Kalman filter
Overview
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
(random variations) and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated values. The Kalman filter has many applications in technology, and is an essential part of space and military technology development.
Unanswered Questions
Discussions
Encyclopedia
In statistics
, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise
(random variations) and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated values. The Kalman filter has many applications in technology, and is an essential part of space and military technology development. A very simple example and perhaps the most commonly used type of Kalman filter is the phaselocked loop
, which is now ubiquitous in FM radios and most electronic communications equipment. Extensions and generalizations to the method have also been developed.
The Kalman filter produces estimates of the true values of measurements and their associated calculated values by predicting a value, estimating the uncertainty of the predicted value, and computing a weighted average
of the predicted value and the measured value. The most weight is given to the value with the least uncertainty. The estimates produced by the method tend to be closer to the true values than the original measurements because the weighted average has a better estimated uncertainty than either of the values that went into the weighted average.
From a theoretical standpoint, the Kalman filter is an algorithm for efficiently doing exact inference in a linear dynamical system
, which is a Bayesian model similar to a hidden Markov model
but where the state space of the latent variable
s is continuous and where all latent and observed variables have a Gaussian distribution (often a multivariate Gaussian distribution).
developed a similar algorithm earlier. Richard S. Bucy of the University of Southern California contributed to the theory, leading to it often being called the KalmanBucy filter.
Stanley F. Schmidt
is generally credited with developing the first implementation of a Kalman filter. It was during a visit by Kalman to the NASA Ames Research Center
that he saw the applicability of his ideas to the 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).
Kalman filters have been vital in the implementation of the navigation systems of U.S. Navy nuclear ballistic missile submarine
s, 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
. It is also used in the guidance and navigation systems of the NASA
Space Shuttle
and the attitude control
and navigation systems of the International Space Station
.
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 L. Stratonovich
. 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.
) that is better than the estimate obtained by using any one measurement alone. As such, it is a common sensor fusion
algorithm.
All measurements and calculations based on models are estimates to some degree. Noisy sensor data, approximations in the equations that describe how a system changes, and external factors that are not accounted for introduce some uncertainty about the inferred values for a system's state. The Kalman filter averages a prediction of a system's state with a 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 in between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that the Kalman filter works recursively
and requires only the last "best guess"  not the entire history  of a system's state to calculate a new state.
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 representation of linear relationships between different state variables (such as position, velocity, and acceleration) in any of the transition models or covariances.
and data fusion
. Typically, realtime systems produce multiple sequential measurements rather than making a single measurement to obtain the state of the system. These multiple measurements are then combined mathematically to generate the system's state at that time instant.
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 always remaining within a few meters of the real position. The truck's position can also be estimated by integrating its speed and direction 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, dead reckoning will provide a very smooth estimate of the truck's position, but it will drift
over time as small errors accumulate. Additionally, the truck is expected to follow the laws of physics, so its position should be expected to change proportionally to its velocity.
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) plus any changes produced by the accelerator pedal and steering wheel. Not only will a new position estimate be calculated, but 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 estimate at high speeds but very certain about the position when moving slowly. 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, if 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 rapidly changing and noisy.
using a Kalman filter can assist computers to track objects in videos with low latency (not to be confused with a low number of latent variables). The tracking of objects is a dynamic problem, using data from sensor and camera images that always suffer from noise. This can sometimes be reduced by using higher quality cameras and sensors but can never be eliminated, so it is often desirable to use a noise reduction method.
The iterative predictorcorrector nature of the Kalman filter can be helpful, because at each time instance only one constraint on the state variable need be considered. This process is repeated, considering a different constraint at every time instance. All the measured data are accumulated over time and help in predicting the state.
Video can also be preprocessed, perhaps using a segmentation technique, to reduce the computation and hence latency.
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, and is an important topic in control theory
and control system
s engineering. Together with the linearquadratic regulator
(LQR), the Kalman filter solves the linearquadraticGaussian control
problem (LQG). The Kalman filter, the linearquadratic regulator and the linearquadraticGaussian controller are solutions to what probably 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 control theory, the Kalman filter is most commonly referred to as linear quadratic estimation (LQE).
In DempsterShafer 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 jointree or Markov tree.
A wide variety of Kalman filters have now been developed, from Kalman's original formulation, now called the "simple" Kalman filter, the KalmanBucy filter, Schmidt's "extended" filter, the "information" filter, and a variety of "squareroot" filters that were developed by Bierman, Thornton and many others. Perhaps the most commonly used type of very simple Kalman filter is the phaselocked 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.
built on linear operators perturbed by Gaussian noise. The state
of the system is represented as a vector
of real number
s. 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). Additionally, the hidden Markov model can represent an arbitrary distribution for the next value of the state variables, in contrast to the Gaussian noise model that is used for the Kalman filter. There is a strong duality
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) and Hamilton (1994), Chapter 13.
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 framework of the Kalman filter. This means specifying the following matrices: F_{k}, the statetransition 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 controlinput model, for each timestep, k, as described below.
The Kalman filter model assumes the true state at time k is evolved from the state at (k − 1) according to
where
At time k an observation (or measurement) z_{k} of the true state x_{k} is made according to
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}.
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, unmodelled 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 unmodelled 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 separating between measurement noise and unmodelled dynamics is a difficult one and is treated in control theory under the framework of robust control
.
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 represents the estimate of at time n given observations up to, and including at time m.
The state of the filter is represented by two variables:
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}).
Innovation or measurement residual
Innovation (or residual) covariance
Optimal Kalman gain
Updated (a posteriori) state estimate
Updated (a posteriori) estimate covariance
The formula for the updated estimate and covariance above is only valid for the optimal Kalman gain. Usage of other gain values require a more complex formula found in the derivations section.
where is the expected value
of , and covariance matrices accurately reflect the covariance of estimates
s of routine operating data to estimate the covariances. The GNU Octave
code used to calculate the noise covariance matrices using the ALS technique is available online under the GNU General Public License
license.
is. We show here how we derive the model from which we create our Kalman filter.
