Observation Theory: Estimating the Unknown

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Week 1: Introduction
Introduction on what is “estimation” and when do we need it? What are the generic sources of uncertainty in observations, and what concepts are needed, e.g. deterministic vs. stochastic parameters, random vs. systematic errors, precision vs. accuracy, bias, and the probability distribution function as a metric of randomness.
In this week and througout the course, all concepts are explained by various practical examples.

Week 2: Mathematical models

Develop a systematic approach to translate real-life problems into mathematical models in the form of observation-equation system including four fundamental blocks: vector of observations, vector of unknown parameters, linear (or linearized) functional relation between observations and unknowns, and stochastic characteristics of observations in the form of dispersion (or covariance matrix) of the observation vector. Discussion on different concepts and models.

Week 3: Least Squares Estimation (LSE)

Given a mathematical model, how to find an estimate that predicts the observations as close as possible? Introduction to (weighted) least squares estimation (WLSE), its mathematical logic and its main properties.

Week 4: Best Linear Unbiased Estimation (BLUE)
How to find the most precise and accurate estimate in linear models? Introduction to the concept of Best Linear Unbiased Estimation (BLUE), its theory and implication, and its relation to other estimators such as WLSE, maximum likelihood, and minimum variance estimators.

Week 5: How precise is the estimate?
Discussion on how the uncertainty/randomness in observations (depicted by a stochastic model) propagates to the uncertainty/randomness of estimates (depicted by probability density function or covariance matrix of estimators). Introduction to the concept of error propagation and its application in specification of the uncertainty/precision of estimates, inferring confidence intervals or statistical tolerance levels of the results, and describing the expected variability of the results of an estimation.

Week 6: Does the estimate make sense?
Introduction to a probabilistic decision making process (or statistical hypothesis testing) in validating the results of estimation in order to avoid wrong decisions/interpretation of the results. Verify the validity of a chosen mathematical model, and how to detect and identify model misspecifications.