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Kalman filter

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Algorithm that estimates unknowns from a series of measurements over
time

[400px-Basic_concept_of_Kalman_filtering]
 
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. x ^ k | k -
1 {\displaystyle {\hat {x}}_{k\mid k-1}} {\hat {x}}_{k\mid k-1}
denotes the estimate of the system's state at time step k before the
k-th measurement y[k] has been taken into account; P k | k - 1 {\
displaystyle P_{k\mid k-1}} {\displaystyle P_{k\mid k-1}} is the
corresponding uncertainty.

In statistics and control theory, Kalman filtering, also known as
linear quadratic estimation (LQE), is an algorithm that uses a series
of measurements observed over time, containing statistical noise and
other inaccuracies, and produces estimates of unknown variables that
tend to be more accurate than those based on a single measurement
alone, by estimating a joint probability distribution over the
variables for each timeframe. The filter is named after Rudolf E.
Kalman, one of the primary developers of its theory.

The Kalman filter has numerous applications in technology. A common
application is for guidance, navigation, and control of vehicles,
particularly aircraft, spacecraft and dynamically positioned ships.^
[1] Furthermore, the Kalman filter is a widely applied concept in
time series analysis used in fields such as signal processing and
econometrics. Kalman filters also are one of the main topics in the
field of robotic motion planning and control and can be used in
trajectory optimization.^[2] The Kalman filter also works for
modeling the central nervous system's control of movement. Due to the
time delay between issuing motor commands and receiving sensory
feedback, use of the Kalman filter supports a realistic model for
making estimates of the current state of the motor system and issuing
updated commands.^[3]

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. The algorithm is recursive. It can run in real
time, using only the present input measurements and the previously
calculated state and its uncertainty matrix; no additional past
information is required.

Optimality of the Kalman filter assumes that the errors are Gaussian.
In the words of Rudolf E. Kalman: "In summary, the following
assumptions are made about random processes: Physical random
phenomena may be thought of as due to primary random sources exciting
dynamic systems. The primary sources are assumed to be independent
gaussian random processes with zero mean; the dynamic systems will be
linear."^[4] Though regardless of Gaussianity, if the process and
measurement covariances are known, the Kalman filter is the best
possible linear estimator in the minimum mean-square-error sense.^[5]

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 hidden Markov model where the state space of the latent
variables is continuous and all latent and observed variables have
Gaussian distributions. Also, Kalman filter has been successfully
used in multi-sensor fusion,^[6] and distributed sensor networks to
develop distributed or consensus Kalman filter.^[7]

[ ]

Contents

  * 1 History
  * 2 Overview of the calculation
  * 3 Example application
  * 4 Technical description and context
  * 5 Underlying dynamical system model
  * 6 Details
      + 6.1 Predict
      + 6.2 Update
      + 6.3 Invariants
      + 6.4 Estimation of the noise covariances Q[k] and R[k]
      + 6.5 Optimality and performance
  * 7 Example application, technical
  * 8 Asymptotic form
  * 9 Derivations
      + 9.1 Deriving the posteriori estimate covariance matrix
      + 9.2 Kalman gain derivation
      + 9.3 Simplification of the posteriori error covariance formula
  * 10 Sensitivity analysis
  * 11 Square root form
  * 12 Parallel form
  * 13 Relationship to recursive Bayesian estimation
  * 14 Marginal likelihood
  * 15 Information filter
  * 16 Fixed-lag smoother
  * 17 Fixed-interval smoothers
      + 17.1 Rauch-Tung-Striebel
      + 17.2 Modified Bryson-Frazier smoother
      + 17.3 Minimum-variance smoother
  * 18 Frequency-weighted Kalman filters
  * 19 Nonlinear filters
      + 19.1 Extended Kalman filter
      + 19.2 Unscented Kalman filter
          o 19.2.1 Sigma points
          o 19.2.2 Predict
          o 19.2.3 Update
  * 20 Kalman-Bucy filter
  * 21 Hybrid Kalman filter
      + 21.1 Initialize
      + 21.2 Predict
      + 21.3 Update
  * 22 Variants for the recovery of sparse signals
  * 23 Applications
  * 24 See also
  * 25 References
  * 26 Further reading
  * 27 External links

History[edit]

The filter is named after Hungarian emigre Rudolf E. Kalman, although
Thorvald Nicolai Thiele^[8]^[9] and Peter Swerling developed a
similar algorithm earlier. Richard S. Bucy of the Johns Hopkins
Applied Physics Laboratory contributed to the theory, leading to it
sometimes being called the Kalman-Bucy filter. Stanley F. Schmidt is
generally credited with developing the first implementation of a
Kalman filter. He realized that the filter could be divided into two
distinct parts, with one part for time periods between sensor outputs
and another part for incorporating measurements.^[10] It was during a
visit by Kalman to the NASA Ames Research Center that Schmidt saw the
applicability of Kalman's ideas to the nonlinear problem of
trajectory estimation for the Apollo program leading to its
incorporation in the Apollo navigation computer. This Kalman filter
was first described and partially developed in technical papers by
Swerling (1958), Kalman (1960) and Kalman and Bucy (1961).

    The Apollo computer used 2k of magnetic core RAM and 36k wire
    rope [...]. The CPU was built from ICs [...]. Clock speed was
    under 100 kHz [...]. The fact that the MIT engineers were able to
    pack such good software (one of the very first applications of
    the Kalman filter) into such a tiny computer is truly remarkable.

    -- Interview with Jack Crenshaw, by Matthew Reed, TRS-80.org
    (2009) [1]

Kalman filters have been vital in the implementation of the
navigation systems of U.S. Navy nuclear ballistic missile submarines,
and in the guidance and navigation systems of cruise missiles such as
the U.S. Navy's Tomahawk missile and the U.S. Air Force's Air
Launched Cruise Missile. They are also used in the guidance and
navigation systems of reusable launch vehicles and the attitude
control and navigation systems of spacecraft which dock at the
International Space Station.^[11]

This digital filter is sometimes called the Stratonovich-Kalman-Bucy
filter because it is a special case of a more general, nonlinear
filter developed somewhat earlier by the Soviet mathematician Ruslan
Stratonovich.^[12]^[13]^[14]^[15] In fact, some of the special case
linear filter's equations appeared in these papers by Stratonovich
that were published before summer 1960, when Kalman met with
Stratonovich during a conference in Moscow.^[16]

Overview of the calculation[edit]

The Kalman filter uses a system's dynamic model (e.g., physical laws
of motion), known control inputs to that system, and multiple
sequential measurements (such as from sensors) to form an estimate of
the system's varying quantities (its state) that is better than the
estimate obtained by using only one measurement alone. As such, it is
a common sensor fusion and data fusion algorithm.

Noisy sensor data, approximations in the equations that describe the
system evolution, and external factors that are not accounted for all
place limits on how well it is possible to determine the system's
state. The Kalman filter deals effectively with the uncertainty due
to noisy sensor data and, to some extent, with random external
factors. The Kalman filter produces an estimate of the state of the
system as an average of the system's predicted state and of the new
measurement using a weighted average. The purpose of the weights is
that values with better (i.e., smaller) estimated uncertainty are
"trusted" more. The weights are calculated from the covariance, a
measure of the estimated uncertainty of the prediction of the
system's state. The result of the weighted average is a new state
estimate that lies between the predicted and measured state, and has
a better estimated uncertainty than either alone. This process is
repeated at every time step, with the new estimate and its covariance
informing the prediction used in the following iteration. This means
that Kalman filter works recursively and requires only the last "best
guess", rather than the entire history, of a system's state to
calculate a new state.

The relative certainty of the measurements and current state estimate
is an important consideration, and it is common to discuss the
response of the filter in terms of the Kalman filter's gain. The
Kalman gain is the relative weight given to the measurements and
current state estimate, and can be "tuned" to achieve a particular
performance. With a high gain, the filter places more weight on the
most recent measurements, and thus follows them more responsively.
With a low gain, the filter follows the model predictions more
closely. At the extremes, a high gain close to one will result in a
more jumpy estimated trajectory, while a low gain close to zero will
smooth out noise but decrease the responsiveness.

When performing the actual calculations for the filter (as discussed
below), the state estimate and covariances are coded into matrices to
handle the multiple dimensions involved in a single set of
calculations. This allows for a representation of linear
relationships between different state variables (such as position,
velocity, and acceleration) in any of the transition models or
covariances.

Example application[edit]

As an example application, consider the problem of determining the
precise location of a truck. The truck can be equipped with a GPS
unit that provides an estimate of the position within a few meters.
The GPS estimate is likely to be noisy; readings 'jump around'
rapidly, though remaining within a few meters of the real position.
In addition, since the truck is expected to follow the laws of
physics, its position can also be estimated by integrating its
velocity over time, determined by keeping track of wheel revolutions
and the angle of the steering wheel. This is a technique known as
dead reckoning. Typically, the dead reckoning will provide a very
smooth estimate of the truck's position, but it will drift over time
as small errors accumulate.

In this example, the Kalman filter can be thought of as operating in
two distinct phases: predict and update. In the prediction phase, the
truck's old position will be modified according to the physical laws
of motion (the dynamic or "state transition" model). Not only will a
new position estimate be calculated, but also a new covariance will
be calculated as well. Perhaps the covariance is proportional to the
speed of the truck because we are more uncertain about the accuracy
of the dead reckoning position estimate at high speeds but very
certain about the position estimate at low speeds. Next, in the
update phase, a measurement of the truck's position is taken from the
GPS unit. Along with this measurement comes some amount of
uncertainty, and its covariance relative to that of the prediction
from the previous phase determines how much the new measurement will
affect the updated prediction. Ideally, as the dead reckoning
estimates tend to drift away from the real position, the GPS
measurement should pull the position estimate back towards the real
position but not disturb it to the point of becoming noisy and
rapidly jumping.

Technical description and context[edit]

The Kalman filter is an efficient recursive filter that estimates the
internal state of a linear dynamic system from a series of noisy
measurements. It is used in a wide range of engineering and
econometric applications from radar and computer vision to estimation
of structural macroeconomic models,^[17]^[18] and is an important
topic in control theory and control systems engineering. Together
with the linear-quadratic regulator (LQR), the Kalman filter solves
the linear-quadratic-Gaussian control problem (LQG). The Kalman
filter, the linear-quadratic regulator, and the
linear-quadratic-Gaussian controller are solutions to what arguably
are the most fundamental problems in control theory.

In most applications, the internal state is much larger (more degrees
of freedom) than the few "observable" parameters which are measured.
However, by combining a series of measurements, the Kalman filter can
estimate the entire internal state.

In the Dempster-Shafer theory, each state equation or observation is
considered a special case of a linear belief function and the Kalman
filter is a special case of combining linear belief functions on a
join-tree or Markov tree. Additional approaches include belief
filters which use Bayes or evidential updates to the state equations.

A wide variety of Kalman filters have now been developed, from
Kalman's original formulation, now called the "simple" Kalman filter,
the Kalman-Bucy filter, Schmidt's "extended" filter, the information
filter, and a variety of "square-root" filters that were developed by
Bierman, Thornton, and many others. Perhaps the most commonly used
type of very simple Kalman filter is the phase-locked loop, which is
now ubiquitous in radios, especially frequency modulation (FM)
radios, television sets, satellite communications receivers, outer
space communications systems, and nearly any other electronic
communications equipment.

Underlying dynamical system model[edit]

[icon] This section needs expansion. You can help by adding to it. (
       August 2011)

Kalman filters are based on linear dynamical systems discretized in
the time domain. They are modeled on a Markov chain built on linear
operators perturbed by errors that may include Gaussian noise. The
state of the system is represented as a vector of real numbers. At
each discrete time increment, a linear operator is applied to the
state to generate the new state, with some noise mixed in, and
optionally some information from the controls on the system if they
are known. Then, another linear operator mixed with more noise
generates the observed outputs from the true ("hidden") state. The
Kalman filter may be regarded as analogous to the hidden Markov
model, with the key difference that the hidden state variables take
values in a continuous space as opposed to a discrete state space as
in the hidden Markov model. There is a strong analogy between the
equations of the Kalman Filter and those of the hidden Markov model.
A review of this and other models is given in Roweis and Ghahramani
(1999),^[19] and Hamilton (1994), Chapter 13.^[20]

In order to use the Kalman filter to estimate the internal state of a
process given only a sequence of noisy observations one must model
the process in accordance with the following framework. This means
specifying the following matrices:

  * F[k], the state-transition model;
  * H[k], the observation model;
  * Q[k], the covariance of the process noise;
  * R[k], the covariance of the observation noise;
  * and sometimes B[k], the control-input model, for each time-step,
    k, as described below.

[631px-Kalman_filter_model_2]
 
Model underlying the Kalman filter. Squares represent matrices.
Ellipses represent multivariate normal distributions (with the mean
and covariance matrix enclosed). Unenclosed values are vectors. In
the simple case, the various matrices are constant with time, and
thus the subscripts are dropped, but the Kalman filter allows any of
them to change each time step.

The Kalman filter model assumes the true state at time k is evolved
from the state at (k - 1) according to

    x k = F k x k - 1 + B k u k + w k {\displaystyle \mathbf {x} _{k}
    =\mathbf {F} _{k}\mathbf {x} _{k-1}+\mathbf {B} _{k}\mathbf {u} _
    {k}+\mathbf {w} _{k}} {\displaystyle \mathbf {x} _{k}=\mathbf {F}
    _{k}\mathbf {x} _{k-1}+\mathbf {B} _{k}\mathbf {u} _{k}+\mathbf
    {w} _{k}}

where

  * F[k] is the state transition model which is applied to the
    previous state x[k-1];
  * B[k] is the control-input model which is applied to the control
    vector u[k];
  * w[k] is the process noise, which is assumed to be drawn from a
    zero mean multivariate normal distribution, N {\displaystyle {\
    mathcal {N}}} {\mathcal {N}}, with covariance, Q[k]: w k ~ N ( 0
    , Q k ) {\displaystyle \mathbf {w} _{k}\sim {\mathcal {N}}\left
    (0,\mathbf {Q} _{k}\right)} {\displaystyle \mathbf {w} _{k}\sim
    {\mathcal {N}}\left(0,\mathbf {Q} _{k}\right)}.

At time k an observation (or measurement) z[k] of the true state x[k]
is made according to

    z k = H k x k + v k {\displaystyle \mathbf {z} _{k}=\mathbf {H} _
    {k}\mathbf {x} _{k}+\mathbf {v} _{k}} {\displaystyle \mathbf {z}
    _{k}=\mathbf {H} _{k}\mathbf {x} _{k}+\mathbf {v} _{k}}

where

  * H[k] is the observation model, which maps the true state space
    into the observed space and
  * v[k] is the observation noise, which is assumed to be zero mean
    Gaussian white noise with covariance R[k]: v k ~ N ( 0 , R k ) {\
    displaystyle \mathbf {v} _{k}\sim {\mathcal {N}}\left(0,\mathbf
    {R} _{k}\right)} {\displaystyle \mathbf {v} _{k}\sim {\mathcal
    {N}}\left(0,\mathbf {R} _{k}\right)}.

The initial state, and the noise vectors at each step {x[0], w[1],
..., w[k], v[1], ... ,v[k]} are all assumed to be mutually
independent.

Many real dynamical systems do not exactly fit this model. In fact,
unmodeled dynamics can seriously degrade the filter performance, even
when it was supposed to work with unknown stochastic signals as
inputs. The reason for this is that the effect of unmodeled dynamics
depends on the input, and, therefore, can bring the estimation
algorithm to instability (it diverges). On the other hand,
independent white noise signals will not make the algorithm diverge.
The problem of distinguishing between measurement noise and unmodeled
dynamics is a difficult one and is treated in control theory under
the framework of robust control.^[21]^[22]

Details[edit]

The Kalman filter is a recursive estimator. This means that only the
estimated state from the previous time step and the current
measurement are needed to compute the estimate for the current state.
In contrast to batch estimation techniques, no history of
observations and/or estimates is required. In what follows, the
notation x ^ n | m {\displaystyle {\hat {\mathbf {x} }}_{n\mid m}} {\
hat {\mathbf {x} }}_{n\mid m} represents the estimate of x {\
displaystyle \mathbf {x} } \mathbf {x} at time n given observations
up to and including at time m <= n.

The state of the filter is represented by two variables:

  * x ^ k | k {\displaystyle {\hat {\mathbf {x} }}_{k\mid k}} {\hat
    {\mathbf {x} }}_{k\mid k}, the a posteriori state estimate at
    time k given observations up to and including at time k;
  * P k | k {\displaystyle \mathbf {P} _{k\mid k}} \mathbf {P} _{k\
    mid k}, the a posteriori estimate covariance matrix (a measure of
    the estimated accuracy of the state estimate).

The Kalman filter can be written as a single equation, however, it is
most often conceptualized as two distinct phases: "Predict" and
"Update". The predict phase uses the state estimate from the previous
timestep to produce an estimate of the state at the current timestep.
This predicted state estimate is also known as the a priori state
estimate because, although it is an estimate of the state at the
current timestep, it does not include observation information from
the current timestep. In the update phase, the current a priori
prediction is combined with current observation information to refine
the state estimate. This improved estimate is termed the a posteriori
state estimate.

Typically, the two phases alternate, with the prediction advancing
the state until the next scheduled observation, and the update
incorporating the observation. However, this is not necessary; if an
observation is unavailable for some reason, the update may be skipped
and multiple prediction steps performed. Likewise, if multiple
independent observations are available at the same time, multiple
update steps may be performed (typically with different observation
matrices H[k]).^[23]^[24]

Predict[edit]

              x ^ k | k - 1 = F k x ^ k - 1 | k - 1 + B k u k {\
Predicted (a  displaystyle {\hat {\mathbf {x} }}_{k\mid k-1}=\mathbf
priori) state {F} _{k}{\hat {\mathbf {x} }}_{k-1\mid k-1}+\mathbf {B}
estimate      _{k}\mathbf {u} _{k}} {\displaystyle {\hat {\mathbf {x}
              }}_{k\mid k-1}=\mathbf {F} _{k}{\hat {\mathbf {x} }}_
              {k-1\mid k-1}+\mathbf {B} _{k}\mathbf {u} _{k}}
              P k | k - 1 = F k P k - 1 | k - 1 F k T + Q k {\
Predicted (a  displaystyle \mathbf {P} _{k\mid k-1}=\mathbf {F} _{k}\
priori)       mathbf {P} _{k-1\mid k-1}\mathbf {F} _{k}^{\textsf {T}}
estimate      +\mathbf {Q} _{k}} {\displaystyle \mathbf {P} _{k\mid
covariance    k-1}=\mathbf {F} _{k}\mathbf {P} _{k-1\mid k-1}\mathbf
              {F} _{k}^{\textsf {T}}+\mathbf {Q} _{k}}

Update[edit]

Innovation or  y ~ k = z k - H k x ^ k | k - 1 {\displaystyle {\tilde
measurement    {\mathbf {y} }}_{k}=\mathbf {z} _{k}-\mathbf {H} _{k}
pre-fit        {\hat {\mathbf {x} }}_{k\mid k-1}} {\displaystyle {\
residual       tilde {\mathbf {y} }}_{k}=\mathbf {z} _{k}-\mathbf {H}
               _{k}{\hat {\mathbf {x} }}_{k\mid k-1}}
               S k = H k P k | k - 1 H k T + R k {\displaystyle \
Innovation (or mathbf {S} _{k}=\mathbf {H} _{k}\mathbf {P} _{k\mid
pre-fit        k-1}\mathbf {H} _{k}^{\textsf {T}}+\mathbf {R} _{k}} 
residual)      {\displaystyle \mathbf {S} _{k}=\mathbf {H} _{k}\
covariance     mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}+
               \mathbf {R} _{k}}
               K k = P k | k - 1 H k T S k - 1 {\displaystyle \mathbf
Optimal Kalman {K} _{k}=\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\
gain           textsf {T}}\mathbf {S} _{k}^{-1}} {\displaystyle \
               mathbf {K} _{k}=\mathbf {P} _{k\mid k-1}\mathbf {H} _
               {k}^{\textsf {T}}\mathbf {S} _{k}^{-1}}
               x ^ k | k = x ^ k | k - 1 + K k y ~ k {\displaystyle
Updated (a     {\hat {\mathbf {x} }}_{k\mid k}={\hat {\mathbf {x} }}_
posteriori)    {k\mid k-1}+\mathbf {K} _{k}{\tilde {\mathbf {y} }}_
state estimate {k}} {\hat {\mathbf {x} }}_{k\mid k}={\hat {\mathbf
               {x} }}_{k\mid k-1}+\mathbf {K} _{k}{\tilde {\mathbf
               {y} }}_{k}
               P k | k = ( I - K k H k ) P k | k - 1 {\displaystyle \
Updated (a     mathbf {P} _{k|k}=\left(\mathbf {I} -\mathbf {K} _{k}\
posteriori)    mathbf {H} _{k}\right)\mathbf {P} _{k|k-1}} {\
estimate       displaystyle \mathbf {P} _{k|k}=\left(\mathbf {I} -\
covariance     mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k|
               k-1}}
               y ~ k | k = z k - H k x ^ k | k {\displaystyle {\tilde
Measurement    {\mathbf {y} }}_{k\mid k}=\mathbf {z} _{k}-\mathbf {H}
post-fit       _{k}{\hat {\mathbf {x} }}_{k\mid k}} {\displaystyle {\
residual       tilde {\mathbf {y} }}_{k\mid k}=\mathbf {z} _{k}-\
               mathbf {H} _{k}{\hat {\mathbf {x} }}_{k\mid k}}

The formula for the updated (a posteriori) estimate covariance above
is valid for the optimal K[k] gain that minimizes the residual error,
in which form it is most widely used in applications. Proof of the
formulae is found in the derivations section, where the formula valid
for any K[k] is also shown.