Since F, H, R and Q are constant, their time indices are dropped.
The position and velocity of the truck are described by the linear state space
where is the velocity, that is, the derivative of position with respect to time.
We assume that between the (k − 1) and k timestep the truck undergoes a constant acceleration of a_{k} that is normally distributed, with mean 0 and standard deviation σ_{a}. From Newton's laws of motion
we conclude that
(note that there is no term since we have no known control inputs) where
and
so that
where and
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 σ_{z}.
where
and
We know the initial starting state of the truck with perfect precision, so we initialize
and to tell the filter that we know the exact position, we give it a zero covariance matrix:
If the initial position and velocity are not known perfectly the covariance matrix should be initialized with a suitably large number, say L, on its diagonal.
The filter will then prefer the information from the first measurements over the information already in the model.
substitute in the definition of
and substitute
and
and by collecting the error vectors we get
Since the measurement error v_{k} is uncorrelated with the other terms, this becomes
by the properties of vector covariance
this becomes
which, using our invariant on P_{kk1} and the definition of R_{k} becomes
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.
estimator. The error in the a posteriori state estimation is
We seek to minimize the expected value of the square of the magnitude of this vector, . This is equivalent to minimizing the trace of the a posteriori estimate covariance matrix . By expanding out the terms in the equation above and collecting, we get:
The trace is minimized when the matrix derivative
is zero:
Solving this for K_{k} yields the Kalman gain:
This gain, which is known as the optimal Kalman gain, is the one that yields MMSE
estimates when used.
Referring back to our expanded formula for the a posteriori error covariance,
we find the last two terms cancel out, giving
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 nonoptimal Kalman gain is deliberately used, this simplification cannot be applied; the a posteriori error covariance formula as derived above must be used.
no longer provides the actual error covariance. In other words, . In most real time applications the covariance matrices that are used in designing the Kalman filter are different from the actual noise covariances matrices. This sensitivity analysis describes the behavior of the estimation error covariance when the noise covariances as well as the system matrices and 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 and respectively, whereas the design values used in the estimator are and respectively. The actual error covariance is denoted by and as computed by the Kalman filter is referred to as the Riccati variable. When and , this means that . While computing the actual error covariance using , substituting for and using the fact that and , results in the following recursive equations for :
. If the process noise covariance Q_{k} is small, roundoff 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 positivedefinite
.
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 UD 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 UD 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 timeconsuming than divisions, while on 21st 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.
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. 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. Any singular covariance matrix is pivoted
so that the first diagonal partition is nonsingular and wellconditioned
. The pivoting algorithm must retain any portion of the innovation covariance matrix directly corresponding to observed statevariables H_{k}·x_{kk1} that are associated with auxiliary observations in
y_{k}. The L·D·L^{T} squareroot filter requires orthogonalization of the observation vector. This may be done with the inverse squareroot of the covariance matrix for the auxiliary variables using Method 2 in Higham (2002, p. 263).
s. The Kalman filter calculates estimates of the true values of measurements 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.
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).
Because of the Markov assumption, the true state is conditionally independent of all earlier states given the immediately previous state.
Similarly the measurement at the kth timestep is dependent only upon the current state and is conditionally independent of all other states given the current state.
Using these assumptions the probability distribution over all states of the hidden Markov model can be written simply as:
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 kth and the probability distribution associated with the previous state, over all possible .
The measurement set up to time t is
The probability distribution of the update is proportional to the product of the measurement likelihood and the predicted state.
The denominator
is a normalization term.
The remaining probability density functions are
Note that 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 given the measurements is the Kalman filter estimate.
vector respectively. These are defined as:
Similarly the predicted covariance and state have equivalent information forms, defined as:
as have the measurement covariance and measurement vector, which are defined as:
The information update now becomes a trivial sum.
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.
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.
Note that if F and Q are time invariant these values can be cached. Note also that F and Q need to be invertible.
where:
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
(random variations) and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated values. The Kalman filter has many applications in technology, and is an essential part of space and military technology development. A very simple example and perhaps the most commonly used type of Kalman filter is the phaselocked loop
Phaselocked loop
A phaselocked loop or phase lock loop is a control system that generates an output signal whose phase is related to the phase of an input "reference" signal. It is an electronic circuit consisting of a variable frequency oscillator and a phase detector...
, which is now ubiquitous in FM radios and most electronic communications equipment. Extensions and generalizations to the method have also been developed.
The Kalman filter produces estimates of the true values of measurements and their associated calculated values by predicting a value, estimating the uncertainty of the predicted value, and computing a weighted average
Weighted mean
The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...
of the predicted value and the measured value. The most weight is given to the value with the least uncertainty. The estimates produced by the method tend to be closer to the true values than the original measurements because the weighted average has a better estimated uncertainty than either of the values that went into the weighted average.
From a theoretical standpoint, the Kalman filter is an algorithm for efficiently doing exact inference in a linear dynamical system
Linear dynamical system
Linear dynamical systems are a special type of dynamical system where the equation governing the system's evolution is linear. While dynamical systems in general do not have closedform solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties...
, which is a Bayesian model similar to a hidden Markov model
Hidden Markov model
A hidden Markov model is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved states. An HMM can be considered as the simplest dynamic Bayesian network. The mathematics behind the HMM was developed by L. E...
but where the state space of the latent variable
Latent variable
In statistics, latent variables , are variables that are not directly observed but are rather inferred from other variables that are observed . Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models...
s is continuous and where all latent and observed variables have a Gaussian distribution (often a multivariate Gaussian distribution).
Naming and historical development
The filter is named after Rudolf E. Kálmán, though Thorvald Nicolai Thiele and Peter SwerlingPeter Swerling
Peter Swerling was one of the most influential radar theoreticians in the second half of the 20th century. He is best known for the class of statistically "fluctuating target" scattering models he developed at the RAND Corporation in the early 1950s to characterize the performance of pulsed radar...
developed a similar algorithm earlier. Richard S. Bucy of the University of Southern California contributed to the theory, leading to it often being called the KalmanBucy filter.
