time-series of population estimates for "study variables" that serve an analyst’s objectives. 

Population statistics use

• Linear estimators for means and totals;
• Synthetic estimators, such as additive margins in statistical tables;
• Pseudo estimators, such as rates of change and other nonlinear functions of primary population statistics. 

In the special case of monitoring terrestrial systems, measurements are field observations or very high resolution remotely sensed data for a probability sample of monitoring sites. 

However, population estimates for study variables might not be sufficiently accurate for the analyst’s objectives.  

Environmetrika develops and applies statistical estimators that improve sample survey estimates for study variables with correlated estimates for "auxiliary variables." 

Measurements of auxiliary variables use relatively inexpensive technologies. Examples include low- and medium-resolution remote sensing, full-coverage predictions with geospatial models, deterministic ecosystem models that predict expected changes over time, administrative records, independent surveys from other sources, and others. Therefore, population estimates for auxiliary variables can be more precise than those for study variables. However, auxiliary variables are imperfectly correlated with study variables, and therefore, they do not directly satisfy the the analyst's objectives. 

Environmetrika develops statistical estimators that use population estimates for auxiliary variables to improve population estimates for study variables, which do directly satisfy the the analyst's objectives. 

These methods replace model-assisted regression estimators with linear transformations of estimators for stochastic systems, where probability sampling is the stochastic process.

• Multi-phase and multi-stage sampling; 
• Pre-stratification and post-stratification; 
• Finite and infinite population models; 
• Equal and unequal inclusion probabilities; 
• Interpenetrating panels, with or without overlap; 
• Multiple independent surveys; 
• Data missing at random; 
• Small domains and small areas; 
• Detection of anomalous prediction residuals; 
• Probability samples of QA/QC data;
• Hypothesis testing and outlier detection; 
• Model-based estimation with diagnostics; 
• Linear and nonlinear estimators; 
• Synthetic and pseudo estimators; 
• Equality and inequality constraints;
• Accuracy assessment for remotely sensed data; and 
• Forecasting
• Robust numerics with numerous variables

Environmetrika develops statistical methods that use a multivariate difference estimator, which is the foundation for model-assisted regression estimators in sample surveys. These methods employ mature numerical methods, such as those developed by system engineers for digital signal processing. 

Environmetrika supports multivariate sample survey estimates for thousands of study variables with thousands of auxiliary variables, including very large covariance matrices. Those estimators are robust, numerically stable and computationally efficient. A step-wise approach accommodates covariance matrices that are ill-conditioned or rank deficient.

This combination of the sample survey and system engineering disciplines produces relatively simple solutions to large and complex analyses.