A more intuitive way to express the Updated state estimate ( x ^ k |
k {\displaystyle {\hat {\mathbf {x} }}_{k\mid k}} {\hat {\mathbf {x}
}}_{k\mid k}) is:

    x ^ k | k = ( I - K k H k ) ( x ^ k | k - 1 ) + ( K k ) ( H k x k
    + v k ) {\displaystyle {\hat {\mathbf {x} }}_{k\mid k}=(\mathbf
    {I} -\mathbf {K} _{k}\mathbf {H} _{k})({\hat {\mathbf {x} }}_{k\
    mid k-1})+(\mathbf {K} _{k})(\mathbf {H} _{k}\mathbf {x} _{k}+\
    mathbf {v} _{k})} {\displaystyle {\hat {\mathbf {x} }}_{k\mid k}=
    (\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k})({\hat {\mathbf
    {x} }}_{k\mid k-1})+(\mathbf {K} _{k})(\mathbf {H} _{k}\mathbf
    {x} _{k}+\mathbf {v} _{k})}

This expression reminds us of a linear interpolation, x = ( 1 - t ) (
a ) + t ( b ) {\displaystyle x=(1-t)(a)+t(b)} {\displaystyle x=(1-t)
(a)+t(b)} for t {\displaystyle t} t between [0,1]. In our case:

  * t {\displaystyle t} t is the Kalman gain( K k {\displaystyle \
    mathbf {K} _{k}} {\displaystyle \mathbf {K} _{k}}), a matrix that
    takes values from 0 {\displaystyle 0} {\displaystyle 0}(high
    error in the sensor) to I {\displaystyle I} I(low error).
  * a {\displaystyle a} a is the value estimated from the model.
  * b {\displaystyle b} b is the value from the measurement.

This expression also resembles the Alpha beta filter update step.

Invariants[edit]

If the model is accurate, and the values for x ^ 0 | 0 {\displaystyle
{\hat {\mathbf {x} }}_{0\mid 0}} {\hat {\mathbf {x} }}_{0\mid 0} and 
P 0 | 0 {\displaystyle \mathbf {P} _{0\mid 0}} \mathbf {P} _{0\mid 0}
accurately reflect the distribution of the initial state values, then
the following invariants are preserved:

    E [?] [ x k - x ^ k | k ] = E [?] [ x k - x ^ k | k - 1 ] = 0 E [?] [ y
    ~ k ] = 0 {\displaystyle {\begin{aligned}\operatorname {E} [\
    mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k}]&=\operatorname
    {E} [\mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k-1}]=0\\\
    operatorname {E} [{\tilde {\mathbf {y} }}_{k}]&=0\end{aligned}}} 
    {\displaystyle {\begin{aligned}\operatorname {E} [\mathbf {x} _
    {k}-{\hat {\mathbf {x} }}_{k\mid k}]&=\operatorname {E} [\mathbf
    {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k-1}]=0\\\operatorname {E}
    [{\tilde {\mathbf {y} }}_{k}]&=0\end{aligned}}}

where E [?] [ x ] {\displaystyle \operatorname {E} [\xi ]} {\
displaystyle \operatorname {E} [\xi ]} is the expected value of x {\
displaystyle \xi } \xi . That is, all estimates have a mean error of
zero.

Also:

    P k | k = cov [?] ( x k - x ^ k | k ) P k | k - 1 = cov [?] ( x k - x
    ^ k | k - 1 ) S k = cov [?] ( y ~ k ) {\displaystyle {\begin
    {aligned}\mathbf {P} _{k\mid k}&=\operatorname {cov} \left(\
    mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k}\right)\\\mathbf
    {P} _{k\mid k-1}&=\operatorname {cov} \left(\mathbf {x} _{k}-{\
    hat {\mathbf {x} }}_{k\mid k-1}\right)\\\mathbf {S} _{k}&=\
    operatorname {cov} \left({\tilde {\mathbf {y} }}_{k}\right)\end
    {aligned}}} {\displaystyle {\begin{aligned}\mathbf {P} _{k\mid k}
    &=\operatorname {cov} \left(\mathbf {x} _{k}-{\hat {\mathbf {x}
    }}_{k\mid k}\right)\\\mathbf {P} _{k\mid k-1}&=\operatorname
    {cov} \left(\mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k-1}\
    right)\\\mathbf {S} _{k}&=\operatorname {cov} \left({\tilde {\
    mathbf {y} }}_{k}\right)\end{aligned}}}

so covariance matrices accurately reflect the covariance of
estimates.

Estimation of the noise covariances Q[k] and R[k][edit]

Practical implementation of the Kalman Filter is often difficult due
to the difficulty of getting a good estimate of the noise covariance
matrices Q[k] and R[k]. Extensive research has been done in this
field to estimate these covariances from data. One practical approach
to do this is the autocovariance least-squares (ALS) technique that
uses the time-lagged autocovariances of routine operating data to
estimate the covariances.^[25]^[26] The GNU Octave and Matlab code
used to calculate the noise covariance matrices using the ALS
technique is available online under the GNU General Public License.^
[27] Field Kalman Filter (FKF), a Bayesian algorithm, which allows
simultaneous estimation of the state, parameters and noise covariance
has been proposed in.^[28] FKF algorithm has a recursive formulation,
good observed convergence and relatively low complexity. This gives a
possibility that the FKF algorithm may be an alternative to the
Autocovariance Least-Squares methods.

Optimality and performance[edit]

It follows from theory that the Kalman filter is the optimal linear
filter in cases where a) the model perfectly matches the real system,
b) the entering noise is white (uncorrelated) and c) the covariances
of the noise are exactly known. Several methods for the noise
covariance estimation have been proposed during past decades,
including ALS, mentioned in the section above. After the covariances
are estimated, it is useful to evaluate the performance of the
filter; i.e., whether it is possible to improve the state estimation
quality. If the Kalman filter works optimally, the innovation
sequence (the output prediction error) is a white noise, therefore
the whiteness property of the innovations measures filter
performance. Several different methods can be used for this purpose.^
[29] If the noise terms are non-Gaussian distributed, methods for
assessing performance of the filter estimate, which use probability
inequalities or large-sample theory, are known in the literature.^
[30]^[31]

Example application, technical[edit]

[220px-Kalman]
 
  Truth;   filtered process;   observations.

Consider a truck on frictionless, straight rails. Initially, the
truck is stationary at position 0, but it is buffeted this way and
that by random uncontrolled forces. We measure the position of the
truck every Dt seconds, but these measurements are imprecise; we want
to maintain a model of the truck's position and velocity. We show
here how we derive the model from which we create our Kalman filter.

Since F , H , R , Q {\displaystyle \mathbf {F} ,\mathbf {H} ,\mathbf
{R} ,\mathbf {Q} } {\displaystyle \mathbf {F} ,\mathbf {H} ,\mathbf
{R} ,\mathbf {Q} } are constant, their time indices are dropped.

The position and velocity of the truck are described by the linear
state space

    x k = [ x x . ] {\displaystyle \mathbf {x} _{k}={\begin{bmatrix}x
    \\{\dot {x}}\end{bmatrix}}} {\displaystyle \mathbf {x} _{k}={\
    begin{bmatrix}x\\{\dot {x}}\end{bmatrix}}}

where x . {\displaystyle {\dot {x}}} {\dot {x}} is the velocity, that
is, the derivative of position with respect to time.

We assume that between the (k - 1) and k timestep uncontrolled forces
cause a constant acceleration of a[k] that is normally distributed,
with mean 0 and standard deviation s[a]. From Newton's laws of motion
we conclude that

    x k = F x k - 1 + G a k {\displaystyle \mathbf {x} _{k}=\mathbf
    {F} \mathbf {x} _{k-1}+\mathbf {G} a_{k}} {\displaystyle \mathbf
    {x} _{k}=\mathbf {F} \mathbf {x} _{k-1}+\mathbf {G} a_{k}}

(there is no B u {\displaystyle \mathbf {B} u} \mathbf {B} u term
since there are no known control inputs. Instead, a[k] is the effect
of an unknown input and G {\displaystyle \mathbf {G} } \mathbf {G} 
applies that effect to the state vector) where

    F = [ 1 D t 0 1 ] G = [ 1 2 D t 2 D t ] {\displaystyle {\begin
    {aligned}\mathbf {F} &={\begin{bmatrix}1&\Delta t\\0&1\end
    {bmatrix}}\\[4pt]\mathbf {G} &={\begin{bmatrix}{\frac {1}{2}}{\
    Delta t}^{2}\\[6pt]\Delta t\end{bmatrix}}\end{aligned}}} {\
    displaystyle {\begin{aligned}\mathbf {F} &={\begin{bmatrix}1&\
    Delta t\\0&1\end{bmatrix}}\\[4pt]\mathbf {G} &={\begin{bmatrix}{\
    frac {1}{2}}{\Delta t}^{2}\\[6pt]\Delta t\end{bmatrix}}\end
    {aligned}}}

so that

    x k = F x k - 1 + w k {\displaystyle \mathbf {x} _{k}=\mathbf {F}
    \mathbf {x} _{k-1}+\mathbf {w} _{k}} {\displaystyle \mathbf {x} _
    {k}=\mathbf {F} \mathbf {x} _{k-1}+\mathbf {w} _{k}}

where

    w k ~ N ( 0 , Q ) Q = G G T s a 2 = [ 1 4 D t 4 1 2 D t 3 1 2 D t
    3 D t 2 ] s a 2 . {\displaystyle {\begin{aligned}\mathbf {w} _{k}
    &\sim N(0,\mathbf {Q} )\\\mathbf {Q} &=\mathbf {G} \mathbf {G} ^
    {\textsf {T}}\sigma _{a}^{2}={\begin{bmatrix}{\frac {1}{4}}{\
    Delta t}^{4}&{\frac {1}{2}}{\Delta t}^{3}\\[6pt]{\frac {1}{2}}{\
    Delta t}^{3}&{\Delta t}^{2}\end{bmatrix}}\sigma _{a}^{2}.\end
    {aligned}}} {\displaystyle {\begin{aligned}\mathbf {w} _{k}&\sim
    N(0,\mathbf {Q} )\\\mathbf {Q} &=\mathbf {G} \mathbf {G} ^{\
    textsf {T}}\sigma _{a}^{2}={\begin{bmatrix}{\frac {1}{4}}{\Delta
    t}^{4}&{\frac {1}{2}}{\Delta t}^{3}\\[6pt]{\frac {1}{2}}{\Delta
    t}^{3}&{\Delta t}^{2}\end{bmatrix}}\sigma _{a}^{2}.\end
    {aligned}}}

The matrix Q {\displaystyle \mathbf {Q} } \mathbf {Q} is not full
rank (it is of rank one if D t [?] 0 {\displaystyle \Delta t\neq 0} {\
displaystyle \Delta t\neq 0}). Hence, the distribution N ( 0 , Q ) {\
displaystyle N(0,\mathbf {Q} )} {\displaystyle N(0,\mathbf {Q} )} is
not absolutely continuous and has no probability density function.
Another way to express this, avoiding explicit degenerate
distributions is given by

    w k ~ G [?] N ( 0 , s a 2 ) . {\displaystyle \mathbf {w} _{k}\sim \
    mathbf {G} \cdot N\left(0,\sigma _{a}^{2}\right).} {\displaystyle
    \mathbf {w} _{k}\sim \mathbf {G} \cdot N\left(0,\sigma _{a}^{2}\
    right).}

At each time step, a noisy measurement of the true position of the
truck is made. Let us suppose the measurement noise v[k] is also
normally distributed, with mean 0 and standard deviation s[z].

    z k = H x k + v k {\displaystyle \mathbf {z} _{k}=\mathbf {Hx} _
    {k}+\mathbf {v} _{k}} {\displaystyle \mathbf {z} _{k}=\mathbf
    {Hx} _{k}+\mathbf {v} _{k}}

where

    H = [ 1 0 ] {\displaystyle \mathbf {H} ={\begin{bmatrix}1&0\end
    {bmatrix}}} \mathbf {H} ={\begin{bmatrix}1&0\end{bmatrix}}

and

    R = E [ v k v k T ] = [ s z 2 ] {\displaystyle \mathbf {R} =\
    mathrm {E} \left[\mathbf {v} _{k}\mathbf {v} _{k}^{\textsf {T}}\
    right]={\begin{bmatrix}\sigma _{z}^{2}\end{bmatrix}}} {\
    displaystyle \mathbf {R} =\mathrm {E} \left[\mathbf {v} _{k}\
    mathbf {v} _{k}^{\textsf {T}}\right]={\begin{bmatrix}\sigma _{z}^
    {2}\end{bmatrix}}}

We know the initial starting state of the truck with perfect
precision, so we initialize

    x ^ 0 | 0 = [ 0 0 ] {\displaystyle {\hat {\mathbf {x} }}_{0\mid
    0}={\begin{bmatrix}0\\0\end{bmatrix}}} {\displaystyle {\hat {\
    mathbf {x} }}_{0\mid 0}={\begin{bmatrix}0\\0\end{bmatrix}}}

and to tell the filter that we know the exact position and velocity,
we give it a zero covariance matrix:

    P 0 | 0 = [ 0 0 0 0 ] {\displaystyle \mathbf {P} _{0\mid 0}={\
    begin{bmatrix}0&0\\0&0\end{bmatrix}}} {\displaystyle \mathbf {P}
    _{0\mid 0}={\begin{bmatrix}0&0\\0&0\end{bmatrix}}}

If the initial position and velocity are not known perfectly, the
covariance matrix should be initialized with suitable variances on
its diagonal:

    P 0 | 0 = [ s x 2 0 0 s x . 2 ] {\displaystyle \mathbf {P} _{0\
    mid 0}={\begin{bmatrix}\sigma _{x}^{2}&0\\0&\sigma _{\dot {x}}^
    {2}\end{bmatrix}}} {\displaystyle \mathbf {P} _{0\mid 0}={\begin
    {bmatrix}\sigma _{x}^{2}&0\\0&\sigma _{\dot {x}}^{2}\end
    {bmatrix}}}

The filter will then prefer the information from the first
measurements over the information already in the model.

Asymptotic form[edit]

For simplicity, assume that the control input u k = 0 {\displaystyle
\mathbf {u} _{k}=\mathbf {0} } {\displaystyle \mathbf {u} _{k}=\
mathbf {0} }. Then the Kalman filter may be written:

    x ^ k | k = F k x ^ k - 1 | k - 1 + K k [ z k - H k F k x ^ k - 1
    | k - 1 ] . {\displaystyle {\hat {\mathbf {x} }}_{k\mid k}=\
    mathbf {F} _{k}{\hat {\mathbf {x} }}_{k-1\mid k-1}+\mathbf {K} _
    {k}[\mathbf {z} _{k}-\mathbf {H} _{k}\mathbf {F} _{k}{\hat {\
    mathbf {x} }}_{k-1\mid k-1}].} {\displaystyle {\hat {\mathbf {x}
    }}_{k\mid k}=\mathbf {F} _{k}{\hat {\mathbf {x} }}_{k-1\mid k-1}+
    \mathbf {K} _{k}[\mathbf {z} _{k}-\mathbf {H} _{k}\mathbf {F} _
    {k}{\hat {\mathbf {x} }}_{k-1\mid k-1}].}

A similar equation holds if we include a non-zero control input. Gain
matrices K k {\displaystyle \mathbf {K} _{k}} {\displaystyle \mathbf
{K} _{k}} evolve independently of the measurements z k {\displaystyle
\mathbf {z} _{k}} \mathbf {z} _{k}. From above, the four equations
needed for updating the Kalman gain are as follows:

    P k | k - 1 = F k P k - 1 | k - 1 F k T + Q k , {\displaystyle \
    mathbf {P} _{k\mid k-1}=\mathbf {F} _{k}\mathbf {P} _{k-1\mid
    k-1}\mathbf {F} _{k}^{\textsf {T}}+\mathbf {Q} _{k},} {\
    displaystyle \mathbf {P} _{k\mid k-1}=\mathbf {F} _{k}\mathbf {P}
    _{k-1\mid k-1}\mathbf {F} _{k}^{\textsf {T}}+\mathbf {Q} _{k},}
    S k = R k + H k P k | k - 1 H k T , {\displaystyle \mathbf {S} _
    {k}=\mathbf {R} _{k}+\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\
    mathbf {H} _{k}^{\textsf {T}},} {\displaystyle \mathbf {S} _{k}=\
    mathbf {R} _{k}+\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\mathbf
    {H} _{k}^{\textsf {T}},}
    K k = P k | k - 1 H k T S k - 1 , {\displaystyle \mathbf {K} _{k}
    =\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf
    {S} _{k}^{-1},} {\displaystyle \mathbf {K} _{k}=\mathbf {P} _{k\
    mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {S} _{k}^{-1},}
    P k | k = ( I - K k H k ) P k | k - 1 . {\displaystyle \mathbf
    {P} _{k|k}=\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\
    right)\mathbf {P} _{k|k-1}.} {\displaystyle \mathbf {P} _{k|k}=\
    left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf
    {P} _{k|k-1}.}

Since the gain matrices depend only on the model, and not the
measurements, they may be computed offline. Convergence of the gain
matrices K k {\displaystyle \mathbf {K} _{k}} {\displaystyle \mathbf
{K} _{k}} to an asymptotic matrix K [?] {\displaystyle \mathbf {K} _{\
infty }} {\displaystyle \mathbf {K} _{\infty }} holds under
conditions established in Walrand and Dimakis .^[32] Simulations
establish the number of steps to convergence. For the moving truck
example described above, with D t = 1 {\displaystyle \Delta t=1} {\
displaystyle \Delta t=1}. and s a 2 = s z 2 = s x 2 = s x . 2 = 1 {\
displaystyle \sigma _{a}^{2}=\sigma _{z}^{2}=\sigma _{x}^{2}=\sigma _
{\dot {x}}^{2}=1} {\displaystyle \sigma _{a}^{2}=\sigma _{z}^{2}=\
sigma _{x}^{2}=\sigma _{\dot {x}}^{2}=1}, simulation shows
convergence in 10 {\displaystyle 10} 10 iterations.

Using the asymptotic gain, and assuming H k {\displaystyle \mathbf
{H} _{k}} {\displaystyle \mathbf {H} _{k}} and F k {\displaystyle \
mathbf {F} _{k}} \mathbf{F}_k are independent of k {\displaystyle k} 
k, the Kalman filter becomes a linear time-invariant filter:

    x ^ k = F x ^ k - 1 + K [?] [ z k - H F x ^ k - 1 ] . {\
    displaystyle {\hat {\mathbf {x} }}_{k}=\mathbf {F} {\hat {\mathbf
    {x} }}_{k-1}+\mathbf {K} _{\infty }[\mathbf {z} _{k}-\mathbf {H}
    \mathbf {F} {\hat {\mathbf {x} }}_{k-1}].} {\displaystyle {\hat
    {\mathbf {x} }}_{k}=\mathbf {F} {\hat {\mathbf {x} }}_{k-1}+\
    mathbf {K} _{\infty }[\mathbf {z} _{k}-\mathbf {H} \mathbf {F} {\
    hat {\mathbf {x} }}_{k-1}].}

The asymptotic gain K [?] {\displaystyle \mathbf {K} _{\infty }} {\
displaystyle \mathbf {K} _{\infty }}, if it exists, can be computed
by first solving the following discrete Riccati equation for the
asymptotic state covariance P [?] {\displaystyle \mathbf {P} _{\infty
}} {\displaystyle \mathbf {P} _{\infty }}:^[32]

    P [?] = F ( P [?] - P [?] H T ( H P [?] H T + R ) - 1 H P [?] ) F T + Q .
    {\displaystyle \mathbf {P} _{\infty }=\mathbf {F} \left(\mathbf
    {P} _{\infty }-\mathbf {P} _{\infty }\mathbf {H} ^{\textsf {T}}\
    left(\mathbf {H} \mathbf {P} _{\infty }\mathbf {H} ^{\textsf {T}}
    +\mathbf {R} \right)^{-1}\mathbf {H} \mathbf {P} _{\infty }\
    right)\mathbf {F} ^{\textsf {T}}+\mathbf {Q} .} {\displaystyle \
    mathbf {P} _{\infty }=\mathbf {F} \left(\mathbf {P} _{\infty }-\
    mathbf {P} _{\infty }\mathbf {H} ^{\textsf {T}}\left(\mathbf {H}
    \mathbf {P} _{\infty }\mathbf {H} ^{\textsf {T}}+\mathbf {R} \
    right)^{-1}\mathbf {H} \mathbf {P} _{\infty }\right)\mathbf {F} ^
    {\textsf {T}}+\mathbf {Q} .}

The asymptotic gain is then computed as before.