Stanley F. Schmidt
Stanley F. Schmidt
Stanley F. Schmidt was born in Hollister, California, on January 21, 1926. He received the B.E.E. degree from Marquette University in 1946, and M.S. and Ph.D...
is generally credited with developing the first implementation of a Kalman filter. It was during a visit by Kalman to the NASA Ames Research Center
NASA Ames Research Center
The Ames Research Center , is one of the United States of America's National Aeronautics and Space Administration 10 major field centers.The centre is located in Moffett Field in California's Silicon Valley, near the hightech companies, entrepreneurial ventures, universities, and other...
that he saw the applicability of his ideas to the problem of trajectory estimation for the Apollo program
Project Apollo
The Apollo program was the spaceflight effort carried out by the United States' National Aeronautics and Space Administration , that landed the first humans on Earth's Moon. Conceived during the Presidency of Dwight D. Eisenhower, Apollo began in earnest after President John F...
, 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).
Kalman filters have been vital in the implementation of the navigation systems of U.S. Navy nuclear ballistic missile submarine
Ballistic missile submarine
A ballistic missile submarine is a submarine equipped to launch ballistic missiles .Description:Ballistic missile submarines are larger than any other type of submarine, in order to accommodate SLBMs such as the Russian R29 or the American Trident...
s, 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
AGM86 ALCM
The Boeing AGM86 ALCM is a U.S. subsonic airlaunched cruise missile built by Boeing Company and operated by the United States Air Force. The missiles were developed to increase the effectiveness and survivability of Boeing B52H Stratofortress bombers...
. It is also used in the guidance and navigation systems of the NASA
NASA
The National Aeronautics and Space Administration is the agency of the United States government that is responsible for the nation's civilian space program and for aeronautics and aerospace research...
Space Shuttle
Space Shuttle
The Space Shuttle was a manned orbital rocket and spacecraft system operated by NASA on 135 missions from 1981 to 2011. The system combined rocket launch, orbital spacecraft, and reentry spaceplane with modular addons...
and the attitude control
Attitude dynamics and control
Spacecraft flight dynamics is the science of space vehicle performance, stability, and control. It requires analysis of the six degrees of freedom of the vehicle's flight, which are similar to those of aircraft: translation in three dimensional axes; and its orientation about the vehicle's center...
and navigation systems of the International Space Station
International Space Station
The International Space Station is a habitable, artificial satellite in low Earth orbit. The ISS follows the Salyut, Almaz, Cosmos, Skylab, and Mir space stations, as the 11th space station launched, not including the Genesis I and II prototypes...
.
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
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Ruslan L. Stratonovich
Ruslan L. Stratonovich
Ruslan Leont'evich Stratonovich was an outstanding physicist, engineer, and probabilist. Professor Stratonovich was born on May 31, 1930 in Moscow, Russia...
. 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.
Overview of the calculation
The Kalman filter uses a system's dynamics model (i.e., physical laws of motion), known control inputs to that system, and measurements (such as from sensors) to form an estimate of the system's varying quantities (its stateState space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by firstorder differential equations...
) that is better than the estimate obtained by using any one measurement alone. As such, it is a common sensor fusion
Sensor fusion
Sensor fusion is the combining of sensory data or data derived from sensory data from disparate sources such that the resulting information is in some sense better than would be possible when these sources were used individually...
algorithm.
All measurements and calculations based on models are estimates to some degree. Noisy sensor data, approximations in the equations that describe how a system changes, and external factors that are not accounted for introduce some uncertainty about the inferred values for a system's state. The Kalman filter averages a prediction of a system's state with a new measurement using a weighted average
Weighted mean
The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...
. 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
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical. Definition :...
, 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 in between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that the Kalman filter works recursively
Recursive filter
In signal processing, a recursive filter is a type of filter which reuses one or more of its outputs as an input. This feedback typically results in an unending impulse response , characterised by either exponentially growing, decaying, or sinusoidal signal output components.However, a recursive...
and requires only the last "best guess"  not the entire history  of a system's state to calculate a new state.
When performing the actual calculations for the filter (as discussed below), the state estimate and covariances are coded into matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
to handle the multiple dimensions involved in a single set of calculations. This allows for representation of linear relationships between different state variables (such as position, velocity, and acceleration) in any of the transition models or covariances.
Example application
The Kalman filter is used in sensor fusionSensor fusion
Sensor fusion is the combining of sensory data or data derived from sensory data from disparate sources such that the resulting information is in some sense better than would be possible when these sources were used individually...
and data fusion
Data fusion
Data fusion, is generally defined as the use of techniques that combine data from multiple sources and gather that information into discrete, actionable items in order to achieve inferences, which will be more efficient and narrowly tailored than if they were achieved by means of disparate...
. Typically, realtime systems produce multiple sequential measurements rather than making a single measurement to obtain the state of the system. These multiple measurements are then combined mathematically to generate the system's state at that time instant.
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 always remaining within a few meters of the real position. The truck's position can also be estimated by integrating its speed and direction over time, determined by keeping track of wheel revolutions and the angle of the steering wheel. This is a technique known as dead reckoning
Dead reckoning
In navigation, dead reckoning is the process of calculating one's current position by using a previously determined position, or fix, and advancing that position based upon known or estimated speeds over elapsed time, and course...
. Typically, dead reckoning will provide a very smooth estimate of the truck's position, but it will drift
Drift (telecommunication)
In telecommunication, a drift is a comparatively longterm change in an attribute, value, or operational parameter of a system or equipment. The drift should be characterized, such as "diurnal frequency drift" and "output level drift." Drift is usually undesirable and unidirectional, but may be...
over time as small errors accumulate. Additionally, the truck is expected to follow the laws of physics, so its position should be expected to change proportionally to its velocity.
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
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
(the dynamic or "state transition" model) plus any changes produced by the accelerator pedal and steering wheel. Not only will a new position estimate be calculated, but 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 estimate at high speeds but very certain about the position when moving slowly. 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, if 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 rapidly changing and noisy.
