# What is Kalman Filter?

The Kalman filter, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing noise (random variations) and other inaccuracies, and produces estimates of unknown variables that tend to be more precise than those based on a single measurement al

#### What is a Kalman Filter and What Can It Do?

A Kalman filter is an optimal estimator- ie infers parameters of interest from indirect, inaccurate and uncertain observations. It is recursive so that new measurements can be processed as they arrive. (cf batch processing where all data must be present).

### Why is Kalman Filtering so popular?

• Good results in practice due to optimality and structure.
• Convenient form for online real time processing.
• Easy to formulate and implement given a basic understanding.
• Measurement equations need not be inverted.

### Examples:

• Determination of planet orbit parameters from limited earth observations.
• Tracking targets - eg aircraft, missiles using RADAR.
• Robot Localisation and Map building from range sensors/ beacons.

 The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate. The estimate is updated using a state transition model and measurements. $\hat{x}_{k|k-1}$ denotes the estimate of the system's state at time step k before the k-th measurement yk has been taken into account; $P_{k|k-1}$ is the corresponding uncertainty.

The algorithm works in a two-step process. In the prediction step, the Kalman filter produces estimates of the current state variables, along with their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some amount of error, including random noise) is observed, these estimates are updated using a weighted average, with more weight being given to estimates with higher certainty. Because of the algorithm's recursive nature, it can run in real time using only the present input measurements and the previously calculated state; no additional past information is required.
From a theoretical standpoint, the main assumption of the Kalman filter is that the underlying system is a linear dynamical system and that all error terms and measurements have a Gaussian distribution (often a multivariate Gaussian distribution).
Extensions and generalizations to the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. The underlying model is a Bayesian model similar to a hidden Markov model but where the state space of the latent variables is continuous and where all latent and observed variables have Gaussian distributions.

### Applications of Kalman Filter

• The Kalman filter has numerous applications in technolog
• Guidance, navigation and control of vehicles, particularly aircraft and spacecraft.
• The Kalman filter is a widely applied concept in time series analysis used in fields such as signal processing and econometrics.