    K [?] = P [?] H T ( H P [?] H T + R ) - 1 . {\displaystyle \mathbf {K}
    _{\infty }=\mathbf {P} _{\infty }\mathbf {H} ^{\textsf {T}}\left
    (\mathbf {H} \mathbf {P} _{\infty }\mathbf {H} ^{\textsf {T}}+\
    mathbf {R} \right)^{-1}.} {\displaystyle \mathbf {K} _{\infty }=\
    mathbf {P} _{\infty }\mathbf {H} ^{\textsf {T}}\left(\mathbf {H}
    \mathbf {P} _{\infty }\mathbf {H} ^{\textsf {T}}+\mathbf {R} \
    right)^{-1}.}

Derivations[edit]

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Deriving the posteriori estimate covariance matrix[edit]

Starting with our invariant on the error covariance P[k | k] as above

    P k | k = cov [?] ( x k - x ^ k | k ) {\displaystyle \mathbf {P} _
    {k\mid k}=\operatorname {cov} \left(\mathbf {x} _{k}-{\hat {\
    mathbf {x} }}_{k\mid k}\right)} {\displaystyle \mathbf {P} _{k\
    mid k}=\operatorname {cov} \left(\mathbf {x} _{k}-{\hat {\mathbf
    {x} }}_{k\mid k}\right)}

substitute in the definition of x ^ k | k {\displaystyle {\hat {\
mathbf {x} }}_{k\mid k}} {\displaystyle {\hat {\mathbf {x} }}_{k\mid
k}}

    P k | k = cov [?] [ x k - ( x ^ k | k - 1 + K k y ~ k ) ] {\
    displaystyle \mathbf {P} _{k\mid k}=\operatorname {cov} \left[\
    mathbf {x} _{k}-\left({\hat {\mathbf {x} }}_{k\mid k-1}+\mathbf
    {K} _{k}{\tilde {\mathbf {y} }}_{k}\right)\right]} {\displaystyle
    \mathbf {P} _{k\mid k}=\operatorname {cov} \left[\mathbf {x} _{k}
    -\left({\hat {\mathbf {x} }}_{k\mid k-1}+\mathbf {K} _{k}{\tilde
    {\mathbf {y} }}_{k}\right)\right]}

and substitute y ~ k {\displaystyle {\tilde {\mathbf {y} }}_{k}} {\
tilde {\mathbf {y} }}_{k}

    P k | k = cov [?] ( x k - [ x ^ k | k - 1 + K k ( z k - H k x ^ k |
    k - 1 ) ] ) {\displaystyle \mathbf {P} _{k\mid k}=\operatorname
    {cov} \left(\mathbf {x} _{k}-\left[{\hat {\mathbf {x} }}_{k\mid
    k-1}+\mathbf {K} _{k}\left(\mathbf {z} _{k}-\mathbf {H} _{k}{\hat
    {\mathbf {x} }}_{k\mid k-1}\right)\right]\right)} {\displaystyle
    \mathbf {P} _{k\mid k}=\operatorname {cov} \left(\mathbf {x} _{k}
    -\left[{\hat {\mathbf {x} }}_{k\mid k-1}+\mathbf {K} _{k}\left(\
    mathbf {z} _{k}-\mathbf {H} _{k}{\hat {\mathbf {x} }}_{k\mid k-1}
    \right)\right]\right)}

and z k {\displaystyle \mathbf {z} _{k}} \mathbf {z} _{k}

    P k | k = cov [?] ( x k - [ x ^ k | k - 1 + K k ( H k x k + v k - H
    k x ^ k | k - 1 ) ] ) {\displaystyle \mathbf {P} _{k\mid k}=\
    operatorname {cov} \left(\mathbf {x} _{k}-\left[{\hat {\mathbf
    {x} }}_{k\mid k-1}+\mathbf {K} _{k}\left(\mathbf {H} _{k}\mathbf
    {x} _{k}+\mathbf {v} _{k}-\mathbf {H} _{k}{\hat {\mathbf {x} }}_
    {k\mid k-1}\right)\right]\right)} {\displaystyle \mathbf {P} _{k\
    mid k}=\operatorname {cov} \left(\mathbf {x} _{k}-\left[{\hat {\
    mathbf {x} }}_{k\mid k-1}+\mathbf {K} _{k}\left(\mathbf {H} _{k}\
    mathbf {x} _{k}+\mathbf {v} _{k}-\mathbf {H} _{k}{\hat {\mathbf
    {x} }}_{k\mid k-1}\right)\right]\right)}

and by collecting the error vectors we get

    P k | k = cov [?] [ ( I - K k H k ) ( x k - x ^ k | k - 1 ) - K k v
    k ] {\displaystyle \mathbf {P} _{k\mid k}=\operatorname {cov} \
    left[\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)\
    left(\mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k-1}\right)-\
    mathbf {K} _{k}\mathbf {v} _{k}\right]} {\displaystyle \mathbf
    {P} _{k\mid k}=\operatorname {cov} \left[\left(\mathbf {I} -\
    mathbf {K} _{k}\mathbf {H} _{k}\right)\left(\mathbf {x} _{k}-{\
    hat {\mathbf {x} }}_{k\mid k-1}\right)-\mathbf {K} _{k}\mathbf
    {v} _{k}\right]}

Since the measurement error v[k] is uncorrelated with the other
terms, this becomes

    P k | k = cov [?] [ ( I - K k H k ) ( x k - x ^ k | k - 1 ) ] + cov
    [?] [ K k v k ] {\displaystyle \mathbf {P} _{k\mid k}=\operatorname
    {cov} \left[\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\
    right)\left(\mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k-1}\
    right)\right]+\operatorname {cov} \left[\mathbf {K} _{k}\mathbf
    {v} _{k}\right]} {\displaystyle \mathbf {P} _{k\mid k}=\
    operatorname {cov} \left[\left(\mathbf {I} -\mathbf {K} _{k}\
    mathbf {H} _{k}\right)\left(\mathbf {x} _{k}-{\hat {\mathbf {x}
    }}_{k\mid k-1}\right)\right]+\operatorname {cov} \left[\mathbf
    {K} _{k}\mathbf {v} _{k}\right]}

by the properties of vector covariance this becomes

    P k | k = ( I - K k H k ) cov [?] ( x k - x ^ k | k - 1 ) ( I - K k
    H k ) T + K k cov [?] ( v k ) K k T {\displaystyle \mathbf {P} _{k\
    mid k}=\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)
    \operatorname {cov} \left(\mathbf {x} _{k}-{\hat {\mathbf {x} }}_
    {k\mid k-1}\right)\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H}
    _{k}\right)^{\textsf {T}}+\mathbf {K} _{k}\operatorname {cov} \
    left(\mathbf {v} _{k}\right)\mathbf {K} _{k}^{\textsf {T}}} {\
    displaystyle \mathbf {P} _{k\mid k}=\left(\mathbf {I} -\mathbf
    {K} _{k}\mathbf {H} _{k}\right)\operatorname {cov} \left(\mathbf
    {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k-1}\right)\left(\mathbf
    {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)^{\textsf {T}}+\
    mathbf {K} _{k}\operatorname {cov} \left(\mathbf {v} _{k}\right)\
    mathbf {K} _{k}^{\textsf {T}}}

which, using our invariant on P[k | k-1] and the definition of R[k]
becomes

    P k | k = ( I - K k H k ) P k | k - 1 ( I - K k H k ) T + K k R k
    K k T {\displaystyle \mathbf {P} _{k\mid k}=\left(\mathbf {I} -\
    mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k\mid k-1}\
    left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)^{\
    textsf {T}}+\mathbf {K} _{k}\mathbf {R} _{k}\mathbf {K} _{k}^{\
    textsf {T}}} {\displaystyle \mathbf {P} _{k\mid k}=\left(\mathbf
    {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k\mid
    k-1}\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)^{\
    textsf {T}}+\mathbf {K} _{k}\mathbf {R} _{k}\mathbf {K} _{k}^{\
    textsf {T}}}

This formula (sometimes known as the Joseph form of the covariance
update equation) is valid for any value of K[k]. It turns out that if
K[k] is the optimal Kalman gain, this can be simplified further as
shown below.

Kalman gain derivation[edit]

The Kalman filter is a minimum mean-square error estimator. The error
in the a posteriori state estimation is

    x k - x ^ k | k {\displaystyle \mathbf {x} _{k}-{\hat {\mathbf
    {x} }}_{k\mid k}} {\displaystyle \mathbf {x} _{k}-{\hat {\mathbf
    {x} }}_{k\mid k}}

We seek to minimize the expected value of the square of the magnitude
of this vector, E [?] [ || x k - x ^ k | k || 2 ] {\displaystyle \
operatorname {E} \left[\left\|\mathbf {x} _{k}-{\hat {\mathbf {x} }}_
{k|k}\right\|^{2}\right]} {\displaystyle \operatorname {E} \left[\
left\|\mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k|k}\right\|^{2}\
right]}. This is equivalent to minimizing the trace of the a
posteriori estimate covariance matrix P k | k {\displaystyle \mathbf
{P} _{k|k}} \mathbf {P} _{k|k}. By expanding out the terms in the
equation above and collecting, we get:

    P k | k = P k | k - 1 - K k H k P k | k - 1 - P k | k - 1 H k T K
    k T + K k ( H k P k | k - 1 H k T + R k ) K k T = P k | k - 1 - K
    k H k P k | k - 1 - P k | k - 1 H k T K k T + K k S k K k T {\
    displaystyle {\begin{aligned}\mathbf {P} _{k\mid k}&=\mathbf {P}
    _{k\mid k-1}-\mathbf {K} _{k}\mathbf {H} _{k}\mathbf {P} _{k\mid
    k-1}-\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\
    mathbf {K} _{k}^{\textsf {T}}+\mathbf {K} _{k}\left(\mathbf {H} _
    {k}\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}+\mathbf
    {R} _{k}\right)\mathbf {K} _{k}^{\textsf {T}}\\[6pt]&=\mathbf {P}
    _{k\mid k-1}-\mathbf {K} _{k}\mathbf {H} _{k}\mathbf {P} _{k\mid
    k-1}-\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\
    mathbf {K} _{k}^{\textsf {T}}+\mathbf {K} _{k}\mathbf {S} _{k}\
    mathbf {K} _{k}^{\textsf {T}}\end{aligned}}} {\displaystyle {\
    begin{aligned}\mathbf {P} _{k\mid k}&=\mathbf {P} _{k\mid k-1}-\
    mathbf {K} _{k}\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}-\mathbf
    {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {K} _{k}^{\
    textsf {T}}+\mathbf {K} _{k}\left(\mathbf {H} _{k}\mathbf {P} _{k
    \mid k-1}\mathbf {H} _{k}^{\textsf {T}}+\mathbf {R} _{k}\right)\
    mathbf {K} _{k}^{\textsf {T}}\\[6pt]&=\mathbf {P} _{k\mid k-1}-\
    mathbf {K} _{k}\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}-\mathbf
    {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {K} _{k}^{\
    textsf {T}}+\mathbf {K} _{k}\mathbf {S} _{k}\mathbf {K} _{k}^{\
    textsf {T}}\end{aligned}}}

The trace is minimized when its matrix derivative with respect to the
gain matrix is zero. Using the gradient matrix rules and the symmetry
of the matrices involved we find that

    [?] tr [?] ( P k | k ) [?] K k = - 2 ( H k P k | k - 1 ) T + 2 K k S k
    = 0. {\displaystyle {\frac {\partial \;\operatorname {tr} (\
    mathbf {P} _{k\mid k})}{\partial \;\mathbf {K} _{k}}}=-2\left(\
    mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\right)^{\textsf {T}}+2\
    mathbf {K} _{k}\mathbf {S} _{k}=0.} {\displaystyle {\frac {\
    partial \;\operatorname {tr} (\mathbf {P} _{k\mid k})}{\partial
    \;\mathbf {K} _{k}}}=-2\left(\mathbf {H} _{k}\mathbf {P} _{k\mid
    k-1}\right)^{\textsf {T}}+2\mathbf {K} _{k}\mathbf {S} _{k}=0.}

Solving this for K[k] yields the Kalman gain:

    K k S k = ( H k P k | k - 1 ) T = P k | k - 1 H k T = K k = P k |
    k - 1 H k T S k - 1 {\displaystyle {\begin{aligned}\mathbf {K} _
    {k}\mathbf {S} _{k}&=\left(\mathbf {H} _{k}\mathbf {P} _{k\mid
    k-1}\right)^{\textsf {T}}=\mathbf {P} _{k\mid k-1}\mathbf {H} _
    {k}^{\textsf {T}}\\\Rightarrow \mathbf {K} _{k}&=\mathbf {P} _{k\
    mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {S} _{k}^{-1}\end
    {aligned}}} {\displaystyle {\begin{aligned}\mathbf {K} _{k}\
    mathbf {S} _{k}&=\left(\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\
    right)^{\textsf {T}}=\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\
    textsf {T}}\\\Rightarrow \mathbf {K} _{k}&=\mathbf {P} _{k\mid
    k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {S} _{k}^{-1}\end
    {aligned}}}

This gain, which is known as the optimal Kalman gain, is the one that
yields MMSE estimates when used.

Simplification of the posteriori error covariance formula[edit]

The formula used to calculate the a posteriori error covariance can
be simplified when the Kalman gain equals the optimal value derived
above. Multiplying both sides of our Kalman gain formula on the right
by S[k]K[k]^T, it follows that

    K k S k K k T = P k | k - 1 H k T K k T {\displaystyle \mathbf
    {K} _{k}\mathbf {S} _{k}\mathbf {K} _{k}^{\textsf {T}}=\mathbf
    {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {K} _{k}^{\
    textsf {T}}} {\displaystyle \mathbf {K} _{k}\mathbf {S} _{k}\
    mathbf {K} _{k}^{\textsf {T}}=\mathbf {P} _{k\mid k-1}\mathbf {H}
    _{k}^{\textsf {T}}\mathbf {K} _{k}^{\textsf {T}}}

Referring back to our expanded formula for the a posteriori error
covariance,

    P k | k = P k | k - 1 - K k H k P k | k - 1 - P k | k - 1 H k T K
    k T + K k S k K k T {\displaystyle \mathbf {P} _{k\mid k}=\mathbf
    {P} _{k\mid k-1}-\mathbf {K} _{k}\mathbf {H} _{k}\mathbf {P} _{k\
    mid k-1}-\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\
    mathbf {K} _{k}^{\textsf {T}}+\mathbf {K} _{k}\mathbf {S} _{k}\
    mathbf {K} _{k}^{\textsf {T}}} {\displaystyle \mathbf {P} _{k\mid
    k}=\mathbf {P} _{k\mid k-1}-\mathbf {K} _{k}\mathbf {H} _{k}\
    mathbf {P} _{k\mid k-1}-\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^
    {\textsf {T}}\mathbf {K} _{k}^{\textsf {T}}+\mathbf {K} _{k}\
    mathbf {S} _{k}\mathbf {K} _{k}^{\textsf {T}}}

we find the last two terms cancel out, giving

    P k | k = P k | k - 1 - K k H k P k | k - 1 = ( I - K k H k ) P k
    | k - 1 . {\displaystyle \mathbf {P} _{k\mid k}=\mathbf {P} _{k\
    mid k-1}-\mathbf {K} _{k}\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}
    =(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k})\mathbf {P} _{k\
    mid k-1}.} {\displaystyle \mathbf {P} _{k\mid k}=\mathbf {P} _{k\
    mid k-1}-\mathbf {K} _{k}\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}
    =(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k})\mathbf {P} _{k\
    mid k-1}.}

This formula is computationally cheaper and thus nearly always used
in practice, but is only correct for the optimal gain. If arithmetic
precision is unusually low causing problems with numerical stability,
or if a non-optimal Kalman gain is deliberately used, this
simplification cannot be applied; the a posteriori error covariance
formula as derived above (Joseph form) must be used.

Sensitivity analysis[edit]

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The Kalman filtering equations provide an estimate of the state x ^ k
| k {\displaystyle {\hat {\mathbf {x} }}_{k\mid k}} {\hat {\mathbf
{x} }}_{k\mid k} and its error covariance P k | k {\displaystyle \
mathbf {P} _{k\mid k}} \mathbf {P} _{k\mid k} recursively. The
estimate and its quality depend on the system parameters and the
noise statistics fed as inputs to the estimator. This section
analyzes the effect of uncertainties in the statistical inputs to the
filter.^[33] In the absence of reliable statistics or the true values
of noise covariance matrices Q k {\displaystyle \mathbf {Q} _{k}} \
mathbf {Q} _{k} and R k {\displaystyle \mathbf {R} _{k}} \mathbf {R}
_{k}, the expression

    P k | k = ( I - K k H k ) P k | k - 1 ( I - K k H k ) T + K k R k
    K k T {\displaystyle \mathbf {P} _{k\mid k}=\left(\mathbf {I} -\
    mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k\mid k-1}\
    left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)^{\
    textsf {T}}+\mathbf {K} _{k}\mathbf {R} _{k}\mathbf {K} _{k}^{\
    textsf {T}}} {\displaystyle \mathbf {P} _{k\mid k}=\left(\mathbf
    {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k\mid
    k-1}\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)^{\
    textsf {T}}+\mathbf {K} _{k}\mathbf {R} _{k}\mathbf {K} _{k}^{\
    textsf {T}}}

no longer provides the actual error covariance. In other words, P k |
k [?] E [ ( x k - x ^ k | k ) ( x k - x ^ k | k ) T ] {\displaystyle \
mathbf {P} _{k\mid k}\neq E\left[\left(\mathbf {x} _{k}-{\hat {\
mathbf {x} }}_{k\mid k}\right)\left(\mathbf {x} _{k}-{\hat {\mathbf
{x} }}_{k\mid k}\right)^{\textsf {T}}\right]} {\displaystyle \mathbf
{P} _{k\mid k}\neq E\left[\left(\mathbf {x} _{k}-{\hat {\mathbf {x}
}}_{k\mid k}\right)\left(\mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\
mid k}\right)^{\textsf {T}}\right]}. In most real-time applications,
the covariance matrices that are used in designing the Kalman filter
are different from the actual (true) noise covariances matrices.^[
citation needed] This sensitivity analysis describes the behavior of
the estimation error covariance when the noise covariances as well as
the system matrices F k {\displaystyle \mathbf {F} _{k}} \mathbf{F}_k
and H k {\displaystyle \mathbf {H} _{k}} {\displaystyle \mathbf {H} _
{k}} that are fed as inputs to the filter are incorrect. Thus, the
sensitivity analysis describes the robustness (or sensitivity) of the
estimator to misspecified statistical and parametric inputs to the
estimator.