Kalman filter in computer vision
Data fusionData fusion
Data fusion, is generally defined as the use of techniques that combine data from multiple sources and gather that information into discrete, actionable items in order to achieve inferences, which will be more efficient and narrowly tailored than if they were achieved by means of disparate...
using a Kalman filter can assist computers to track objects in videos with low latency (not to be confused with a low number of latent variables). The tracking of objects is a dynamic problem, using data from sensor and camera images that always suffer from noise. This can sometimes be reduced by using higher quality cameras and sensors but can never be eliminated, so it is often desirable to use a noise reduction method.
The iterative predictorcorrector nature of the Kalman filter can be helpful, because at each time instance only one constraint on the state variable need be considered. This process is repeated, considering a different constraint at every time instance. All the measured data are accumulated over time and help in predicting the state.
Video can also be preprocessed, perhaps using a segmentation technique, to reduce the computation and hence latency.
Technical description and context
The Kalman filter is an efficient recursive filterRecursive filter
In signal processing, a recursive filter is a type of filter which reuses one or more of its outputs as an input. This feedback typically results in an unending impulse response , characterised by either exponentially growing, decaying, or sinusoidal signal output components.However, a recursive...
that estimates
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....
the internal state of a linear dynamic system
Linear dynamical system
Linear dynamical systems are a special type of dynamical system where the equation governing the system's evolution is linear. While dynamical systems in general do not have closedform solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties...
from a series of noisy
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
measurements. It is used in a wide range of engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
and econometric applications from radar
Radar
Radar is an objectdetection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
and computer vision
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, highdimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
to estimation of structural macroeconomic models, and is an important topic in control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
and control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...
s engineering. Together with the linearquadratic regulator
Linearquadratic regulator
The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic functional is called the LQ problem...
(LQR), the Kalman filter solves the linearquadraticGaussian control
LinearquadraticGaussian control
In control theory, the linearquadraticGaussian control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, having incomplete state information and undergoing control subject to quadratic costs...
problem (LQG). The Kalman filter, the linearquadratic regulator and the linearquadraticGaussian controller are solutions to what probably are the most fundamental problems in control theory.
In most applications, the internal state is much larger (more degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
) 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 control theory, the Kalman filter is most commonly referred to as linear quadratic estimation (LQE).
In DempsterShafer theory
DempsterShafer theory
The Dempster–Shafer theory is a mathematical theory of evidence. It allows one to combine evidence from different sources and arrive at a degree of belief that takes into account all the available evidence. The theory was first developed by Arthur P...
, each state equation or observation is considered a special case of a Linear belief function
Linear Belief Function
Linear Belief Function is an extension of the DempsterShafer theory of belief functions to the case when variables of interest are continuous. Examples of such variables include financial asset prices, portfolio performance, and other antecedent and consequent variables. The theory was originally...
and the Kalman filter is a special case of combining linear belief functions on a jointree or Markov tree.
A wide variety of Kalman filters have now been developed, from Kalman's original formulation, now called the "simple" Kalman filter, the KalmanBucy filter, Schmidt's "extended" filter, the "information" filter, and a variety of "squareroot" filters that were developed by Bierman, Thornton and many others. Perhaps the most commonly used type of very simple Kalman filter is the phaselocked loop
Phaselocked loop
A phaselocked loop or phase lock loop is a control system that generates an output signal whose phase is related to the phase of an input "reference" signal. It is an electronic circuit consisting of a variable frequency oscillator and a phase detector...
, which is now ubiquitous in radios, especially frequency modulation
Frequency modulation
In telecommunications and signal processing, frequency modulation conveys information over a carrier wave by varying its instantaneous frequency. This contrasts with amplitude modulation, in which the amplitude of the carrier is varied while its frequency remains constant...
(FM) radios, television sets, satellite communications receivers, outer space communications systems, and nearly any other electronic
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
communications equipment.
Underlying dynamic system model
Kalman filters are based on linear dynamic systems discretized in the time domain. They are modelled on a Markov chainMarkov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
built on linear operators perturbed by Gaussian noise. The state
State space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by firstorder differential equations...
of the system is represented as a vector
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s. At each discrete time
Discrete time
Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...
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). Additionally, the hidden Markov model can represent an arbitrary distribution for the next value of the state variables, in contrast to the Gaussian noise model that is used for the Kalman filter. There is a strong duality
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a onetoone fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
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) and Hamilton (1994), Chapter 13.
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 framework of the Kalman filter. This means specifying the following matrices: F_{k}, the statetransition 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 controlinput model, for each timestep, k, as described below.
The Kalman filter model assumes the true state at time k is evolved from the state at (k − 1) according to
where
 F_{k} is the state transition model which is applied to the previous state x_{k−1};
 B_{k} is the controlinput 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 with covarianceCovariance matrixIn probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...
Q_{k}.
At time k an observation (or measurement) z_{k} of the true state x_{k} is made according to
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}.
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
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
.
Many real dynamical systems do not exactly fit this model. In fact, unmodelled 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 unmodelled 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 separating between measurement noise and unmodelled dynamics is a difficult one and is treated in control theory under the framework of robust control
Robust control
Robust control is a branch of control theory that explicitly deals with uncertainty in its approach to controller design. Robust control methods are designed to function properly so long as uncertain parameters or disturbances are within some set...
.
The Kalman filter
The Kalman filter is a recursiveInfinite impulse response
Infinite impulse response is a property of signal processing systems. Systems with this property are known as IIR systems or, when dealing with filter systems, as IIR filters. IIR systems have an impulse response function that is nonzero over an infinite length of time...
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 represents the estimate of at time n given observations up to, and including at time m.
The state of the filter is represented by two variables:
 , the a posterioriA PosterioriApart from the album, some additional remixes were released exclusively through the iTunes Store. They are:*"Eppur si muove" – 6:39*"Dreaming of Andromeda" Apart from the album, some additional remixes were released exclusively through the iTunes Store. They are:*"Eppur si muove" (Tocadisco...
state estimate at time k given observations up to and including at time k;  , the a posteriori error covariance matrix (a measure of the estimated accuracyAccuracy and precisionIn the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...
