Starting point is a system with a state (s,y) where s denotes the inner state and y can be observed.

Further we assume that the system obeys the known update equation below:

s_t+1 = A s_t + W_t

y_t+1 = B s_t+1 + N_t

The Kalman Filter is used to predict the inner state s for a sequence of observations y based on the maximum likelihood approach.

Both s and y are disturbed through white gaussian noise. We now assume that the joint likelihood of s and y can be modeled by a gaussian distribution.

The likelihood that the system is at time t in a specific state is described by a probability density distribution f_t(s,y). Now we assume that the distribution f_t(s,y) is a multivariate gaussian distribution.

„ s_t… expected mean value for s assuming the distribution f_t(s,y)

„ y_t… expected mean value for y assuming the distribution f_t(s,y)

v^t_ss expected mean value for „ s_t s_t… assuming the distribution f_t(s,y)

v^t_yy expected mean value for „ y_t y_t… assuming the distribution f_t(s,y)

v^t_sy expected mean value for „ s_t y_t… assuming the distribution f_t(s,y)

v^t_ys expected mean value for „ y_t s_t… assuming the distribution f_t(s,y)

The two dimensional case f_t(s,y) = const delivers ellipses as depicted in the figure. We get ellipsoides for the three dimensional case.

The covariance

can be represented by a product of the standard deviations s _s and s _y weighted by the correlation r . The correlation itself is a function of the angle a in order to close the gap between the graphical representation and the common representation of the distribution density function f(s,y).

This step delivers the mean and variance for the most likely distribution of s based on a measured y which represents the update for the most likely mean of y_t. The distribution f_t(s,y) can be reduced to the conditional distribution f_t(s|y).assuming that y is known.

The advantage of the conditional distribution is that it requires less computational effort. However, its derivation is not trivial.

The factor K denotes the Kalman Gain because it determines the influence of the measured value y onto the predicted inner state s.

The Kalman Gain is related to the angle b in the graphical represenation by the following equation:

This step delivers the prediction for t+1 with respect to state s_t+1.

In order to setup the procedure, firstly a starting value for y,s together with a distribution is assumed. A first measure value of y is used to get the most likely value for s. Based on this update for s, its value is estimated for t+1 employing the transfer function of the system. In addition also the variance is predicted leading to a predicted distribution. This process of updates using measured values and predictions using the transfer function can be repeated iteratively.