This discussion is limited to the error sensitivity analysis for the
case of statistical uncertainties. Here the actual noise covariances
are denoted by Q k a {\displaystyle \mathbf {Q} _{k}^{a}} {\
displaystyle \mathbf {Q} _{k}^{a}} and R k a {\displaystyle \mathbf
{R} _{k}^{a}} {\displaystyle \mathbf {R} _{k}^{a}} respectively,
whereas the design values used in the estimator are Q k {\
displaystyle \mathbf {Q} _{k}} \mathbf {Q} _{k} and R k {\
displaystyle \mathbf {R} _{k}} \mathbf {R} _{k} respectively. The
actual error covariance is denoted by P k | k a {\displaystyle \
mathbf {P} _{k\mid k}^{a}} {\displaystyle \mathbf {P} _{k\mid k}^{a}}
and P k | k {\displaystyle \mathbf {P} _{k\mid k}} {\displaystyle \
mathbf {P} _{k\mid k}} as computed by the Kalman filter is referred
to as the Riccati variable. When Q k [?] Q k a {\displaystyle \mathbf
{Q} _{k}\equiv \mathbf {Q} _{k}^{a}} {\displaystyle \mathbf {Q} _{k}\
equiv \mathbf {Q} _{k}^{a}} and R k [?] R k a {\displaystyle \mathbf
{R} _{k}\equiv \mathbf {R} _{k}^{a}} {\displaystyle \mathbf {R} _{k}\
equiv \mathbf {R} _{k}^{a}}, this means that P k | k = P k | k a {\
displaystyle \mathbf {P} _{k\mid k}=\mathbf {P} _{k\mid k}^{a}} {\
displaystyle \mathbf {P} _{k\mid k}=\mathbf {P} _{k\mid k}^{a}}.
While computing the actual error covariance using P k | k a = E [ ( x
k - x ^ k | k ) ( x k - x ^ k | k ) T ] {\displaystyle \mathbf {P} _
{k\mid k}^{a}=E\left[\left(\mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\
mid k}\right)\left(\mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k}\
right)^{\textsf {T}}\right]} {\displaystyle \mathbf {P} _{k\mid k}^
{a}=E\left[\left(\mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k}\
right)\left(\mathbf {x} _{k}-{\hat {\mathbf {x} }}_{k\mid k}\right)^
{\textsf {T}}\right]}, substituting for x ^ k | k {\displaystyle {\
widehat {\mathbf {x} }}_{k\mid k}} {\displaystyle {\widehat {\mathbf
{x} }}_{k\mid k}} and using the fact that E [ w k w k T ] = Q k a {\
displaystyle E\left[\mathbf {w} _{k}\mathbf {w} _{k}^{\textsf {T}}\
right]=\mathbf {Q} _{k}^{a}} {\displaystyle E\left[\mathbf {w} _{k}\
mathbf {w} _{k}^{\textsf {T}}\right]=\mathbf {Q} _{k}^{a}} and E [ v
k v k T ] = R k a {\displaystyle E\left[\mathbf {v} _{k}\mathbf {v} _
{k}^{\textsf {T}}\right]=\mathbf {R} _{k}^{a}} {\displaystyle E\left
[\mathbf {v} _{k}\mathbf {v} _{k}^{\textsf {T}}\right]=\mathbf {R} _
{k}^{a}}, results in the following recursive equations for P k | k a
{\displaystyle \mathbf {P} _{k\mid k}^{a}} {\displaystyle \mathbf {P}
_{k\mid k}^{a}} :

    P k | k - 1 a = F k P k - 1 | k - 1 a F k T + Q k a {\
    displaystyle \mathbf {P} _{k\mid k-1}^{a}=\mathbf {F} _{k}\mathbf
    {P} _{k-1\mid k-1}^{a}\mathbf {F} _{k}^{\textsf {T}}+\mathbf {Q}
    _{k}^{a}} {\displaystyle \mathbf {P} _{k\mid k-1}^{a}=\mathbf {F}
    _{k}\mathbf {P} _{k-1\mid k-1}^{a}\mathbf {F} _{k}^{\textsf {T}}+
    \mathbf {Q} _{k}^{a}}

and

    P k | k a = ( I - K k H k ) P k | k - 1 a ( I - K k H k ) T + K k
    R k a K k T {\displaystyle \mathbf {P} _{k\mid k}^{a}=\left(\
    mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _
    {k\mid k-1}^{a}\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _
    {k}\right)^{\textsf {T}}+\mathbf {K} _{k}\mathbf {R} _{k}^{a}\
    mathbf {K} _{k}^{\textsf {T}}} {\displaystyle \mathbf {P} _{k\mid
    k}^{a}=\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)
    \mathbf {P} _{k\mid k-1}^{a}\left(\mathbf {I} -\mathbf {K} _{k}\
    mathbf {H} _{k}\right)^{\textsf {T}}+\mathbf {K} _{k}\mathbf {R}
    _{k}^{a}\mathbf {K} _{k}^{\textsf {T}}}

While computing P k | k {\displaystyle \mathbf {P} _{k\mid k}} {\
displaystyle \mathbf {P} _{k\mid k}}, by design the filter implicitly
assumes that E [ w k w k T ] = Q k {\displaystyle E\left[\mathbf {w}
_{k}\mathbf {w} _{k}^{\textsf {T}}\right]=\mathbf {Q} _{k}} {\
displaystyle E\left[\mathbf {w} _{k}\mathbf {w} _{k}^{\textsf {T}}\
right]=\mathbf {Q} _{k}} and E [ v k v k T ] = R k {\displaystyle E\
left[\mathbf {v} _{k}\mathbf {v} _{k}^{\textsf {T}}\right]=\mathbf
{R} _{k}} {\displaystyle E\left[\mathbf {v} _{k}\mathbf {v} _{k}^{\
textsf {T}}\right]=\mathbf {R} _{k}}. The recursive expressions for P
k | k a {\displaystyle \mathbf {P} _{k\mid k}^{a}} {\displaystyle \
mathbf {P} _{k\mid k}^{a}} and P k | k {\displaystyle \mathbf {P} _{k
\mid k}} {\displaystyle \mathbf {P} _{k\mid k}} are identical except
for the presence of Q k a {\displaystyle \mathbf {Q} _{k}^{a}} {\
displaystyle \mathbf {Q} _{k}^{a}} and R k a {\displaystyle \mathbf
{R} _{k}^{a}} {\displaystyle \mathbf {R} _{k}^{a}} in place of the
design values Q k {\displaystyle \mathbf {Q} _{k}} \mathbf {Q} _{k}
and R k {\displaystyle \mathbf {R} _{k}} \mathbf {R} _{k}
respectively. Researches have been done to analyze Kalman filter
system's robustness.^[34]

Square root form[edit]

One problem with the Kalman filter is its numerical stability. If the
process noise covariance Q[k] is small, round-off error often causes
a small positive eigenvalue to be computed as a negative number. This
renders the numerical representation of the state covariance matrix P
indefinite, while its true form is positive-definite.

Positive definite matrices have the property that they have a
triangular matrix square root P = S*S^T. This can be computed
efficiently using the Cholesky factorization algorithm, but more
importantly, if the covariance is kept in this form, it can never
have a negative diagonal or become asymmetric. An equivalent form,
which avoids many of the square root operations required by the
matrix square root yet preserves the desirable numerical properties,
is the U-D decomposition form, P = U*D*U^T, where U is a unit
triangular matrix (with unit diagonal), and D is a diagonal matrix.

Between the two, the U-D factorization uses the same amount of
storage, and somewhat less computation, and is the most commonly used
square root form. (Early literature on the relative efficiency is
somewhat misleading, as it assumed that square roots were much more
time-consuming than divisions,^[35]^:69 while on 21-st century
computers they are only slightly more expensive.)

Efficient algorithms for the Kalman prediction and update steps in
the square root form were developed by G. J. Bierman and C. L.
Thornton.^[35]^[36]

The L*D*L^T decomposition of the innovation covariance matrix S[k] is
the basis for another type of numerically efficient and robust square
root filter.^[37] The algorithm starts with the LU decomposition as
implemented in the Linear Algebra PACKage (LAPACK). These results are
further factored into the L*D*L^T structure with methods given by
Golub and Van Loan (algorithm 4.1.2) for a symmetric nonsingular
matrix.^[38] Any singular covariance matrix is pivoted so that the
first diagonal partition is nonsingular and well-conditioned. The
pivoting algorithm must retain any portion of the innovation
covariance matrix directly corresponding to observed state-variables
H[k]*x[k|k-1] that are associated with auxiliary observations in y
[k]. The l*d*l^t square-root filter requires orthogonalization of the
observation vector.^[36]^[37] This may be done with the inverse
square-root of the covariance matrix for the auxiliary variables
using Method 2 in Higham (2002, p. 263).^[39]

Parallel form[edit]

The Kalman filter is efficient for sequential data processing on
central processing units (CPUs), but in its original form it is
inefficient on parallel architectures such as graphics processing
units (GPUs). It is however possible to express the filter-update
routine in terms of an associative operator using the formulation in
Sarkka (2021).^[40] The filter solution is then be retrieved by the
use of a prefix sum algorithm which can be efficiently implemented on
GPU.^[41] This reduces the computational complexity from O ( N ) {\
displaystyle O(N)} O(N) in the number of time steps to O ( log [?] ( N
) ) {\displaystyle O(\log(N))} {\displaystyle O(\log(N))}.

Relationship to recursive Bayesian estimation[edit]

The Kalman filter can be presented as one of the simplest dynamic
Bayesian networks. The Kalman filter calculates estimates of the true
values of states recursively over time using incoming measurements
and a mathematical process model. Similarly, recursive Bayesian
estimation calculates estimates of an unknown probability density
function (PDF) recursively over time using incoming measurements and
a mathematical process model.^[42]

In recursive Bayesian estimation, the true state is assumed to be an
unobserved Markov process, and the measurements are the observed
states of a hidden Markov model (HMM).

hidden markov model

because of the Markov assumption, the true state is conditionally
independent of all earlier states given the immediately previous
state.

    p ( x k | x 0 , ... , x k - 1 ) = p ( x k | x k - 1 ) {\
    displaystyle p(\mathbf {x} _{k}\mid \mathbf {x} _{0},\dots ,\
    mathbf {x} _{k-1})=p(\mathbf {x} _{k}\mid \mathbf {x} _{k-1})} {\
    displaystyle p(\mathbf {x} _{k}\mid \mathbf {x} _{0},\dots ,\
    mathbf {x} _{k-1})=p(\mathbf {x} _{k}\mid \mathbf {x} _{k-1})}

Similarly, the measurement at the k-th timestep is dependent only
upon the current state and is conditionally independent of all other
states given the current state.

    p ( z k | x 0 , ... , x k ) = p ( z k | x k ) {\displaystyle p(\
    mathbf {z} _{k}\mid \mathbf {x} _{0},\dots ,\mathbf {x} _{k})=p(\
    mathbf {z} _{k}\mid \mathbf {x} _{k})} {\displaystyle p(\mathbf
    {z} _{k}\mid \mathbf {x} _{0},\dots ,\mathbf {x} _{k})=p(\mathbf
    {z} _{k}\mid \mathbf {x} _{k})}

Using these assumptions the probability distribution over all states
of the hidden Markov model can be written simply as:

    p ( x 0 , ... , x k , z 1 , ... , z k ) = p ( x 0 ) [?] i = 1 k p ( z i
    | x i ) p ( x i | x i - 1 ) {\displaystyle p\left(\mathbf {x} _
    {0},\dots ,\mathbf {x} _{k},\mathbf {z} _{1},\dots ,\mathbf {z} _
    {k}\right)=p\left(\mathbf {x} _{0}\right)\prod _{i=1}^{k}p\left(\
    mathbf {z} _{i}\mid \mathbf {x} _{i}\right)p\left(\mathbf {x} _
    {i}\mid \mathbf {x} _{i-1}\right)} {\displaystyle p\left(\mathbf
    {x} _{0},\dots ,\mathbf {x} _{k},\mathbf {z} _{1},\dots ,\mathbf
    {z} _{k}\right)=p\left(\mathbf {x} _{0}\right)\prod _{i=1}^{k}p\
    left(\mathbf {z} _{i}\mid \mathbf {x} _{i}\right)p\left(\mathbf
    {x} _{i}\mid \mathbf {x} _{i-1}\right)}

However, when the Kalman filter is used to estimate the state x, the
probability distribution of interest is that associated with the
current states conditioned on the measurements up to the current
timestep. This is achieved by marginalizing out the previous states
and dividing by the probability of the measurement set.

This leads to the predict and update steps of the Kalman filter
written probabilistically. The probability distribution associated
with the predicted state is the sum (integral) of the products of the
probability distribution associated with the transition from the (k
 - 1)-th timestep to the k-th and the probability distribution
associated with the previous state, over all possible x k - 1 {\
displaystyle x_{k-1}} x_{k-1}.

    p ( x k | Z k - 1 ) = [?] p ( x k | x k - 1 ) p ( x k - 1 | Z k - 1
    ) d x k - 1 {\displaystyle p\left(\mathbf {x} _{k}\mid \mathbf
    {Z} _{k-1}\right)=\int p\left(\mathbf {x} _{k}\mid \mathbf {x} _
    {k-1}\right)p\left(\mathbf {x} _{k-1}\mid \mathbf {Z} _{k-1}\
    right)\,d\mathbf {x} _{k-1}} {\displaystyle p\left(\mathbf {x} _
    {k}\mid \mathbf {Z} _{k-1}\right)=\int p\left(\mathbf {x} _{k}\
    mid \mathbf {x} _{k-1}\right)p\left(\mathbf {x} _{k-1}\mid \
    mathbf {Z} _{k-1}\right)\,d\mathbf {x} _{k-1}}

The measurement set up to time t is

    Z t = { z 1 , ... , z t } {\displaystyle \mathbf {Z} _{t}=\left\{\
    mathbf {z} _{1},\dots ,\mathbf {z} _{t}\right\}} {\displaystyle \
    mathbf {Z} _{t}=\left\{\mathbf {z} _{1},\dots ,\mathbf {z} _{t}\
    right\}}

The probability distribution of the update is proportional to the
product of the measurement likelihood and the predicted state.

    p ( x k | Z k ) = p ( z k | x k ) p ( x k | Z k - 1 ) p ( z k | Z
    k - 1 ) {\displaystyle p\left(\mathbf {x} _{k}\mid \mathbf {Z} _
    {k}\right)={\frac {p\left(\mathbf {z} _{k}\mid \mathbf {x} _{k}\
    right)p\left(\mathbf {x} _{k}\mid \mathbf {Z} _{k-1}\right)}{p\
    left(\mathbf {z} _{k}\mid \mathbf {Z} _{k-1}\right)}}} {\
    displaystyle p\left(\mathbf {x} _{k}\mid \mathbf {Z} _{k}\right)=
    {\frac {p\left(\mathbf {z} _{k}\mid \mathbf {x} _{k}\right)p\left
    (\mathbf {x} _{k}\mid \mathbf {Z} _{k-1}\right)}{p\left(\mathbf
    {z} _{k}\mid \mathbf {Z} _{k-1}\right)}}}

The denominator

    p ( z k | Z k - 1 ) = [?] p ( z k | x k ) p ( x k | Z k - 1 ) d x k
    {\displaystyle p\left(\mathbf {z} _{k}\mid \mathbf {Z} _{k-1}\
    right)=\int p\left(\mathbf {z} _{k}\mid \mathbf {x} _{k}\right)p\
    left(\mathbf {x} _{k}\mid \mathbf {Z} _{k-1}\right)\,d\mathbf {x}
    _{k}} {\displaystyle p\left(\mathbf {z} _{k}\mid \mathbf {Z} _
    {k-1}\right)=\int p\left(\mathbf {z} _{k}\mid \mathbf {x} _{k}\
    right)p\left(\mathbf {x} _{k}\mid \mathbf {Z} _{k-1}\right)\,d\
    mathbf {x} _{k}}

is a normalization term.

The remaining probability density functions are

    p ( x k | x k - 1 ) = N ( F k x k - 1 , Q k ) p ( z k | x k ) = N
    ( H k x k , R k ) p ( x k - 1 | Z k - 1 ) = N ( x ^ k - 1 , P k -
    1 ) {\displaystyle {\begin{aligned}p\left(\mathbf {x} _{k}\mid \
    mathbf {x} _{k-1}\right)&={\mathcal {N}}\left(\mathbf {F} _{k}\
    mathbf {x} _{k-1},\mathbf {Q} _{k}\right)\\p\left(\mathbf {z} _
    {k}\mid \mathbf {x} _{k}\right)&={\mathcal {N}}\left(\mathbf {H}
    _{k}\mathbf {x} _{k},\mathbf {R} _{k}\right)\\p\left(\mathbf {x}
    _{k-1}\mid \mathbf {Z} _{k-1}\right)&={\mathcal {N}}\left({\hat
    {\mathbf {x} }}_{k-1},\mathbf {P} _{k-1}\right)\end{aligned}}} {\
    displaystyle {\begin{aligned}p\left(\mathbf {x} _{k}\mid \mathbf
    {x} _{k-1}\right)&={\mathcal {N}}\left(\mathbf {F} _{k}\mathbf
    {x} _{k-1},\mathbf {Q} _{k}\right)\\p\left(\mathbf {z} _{k}\mid \
    mathbf {x} _{k}\right)&={\mathcal {N}}\left(\mathbf {H} _{k}\
    mathbf {x} _{k},\mathbf {R} _{k}\right)\\p\left(\mathbf {x} _
    {k-1}\mid \mathbf {Z} _{k-1}\right)&={\mathcal {N}}\left({\hat {\
    mathbf {x} }}_{k-1},\mathbf {P} _{k-1}\right)\end{aligned}}}

The PDF at the previous timestep is inductively assumed to be the
estimated state and covariance. This is justified because, as an
optimal estimator, the Kalman filter makes best use of the
measurements, therefore the PDF for x k {\displaystyle \mathbf {x} _
{k}} \mathbf {x} _{k} given the measurements Z k {\displaystyle \
mathbf {Z} _{k}} \mathbf {Z} _{k} is the Kalman filter estimate.

Marginal likelihood[edit]

Related to the recursive Bayesian interpretation described above, the
Kalman filter can be viewed as a generative model, i.e., a process
for generating a stream of random observations z = (z[0], z[1], z[2],
...). Specifically, the process is

 1. Sample a hidden state x 0 {\displaystyle \mathbf {x} _{0}} \
    mathbf {x} _{0} from the Gaussian prior distribution p ( x 0 ) =
    N ( x ^ 0 | 0 , P 0 | 0 ) {\displaystyle p\left(\mathbf {x} _{0}\
    right)={\mathcal {N}}\left({\hat {\mathbf {x} }}_{0\mid 0},\
    mathbf {P} _{0\mid 0}\right)} {\displaystyle p\left(\mathbf {x} _
    {0}\right)={\mathcal {N}}\left({\hat {\mathbf {x} }}_{0\mid 0},\
    mathbf {P} _{0\mid 0}\right)}.
 2. Sample an observation z 0 {\displaystyle \mathbf {z} _{0}} \
    mathbf {z} _{0} from the observation model p ( z 0 | x 0 ) = N (
    H 0 x 0 , R 0 ) {\displaystyle p\left(\mathbf {z} _{0}\mid \
    mathbf {x} _{0}\right)={\mathcal {N}}\left(\mathbf {H} _{0}\
    mathbf {x} _{0},\mathbf {R} _{0}\right)} {\displaystyle p\left(\
    mathbf {z} _{0}\mid \mathbf {x} _{0}\right)={\mathcal {N}}\left(\
    mathbf {H} _{0}\mathbf {x} _{0},\mathbf {R} _{0}\right)}.
 3. For k = 1 , 2 , 3 , ... {\displaystyle k=1,2,3,\ldots } {\
    displaystyle k=1,2,3,\ldots }, do
     1. Sample the next hidden state x k {\displaystyle \mathbf {x} _
        {k}} \mathbf {x} _{k} from the transition model p ( x k | x k
        - 1 ) = N ( F k x k - 1 + B k u k , Q k ) . {\displaystyle p\
        left(\mathbf {x} _{k}\mid \mathbf {x} _{k-1}\right)={\mathcal
        {N}}\left(\mathbf {F} _{k}\mathbf {x} _{k-1}+\mathbf {B} _{k}
        \mathbf {u} _{k},\mathbf {Q} _{k}\right).} {\displaystyle p\
        left(\mathbf {x} _{k}\mid \mathbf {x} _{k-1}\right)={\mathcal
        {N}}\left(\mathbf {F} _{k}\mathbf {x} _{k-1}+\mathbf {B} _{k}
        \mathbf {u} _{k},\mathbf {Q} _{k}\right).}
     2. Sample an observation z k {\displaystyle \mathbf {z} _{k}} \
        mathbf {z} _{k} from the observation model p ( z k | x k ) =
        N ( H k x k , R k ) . {\displaystyle p\left(\mathbf {z} _{k}\
        mid \mathbf {x} _{k}\right)={\mathcal {N}}\left(\mathbf {H} _
        {k}\mathbf {x} _{k},\mathbf {R} _{k}\right).} {\displaystyle
        p\left(\mathbf {z} _{k}\mid \mathbf {x} _{k}\right)={\mathcal
        {N}}\left(\mathbf {H} _{k}\mathbf {x} _{k},\mathbf {R} _{k}\
        right).}

This process has identical structure to the hidden Markov model,
except that the discrete state and observations are replaced with
continuous variables sampled from Gaussian distributions.

In some applications, it is useful to compute the probability that a
Kalman filter with a given set of parameters (prior distribution,
transition and observation models, and control inputs) would generate
a particular observed signal. This probability is known as the
marginal likelihood because it integrates over ("marginalizes out")
the values of the hidden state variables, so it can be computed using
only the observed signal. The marginal likelihood can be useful to
evaluate different parameter choices, or to compare the Kalman filter
against other models using Bayesian model comparison.