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}).
Predict
Predicted (a priori) state estimate  
Predicted (a priori) estimate covariance 
Update
Innovation or measurement residual
Innovation (or residual) covariance
Optimal Kalman gain
Updated (a posteriori) state estimate
Updated (a posteriori) estimate covariance
The formula for the updated estimate and covariance above is only valid for the optimal Kalman gain. Usage of other gain values require a more complex formula found in the derivations section.
Invariants
If the model is accurate, and the values for and accurately reflect the distribution of the initial state values, then the following invariants are preserved: (all estimates have a mean error of zero)where is the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of , and covariance matrices accurately reflect the covariance of estimates
Estimation of the noise covariances Q_{k} and R_{k}
Practical implementation of the Kalman Filter is often difficult due to the inability in 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 of the more promising approaches to doing this is called the Autocovariance LeastSquares (ALS) technique that uses autocovarianceAutocovariance
In statistics, given a real stochastic process X, the autocovariance is the covariance of the variable with itself, i.e. the variance of the variable against a timeshifted version of itself...
s of routine operating data to estimate the covariances. The GNU Octave
GNU Octave
GNU Octave is a highlevel language, primarily intended for numerical computations. It provides a convenient commandline interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...
code used to calculate the noise covariance matrices using the ALS technique is available online under the GNU General Public License
GNU General Public License
The GNU General Public License is the most widely used free software license, originally written by Richard Stallman for the GNU Project....
license.
Example application, technical
Consider a truck on perfectly frictionless, infinitely long straight rails. Initially the truck is stationary at position 0, but it is buffeted this way and that by random acceleration. We measure the position of the truck every Δt seconds, but these measurements are imprecise; we want to maintain a model of where the truck is and what its velocityVelocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
is. We show here how we derive the model from which we create our Kalman filter.
Since F, H, R and Q are constant, their time indices are dropped.
The position and velocity of the truck are described by the linear state space
where is the velocity, that is, the derivative of position with respect to time.
We assume that between the (k − 1) and k timestep the truck undergoes a constant acceleration of a_{k} that is normally distributed, with mean 0 and standard deviation σ_{a}. From Newton's laws of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
we conclude that
(note that there is no term since we have no known control inputs) where
and
so that
where and
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 σ_{z}.
where
and
We know the initial starting state of the truck with perfect precision, so we initialize
and to tell the filter that we know the exact position, we give it a zero covariance matrix:
If the initial position and velocity are not known perfectly the covariance matrix should be initialized with a suitably large number, say L, on its diagonal.
The filter will then prefer the information from the first measurements over the information already in the model.
Deriving the a posteriori estimate covariance matrix
Starting with our invariant on the error covariance P_{kk} as abovesubstitute in the definition of
and substitute
and
and by collecting the error vectors we get
Since the measurement error v_{k} is uncorrelated with the other terms, this becomes
by the properties of vector covariance
Covariance matrix
In probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...
this becomes
which, using our invariant on P_{kk1} and the definition of R_{k} becomes
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
The Kalman filter is a minimum meansquare errorMinimum meansquare error
In statistics and signal processing, a minimum mean square error estimator describes the approach which minimizes the mean square error , which is a common measure of estimator quality....
estimator. The error in the a posteriori state estimation is
We seek to minimize the expected value of the square of the magnitude of this vector, . This is equivalent to minimizing the trace of the a posteriori estimate covariance matrix . By expanding out the terms in the equation above and collecting, we get:
The trace is minimized when the matrix derivative
Matrix calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices, where it defines the matrix derivative. This notation was to describe systems of differential equations, and taking derivatives of matrixvalued functions with respect...
is zero:
Solving this for K_{k} yields the Kalman gain:
This gain, which is known as the optimal Kalman gain, is the one that yields MMSE
Minimum meansquare error
In statistics and signal processing, a minimum mean square error estimator describes the approach which minimizes the mean square error , which is a common measure of estimator quality....
estimates when used.
Simplification of the a posteriori error covariance formula
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 thatReferring back to our expanded formula for the a posteriori error covariance,
we find the last two terms cancel out, giving
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
Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....
, or if a nonoptimal Kalman gain is deliberately used, this simplification cannot be applied; the a posteriori error covariance formula as derived above must be used.
Sensitivity analysis
The Kalman filtering equations provide an estimate of the state and its error covariance 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. In the absence of reliable statistics or the true values of noise covariance matrices and , the expressionno longer provides the actual error covariance. In other words, . In most real time applications the covariance matrices that are used in designing the Kalman filter are different from the actual noise covariances matrices. This sensitivity analysis describes the behavior of the estimation error covariance when the noise covariances as well as the system matrices and 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 and respectively, whereas the design values used in the estimator are and respectively. The actual error covariance is denoted by and as computed by the Kalman filter is referred to as the Riccati variable. When and , this means that . While computing the actual error covariance using , substituting for and using the fact that and , results in the following recursive equations for :
 While computing , by design the filter implicitly assumes that and .
 The recursive expressions for and are identical except for the presence of and in place of the design values and respectively.
Square root form
One problem with the Kalman filter is its numerical stabilityNumerical stability
In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....
. If the process noise covariance Q_{k} is small, roundoff 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 positivedefinite
Positivedefinite matrix
In linear algebra, a positivedefinite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positivedefinite symmetric bilinear form ....
.
Positive definite matrices have the property that they have a triangular matrix
Triangular matrix
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where either all the entries below or all the entries above the main diagonal are zero...
square root
Square root of a matrix
In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product B · B is equal to A.Properties:...
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
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
operations required by the matrix square root yet preserves the desirable numerical properties, is the UD 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 UD 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 timeconsuming than divisions, while on 21st 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.
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. The algorithm starts with the LU decomposition as implemented in the Linear Algebra PACKage (LAPACK
LAPACK
External links:* : a modern replacement for PLAPACK and ScaLAPACK* on Netlib.org* * * : a modern replacement for LAPACK that is MultiGPU ready* on Sourceforge.net* * optimized LAPACK for Solaris OS on SPARC/x86/x64 and Linux* * *...
