Skip to content

Kalman Filter (Equations)

Statistical Glossary

Kalman Filter (Equations):

The basic mathematics behind the idea of Kalman filter may be described as follows – Consider, for example, a Markov chain – i.e. a random series with Markov property – described by the following equation:



  • Math image – is the value of the vector-values Markov chain Math image at the moment Math image of discrete time Math image ;
  • Math image is a known matrix which describes regular causal link between the current state Math image and the next state Math image ;
  • Math image is a zero-mean random noise with known covariance matrix.

The Kalman filter for time series Math image specified by equation (1) may be described by the following recursive expression:



  • Math image is the output of the Kalman filter, which is normally used as a predictor for Math image ;
  • Math image is the “Kalman gain”.

The expression (2) is only one of several mathematically- equivalent forms. It shows that the Kalman filter output Math image is a sum of two terms. The first term ( Math image ) is a dynamic prognosis based on the known transition matrix Math image of the Markov chain (1) ; the second term is a linear function of the prediction error ( Math image ) from the previous step.

The theory of Kalman filtering specifies expressions for calculation of the optimal matrix Math image in equation (2) from the known covariance matrix Math image of the random vector Math image in the model (1) .

Besides the simplest case described by (1-2) , the classical theory of Kalman filters covers more complex settings, e.g. when the state vector Math image is not observable – i.e. only a linear function Math image of Math image is known


  • Math image is a known matrix;
  • Math image is the observation noise with known covariance matrix.

There are also generalizations of the Kalman filter theory for continuous time Math image , expressed in terms of differential equations (not difference equations, like above); and there are extensions to non-linear filtering.

Browse Other Glossary Entries

Test Yourself

Planning on taking an introductory statistics course, but not sure if you need to start at the beginning? Review the course description for each of our introductory statistics courses and estimate which best matches your level, then take the self test for that course. If you get all or almost all the questions correct, move on and take the next test.

Data Analytics

Considering becoming adata scientist, customer analyst or our data science certificate program?

Analytics Quiz

Advanced Statistics Quiz

Statistics Quiz


Looking at statistics for graduate programs or to enhance your foundational knowledge?

Statistics 1 Quiz

Regression Quiz

Regression Quiz


Entering the biostatistics field? Test your skill here.

Biostatistics Quiz

Advanced Statistics Quiz

Statistics 2 Quiz

Stay Informed

Our Blog

Read up on our latest blogs


Learn about our certificate programs


Find the right course for you

Contact Us

We'd love to answer your questions

Our mentors and academic advisors are standing by to help guide you towards the courses or program that makes the most sense for you and your goals.

300 W Main St STE 301, Charlottesville, VA 22903

(434) 973-7673

By submitting your information, you agree to receive email communications from All information submitted is subject to our privacy policy. You may opt out of receiving communications at any time.