It is straightforward to compute the marginal likelihood as a side
effect of the recursive filtering computation. By the chain rule, the
likelihood can be factored as the product of the probability of each
observation given previous observations,

    p ( z ) = [?] k = 0 T p ( z k | z k - 1 , ... , z 0 ) {\displaystyle
    p(\mathbf {z} )=\prod _{k=0}^{T}p\left(\mathbf {z} _{k}\mid \
    mathbf {z} _{k-1},\ldots ,\mathbf {z} _{0}\right)} {\displaystyle
    p(\mathbf {z} )=\prod _{k=0}^{T}p\left(\mathbf {z} _{k}\mid \
    mathbf {z} _{k-1},\ldots ,\mathbf {z} _{0}\right)},

and because the Kalman filter describes a Markov process, all
relevant information from previous observations is contained in the
current state estimate x ^ k | k - 1 , P k | k - 1 . {\displaystyle
{\hat {\mathbf {x} }}_{k\mid k-1},\mathbf {P} _{k\mid k-1}.} {\
displaystyle {\hat {\mathbf {x} }}_{k\mid k-1},\mathbf {P} _{k\mid
k-1}.} Thus the marginal likelihood is given by

    p ( z ) = [?] k = 0 T [?] p ( z k | x k ) p ( x k | z k - 1 , ... , z 0
    ) d x k = [?] k = 0 T [?] N ( z k ; H k x k , R k ) N ( x k ; x ^ k |
    k - 1 , P k | k - 1 ) d x k = [?] k = 0 T N ( z k ; H k x ^ k | k -
    1 , R k + H k P k | k - 1 H k T ) = [?] k = 0 T N ( z k ; H k x ^ k
    | k - 1 , S k ) , {\displaystyle {\begin{aligned}p(\mathbf {z} )&
    =\prod _{k=0}^{T}\int p\left(\mathbf {z} _{k}\mid \mathbf {x} _
    {k}\right)p\left(\mathbf {x} _{k}\mid \mathbf {z} _{k-1},\ldots ,
    \mathbf {z} _{0}\right)d\mathbf {x} _{k}\\&=\prod _{k=0}^{T}\int
    {\mathcal {N}}\left(\mathbf {z} _{k};\mathbf {H} _{k}\mathbf {x}
    _{k},\mathbf {R} _{k}\right){\mathcal {N}}\left(\mathbf {x} _{k};
    {\hat {\mathbf {x} }}_{k\mid k-1},\mathbf {P} _{k\mid k-1}\right)
    d\mathbf {x} _{k}\\&=\prod _{k=0}^{T}{\mathcal {N}}\left(\mathbf
    {z} _{k};\mathbf {H} _{k}{\hat {\mathbf {x} }}_{k\mid k-1},\
    mathbf {R} _{k}+\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\mathbf
    {H} _{k}^{\textsf {T}}\right)\\&=\prod _{k=0}^{T}{\mathcal {N}}\
    left(\mathbf {z} _{k};\mathbf {H} _{k}{\hat {\mathbf {x} }}_{k\
    mid k-1},\mathbf {S} _{k}\right),\end{aligned}}} {\displaystyle
    {\begin{aligned}p(\mathbf {z} )&=\prod _{k=0}^{T}\int p\left(\
    mathbf {z} _{k}\mid \mathbf {x} _{k}\right)p\left(\mathbf {x} _
    {k}\mid \mathbf {z} _{k-1},\ldots ,\mathbf {z} _{0}\right)d\
    mathbf {x} _{k}\\&=\prod _{k=0}^{T}\int {\mathcal {N}}\left(\
    mathbf {z} _{k};\mathbf {H} _{k}\mathbf {x} _{k},\mathbf {R} _{k}
    \right){\mathcal {N}}\left(\mathbf {x} _{k};{\hat {\mathbf {x} }}
    _{k\mid k-1},\mathbf {P} _{k\mid k-1}\right)d\mathbf {x} _{k}\\&=
    \prod _{k=0}^{T}{\mathcal {N}}\left(\mathbf {z} _{k};\mathbf {H}
    _{k}{\hat {\mathbf {x} }}_{k\mid k-1},\mathbf {R} _{k}+\mathbf
    {H} _{k}\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\
    right)\\&=\prod _{k=0}^{T}{\mathcal {N}}\left(\mathbf {z} _{k};\
    mathbf {H} _{k}{\hat {\mathbf {x} }}_{k\mid k-1},\mathbf {S} _{k}
    \right),\end{aligned}}}

i.e., a product of Gaussian densities, each corresponding to the
density of one observation z[k] under the current filtering
distribution H k x ^ k | k - 1 , S k {\displaystyle \mathbf {H} _{k}
{\hat {\mathbf {x} }}_{k\mid k-1},\mathbf {S} _{k}} {\displaystyle \
mathbf {H} _{k}{\hat {\mathbf {x} }}_{k\mid k-1},\mathbf {S} _{k}}.
This can easily be computed as a simple recursive update; however, to
avoid numeric underflow, in a practical implementation it is usually
desirable to compute the log marginal likelihood  = log [?] p ( z ) {\
displaystyle \ell =\log p(\mathbf {z} )} \ell =\log p(\mathbf {z} )
instead. Adopting the convention  ( - 1 ) = 0 {\displaystyle \ell ^
{(-1)}=0} \ell ^{(-1)}=0, this can be done via the recursive update
rule

     ( k ) =  ( k - 1 ) - 1 2 ( y ~ k T S k - 1 y ~ k + log [?] | S k
    | + d y log [?] 2 p ) , {\displaystyle \ell ^{(k)}=\ell ^{(k-1)}-{\
    frac {1}{2}}\left({\tilde {\mathbf {y} }}_{k}^{\textsf {T}}\
    mathbf {S} _{k}^{-1}{\tilde {\mathbf {y} }}_{k}+\log \left|\
    mathbf {S} _{k}\right|+d_{y}\log 2\pi \right),} {\displaystyle \
    ell ^{(k)}=\ell ^{(k-1)}-{\frac {1}{2}}\left({\tilde {\mathbf {y}
    }}_{k}^{\textsf {T}}\mathbf {S} _{k}^{-1}{\tilde {\mathbf {y} }}_
    {k}+\log \left|\mathbf {S} _{k}\right|+d_{y}\log 2\pi \right),}

where d y {\displaystyle d_{y}} d_{y} is the dimension of the
measurement vector.^[43]

An important application where such a (log) likelihood of the
observations (given the filter parameters) is used is multi-target
tracking. For example, consider an object tracking scenario where a
stream of observations is the input, however, it is unknown how many
objects are in the scene (or, the number of objects is known but is
greater than one). In such a scenario, it can be unknown apriori
which observations/measurements were generated by which object. A
multiple hypothesis tracker (MHT) typically will form different track
association hypotheses, where each hypothesis can be viewed as a
Kalman filter (in the linear Gaussian case) with a specific set of
parameters associated with the hypothesized object. Thus, it is
important to compute the likelihood of the observations for the
different hypotheses under consideration, such that the most-likely
one can be found.

Information filter[edit]

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In the information filter, or inverse covariance filter, the
estimated covariance and estimated state are replaced by the
information matrix and information vector respectively. These are
defined as:

    Y k | k = P k | k - 1 y ^ k | k = P k | k - 1 x ^ k | k {\
    displaystyle {\begin{aligned}\mathbf {Y} _{k\mid k}&=\mathbf {P}
    _{k\mid k}^{-1}\\{\hat {\mathbf {y} }}_{k\mid k}&=\mathbf {P} _{k
    \mid k}^{-1}{\hat {\mathbf {x} }}_{k\mid k}\end{aligned}}} {\
    displaystyle {\begin{aligned}\mathbf {Y} _{k\mid k}&=\mathbf {P}
    _{k\mid k}^{-1}\\{\hat {\mathbf {y} }}_{k\mid k}&=\mathbf {P} _{k
    \mid k}^{-1}{\hat {\mathbf {x} }}_{k\mid k}\end{aligned}}}

Similarly the predicted covariance and state have equivalent
information forms, defined as:

    Y k | k - 1 = P k | k - 1 - 1 y ^ k | k - 1 = P k | k - 1 - 1 x ^
    k | k - 1 {\displaystyle {\begin{aligned}\mathbf {Y} _{k\mid k-1}
    &=\mathbf {P} _{k\mid k-1}^{-1}\\{\hat {\mathbf {y} }}_{k\mid
    k-1}&=\mathbf {P} _{k\mid k-1}^{-1}{\hat {\mathbf {x} }}_{k\mid
    k-1}\end{aligned}}} {\displaystyle {\begin{aligned}\mathbf {Y} _
    {k\mid k-1}&=\mathbf {P} _{k\mid k-1}^{-1}\\{\hat {\mathbf {y} }}
    _{k\mid k-1}&=\mathbf {P} _{k\mid k-1}^{-1}{\hat {\mathbf {x} }}_
    {k\mid k-1}\end{aligned}}}

as have the measurement covariance and measurement vector, which are
defined as:

    I k = H k T R k - 1 H k i k = H k T R k - 1 z k {\displaystyle {\
    begin{aligned}\mathbf {I} _{k}&=\mathbf {H} _{k}^{\textsf {T}}\
    mathbf {R} _{k}^{-1}\mathbf {H} _{k}\\\mathbf {i} _{k}&=\mathbf
    {H} _{k}^{\textsf {T}}\mathbf {R} _{k}^{-1}\mathbf {z} _{k}\end
    {aligned}}} {\displaystyle {\begin{aligned}\mathbf {I} _{k}&=\
    mathbf {H} _{k}^{\textsf {T}}\mathbf {R} _{k}^{-1}\mathbf {H} _
    {k}\\\mathbf {i} _{k}&=\mathbf {H} _{k}^{\textsf {T}}\mathbf {R}
    _{k}^{-1}\mathbf {z} _{k}\end{aligned}}}

The information update now becomes a trivial sum.^[44]

    Y k | k = Y k | k - 1 + I k y ^ k | k = y ^ k | k - 1 + i k {\
    displaystyle {\begin{aligned}\mathbf {Y} _{k\mid k}&=\mathbf {Y}
    _{k\mid k-1}+\mathbf {I} _{k}\\{\hat {\mathbf {y} }}_{k\mid k}&=
    {\hat {\mathbf {y} }}_{k\mid k-1}+\mathbf {i} _{k}\end{aligned}}}
    {\displaystyle {\begin{aligned}\mathbf {Y} _{k\mid k}&=\mathbf
    {Y} _{k\mid k-1}+\mathbf {I} _{k}\\{\hat {\mathbf {y} }}_{k\mid
    k}&={\hat {\mathbf {y} }}_{k\mid k-1}+\mathbf {i} _{k}\end
    {aligned}}}

The main advantage of the information filter is that N measurements
can be filtered at each timestep simply by summing their information
matrices and vectors.

    Y k | k = Y k | k - 1 + [?] j = 1 N I k , j y ^ k | k = y ^ k | k -
    1 + [?] j = 1 N i k , j {\displaystyle {\begin{aligned}\mathbf {Y}
    _{k\mid k}&=\mathbf {Y} _{k\mid k-1}+\sum _{j=1}^{N}\mathbf {I} _
    {k,j}\\{\hat {\mathbf {y} }}_{k\mid k}&={\hat {\mathbf {y} }}_{k\
    mid k-1}+\sum _{j=1}^{N}\mathbf {i} _{k,j}\end{aligned}}} {\
    displaystyle {\begin{aligned}\mathbf {Y} _{k\mid k}&=\mathbf {Y}
    _{k\mid k-1}+\sum _{j=1}^{N}\mathbf {I} _{k,j}\\{\hat {\mathbf
    {y} }}_{k\mid k}&={\hat {\mathbf {y} }}_{k\mid k-1}+\sum _{j=1}^
    {N}\mathbf {i} _{k,j}\end{aligned}}}

To predict the information filter the information matrix and vector
can be converted back to their state space equivalents, or
alternatively the information space prediction can be used.^[44]

    M k = [ F k - 1 ] T Y k - 1 | k - 1 F k - 1 C k = M k [ M k + Q k
    - 1 ] - 1 L k = I - C k Y k | k - 1 = L k M k L k T + C k Q k - 1
    C k T y ^ k | k - 1 = L k [ F k - 1 ] T y ^ k - 1 | k - 1 {\
    displaystyle {\begin{aligned}\mathbf {M} _{k}&=\left[\mathbf {F}
    _{k}^{-1}\right]^{\textsf {T}}\mathbf {Y} _{k-1\mid k-1}\mathbf
    {F} _{k}^{-1}\\\mathbf {C} _{k}&=\mathbf {M} _{k}\left[\mathbf
    {M} _{k}+\mathbf {Q} _{k}^{-1}\right]^{-1}\\\mathbf {L} _{k}&=\
    mathbf {I} -\mathbf {C} _{k}\\\mathbf {Y} _{k\mid k-1}&=\mathbf
    {L} _{k}\mathbf {M} _{k}\mathbf {L} _{k}^{\textsf {T}}+\mathbf
    {C} _{k}\mathbf {Q} _{k}^{-1}\mathbf {C} _{k}^{\textsf {T}}\\{\
    hat {\mathbf {y} }}_{k\mid k-1}&=\mathbf {L} _{k}\left[\mathbf
    {F} _{k}^{-1}\right]^{\textsf {T}}{\hat {\mathbf {y} }}_{k-1\mid
    k-1}\end{aligned}}} {\displaystyle {\begin{aligned}\mathbf {M} _
    {k}&=\left[\mathbf {F} _{k}^{-1}\right]^{\textsf {T}}\mathbf {Y}
    _{k-1\mid k-1}\mathbf {F} _{k}^{-1}\\\mathbf {C} _{k}&=\mathbf
    {M} _{k}\left[\mathbf {M} _{k}+\mathbf {Q} _{k}^{-1}\right]^{-1}\
    \\mathbf {L} _{k}&=\mathbf {I} -\mathbf {C} _{k}\\\mathbf {Y} _{k
    \mid k-1}&=\mathbf {L} _{k}\mathbf {M} _{k}\mathbf {L} _{k}^{\
    textsf {T}}+\mathbf {C} _{k}\mathbf {Q} _{k}^{-1}\mathbf {C} _{k}
    ^{\textsf {T}}\\{\hat {\mathbf {y} }}_{k\mid k-1}&=\mathbf {L} _
    {k}\left[\mathbf {F} _{k}^{-1}\right]^{\textsf {T}}{\hat {\mathbf
    {y} }}_{k-1\mid k-1}\end{aligned}}}

If F and Q are time invariant these values can be cached, and F and Q
need to be invertible.

Fixed-lag smoother[edit]

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The optimal fixed-lag smoother provides the optimal estimate of x ^ k
- N | k {\displaystyle {\hat {\mathbf {x} }}_{k-N\mid k}} {\
displaystyle {\hat {\mathbf {x} }}_{k-N\mid k}} for a given fixed-lag
N {\displaystyle N} N using the measurements from z 1 {\displaystyle
\mathbf {z} _{1}} {\displaystyle \mathbf {z} _{1}} to z k {\
displaystyle \mathbf {z} _{k}} \mathbf {z} _{k}.^[45] It can be
derived using the previous theory via an augmented state, and the
main equation of the filter is the following:

    [ x ^ t | t x ^ t - 1 | t [?] x ^ t - N + 1 | t ] = [ I 0 [?] 0 ] x ^
    t | t - 1 + [ 0 ... 0 I 0 [?] [?] [?] [?] 0 ... I ] [ x ^ t - 1 | t - 1 x ^ t
    - 2 | t - 1 [?] x ^ t - N + 1 | t - 1 ] + [ K ( 0 ) K ( 1 ) [?] K ( N
    - 1 ) ] y t | t - 1 {\displaystyle {\begin{bmatrix}{\hat {\mathbf
    {x} }}_{t\mid t}\\{\hat {\mathbf {x} }}_{t-1\mid t}\\\vdots \\{\
    hat {\mathbf {x} }}_{t-N+1\mid t}\\\end{bmatrix}}={\begin
    {bmatrix}\mathbf {I} \\0\\\vdots \\0\\\end{bmatrix}}{\hat {\
    mathbf {x} }}_{t\mid t-1}+{\begin{bmatrix}0&\ldots &0\\\mathbf
    {I} &0&\vdots \\\vdots &\ddots &\vdots \\0&\ldots &\mathbf {I} \\
    \end{bmatrix}}{\begin{bmatrix}{\hat {\mathbf {x} }}_{t-1\mid t-1}
    \\{\hat {\mathbf {x} }}_{t-2\mid t-1}\\\vdots \\{\hat {\mathbf
    {x} }}_{t-N+1\mid t-1}\\\end{bmatrix}}+{\begin{bmatrix}\mathbf
    {K} ^{(0)}\\\mathbf {K} ^{(1)}\\\vdots \\\mathbf {K} ^{(N-1)}\\\
    end{bmatrix}}\mathbf {y} _{t\mid t-1}} {\displaystyle {\begin
    {bmatrix}{\hat {\mathbf {x} }}_{t\mid t}\\{\hat {\mathbf {x} }}_
    {t-1\mid t}\\\vdots \\{\hat {\mathbf {x} }}_{t-N+1\mid t}\\\end
    {bmatrix}}={\begin{bmatrix}\mathbf {I} \\0\\\vdots \\0\\\end
    {bmatrix}}{\hat {\mathbf {x} }}_{t\mid t-1}+{\begin{bmatrix}0&\
    ldots &0\\\mathbf {I} &0&\vdots \\\vdots &\ddots &\vdots \\0&\
    ldots &\mathbf {I} \\\end{bmatrix}}{\begin{bmatrix}{\hat {\mathbf
    {x} }}_{t-1\mid t-1}\\{\hat {\mathbf {x} }}_{t-2\mid t-1}\\\vdots
    \\{\hat {\mathbf {x} }}_{t-N+1\mid t-1}\\\end{bmatrix}}+{\begin
    {bmatrix}\mathbf {K} ^{(0)}\\\mathbf {K} ^{(1)}\\\vdots \\\mathbf
    {K} ^{(N-1)}\\\end{bmatrix}}\mathbf {y} _{t\mid t-1}}

where:

  * x ^ t | t - 1 {\displaystyle {\hat {\mathbf {x} }}_{t\mid t-1}} 
    {\displaystyle {\hat {\mathbf {x} }}_{t\mid t-1}} is estimated
    via a standard Kalman filter;
  * y t | t - 1 = z t - H x ^ t | t - 1 {\displaystyle \mathbf {y} _
    {t\mid t-1}=\mathbf {z} _{t}-\mathbf {H} {\hat {\mathbf {x} }}_{t
    \mid t-1}} {\displaystyle \mathbf {y} _{t\mid t-1}=\mathbf {z} _
    {t}-\mathbf {H} {\hat {\mathbf {x} }}_{t\mid t-1}} is the
    innovation produced considering the estimate of the standard
    Kalman filter;
  * the various x ^ t - i | t {\displaystyle {\hat {\mathbf {x} }}_
    {t-i\mid t}} {\displaystyle {\hat {\mathbf {x} }}_{t-i\mid t}}
    with i = 1 , ... , N - 1 {\displaystyle i=1,\ldots ,N-1} {\
    displaystyle i=1,\ldots ,N-1} are new variables; i.e., they do
    not appear in the standard Kalman filter;
  * the gains are computed via the following scheme:

        K ( i + 1 ) = P ( i ) H T [ H P H T + R ] - 1 {\displaystyle
        \mathbf {K} ^{(i+1)}=\mathbf {P} ^{(i)}\mathbf {H} ^{\textsf
        {T}}\left[\mathbf {H} \mathbf {P} \mathbf {H} ^{\textsf {T}}+
        \mathbf {R} \right]^{-1}} {\displaystyle \mathbf {K} ^{(i+1)}
        =\mathbf {P} ^{(i)}\mathbf {H} ^{\textsf {T}}\left[\mathbf
        {H} \mathbf {P} \mathbf {H} ^{\textsf {T}}+\mathbf {R} \
        right]^{-1}}

    and

        P ( i ) = P [ ( F - K H ) T ] i {\displaystyle \mathbf {P} ^
        {(i)}=\mathbf {P} \left[\left(\mathbf {F} -\mathbf {K} \
        mathbf {H} \right)^{\textsf {T}}\right]^{i}} {\displaystyle \
        mathbf {P} ^{(i)}=\mathbf {P} \left[\left(\mathbf {F} -\
        mathbf {K} \mathbf {H} \right)^{\textsf {T}}\right]^{i}}

    where P {\displaystyle \mathbf {P} } {\mathbf {P}} and K {\
    displaystyle \mathbf {K} } \mathbf{K} are the prediction error
    covariance and the gains of the standard Kalman filter (i.e., P t
    | t - 1 {\displaystyle \mathbf {P} _{t\mid t-1}} {\displaystyle \
    mathbf {P} _{t\mid t-1}}).