). 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. Any singular covariance matrix is pivoted
Pivot element
The pivot or pivot element is the element of a matrix, an array, or some other kind of finite set, which is selected first by an algorithm , to do certain calculations...
so that the first diagonal partition is nonsingular and wellconditioned
Condition number
In the field of numerical analysis, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument...
. The pivoting algorithm must retain any portion of the innovation covariance matrix directly corresponding to observed statevariables H_{k}·x_{kk1} that are associated with auxiliary observations in
y_{k}. The L·D·L^{T} squareroot filter requires orthogonalization of the observation vector. This may be done with the inverse squareroot of the covariance matrix for the auxiliary variables using Method 2 in Higham (2002, p. 263).
Relationship to recursive Bayesian estimation
The Kalman filter can be considered to be one of the most simple dynamic Bayesian networkDynamic Bayesian network
A dynamic Bayesian network is a Bayesian network that represents sequences of variables. These sequences are often timeseries or sequences of symbols . The hidden Markov model can be considered as a simple dynamic Bayesian network. References :* , Zoubin Ghahramani, Lecture Notes In Computer...
s. The Kalman filter calculates estimates of the true values of measurements recursively over time using incoming measurements and a mathematical process model. Similarly, recursive Bayesian estimation
Recursive Bayesian estimation
Recursive Bayesian estimation, also known as a Bayes filter, is a general probabilistic approach for estimating an unknown probability density function recursively over time using incoming measurements and a mathematical process model.In robotics:...
calculates estimates
Density estimation
In probability and statistics,density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function...
of an unknown probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
(PDF) recursively over time using incoming measurements and a mathematical process model.
In recursive Bayesian estimation, the true state is assumed to be an unobserved Markov process
Markov process
In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a timevarying random phenomenon for which a specific property holds...
, and the measurements are the observed states of a hidden Markov model (HMM).
Because of the Markov assumption, the true state is conditionally independent of all earlier states given the immediately previous state.
Similarly the measurement at the kth timestep is dependent only upon the current state and is conditionally independent of all other states given the current state.
Using these assumptions the probability distribution over all states of the hidden Markov model can be written simply as:
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 kth and the probability distribution associated with the previous state, over all possible .
The measurement set up to time t is
The probability distribution of the update is proportional to the product of the measurement likelihood and the predicted state.
The denominator
is a normalization term.
The remaining probability density functions are
Note that 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 given the measurements is the Kalman filter estimate.
Information filter
In the information filter, or inverse covariance filter, the estimated covariance and estimated state are replaced by the information matrix and informationFisher information
In mathematical statistics and information theory, the Fisher information is the variance of the score. In Bayesian statistics, the asymptotic distribution of the posterior mode depends on the Fisher information and not on the prior...
vector respectively. These are defined as:
Similarly the predicted covariance and state have equivalent information forms, defined as:
as have the measurement covariance and measurement vector, which are defined as:
The information update now becomes a trivial sum.
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.
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.
Note that if F and Q are time invariant these values can be cached. Note also that F and Q need to be invertible.
Fixedlag smoother
The optimal fixedlag smoother provides the optimal estimate of for a given fixedlag using the measurements from to . It can be derived using the previous theory via an augmented state, and the main equation of the filter is the following:where:
 is estimated via a standard Kalman filter;
 is the innovation produced considering the estimate of the standard Kalman filter;
 the various with are new variables, i.e. they do not appear in the standard Kalman filter;
 the gains are computed via the following scheme:


 and

 where and are the prediction error covariance and the gains of the standard Kalman filter (i.e., ).
If the estimation error covariance is defined so that
then we have that the improvement on the estimation of is given by:
Fixedinterval smoothers
The optimal fixedinterval smoother provides the optimal estimate of () using the measurements from a fixed interval to . This is also called "Kalman Smoothing". There are several smoothing algorithms in common use.
Rauch–Tung–Striebel
The Rauch–Tung–Striebel (RTS) smoother is an efficient twopass algorithm for fixed interval smoothing.
The main equations of the smoother are the following (assuming ): forward pass: regular Kalman filter algorithm
 backward pass:
, where
Modified Bryson–Frazier smoother
An alternative to the RTS algorithm is the modified Bryson–Frazier (MBF) fixed interval smoother developed by Bierman. 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
where is the residual covariance and . The smoothed state and covariance can then be found by substitution in the equations
or
.
An important advantage of the MBF is that it does not require finding the inverse of the covariance matrix.
Nonlinear filters
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.
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.
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
When the state transition and observation models – that is, the predict and update functions and (see above) – are highly nonlinear, the extended Kalman filter can give particularly poor performance. This is because the covariance is propagated through linearization of the underlying nonlinear model. The unscented Kalman filter (UKF) uses a deterministic sampling technique known as the unscented transformUnscented transformThe Unscented Transform is a mathematical function used to estimate the result of applying a given nonlinear transformation to a probability distribution that is characterized only in terms of a finite set of statistics...
to pick a minimal set of sample points (called sigma points) around the mean. These sigma points are then propagated through the nonlinear functions, from which the mean and covariance of the estimate are then recovered. The result is a filter which more accurately captures the true mean and covariance. (This can be verified using Monte Carlo sampling or through a Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
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).
Predict
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.
The estimated state and covariance are augmented with the mean and covariance of the process noise.
A set of 2L+1 sigma points is derived from the augmented state and covariance where L is the dimension of the augmented state.
where
is the ith column of the matrix square root of
using the definition: square root A of matrix B satisfies
.
The matrix square root should be calculated using numerically efficient and stable methods such as the Cholesky decompositionCholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a Hermitian, positivedefinite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by AndréLouis Cholesky for real matrices...
.
The sigma points are propagated through the transition function f.
where . The weighted sigma points are recombined to produce the predicted state and covariance.
where the weights for the state and covariance are given by:
and control the spread of the sigma points. is related to the distribution of .
Normal values are , and . If the true distribution of is Gaussian, is optimal.
