System's parameters or state: The internal variables or parameters that describe or characterize the system at any given time. For example, in a robot, the state might include position, velocity, and orientation. In a temperature control system it could include the current temperature and the rate of temperature change.
Measurements: These are the data points obtained from sensors or other sources. Measurements are typically noisy or incomplete and optimal estimation methods aim to reduce the effect of this noise.
Noise and Uncertainty: In most practical systems measurements are not perfect. Noise can come from various sources like sensor inaccuracies, environmental interactions or process variations. Estimation methods must account for this uncertainty to provide accurate estimates.
Errors: The error between the true or expected values of the system's parameters/state and the estimated parameters/state.
Optimal estimation aims to produce the best possible estimate of the system's parameters/state, given the noisy measurements. The "best" estimate is typically defined as the one that minimizes the estimation error, often quantified in terms of variance or mean squared error.
Weighted Least Squares Estimation (WLS): This method minimizes the sum of the squared differences between the measured values and the estimated values. It's often used when the system is linear, and the errors are independent and identically distributed.
Maximum Likelihood Estimation (MLE): In this approach, we choose the estimate that maximizes the likelihood of the observed measurements given a model of the system. MLE is particularly useful when the probability distribution of the noise is known.
Kalman Filter (KF): The Kalman filter is a recursive algorithm used for optimal estimation in linear systems with Gaussian noise. It combines the predicted state (from a model of the system) and measurements to provide an estimate that minimizes the expected error covariance. The Kalman filter is widely used in applications like navigation, robotics, and aerospace.
Extended Kalman Filter (EKF): This is an extension of the Kalman filter used for nonlinear systems. It linearizes the system around the current estimate, making it applicable to more complex nonlinear systems.
Optimal estimation is used in many fields, just to cite few:
Navigation and Tracking: Estimating the position, velocity, and other states of moving objects like vehicles, drones, or satellites. The Kalman filter, for example, is often used in GPS systems to estimate a vehicle's position accurately.
Robotics: In robot navigation, optimal estimation helps estimate the robot's position and orientation (often using sensors like lidar, accelerometers, and cameras) to ensure accurate movement and decision-making.
Aerospace: Estimating flight parameters such as altitude, speed and direction using noisy sensor data, enabling autopilot systems to make optimal decisions.
Econometrics and Finance: Estimating economic variables like inflation or stock market behavior from noisy data providing forecasts and decision support.
Signal Processing: Enhancing the quality of signals by filtering out noise, such as in audio or image processing, to provide a clearer and more accurate representation of the underlying data.