If the estimation error covariance is defined so that

    P i := E [ ( x t - i - x ^ t - i | t ) * ( x t - i - x ^ t - i |
    t ) | z 1 ... z t ] , {\displaystyle \mathbf {P} _{i}:=E\left[\left
    (\mathbf {x} _{t-i}-{\hat {\mathbf {x} }}_{t-i\mid t}\right)^{*}\
    left(\mathbf {x} _{t-i}-{\hat {\mathbf {x} }}_{t-i\mid t}\right)\
    mid z_{1}\ldots z_{t}\right],} {\displaystyle \mathbf {P} _{i}:=E
    \left[\left(\mathbf {x} _{t-i}-{\hat {\mathbf {x} }}_{t-i\mid t}\
    right)^{*}\left(\mathbf {x} _{t-i}-{\hat {\mathbf {x} }}_{t-i\mid
    t}\right)\mid z_{1}\ldots z_{t}\right],}

then we have that the improvement on the estimation of x t - i {\
displaystyle \mathbf {x} _{t-i}} {\displaystyle \mathbf {x} _{t-i}}
is given by:

    P - P i = [?] j = 0 i [ P ( j ) H T ( H P H T + R ) - 1 H ( P ( i )
    ) T ] {\displaystyle \mathbf {P} -\mathbf {P} _{i}=\sum _{j=0}^
    {i}\left[\mathbf {P} ^{(j)}\mathbf {H} ^{\textsf {T}}\left(\
    mathbf {H} \mathbf {P} \mathbf {H} ^{\textsf {T}}+\mathbf {R} \
    right)^{-1}\mathbf {H} \left(\mathbf {P} ^{(i)}\right)^{\textsf
    {T}}\right]} {\displaystyle \mathbf {P} -\mathbf {P} _{i}=\sum _
    {j=0}^{i}\left[\mathbf {P} ^{(j)}\mathbf {H} ^{\textsf {T}}\left
    (\mathbf {H} \mathbf {P} \mathbf {H} ^{\textsf {T}}+\mathbf {R} \
    right)^{-1}\mathbf {H} \left(\mathbf {P} ^{(i)}\right)^{\textsf
    {T}}\right]}

Fixed-interval smoothers[edit]

The optimal fixed-interval smoother provides the optimal estimate of 
x ^ k | n {\displaystyle {\hat {\mathbf {x} }}_{k\mid n}} {\
displaystyle {\hat {\mathbf {x} }}_{k\mid n}} ( k < n {\displaystyle
k<n} k<n) using the measurements from a fixed interval z 1 {\
displaystyle \mathbf {z} _{1}} {\displaystyle \mathbf {z} _{1}} to z
n {\displaystyle \mathbf {z} _{n}} {\mathbf {z}}_{n}. This is also
called "Kalman Smoothing". There are several smoothing algorithms in
common use.

Rauch-Tung-Striebel[edit]

The Rauch-Tung-Striebel (RTS) smoother is an efficient two-pass
algorithm for fixed interval smoothing.^[46]

The forward pass is the same as the regular Kalman filter algorithm.
These filtered a-priori and a-posteriori state estimates x ^ k | k -
1 {\displaystyle {\hat {\mathbf {x} }}_{k\mid k-1}} {\displaystyle {\
hat {\mathbf {x} }}_{k\mid k-1}}, x ^ k | k {\displaystyle {\hat {\
mathbf {x} }}_{k\mid k}} {\displaystyle {\hat {\mathbf {x} }}_{k\mid
k}} and covariances P k | k - 1 {\displaystyle \mathbf {P} _{k\mid
k-1}} {\displaystyle \mathbf {P} _{k\mid k-1}}, P k | k {\
displaystyle \mathbf {P} _{k\mid k}} {\displaystyle \mathbf {P} _{k\
mid k}} are saved for use in the backwards pass.

In the backwards pass, we compute the smoothed state estimates x ^ k
| n {\displaystyle {\hat {\mathbf {x} }}_{k\mid n}} {\displaystyle {\
hat {\mathbf {x} }}_{k\mid n}} and covariances P k | n {\displaystyle
\mathbf {P} _{k\mid n}} {\displaystyle \mathbf {P} _{k\mid n}}. We
start at the last time step and proceed backwards in time using the
following recursive equations:

    x ^ k | n = x ^ k | k + C k ( x ^ k + 1 | n - x ^ k + 1 | k ) P k
    | n = P k | k + C k ( P k + 1 | n - P k + 1 | k ) C k T {\
    displaystyle {\begin{aligned}{\hat {\mathbf {x} }}_{k\mid n}&={\
    hat {\mathbf {x} }}_{k\mid k}+\mathbf {C} _{k}\left({\hat {\
    mathbf {x} }}_{k+1\mid n}-{\hat {\mathbf {x} }}_{k+1\mid k}\
    right)\\\mathbf {P} _{k\mid n}&=\mathbf {P} _{k\mid k}+\mathbf
    {C} _{k}\left(\mathbf {P} _{k+1\mid n}-\mathbf {P} _{k+1\mid k}\
    right)\mathbf {C} _{k}^{\textsf {T}}\end{aligned}}} {\
    displaystyle {\begin{aligned}{\hat {\mathbf {x} }}_{k\mid n}&={\
    hat {\mathbf {x} }}_{k\mid k}+\mathbf {C} _{k}\left({\hat {\
    mathbf {x} }}_{k+1\mid n}-{\hat {\mathbf {x} }}_{k+1\mid k}\
    right)\\\mathbf {P} _{k\mid n}&=\mathbf {P} _{k\mid k}+\mathbf
    {C} _{k}\left(\mathbf {P} _{k+1\mid n}-\mathbf {P} _{k+1\mid k}\
    right)\mathbf {C} _{k}^{\textsf {T}}\end{aligned}}}

where

    C k = P k | k F k + 1 T P k + 1 | k - 1 . {\displaystyle \mathbf
    {C} _{k}=\mathbf {P} _{k\mid k}\mathbf {F} _{k+1}^{\textsf {T}}\
    mathbf {P} _{k+1\mid k}^{-1}.} {\displaystyle \mathbf {C} _{k}=\
    mathbf {P} _{k\mid k}\mathbf {F} _{k+1}^{\textsf {T}}\mathbf {P}
    _{k+1\mid k}^{-1}.}

x k | k {\displaystyle \mathbf {x} _{k\mid k}} {\displaystyle \mathbf
{x} _{k\mid k}} is the a-posteriori state estimate of timestep k {\
displaystyle k} k and x k + 1 | k {\displaystyle \mathbf {x} _{k+1\
mid k}} {\displaystyle \mathbf {x} _{k+1\mid k}} is the a-priori
state estimate of timestep k + 1 {\displaystyle k+1} k+1. The same
notation applies to the covariance.

Modified Bryson-Frazier smoother[edit]

An alternative to the RTS algorithm is the modified Bryson-Frazier
(MBF) fixed interval smoother developed by Bierman.^[36] This also
uses a backward pass that processes data saved from the Kalman filter
forward pass. The equations for the backward pass involve the
recursive computation of data which are used at each observation time
to compute the smoothed state and covariance.

The recursive equations are

    L ~ k = H k T S k - 1 H k + C ^ k T L ^ k C ^ k L ^ k - 1 = F k T
    L ~ k F k L ^ n = 0 l ~ k = - H k T S k - 1 y k + C ^ k T l ^ k l
    ^ k - 1 = F k T l ~ k l ^ n = 0 {\displaystyle {\begin{aligned}{\
    tilde {\Lambda }}_{k}&=\mathbf {H} _{k}^{\textsf {T}}\mathbf {S}
    _{k}^{-1}\mathbf {H} _{k}+{\hat {\mathbf {C} }}_{k}^{\textsf {T}}
    {\hat {\Lambda }}_{k}{\hat {\mathbf {C} }}_{k}\\{\hat {\Lambda }}
    _{k-1}&=\mathbf {F} _{k}^{\textsf {T}}{\tilde {\Lambda }}_{k}\
    mathbf {F} _{k}\\{\hat {\Lambda }}_{n}&=0\\{\tilde {\lambda }}_
    {k}&=-\mathbf {H} _{k}^{\textsf {T}}\mathbf {S} _{k}^{-1}\mathbf
    {y} _{k}+{\hat {\mathbf {C} }}_{k}^{\textsf {T}}{\hat {\lambda }}
    _{k}\\{\hat {\lambda }}_{k-1}&=\mathbf {F} _{k}^{\textsf {T}}{\
    tilde {\lambda }}_{k}\\{\hat {\lambda }}_{n}&=0\end{aligned}}} {\
    displaystyle {\begin{aligned}{\tilde {\Lambda }}_{k}&=\mathbf {H}
    _{k}^{\textsf {T}}\mathbf {S} _{k}^{-1}\mathbf {H} _{k}+{\hat {\
    mathbf {C} }}_{k}^{\textsf {T}}{\hat {\Lambda }}_{k}{\hat {\
    mathbf {C} }}_{k}\\{\hat {\Lambda }}_{k-1}&=\mathbf {F} _{k}^{\
    textsf {T}}{\tilde {\Lambda }}_{k}\mathbf {F} _{k}\\{\hat {\
    Lambda }}_{n}&=0\\{\tilde {\lambda }}_{k}&=-\mathbf {H} _{k}^{\
    textsf {T}}\mathbf {S} _{k}^{-1}\mathbf {y} _{k}+{\hat {\mathbf
    {C} }}_{k}^{\textsf {T}}{\hat {\lambda }}_{k}\\{\hat {\lambda }}_
    {k-1}&=\mathbf {F} _{k}^{\textsf {T}}{\tilde {\lambda }}_{k}\\{\
    hat {\lambda }}_{n}&=0\end{aligned}}}

where S k {\displaystyle \mathbf {S} _{k}} {\mathbf {S}}_{k} is the
residual covariance and C ^ k = I - K k H k {\displaystyle {\hat {\
mathbf {C} }}_{k}=\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}} {\
displaystyle {\hat {\mathbf {C} }}_{k}=\mathbf {I} -\mathbf {K} _{k}\
mathbf {H} _{k}}. The smoothed state and covariance can then be found
by substitution in the equations

    P k | n = P k | k - P k | k L ^ k P k | k x k | n = x k | k - P k
    | k l ^ k {\displaystyle {\begin{aligned}\mathbf {P} _{k\mid n}&=
    \mathbf {P} _{k\mid k}-\mathbf {P} _{k\mid k}{\hat {\Lambda }}_
    {k}\mathbf {P} _{k\mid k}\\\mathbf {x} _{k\mid n}&=\mathbf {x} _
    {k\mid k}-\mathbf {P} _{k\mid k}{\hat {\lambda }}_{k}\end
    {aligned}}} {\displaystyle {\begin{aligned}\mathbf {P} _{k\mid n}
    &=\mathbf {P} _{k\mid k}-\mathbf {P} _{k\mid k}{\hat {\Lambda }}_
    {k}\mathbf {P} _{k\mid k}\\\mathbf {x} _{k\mid n}&=\mathbf {x} _
    {k\mid k}-\mathbf {P} _{k\mid k}{\hat {\lambda }}_{k}\end
    {aligned}}}

or

    P k | n = P k | k - 1 - P k | k - 1 L ~ k P k | k - 1 x k | n = x
    k | k - 1 - P k | k - 1 l ~ k . {\displaystyle {\begin{aligned}\
    mathbf {P} _{k\mid n}&=\mathbf {P} _{k\mid k-1}-\mathbf {P} _{k\
    mid k-1}{\tilde {\Lambda }}_{k}\mathbf {P} _{k\mid k-1}\\\mathbf
    {x} _{k\mid n}&=\mathbf {x} _{k\mid k-1}-\mathbf {P} _{k\mid k-1}
    {\tilde {\lambda }}_{k}.\end{aligned}}} {\displaystyle {\begin
    {aligned}\mathbf {P} _{k\mid n}&=\mathbf {P} _{k\mid k-1}-\mathbf
    {P} _{k\mid k-1}{\tilde {\Lambda }}_{k}\mathbf {P} _{k\mid k-1}\\
    \mathbf {x} _{k\mid n}&=\mathbf {x} _{k\mid k-1}-\mathbf {P} _{k\
    mid k-1}{\tilde {\lambda }}_{k}.\end{aligned}}}

An important advantage of the MBF is that it does not require finding
the inverse of the covariance matrix.

Minimum-variance smoother[edit]

The minimum-variance smoother can attain the best-possible error
performance, provided that the models are linear, their parameters
and the noise statistics are known precisely.^[47] This smoother is a
time-varying state-space generalization of the optimal non-causal
Wiener filter.

The smoother calculations are done in two passes. The forward
calculations involve a one-step-ahead predictor and are given by

    x ^ k + 1 | k = ( F k - K k H k ) x ^ k | k - 1 + K k z k a k = -
    S k - 1 2 H k x ^ k | k - 1 + S k - 1 2 z k {\displaystyle {\
    begin{aligned}{\hat {\mathbf {x} }}_{k+1\mid k}&=(\mathbf {F} _
    {k}-\mathbf {K} _{k}\mathbf {H} _{k}){\hat {\mathbf {x} }}_{k\mid
    k-1}+\mathbf {K} _{k}\mathbf {z} _{k}\\\alpha _{k}&=-\mathbf {S}
    _{k}^{-{\frac {1}{2}}}\mathbf {H} _{k}{\hat {\mathbf {x} }}_{k\
    mid k-1}+\mathbf {S} _{k}^{-{\frac {1}{2}}}\mathbf {z} _{k}\end
    {aligned}}} {\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}_
    {k+1\mid k}&=(\mathbf {F} _{k}-\mathbf {K} _{k}\mathbf {H} _{k})
    {\hat {\mathbf {x} }}_{k\mid k-1}+\mathbf {K} _{k}\mathbf {z} _
    {k}\\\alpha _{k}&=-\mathbf {S} _{k}^{-{\frac {1}{2}}}\mathbf {H}
    _{k}{\hat {\mathbf {x} }}_{k\mid k-1}+\mathbf {S} _{k}^{-{\frac
    {1}{2}}}\mathbf {z} _{k}\end{aligned}}}

The above system is known as the inverse Wiener-Hopf factor. The
backward recursion is the adjoint of the above forward system. The
result of the backward pass b k {\displaystyle \beta _{k}} \beta _{k}
may be calculated by operating the forward equations on the
time-reversed a k {\displaystyle \alpha _{k}} \alpha _{k} and time
reversing the result. In the case of output estimation, the smoothed
estimate is given by

    y ^ k | N = z k - R k b k {\displaystyle {\hat {\mathbf {y} }}_{k
    \mid N}=\mathbf {z} _{k}-\mathbf {R} _{k}\beta _{k}} {\
    displaystyle {\hat {\mathbf {y} }}_{k\mid N}=\mathbf {z} _{k}-\
    mathbf {R} _{k}\beta _{k}}

Taking the causal part of this minimum-variance smoother yields

    y ^ k | k = z k - R k S k - 1 2 a k {\displaystyle {\hat {\mathbf
    {y} }}_{k\mid k}=\mathbf {z} _{k}-\mathbf {R} _{k}\mathbf {S} _
    {k}^{-{\frac {1}{2}}}\alpha _{k}} {\displaystyle {\hat {\mathbf
    {y} }}_{k\mid k}=\mathbf {z} _{k}-\mathbf {R} _{k}\mathbf {S} _
    {k}^{-{\frac {1}{2}}}\alpha _{k}}

which is identical to the minimum-variance Kalman filter. The above
solutions minimize the variance of the output estimation error. Note
that the Rauch-Tung-Striebel smoother derivation assumes that the
underlying distributions are Gaussian, whereas the minimum-variance
solutions do not. Optimal smoothers for state estimation and input
estimation can be constructed similarly.

A continuous-time version of the above smoother is described in.^[48]
^[49]

Expectation-maximization algorithms may be employed to calculate
approximate maximum likelihood estimates of unknown state-space
parameters within minimum-variance filters and smoothers. Often
uncertainties remain within problem assumptions. A smoother that
accommodates uncertainties can be designed by adding a positive
definite term to the Riccati equation.^[50]

In cases where the models are nonlinear, step-wise linearizations may
be within the minimum-variance filter and smoother recursions (
extended Kalman filtering).

Frequency-weighted Kalman filters[edit]

Pioneering research on the perception of sounds at different
frequencies was conducted by Fletcher and Munson in the 1930s. Their
work led to a standard way of weighting measured sound levels within
investigations of industrial noise and hearing loss. Frequency
weightings have since been used within filter and controller designs
to manage performance within bands of interest.

Typically, a frequency shaping function is used to weight the average
power of the error spectral density in a specified frequency band.
Let y - y ^ {\displaystyle \mathbf {y} -{\hat {\mathbf {y} }}} {\
displaystyle \mathbf {y} -{\hat {\mathbf {y} }}} denote the output
estimation error exhibited by a conventional Kalman filter. Also, let
W {\displaystyle \mathbf {W} } \mathbf {W} denote a causal frequency
weighting transfer function. The optimum solution which minimizes the
variance of W ( y - y ^ ) {\displaystyle \mathbf {W} \left(\mathbf
{y} -{\hat {\mathbf {y} }}\right)} {\displaystyle \mathbf {W} \left(\
mathbf {y} -{\hat {\mathbf {y} }}\right)} arises by simply
constructing W - 1 y ^ {\displaystyle \mathbf {W} ^{-1}{\hat {\mathbf
{y} }}} {\displaystyle \mathbf {W} ^{-1}{\hat {\mathbf {y} }}}.

The design of W {\displaystyle \mathbf {W} } \mathbf {W} remains an
open question. One way of proceeding is to identify a system which
generates the estimation error and setting W {\displaystyle \mathbf
{W} } \mathbf {W} equal to the inverse of that system.^[51] This
procedure may be iterated to obtain mean-square error improvement at
the cost of increased filter order. The same technique can be applied
to smoothers.

Nonlinear filters[edit]

The basic Kalman filter is limited to a linear assumption. More
complex systems, however, can be nonlinear. The nonlinearity can be
associated either with the process model or with the observation
model or with both.

The most common variants of Kalman filters for non-linear systems are
the Extended Kalman Filter and Unscented Kalman filter. The
suitability of which filter to use depends on the non-linearity
indices of the process and observation model.^[52]

Extended Kalman filter[edit]

Main article: Extended Kalman filter

In the extended Kalman filter (EKF), the state transition and
observation models need not be linear functions of the state but may
instead be nonlinear functions. These functions are of differentiable
type.

    x k = f ( x k - 1 , u k ) + w k z k = h ( x k ) + v k {\
    displaystyle {\begin{aligned}\mathbf {x} _{k}&=f(\mathbf {x} _
    {k-1},\mathbf {u} _{k})+\mathbf {w} _{k}\\\mathbf {z} _{k}&=h(\
    mathbf {x} _{k})+\mathbf {v} _{k}\end{aligned}}} {\displaystyle
    {\begin{aligned}\mathbf {x} _{k}&=f(\mathbf {x} _{k-1},\mathbf
    {u} _{k})+\mathbf {w} _{k}\\\mathbf {z} _{k}&=h(\mathbf {x} _{k})
    +\mathbf {v} _{k}\end{aligned}}}

The function f can be used to compute the predicted state from the
previous estimate and similarly the function h can be used to compute
the predicted measurement from the predicted state. However, f and h
cannot be applied to the covariance directly. Instead a matrix of
partial derivatives (the Jacobian) is computed.

At each timestep the Jacobian is evaluated with current predicted
states. These matrices can be used in the Kalman filter equations.
This process essentially linearizes the nonlinear function around the
current estimate.

Unscented Kalman filter[edit]

When the state transition and observation models--that is, the predict
and update functions f {\displaystyle f} f and h {\displaystyle h} h
--are highly nonlinear, the extended Kalman filter can give
particularly poor performance.^[53] This is because the covariance is
propagated through linearization of the underlying nonlinear model.
The unscented Kalman filter (UKF) ^[53] uses a deterministic sampling
technique known as the unscented transformation (UT) to pick a
minimal set of sample points (called sigma points) around the mean.
The sigma points are then propagated through the nonlinear functions,
from which a new mean and covariance estimate are then formed. The
resulting filter depends on how the transformed statistics of the UT
are calculated and which set of sigma points are used. It should be
remarked that it is always possible to construct new UKFs in a
consistent way.^[54] For certain systems, the resulting UKF more
accurately estimates the true mean and covariance.^[55] This can be
verified with Monte Carlo sampling or Taylor series expansion of the
posterior statistics. In addition, this technique removes the
requirement to explicitly calculate Jacobians, which for complex
functions can be a difficult task in itself (i.e., requiring
complicated derivatives if done analytically or being computationally
costly if done numerically), if not impossible (if those functions
are not differentiable).