Update
The predicted state and covariance are augmented as before, except now with the mean and covariance of the measurement noise.
As before, a set of 2L + 1 sigma points is derived from the augmented state and covariance where L is the dimension of the augmented state.
Alternatively if the UKF prediction has been used the sigma points themselves can be augmented along the following lines
where
The sigma points are projected through the observation function h.
The weighted sigma points are recombined to produce the predicted measurement and predicted measurement covariance.
The statemeasurement crosscovariance matrix,
is used to compute the UKF Kalman gain.
As with the Kalman filter, the updated state is the predicted state plus the innovation weighted by the Kalman gain,
And the updated covariance is the predicted covariance, minus the predicted measurement covariance, weighted by the Kalman gain.
Kalman–Bucy filter
The Kalman–Bucy filter (named after Richard Snowden Bucy) is a continuous time version of the Kalman filter.
It is based on the state space model
where the covariances of the noise terms and are given by and , respectively.
The filter consists of two differential equations, one for the state estimate and one for the covariance:
where the Kalman gain is given by
Note that in this expression for the covariance of the observation noise represents at the same time the covariance of the prediction error (or innovation) ; these covariances are equal only in the case of continuous time.
The distinction between the prediction and update steps of discretetime Kalman filtering does not exist in continuous time.
The second differential equation, for the covariance, is an example of a Riccati equationRiccati equationIn mathematics, a Riccati equation is any ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form y' = q_0 + q_1 \, y + q_2 \, y^2...
.
Hybrid Kalman filter
Most physical systems are represented as continuoustime models while discretetime measurements are frequently taken for state estimation via a digital processor. Therefore, the system model and measurement model are given by
where.
Initialize
Predict
The prediction equations are derived from those of continuoustime Kalman filter without update from measurements, i.e., . 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.
Update
The update equations are identical to those of discretetime Kalman filter.
Kalman filter variants for the recovery of sparse signals
Recently the traditional Kalman filter has been employed for the recovery of sparse, possibly dynamic, signals from
noisy observations. Both works and utilize notions from the theory of compressed sensingCompressed sensingCompressed sensing, also known as compressive sensing, compressive sampling and sparse sampling, is a technique for finding sparse solutions to underdetermined linear systems...
/sampling, such as the restricted isometry property and related probabilistic recovery arguments, for sequentially estimating the sparse state in intrinsically lowdimensional systems.
Applications
 Attitude and Heading Reference SystemsAttitude and Heading Reference SystemsAn attitude heading reference system consists of sensors on three axes that provide heading, attitude and yaw information for aircraft. They are designed to replace traditional mechanical gyroscopic flight instruments and provide superior reliability and accuracy.AHRS consist of either solidstate...
 AutopilotAutopilotAn autopilot is a mechanical, electrical, or hydraulic system used to guide a vehicle without assistance from a human being. An autopilot can refer specifically to aircraft, selfsteering gear for boats, or auto guidance of space craft and missiles...
 Battery state of charge (SoC) estimation http://dx.doi.org/10.1016/j.jpowsour.2007.04.011http://dx.doi.org/10.1016/j.enconman.2007.05.017
 Braincomputer interfaceBraincomputer interfaceA brain–computer interface , sometimes called a direct neural interface or a brain–machine interface , is a direct communication pathway between the brain and an external device...
 Chaotic signals
 Dynamic positioningDynamic positioningDynamic positioning is a computer controlled system to automatically maintain a vessel's position and heading by using its own propellers and thrusters...
 EconomicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, in particular macroeconomicsMacroeconomicsMacroeconomics is a branch of economics dealing with the performance, structure, behavior, and decisionmaking of the whole economy. This includes a national, regional, or global economy...
, time seriesTime seriesIn statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
, and econometricsEconometricsEconometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...  Inertial guidance system
 Orbit DeterminationOrbit determinationOrbit determination is a branch of astronomy specialised in calculating, and hence predicting, the orbits of objects such as moons, planets, and spacecraft . These orbits could be orbiting the Earth, or other bodies...
 Radar trackerRadar trackerA radar tracker is a component of a radar system, or an associated command and control system, that associates consecutive radar observations of the same target into tracks...
 Satellite navigation systems
 SeismologySeismologySeismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planetlike bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...
http://adsabs.harvard.edu/abs/2008AGUFM.G43B..01B  Simultaneous localization and mappingSimultaneous localization and mappingSimultaneous localization and mapping is a technique used by robots and autonomous vehicles to build up a map within an unknown environment , or to update a map within a known environment , while at the same time keeping track of their current location. Operational definition :Maps are used...
 Speech enhancementSpeech enhancementSpeech enhancement aims to improve speech quality by using various algorithms.The objective of enhancement is improvement in intelligibility and/or overall perceptual quality of degraded speech signal using audio signal processing techniques....
 Weather forecastingWeather forecastingWeather forecasting is the application of science and technology to predict the state of the atmosphere for a given location. Human beings have attempted to predict the weather informally for millennia, and formally since the nineteenth century...
 Navigation systemNavigation systemA navigation system is a system that aids is navigation. Navigation systems may be entirely on board a vehicle or vessel, or they may be located elsewhere and communicate via radio or other signals with a vehicle or vessel, or they may use a combination of these methods.Navigation systems may be...
 3D modeling3D modelingIn 3D computer graphics, 3D modeling is the process of developing a mathematical representation of any threedimensional surface of object via specialized software. The product is called a 3D model...
See also
 Alpha beta filterAlpha beta filterAn alpha beta filter is a simplified form of observer for estimation, data smoothing and control applications. It is closely related to Kalman filters and to linear state observers used in control theory...
 Covariance intersectionCovariance intersectionCovariance intersection is an algorithm for combining two or more estimates of state variables in a Kalman filter when the correlation between them is unknown.Specification:...
 Data assimilationData assimilationApplications of data assimilation arise in many fields of geosciences, perhaps most importantly in weather forecasting and hydrology. Data assimilation proceeds by analysis cycles...
 Ensemble Kalman filterEnsemble Kalman filterThe ensemble Kalman filter is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models...
 Extended Kalman filterExtended Kalman filterIn estimation theory, the extended Kalman filter is the nonlinear version of the Kalman filter which linearizes about the current mean and covariance...
 Invariant extended Kalman filterInvariant extended Kalman filterThe invariant extended Kalman filter is a new version of the extended Kalman filter for nonlinear systems possessing symmetries . It combines the advantages of both the EKF and the recently introduced symmetrypreserving filters...
 Fast Kalman filterFast Kalman filterThe fast Kalman filter , devised by Antti Lange , is an extension of the HelmertWolf blocking method from geodesy to realtime applications of Kalman filtering such as satellite imaging of the Earth...
 Compare with: Wiener filterWiener filterIn signal processing, the Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published in 1949. Its purpose is to reduce the amount of noise present in a signal by comparison with an estimation of the desired noiseless signal. The discretetime equivalent of Wiener's work was...
, and the multimodal Particle filterParticle filterIn statistics, particle filters, also known as Sequential Monte Carlo methods , are sophisticated model estimation techniques based on simulation...
estimator.  Filtering problem (stochastic processes)Filtering problem (stochastic processes)In the theory of stochastic processes, the filtering problem is a mathematical model for a number of filtering problems in signal processing and the like. The general idea is to form some kind of "best estimate" for the true value of some system, given only some observations of that system...
 Kernel adaptive filterKernel adaptive filterKernel adaptive filtering is an adaptive filtering technique for general nonlinear problems. It is a natural generalization of linear adaptive filtering in reproducing kernel Hilbert spaces. Kernel adaptive filters are online kernel methods, closely related to some artificial neural networks such...
 Nonlinear filterNonlinear filterA nonlinear filter is a signalprocessing device whose output is not a linear function of its input. Terminology concerning the filtering problem may refer to the time domain showing of the signal or to the frequency domain representation of the signal. When referring to filters with adjectives...
 Predictor corrector
 Recursive least squares
 Sliding mode controlSliding mode controlIn control theory, sliding mode control, or SMC, is a nonlinear control method that alters the dynamics of a nonlinear system by application of a discontinuous control signal that forces the system to "slide" along a crosssection of the system's normal behavior. The statefeedback control law is...
– describes a sliding mode observer that has similar noise performance to the Kalman filter  Separation principleSeparation principleIn control theory, a separation principle, more formally known as a principle of separation of estimation and control, states that under some assumptions the problem of designing an optimal feedback controller for a stochastic system can be solved by designing an optimal observer for the state of...
 Zakai equationZakai equationIn filtering theory the Zakai equation is a linear recursive filtering equation for the unnormalized density of a hidden state. In contrast, the Kushner equation gives a nonlinear recursive equation for the normalized density of the hidden state...
 Stochastic differential equationStochastic differential equationA stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....
s  Volterra seriesVolterra SeriesThe Volterra series is a model for nonlinear behavior similar to the Taylor series. It differs from the Taylor series in its ability to capture 'memory' effects. The Taylor series can be used to approximate the response of a nonlinear system to a given input if the output of this system depends...
Further reading
 Thomas KailathThomas KailathThomas Kailath is an Indian electrical engineer, information theorist, control engineer, entrepreneur and the Hitachi America Professor of Engineering, Emeritus, at Stanford University...
, Ali H. SayedAli H. SayedAli H. Sayed is Professor of Electrical Engineering at the University of California, Los Angeles , where he teaches and conducts research on Adaptation, Learning, Statistical Signal Processing, and Signal Processing for Communications. He is the Director of the UCLA Adaptive Systems Laboratory...
, and Babak HassibiBabak HassibiBabak Hassibi is an IranianAmerican electrical engineer who is currently professor of Electrical Engineering and head of the Department of Electrical Engineering at the California Institute of Technology ....
, Linear Estimation, PrenticeHall, NJ, 2000, ISBN 9780130224644.
 Ali H. SayedAli H. SayedAli H. Sayed is Professor of Electrical Engineering at the University of California, Los Angeles , where he teaches and conducts research on Adaptation, Learning, Statistical Signal Processing, and Signal Processing for Communications. He is the Director of the UCLA Adaptive Systems Laboratory...
, Adaptive Filters, Wiley, NJ, 2008, ISBN 9780470253885.
External links
 A New Approach to Linear Filtering and Prediction Problems, by R. E. Kalman, 1960
 Kalman–Bucy Filter, a good derivation of the Kalman–Bucy Filter
 MIT Video Lecture on the Kalman filter
 An Introduction to the Kalman Filter, SIGGRAPH 2001 Course, Greg Welch and Gary Bishop
 Kalman filtering chapter from Stochastic Models, Estimation, and Control, vol. 1, by Peter S. Maybeck
 Kalman Filter webpage, with lots of links
 Kalman Filtering
 Kalman Filters, thorough introduction to several types, together with applications to Robot Localization
 Kalman filters used in Weather models, SIAM News, Volume 36, Number 8, October 2003.
 Critical Evaluation of Extended Kalman Filtering and MovingHorizon Estimation, Ind. Eng. Chem. Res., 44 (8), 2451–2460, 2005.
 Source code for the propeller microprocessor: Well documented source code written for the Parallax propeller processor.
 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 of Kalman Filtering applied to Simultaneous Localization and Mapping: Vehicle moving in 1D, 2D and 3D
 Derivation of a 6D EKF solution to Simultaneous Localization and Mapping (In old version PDF). See also the tutorial on implementing a Kalman Filter with the MRPT C++ librariesMobile Robot Programming ToolkitThe Mobile Robot Programming Toolkit is a crossplatform and open source C++ library aimed to help robotics researchers to design and implement algorithms related to Simultaneous Localization and Mapping , computer vision and motion planning...
.  The Kalman Filter Explained A very simple tutorial.
 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 exponentialaffine term structure models by kalman filter" published by Review of Quantitative Finance and Accounting in 1999.
 Extended Kalman Filters explained in the context of Simulation, Estimation, Control, and Optimization

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