Sigma points[edit]

For a random vector x = ( x 1 , ... , x L ) {\displaystyle \mathbf {x}
=(x_{1},\dots ,x_{L})} {\displaystyle \mathbf {x} =(x_{1},\dots ,x_
{L})}, sigma points are any set of vectors

    { s 0 , ... , s N } = { ( s 0 , 1 s 0 , 2 ... s 0 , L ) , ... , ( s N ,
    1 s N , 2 ... s N , L ) } {\displaystyle \{\mathbf {s} _{0},\dots ,
    \mathbf {s} _{N}\}={\bigl \{}{\begin{pmatrix}s_{0,1}&s_{0,2}&\
    ldots &s_{0,L}\end{pmatrix}},\dots ,{\begin{pmatrix}s_{N,1}&s_
    {N,2}&\ldots &s_{N,L}\end{pmatrix}}{\bigr \}}} {\displaystyle \{\
    mathbf {s} _{0},\dots ,\mathbf {s} _{N}\}={\bigl \{}{\begin
    {pmatrix}s_{0,1}&s_{0,2}&\ldots &s_{0,L}\end{pmatrix}},\dots ,{\
    begin{pmatrix}s_{N,1}&s_{N,2}&\ldots &s_{N,L}\end{pmatrix}}{\bigr
    \}}}

attributed with

  * first-order weights W 0 a , ... , W N a {\displaystyle W_{0}^{a},\
    dots ,W_{N}^{a}} {\displaystyle W_{0}^{a},\dots ,W_{N}^{a}} that
    fulfill

 1. [?] j = 0 N W j a = 1 {\displaystyle \sum _{j=0}^{N}W_{j}^{a}=1} {\
    displaystyle \sum _{j=0}^{N}W_{j}^{a}=1}
 2. for all i = 1 , ... , L {\displaystyle i=1,\dots ,L} {\displaystyle
    i=1,\dots ,L}: E [ x i ] = [?] j = 0 N W j a s j , i {\displaystyle
    E[x_{i}]=\sum _{j=0}^{N}W_{j}^{a}s_{j,i}} {\displaystyle E[x_{i}]
    =\sum _{j=0}^{N}W_{j}^{a}s_{j,i}}

  * second-order weights W 0 c , ... , W N c {\displaystyle W_{0}^{c},\
    dots ,W_{N}^{c}} {\displaystyle W_{0}^{c},\dots ,W_{N}^{c}} that
    fulfill

 1. [?] j = 0 N W j c = 1 {\displaystyle \sum _{j=0}^{N}W_{j}^{c}=1} {\
    displaystyle \sum _{j=0}^{N}W_{j}^{c}=1}
 2. for all pairs ( i , l ) [?] { 1 , ... , L } 2 : E [ x i x l ] = [?] j =
    0 N W j c s j , i s j , l {\displaystyle (i,l)\in \{1,\dots ,L\}^
    {2}:E[x_{i}x_{l}]=\sum _{j=0}^{N}W_{j}^{c}s_{j,i}s_{j,l}} {\
    displaystyle (i,l)\in \{1,\dots ,L\}^{2}:E[x_{i}x_{l}]=\sum _{j=
    0}^{N}W_{j}^{c}s_{j,i}s_{j,l}}.

A simple choice of sigma points and weights for x k - 1 | k - 1 {\
displaystyle \mathbf {x} _{k-1\mid k-1}} {\displaystyle \mathbf {x} _
{k-1\mid k-1}} in the UKF algorithm is

    s 0 = x ^ k - 1 | k - 1 - 1 < W 0 a = W 0 c < 1 s j = x ^ k - 1 |
    k - 1 + L 1 - W 0 A j , j = 1 , ... , L s L + j = x ^ k - 1 | k - 1
    - L 1 - W 0 A j , j = 1 , ... , L W j a = W j c = 1 - W 0 2 L , j =
    1 , ... , 2 L {\displaystyle {\begin{aligned}\mathbf {s} _{0}&={\
    hat {\mathbf {x} }}_{k-1\mid k-1}\\-1&<W_{0}^{a}=W_{0}^{c}<1\\\
    mathbf {s} _{j}&={\hat {\mathbf {x} }}_{k-1\mid k-1}+{\sqrt {\
    frac {L}{1-W_{0}}}}\mathbf {A} _{j},\quad j=1,\dots ,L\\\mathbf
    {s} _{L+j}&={\hat {\mathbf {x} }}_{k-1\mid k-1}-{\sqrt {\frac {L}
    {1-W_{0}}}}\mathbf {A} _{j},\quad j=1,\dots ,L\\W_{j}^{a}&=W_{j}^
    {c}={\frac {1-W_{0}}{2L}},\quad j=1,\dots ,2L\end{aligned}}} {\
    displaystyle {\begin{aligned}\mathbf {s} _{0}&={\hat {\mathbf {x}
    }}_{k-1\mid k-1}\\-1&<W_{0}^{a}=W_{0}^{c}<1\\\mathbf {s} _{j}&={\
    hat {\mathbf {x} }}_{k-1\mid k-1}+{\sqrt {\frac {L}{1-W_{0}}}}\
    mathbf {A} _{j},\quad j=1,\dots ,L\\\mathbf {s} _{L+j}&={\hat {\
    mathbf {x} }}_{k-1\mid k-1}-{\sqrt {\frac {L}{1-W_{0}}}}\mathbf
    {A} _{j},\quad j=1,\dots ,L\\W_{j}^{a}&=W_{j}^{c}={\frac {1-W_
    {0}}{2L}},\quad j=1,\dots ,2L\end{aligned}}}

where x ^ k - 1 | k - 1 {\displaystyle {\hat {\mathbf {x} }}_{k-1\mid
k-1}} {\displaystyle {\hat {\mathbf {x} }}_{k-1\mid k-1}} is the mean
estimate of x k - 1 | k - 1 {\displaystyle \mathbf {x} _{k-1\mid
k-1}} {\displaystyle \mathbf {x} _{k-1\mid k-1}}. The vector A j {\
displaystyle \mathbf {A} _{j}} {\displaystyle \mathbf {A} _{j}} is
the jth column of A {\displaystyle \mathbf {A} } \mathbf {A} where P
k - 1 | k - 1 = A A T {\displaystyle \mathbf {P} _{k-1\mid k-1}=\
mathbf {AA} ^{\textsf {T}}} {\displaystyle \mathbf {P} _{k-1\mid k-1}
=\mathbf {AA} ^{\textsf {T}}}. The matrix A {\displaystyle \mathbf
{A} } \mathbf {A} should be calculated using numerically efficient
and stable methods such as the Cholesky decomposition. The weight of
the mean value, W 0 {\displaystyle W_{0}} W_{0}, can be chosen
arbitrarily.

Another popular parameterization (which generalizes the above) is

    s 0 = x ^ k - 1 | k - 1 W 0 a = a 2 k - L a 2 k W 0 c = W 0 a + 1
    - a 2 + b s j = x ^ k - 1 | k - 1 + a k A j , j = 1 , ... , L s L +
    j = x ^ k - 1 | k - 1 - a k A j , j = 1 , ... , L W j a = W j c = 1
    2 a 2 k , j = 1 , ... , 2 L . {\displaystyle {\begin{aligned}\
    mathbf {s} _{0}&={\hat {\mathbf {x} }}_{k-1\mid k-1}\\W_{0}^{a}&=
    {\frac {\alpha ^{2}\kappa -L}{\alpha ^{2}\kappa }}\\W_{0}^{c}&=W_
    {0}^{a}+1-\alpha ^{2}+\beta \\\mathbf {s} _{j}&={\hat {\mathbf
    {x} }}_{k-1\mid k-1}+\alpha {\sqrt {\kappa }}\mathbf {A} _{j},\
    quad j=1,\dots ,L\\\mathbf {s} _{L+j}&={\hat {\mathbf {x} }}_{k-1
    \mid k-1}-\alpha {\sqrt {\kappa }}\mathbf {A} _{j},\quad j=1,\
    dots ,L\\W_{j}^{a}&=W_{j}^{c}={\frac {1}{2\alpha ^{2}\kappa }},\
    quad j=1,\dots ,2L.\end{aligned}}} {\displaystyle {\begin
    {aligned}\mathbf {s} _{0}&={\hat {\mathbf {x} }}_{k-1\mid k-1}\\
    W_{0}^{a}&={\frac {\alpha ^{2}\kappa -L}{\alpha ^{2}\kappa }}\\W_
    {0}^{c}&=W_{0}^{a}+1-\alpha ^{2}+\beta \\\mathbf {s} _{j}&={\hat
    {\mathbf {x} }}_{k-1\mid k-1}+\alpha {\sqrt {\kappa }}\mathbf {A}
    _{j},\quad j=1,\dots ,L\\\mathbf {s} _{L+j}&={\hat {\mathbf {x}
    }}_{k-1\mid k-1}-\alpha {\sqrt {\kappa }}\mathbf {A} _{j},\quad j
    =1,\dots ,L\\W_{j}^{a}&=W_{j}^{c}={\frac {1}{2\alpha ^{2}\kappa
    }},\quad j=1,\dots ,2L.\end{aligned}}}

a {\displaystyle \alpha } \alpha and k {\displaystyle \kappa } \kappa
control the spread of the sigma points. b {\displaystyle \beta } \
beta is related to the distribution of x {\displaystyle x} x.

Appropriate values depend on the problem at hand, but a typical
recommendation is a = 10 - 3 {\displaystyle \alpha =10^{-3}} {\
displaystyle \alpha =10^{-3}}, k = 1 {\displaystyle \kappa =1} \kappa
=1, and b = 2 {\displaystyle \beta =2} {\displaystyle \beta =2}.
However, a larger value of a {\displaystyle \alpha } \alpha (e.g., a
= 1 {\displaystyle \alpha =1} \alpha =1) may be beneficial in order
to better capture the spread of the distribution and possible
nonlinearities.^[56] If the true distribution of x {\displaystyle x} 
x is Gaussian, b = 2 {\displaystyle \beta =2} {\displaystyle \beta =
2} is optimal.^[57]

Predict[edit]

As with the EKF, the UKF prediction can be used independently from
the UKF update, in combination with a linear (or indeed EKF) update,
or vice versa.

Given estimates of the mean and covariance, x ^ k - 1 | k - 1 {\
displaystyle {\hat {\mathbf {x} }}_{k-1\mid k-1}} {\displaystyle {\
hat {\mathbf {x} }}_{k-1\mid k-1}} and P k - 1 | k - 1 {\displaystyle
\mathbf {P} _{k-1\mid k-1}} {\displaystyle \mathbf {P} _{k-1\mid
k-1}}, one obtains N = 2 L + 1 {\displaystyle N=2L+1} {\displaystyle
N=2L+1} sigma points as described in the section above. The sigma
points are propagated through the transition function f.

    x j = f ( s j ) j = 0 , ... , 2 L {\displaystyle \mathbf {x} _{j}=f
    \left(\mathbf {s} _{j}\right)\quad j=0,\dots ,2L} {\displaystyle
    \mathbf {x} _{j}=f\left(\mathbf {s} _{j}\right)\quad j=0,\dots
    ,2L}.

The propagated sigma points are weighed to produce the predicted mean
and covariance.

    x ^ k | k - 1 = [?] j = 0 2 L W j a x j P k | k - 1 = [?] j = 0 2 L W
    j c ( x j - x ^ k | k - 1 ) ( x j - x ^ k | k - 1 ) T + Q k {\
    displaystyle {\begin{aligned}{\hat {\mathbf {x} }}_{k\mid k-1}&=\
    sum _{j=0}^{2L}W_{j}^{a}\mathbf {x} _{j}\\\mathbf {P} _{k\mid
    k-1}&=\sum _{j=0}^{2L}W_{j}^{c}\left(\mathbf {x} _{j}-{\hat {\
    mathbf {x} }}_{k\mid k-1}\right)\left(\mathbf {x} _{j}-{\hat {\
    mathbf {x} }}_{k\mid k-1}\right)^{\textsf {T}}+\mathbf {Q} _{k}\
    end{aligned}}} {\displaystyle {\begin{aligned}{\hat {\mathbf {x}
    }}_{k\mid k-1}&=\sum _{j=0}^{2L}W_{j}^{a}\mathbf {x} _{j}\\\
    mathbf {P} _{k\mid k-1}&=\sum _{j=0}^{2L}W_{j}^{c}\left(\mathbf
    {x} _{j}-{\hat {\mathbf {x} }}_{k\mid k-1}\right)\left(\mathbf
    {x} _{j}-{\hat {\mathbf {x} }}_{k\mid k-1}\right)^{\textsf {T}}+\
    mathbf {Q} _{k}\end{aligned}}}

where W j a {\displaystyle W_{j}^{a}} {\displaystyle W_{j}^{a}} are
the first-order weights of the original sigma points, and W j c {\
displaystyle W_{j}^{c}} {\displaystyle W_{j}^{c}} are the
second-order weights. The matrix Q k {\displaystyle \mathbf {Q} _{k}}
{\displaystyle \mathbf {Q} _{k}} is the covariance of the transition
noise, w k {\displaystyle \mathbf {w} _{k}} {\mathbf {w}}_{k}.

Update[edit]

Given prediction estimates x ^ k | k - 1 {\displaystyle {\hat {\
mathbf {x} }}_{k\mid k-1}} {\displaystyle {\hat {\mathbf {x} }}_{k\
mid k-1}} and P k | k - 1 {\displaystyle \mathbf {P} _{k\mid k-1}} {\
displaystyle \mathbf {P} _{k\mid k-1}}, a new set of N = 2 L + 1 {\
displaystyle N=2L+1} {\displaystyle N=2L+1} sigma points s 0 , ... , s
2 L {\displaystyle \mathbf {s} _{0},\dots ,\mathbf {s} _{2L}} {\
displaystyle \mathbf {s} _{0},\dots ,\mathbf {s} _{2L}} with
corresponding first-order weights W 0 a , ... W 2 L a {\displaystyle W_
{0}^{a},\dots W_{2L}^{a}} {\displaystyle W_{0}^{a},\dots W_{2L}^{a}}
and second-order weights W 0 c , ... , W 2 L c {\displaystyle W_{0}^
{c},\dots ,W_{2L}^{c}} {\displaystyle W_{0}^{c},\dots ,W_{2L}^{c}} is
calculated.^[58] These sigma points are transformed through h {\
displaystyle h} h.

    z j = h ( s j ) , j = 0 , 1 , ... , 2 L {\displaystyle \mathbf {z}
    _{j}=h(\mathbf {s} _{j}),\,\,j=0,1,\dots ,2L} {\displaystyle \
    mathbf {z} _{j}=h(\mathbf {s} _{j}),\,\,j=0,1,\dots ,2L}.

Then the empirical mean and covariance of the transformed points are
calculated.

    z ^ = [?] j = 0 2 L W j a z j S ^ k = [?] j = 0 2 L W j c ( z j - z ^
    ) ( z j - z ^ ) T + R k {\displaystyle {\begin{aligned}{\hat {\
    mathbf {z} }}&=\sum _{j=0}^{2L}W_{j}^{a}\mathbf {z} _{j}\\[6pt]{\
    hat {\mathbf {S} }}_{k}&=\sum _{j=0}^{2L}W_{j}^{c}(\mathbf {z} _
    {j}-{\hat {\mathbf {z} }})(\mathbf {z} _{j}-{\hat {\mathbf {z}
    }})^{\textsf {T}}+\mathbf {R} _{k}\end{aligned}}} {\displaystyle
    {\begin{aligned}{\hat {\mathbf {z} }}&=\sum _{j=0}^{2L}W_{j}^{a}\
    mathbf {z} _{j}\\[6pt]{\hat {\mathbf {S} }}_{k}&=\sum _{j=0}^{2L}
    W_{j}^{c}(\mathbf {z} _{j}-{\hat {\mathbf {z} }})(\mathbf {z} _
    {j}-{\hat {\mathbf {z} }})^{\textsf {T}}+\mathbf {R} _{k}\end
    {aligned}}}

where R k {\displaystyle \mathbf {R} _{k}} \mathbf {R} _{k} is the
covariance matrix of the observation noise, v k {\displaystyle \
mathbf {v} _{k}} {\mathbf {v}}_{k}. Additionally, the cross
covariance matrix is also needed

    C s z = [?] j = 0 2 L W j c ( s j - x ^ k | k - 1 ) ( z j - z ^ ) T
    {\displaystyle {\begin{aligned}\mathbf {C_{sz}} &=\sum _{j=0}^
    {2L}W_{j}^{c}(\mathbf {s} _{j}-{\hat {\mathbf {x} }}_{k|k-1})(\
    mathbf {z} _{j}-{\hat {\mathbf {z} }})^{\textsf {T}}\end
    {aligned}}} {\displaystyle {\begin{aligned}\mathbf {C_{sz}} &=\
    sum _{j=0}^{2L}W_{j}^{c}(\mathbf {s} _{j}-{\hat {\mathbf {x} }}_
    {k|k-1})(\mathbf {z} _{j}-{\hat {\mathbf {z} }})^{\textsf {T}}\
    end{aligned}}}

where s j {\displaystyle \mathbf {s} _{j}} {\mathbf {s}}_{j} are the
untransformed sigma points created from x ^ k | k - 1 {\displaystyle
{\hat {\mathbf {x} }}_{k\mid k-1}} {\displaystyle {\hat {\mathbf {x}
}}_{k\mid k-1}} and P k | k - 1 {\displaystyle \mathbf {P} _{k\mid
k-1}} {\displaystyle \mathbf {P} _{k\mid k-1}}.

The Kalman gain is

    K k = C s z S ^ k - 1 . {\displaystyle {\begin{aligned}\mathbf
    {K} _{k}=\mathbf {C_{sz}} {\hat {\mathbf {S} }}_{k}^{-1}.\end
    {aligned}}} {\displaystyle {\begin{aligned}\mathbf {K} _{k}=\
    mathbf {C_{sz}} {\hat {\mathbf {S} }}_{k}^{-1}.\end{aligned}}}

The updated mean and covariance estimates are

    x ^ k | k = x ^ k | k - 1 + K k ( z k - z ^ ) P k | k = P k | k -
    1 - K k S ^ k K k T . {\displaystyle {\begin{aligned}{\hat {\
    mathbf {x} }}_{k\mid k}&={\hat {\mathbf {x} }}_{k|k-1}+\mathbf
    {K} _{k}(\mathbf {z} _{k}-{\hat {\mathbf {z} }})\\\mathbf {P} _{k
    \mid k}&=\mathbf {P} _{k\mid k-1}-\mathbf {K} _{k}{\hat {\mathbf
    {S} }}_{k}\mathbf {K} _{k}^{\textsf {T}}.\end{aligned}}} {\
    displaystyle {\begin{aligned}{\hat {\mathbf {x} }}_{k\mid k}&={\
    hat {\mathbf {x} }}_{k|k-1}+\mathbf {K} _{k}(\mathbf {z} _{k}-{\
    hat {\mathbf {z} }})\\\mathbf {P} _{k\mid k}&=\mathbf {P} _{k\mid
    k-1}-\mathbf {K} _{k}{\hat {\mathbf {S} }}_{k}\mathbf {K} _{k}^{\
    textsf {T}}.\end{aligned}}}

Kalman-Bucy filter[edit]

The Kalman-Bucy filter (named after Richard Snowden Bucy) is a
continuous time version of the Kalman filter.^[59]^[60]

It is based on the state space model

    d d t x ( t ) = F ( t ) x ( t ) + B ( t ) u ( t ) + w ( t ) z ( t
    ) = H ( t ) x ( t ) + v ( t ) {\displaystyle {\begin{aligned}{\
    frac {d}{dt}}\mathbf {x} (t)&=\mathbf {F} (t)\mathbf {x} (t)+\
    mathbf {B} (t)\mathbf {u} (t)+\mathbf {w} (t)\\\mathbf {z} (t)&=\
    mathbf {H} (t)\mathbf {x} (t)+\mathbf {v} (t)\end{aligned}}} {\
    displaystyle {\begin{aligned}{\frac {d}{dt}}\mathbf {x} (t)&=\
    mathbf {F} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t)+\
    mathbf {w} (t)\\\mathbf {z} (t)&=\mathbf {H} (t)\mathbf {x} (t)+\
    mathbf {v} (t)\end{aligned}}}

where Q ( t ) {\displaystyle \mathbf {Q} (t)} \mathbf {Q} (t) and R (
t ) {\displaystyle \mathbf {R} (t)} \mathbf {R} (t) represent the
intensities (or, more accurately: the Power Spectral Density - PSD -
matrices) of the two white noise terms w ( t ) {\displaystyle \mathbf
{w} (t)} \mathbf {w} (t) and v ( t ) {\displaystyle \mathbf {v} (t)} 
\mathbf {v} (t), respectively.

The filter consists of two differential equations, one for the state
estimate and one for the covariance:

    d d t x ^ ( t ) = F ( t ) x ^ ( t ) + B ( t ) u ( t ) + K ( t ) (
    z ( t ) - H ( t ) x ^ ( t ) ) d d t P ( t ) = F ( t ) P ( t ) + P
    ( t ) F T ( t ) + Q ( t ) - K ( t ) R ( t ) K T ( t ) {\
    displaystyle {\begin{aligned}{\frac {d}{dt}}{\hat {\mathbf {x} }}
    (t)&=\mathbf {F} (t){\hat {\mathbf {x} }}(t)+\mathbf {B} (t)\
    mathbf {u} (t)+\mathbf {K} (t)\left(\mathbf {z} (t)-\mathbf {H}
    (t){\hat {\mathbf {x} }}(t)\right)\\{\frac {d}{dt}}\mathbf {P}
    (t)&=\mathbf {F} (t)\mathbf {P} (t)+\mathbf {P} (t)\mathbf {F} ^
    {\textsf {T}}(t)+\mathbf {Q} (t)-\mathbf {K} (t)\mathbf {R} (t)\
    mathbf {K} ^{\textsf {T}}(t)\end{aligned}}} {\displaystyle {\
    begin{aligned}{\frac {d}{dt}}{\hat {\mathbf {x} }}(t)&=\mathbf
    {F} (t){\hat {\mathbf {x} }}(t)+\mathbf {B} (t)\mathbf {u} (t)+\
    mathbf {K} (t)\left(\mathbf {z} (t)-\mathbf {H} (t){\hat {\mathbf
    {x} }}(t)\right)\\{\frac {d}{dt}}\mathbf {P} (t)&=\mathbf {F} (t)
    \mathbf {P} (t)+\mathbf {P} (t)\mathbf {F} ^{\textsf {T}}(t)+\
    mathbf {Q} (t)-\mathbf {K} (t)\mathbf {R} (t)\mathbf {K} ^{\
    textsf {T}}(t)\end{aligned}}}

where the Kalman gain is given by

    K ( t ) = P ( t ) H T ( t ) R - 1 ( t ) {\displaystyle \mathbf
    {K} (t)=\mathbf {P} (t)\mathbf {H} ^{\textsf {T}}(t)\mathbf {R} ^
    {-1}(t)} {\displaystyle \mathbf {K} (t)=\mathbf {P} (t)\mathbf
    {H} ^{\textsf {T}}(t)\mathbf {R} ^{-1}(t)}

Note that in this expression for K ( t ) {\displaystyle \mathbf {K}
(t)} \mathbf {K} (t) the covariance of the observation noise R ( t )
{\displaystyle \mathbf {R} (t)} \mathbf {R} (t) represents at the
same time the covariance of the prediction error (or innovation) y ~
( t ) = z ( t ) - H ( t ) x ^ ( t ) {\displaystyle {\tilde {\mathbf
{y} }}(t)=\mathbf {z} (t)-\mathbf {H} (t){\hat {\mathbf {x} }}(t)} {\
displaystyle {\tilde {\mathbf {y} }}(t)=\mathbf {z} (t)-\mathbf {H}
(t){\hat {\mathbf {x} }}(t)}; these covariances are equal only in the
case of continuous time.^[61]

The distinction between the prediction and update steps of
discrete-time Kalman filtering does not exist in continuous time.

The second differential equation, for the covariance, is an example
of a Riccati equation. Nonlinear generalizations to Kalman-Bucy
filters include continuous time extended Kalman filter.

Hybrid Kalman filter[edit]

Most physical systems are represented as continuous-time models while
discrete-time measurements are frequently taken for state estimation
via a digital processor. Therefore, the system model and measurement
model are given by

    x . ( t ) = F ( t ) x ( t ) + B ( t ) u ( t ) + w ( t ) , w ( t )
    ~ N ( 0 , Q ( t ) ) z k = H k x k + v k , v k ~ N ( 0 , R k ) {\
    displaystyle {\begin{aligned}{\dot {\mathbf {x} }}(t)&=\mathbf
    {F} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t)+\mathbf {w}
    (t),&\mathbf {w} (t)&\sim N\left(\mathbf {0} ,\mathbf {Q} (t)\
    right)\\\mathbf {z} _{k}&=\mathbf {H} _{k}\mathbf {x} _{k}+\
    mathbf {v} _{k},&\mathbf {v} _{k}&\sim N(\mathbf {0} ,\mathbf {R}
    _{k})\end{aligned}}} {\displaystyle {\begin{aligned}{\dot {\
    mathbf {x} }}(t)&=\mathbf {F} (t)\mathbf {x} (t)+\mathbf {B} (t)\
    mathbf {u} (t)+\mathbf {w} (t),&\mathbf {w} (t)&\sim N\left(\
    mathbf {0} ,\mathbf {Q} (t)\right)\\\mathbf {z} _{k}&=\mathbf {H}
    _{k}\mathbf {x} _{k}+\mathbf {v} _{k},&\mathbf {v} _{k}&\sim N(\
    mathbf {0} ,\mathbf {R} _{k})\end{aligned}}}

where

    x k = x ( t k ) {\displaystyle \mathbf {x} _{k}=\mathbf {x} (t_
    {k})} {\displaystyle \mathbf {x} _{k}=\mathbf {x} (t_{k})}.

Initialize[edit]

    x ^ 0 | 0 = E [ x ( t 0 ) ] , P 0 | 0 = Var [?] [ x ( t 0 ) ] {\
    displaystyle {\hat {\mathbf {x} }}_{0\mid 0}=E\left[\mathbf {x}
    (t_{0})\right],\mathbf {P} _{0\mid 0}=\operatorname {Var} \left[\
    mathbf {x} \left(t_{0}\right)\right]} {\displaystyle {\hat {\
    mathbf {x} }}_{0\mid 0}=E\left[\mathbf {x} (t_{0})\right],\mathbf
    {P} _{0\mid 0}=\operatorname {Var} \left[\mathbf {x} \left(t_{0}\
    right)\right]}

Predict[edit]

    x ^ . ( t ) = F ( t ) x ^ ( t ) + B ( t ) u ( t ) , with  x ^ ( t
    k - 1 ) = x ^ k - 1 | k - 1 = x ^ k | k - 1 = x ^ ( t k ) P . ( t
    ) = F ( t ) P ( t ) + P ( t ) F ( t ) T + Q ( t ) , with  P ( t k
    - 1 ) = P k - 1 | k - 1 = P k | k - 1 = P ( t k ) {\displaystyle
    {\begin{aligned}{\dot {\hat {\mathbf {x} }}}(t)&=\mathbf {F} (t)
    {\hat {\mathbf {x} }}(t)+\mathbf {B} (t)\mathbf {u} (t){\text{,
    with }}{\hat {\mathbf {x} }}\left(t_{k-1}\right)={\hat {\mathbf
    {x} }}_{k-1\mid k-1}\\\Rightarrow {\hat {\mathbf {x} }}_{k\mid
    k-1}&={\hat {\mathbf {x} }}\left(t_{k}\right)\\{\dot {\mathbf {P}
    }}(t)&=\mathbf {F} (t)\mathbf {P} (t)+\mathbf {P} (t)\mathbf {F}
    (t)^{\textsf {T}}+\mathbf {Q} (t){\text{, with }}\mathbf {P} \
    left(t_{k-1}\right)=\mathbf {P} _{k-1\mid k-1}\\\Rightarrow \
    mathbf {P} _{k\mid k-1}&=\mathbf {P} \left(t_{k}\right)\end
    {aligned}}} {\displaystyle {\begin{aligned}{\dot {\hat {\mathbf
    {x} }}}(t)&=\mathbf {F} (t){\hat {\mathbf {x} }}(t)+\mathbf {B}
    (t)\mathbf {u} (t){\text{, with }}{\hat {\mathbf {x} }}\left(t_
    {k-1}\right)={\hat {\mathbf {x} }}_{k-1\mid k-1}\\\Rightarrow {\
    hat {\mathbf {x} }}_{k\mid k-1}&={\hat {\mathbf {x} }}\left(t_{k}
    \right)\\{\dot {\mathbf {P} }}(t)&=\mathbf {F} (t)\mathbf {P} (t)
    +\mathbf {P} (t)\mathbf {F} (t)^{\textsf {T}}+\mathbf {Q} (t){\
    text{, with }}\mathbf {P} \left(t_{k-1}\right)=\mathbf {P} _{k-1\
    mid k-1}\\\Rightarrow \mathbf {P} _{k\mid k-1}&=\mathbf {P} \left
    (t_{k}\right)\end{aligned}}}

The prediction equations are derived from those of continuous-time
Kalman filter without update from measurements, i.e., K ( t ) = 0 {\
displaystyle \mathbf {K} (t)=0} {\displaystyle \mathbf {K} (t)=0}.
The predicted state and covariance are calculated respectively by
solving a set of differential equations with the initial value equal
to the estimate at the previous step.

In the case of linear time invariant systems, the continuous time
dynamics can be exactly discretized into a discrete time system using
matrix exponentials.

Update[edit]

    K k = P k | k - 1 H k T ( H k P k | k - 1 H k T + R k ) - 1 x ^ k
    | k = x ^ k | k - 1 + K k ( z k - H k x ^ k | k - 1 ) P k | k = (
    I - K k H k ) P k | k - 1 {\displaystyle {\begin{aligned}\mathbf
    {K} _{k}&=\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\
    left(\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\
    textsf {T}}+\mathbf {R} _{k}\right)^{-1}\\{\hat {\mathbf {x} }}_
    {k\mid k}&={\hat {\mathbf {x} }}_{k\mid k-1}+\mathbf {K} _{k}\
    left(\mathbf {z} _{k}-\mathbf {H} _{k}{\hat {\mathbf {x} }}_{k\
    mid k-1}\right)\\\mathbf {P} _{k\mid k}&=\left(\mathbf {I} -\
    mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k\mid k-1}\
    end{aligned}}} {\displaystyle {\begin{aligned}\mathbf {K} _{k}&=\
    mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\left(\
    mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf
    {T}}+\mathbf {R} _{k}\right)^{-1}\\{\hat {\mathbf {x} }}_{k\mid
    k}&={\hat {\mathbf {x} }}_{k\mid k-1}+\mathbf {K} _{k}\left(\
    mathbf {z} _{k}-\mathbf {H} _{k}{\hat {\mathbf {x} }}_{k\mid k-1}
    \right)\\\mathbf {P} _{k\mid k}&=\left(\mathbf {I} -\mathbf {K} _
    {k}\mathbf {H} _{k}\right)\mathbf {P} _{k\mid k-1}\end{aligned}}}

The update equations are identical to those of the discrete-time
Kalman filter.

Variants for the recovery of sparse signals[edit]

The traditional Kalman filter has also been employed for the recovery
of sparse, possibly dynamic, signals from noisy observations. Recent
works^[62]^[63]^[64] utilize notions from the theory of compressed
sensing/sampling, such as the restricted isometry property and
related probabilistic recovery arguments, for sequentially estimating
the sparse state in intrinsically low-dimensional systems.

Applications[edit]

  * Attitude and heading reference systems
  * Autopilot
  * Battery state of charge (SoC) estimation^[65]^[66]
  * Brain-computer interfaces^[67]
  * Chaotic signals
  * Tracking and vertex fitting of charged particles in particle
    detectors^[68]
  * Tracking of objects in computer vision
  * Dynamic positioning in shipping
  * Economics, in particular macroeconomics, time series analysis,
    and econometrics^[69]
  * Inertial guidance system
  * Nuclear medicine - single photon emission computed tomography
    image restoration^[70]
  * Orbit determination
  * Power system state estimation
  * Radar tracker
  * Satellite navigation systems
  * Seismology^[71]
  * Sensorless control of AC motor variable-frequency drives
  * Simultaneous localization and mapping
  * Speech enhancement
  * Visual odometry
  * Weather forecasting
  * Navigation system
  * 3D modeling
  * Structural health monitoring
  * Human sensorimotor processing^[72]

See also[edit]

  * Alpha beta filter
  * Covariance intersection
  * Ensemble Kalman filter
  * Fast Kalman filter
  * Filtering problem (stochastic processes)
  * Generalized filtering
  * Invariant extended Kalman filter
  * Kernel adaptive filter
  * Masreliez's theorem
  * Moving horizon estimation
  * Particle filter estimator
  * PID controller
  * Predictor-corrector method
  * Recursive least squares filter
  * Schmidt-Kalman filter
  * Separation principle
  * Sliding mode control
  * State-transition matrix
  * Stochastic differential equations
  * Switching Kalman filter

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63. ^ Carmi, Avishy; Gurfil, Pini; Kanevsky, Dimitri (2010). "Methods
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Further reading[edit]

       This further reading section may contain inappropriate or
       excessive suggestions that may not follow Wikipedia's
       guidelines. Please ensure that only a reasonable number of
       balanced, topical, reliable, and notable further reading
[40px] suggestions are given; removing less relevant or redundant
       publications with the same point of view where appropriate.
       Consider utilising appropriate texts as inline sources or
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       how and when to remove this template message)

  * Einicke, G.A. (2019). Smoothing, Filtering and Prediction:
    Estimating the Past, Present and Future (2nd ed.). Amazon Prime
    Publishing. ISBN 978-0-6485115-0-2.
  * Jinya Su; Baibing Li; Wen-Hua Chen (2015). "On existence,
    optimality and asymptotic stability of the Kalman filter with
    partially observed inputs". Automatica. 53: 149-154. doi:10.1016/
    j.automatica.2014.12.044.
  * Gelb, A. (1974). Applied Optimal Estimation. MIT Press.
  * Kalman, R.E. (1960). "A new approach to linear filtering and
    prediction problems" (PDF). Journal of Basic Engineering. 82 (1):
    35-45. doi:10.1115/1.3662552. Archived from the original (PDF) on
    2008-05-29. Retrieved 2008-05-03.
  * Kalman, R.E.; Bucy, R.S. (1961). "New Results in Linear Filtering
    and Prediction Theory". Journal of Basic Engineering. 83: 95-108.
    CiteSeerX 10.1.1.361.6851. doi:10.1115/1.3658902.
  * Harvey, A.C. (1990). Forecasting, Structural Time Series Models
    and the Kalman Filter. Cambridge University Press. ISBN 
    9780521405737.
  * Roweis, S.; Ghahramani, Z. (1999). "A Unifying Review of Linear
    Gaussian Models" (PDF). Neural Computation. 11 (2): 305-345. doi:
    10.1162/089976699300016674. PMID 9950734. S2CID 2590898.
  * Simon, D. (2006). Optimal State Estimation: Kalman, H Infinity,
    and Nonlinear Approaches. Wiley-Interscience.
  * Stengel, R.F. (1994). Optimal Control and Estimation. Dover
    Publications. ISBN 978-0-486-68200-6.
  * Warwick, K. (1987). "Optimal observers for ARMA models".
    International Journal of Control. 46 (5): 1493-1503. doi:10.1080/
    00207178708933989.
  * Bierman, G.J. (1977). Factorization Methods for Discrete
    Sequential Estimation. Mathematics in Science and Engineering.
    128. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-44981-4.
  * Bozic, S.M. (1994). Digital and Kalman filtering.
    Butterworth-Heinemann.
  * Haykin, S. (2002). Adaptive Filter Theory. Prentice Hall.
  * Liu, W.; Principe, J.C. and Haykin, S. (2010). Kernel Adaptive
    Filtering: A Comprehensive Introduction. John Wiley.CS1 maint:
    multiple names: authors list (link)
  * Manolakis, D.G. (1999). Statistical and Adaptive signal
    processing. Artech House.
  * Welch, Greg; Bishop, Gary (1997). "SCAAT: incremental tracking
    with incomplete information" (PDF). SIGGRAPH '97 Proceedings of
    the 24th annual conference on Computer graphics and interactive
    techniques. ACM Press/Addison-Wesley Publishing Co. pp. 333-344.
    doi:10.1145/258734.258876. ISBN 978-0-89791-896-1. S2CID 1512754.
  * Jazwinski, Andrew H. (1970). Stochastic Processes and Filtering.
    Mathematics in Science and Engineering. New York: Academic Press.
    p. 376. ISBN 978-0-12-381550-7.
  * Maybeck, Peter S. (1979). "Chapter 1" (PDF). Stochastic Models,
    Estimation, and Control. Mathematics in Science and Engineering.
    141-1. New York: Academic Press. ISBN 978-0-12-480701-3.
  * Moriya, N. (2011). Primer to Kalman Filtering: A Physicist
    Perspective. New York: Nova Science Publishers, Inc. ISBN 
    978-1-61668-311-5.
  * Dunik, J.; Simandl M.; Straka O. (2009). "Methods for Estimating
    State and Measurement Noise Covariance Matrices: Aspects and
    Comparison". 15th IFAC Symposium on System Identification, 2009.
    15th IFAC Symposium on System Identification, 2009. France.
    pp. 372-377. doi:10.3182/20090706-3-FR-2004.00061. ISBN 
    978-3-902661-47-0.
  * Chui, Charles K.; Chen, Guanrong (2009). Kalman Filtering with
    Real-Time Applications. Springer Series in Information Sciences.
    17 (4th ed.). New York: Springer. p. 229. ISBN 978-3-540-87848-3.
  * Spivey, Ben; Hedengren, J. D. and Edgar, T. F. (2010).
    "Constrained Nonlinear Estimation for Industrial Process
    Fouling". Industrial & Engineering Chemistry Research. 49 (17):
    7824-7831. doi:10.1021/ie9018116.CS1 maint: multiple names:
    authors list (link)
  * Thomas Kailath; Ali H. Sayed; Babak Hassibi (2000). Linear
    Estimation. NJ: Prentice-Hall. ISBN 978-0-13-022464-4.
  * Ali H. Sayed (2008). Adaptive Filters. NJ: Wiley. ISBN 
    978-0-470-25388-5.CS1 maint: uses authors parameter (link)

External links[edit]

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[40px] article by removing excessive or inappropriate external links,
       and converting useful links where appropriate into footnote
       references. (June 2015) (Learn how and when to remove this
       template message)

  * A New Approach to Linear Filtering and Prediction Problems, by R.
    E. Kalman, 1960
  * Kalman and Bayesian Filters in Python. Open source Kalman
    filtering textbook.
  * How a Kalman filter works, in pictures. Illuminates the Kalman
    filter with pictures and colors
  * Kalman-Bucy Filter, a derivation of the Kalman-Bucy Filter
  * MIT Video Lecture on the Kalman filter on YouTube
  * Kalman filter in Javascript. Open source Kalman filter library
    for node.js and the web browser.
  * An Introduction to the Kalman Filter, SIGGRAPH 2001 Course, Greg
    Welch and Gary Bishop
  * Kalman Filter webpage, with many links
  * Kalman Filter Explained Simply, Step-by-Step Tutorial of the
    Kalman Filter with Equations
  * "Kalman filters used in Weather models" (PDF). SIAM News. 36 (8).
    October 2003. Archived from the original (PDF) on 2011-05-17.
    Retrieved 2007-01-27.
  * Haseltine, Eric L.; Rawlings, James B. (2005). "Critical
    Evaluation of Extended Kalman Filtering and Moving-Horizon
    Estimation". Industrial & Engineering Chemistry Research. 44 (8):
    2451. doi:10.1021/ie034308l.
  * Gerald J. Bierman's Estimation Subroutine Library: Corresponds to
    the code in the research monograph "Factorization Methods for
    Discrete Sequential Estimation" originally published by Academic
    Press in 1977. Republished by Dover.
  * Matlab Toolbox implementing parts of Gerald J. Bierman's
    Estimation Subroutine Library: UD / UDU' and LD / LDL'
    factorization with associated time and measurement updates making
    up the Kalman filter.
  * Matlab Toolbox of Kalman Filtering applied to Simultaneous
    Localization and Mapping: Vehicle moving in 1D, 2D and 3D
  * The Kalman Filter in Reproducing Kernel Hilbert Spaces A
    comprehensive introduction.
  * Matlab code to estimate Cox-Ingersoll-Ross interest rate model
    with Kalman Filter: Corresponds to the paper "estimating and
    testing exponential-affine term structure models by kalman
    filter" published by Review of Quantitative Finance and
    Accounting in 1999.
  * Online demo of the Kalman Filter. Demonstration of Kalman Filter
    (and other data assimilation methods) using twin experiments.
  * Botella, Guillermo; Martin h., Jose Antonio; Santos, Matilde;
    Meyer-Baese, Uwe (2011). "FPGA-Based Multimodal Embedded Sensor
    System Integrating Low- and Mid-Level Vision". Sensors. 11 (12):
    1251-1259. doi:10.3390/s110808164. PMC 3231703. PMID 22164069.
  * Examples and how-to on using Kalman Filters with MATLAB A
    Tutorial on Filtering and Estimation
  * Explaining Filtering (Estimation) in One Hour, Ten Minutes, One
    Minute, and One Sentence by Yu-Chi Ho
  * Simo Sarkka (2013). "Bayesian Filtering and Smoothing". Cambridge
    University Press. Full text available on author's webpage https:/
    /users.aalto.fi/~ssarkka/.

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