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Kalman filter : In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán. Kalman filtering has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically. Furthermore, Kalman filtering is much applied in time series analysis tasks such as signal processing and econometrics. Kalman filtering is also important for robotic motion planning and control, and can be used for trajectory optimization. Kalman filtering also works for modeling the central nervous system's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback, the use of Kalman filters provides a realistic model for making estimates of the current state of a motor system and issuing updated commands. The algorithm works via a two-phase process: a prediction phase and an update phase. In the prediction phase, the Kalman filter produces estimates of the current state variables, including their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some error, including random noise) is observed, these estimates are updated using a weighted average, with more weight given to estimates with greater certainty. The algorithm is recursive. It can operate in real time, using only the present input measurements and the state calculated previously and its uncertainty matrix; no additional past information is required. Optimality of Kalman filtering assumes that errors have a normal (Gaussian) distribution. In the words of Rudolf E. Kálmán: "The following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear." Regardless of Gaussianity, however, if the process and measurement covariances are known, then the Kalman filter is the best possible linear estimator in the minimum mean-square-error sense, although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian. Extensions and generalizations of the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. The basis is a hidden Markov model such that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Kalman filtering has been used successfully in multi-sensor fusion, and distributed sensor networks to develop distributed or consensus Kalman filtering.
Kalman filter : The filtering method is named for Hungarian émigré Rudolf E. Kálmán, although Thorvald Nicolai Thiele and Peter Swerling developed a similar algorithm earlier. Richard S. Bucy of the Johns Hopkins Applied Physics Laboratory contributed to the theory, causing it to be known sometimes as Kalman–Bucy filtering. Kalman was inspired to derive the Kalman filter by applying state variables to the Wiener filtering problem. Stanley F. Schmidt is generally credited with developing the first implementation of a Kalman filter. He realized that the filter could be divided into two distinct parts, with one part for time periods between sensor outputs and another part for incorporating measurements. It was during a visit by Kálmán to the NASA Ames Research Center that Schmidt saw the applicability of Kálmán's ideas to the nonlinear problem of trajectory estimation for the Apollo program resulting in its incorporation in the Apollo navigation computer.: 16 This digital filter is sometimes termed the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, nonlinear filter developed by the Soviet mathematician Ruslan Stratonovich. In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before the summer of 1961, when Kalman met with Stratonovich during a conference in Moscow. This Kalman filtering was first described and developed partially in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961). The Apollo computer used 2k of magnetic core RAM and 36k wire rope [...]. The CPU was built from ICs [...]. Clock speed was under 100 kHz [...]. The fact that the MIT engineers were able to pack such good software (one of the very first applications of the Kalman filter) into such a tiny computer is truly remarkable. Kalman filters have been vital in the implementation of the navigation systems of U.S. Navy nuclear ballistic missile submarines, and in the guidance and navigation systems of cruise missiles such as the U.S. Navy's Tomahawk missile and the U.S. Air Force's Air Launched Cruise Missile. They are also used in the guidance and navigation systems of reusable launch vehicles and the attitude control and navigation systems of spacecraft which dock at the International Space Station.
Kalman filter : Kalman filtering uses a system's dynamic model (e.g., physical laws of motion), known control inputs to that system, and multiple sequential measurements (such as from sensors) to form an estimate of the system's varying quantities (its state) that is better than the estimate obtained by using only one measurement alone. As such, it is a common sensor fusion and data fusion algorithm. Noisy sensor data, approximations in the equations that describe the system evolution, and external factors that are not accounted for, all limit how well it is possible to determine the system's state. The Kalman filter deals effectively with the uncertainty due to noisy sensor data and, to some extent, with random external factors. The Kalman filter produces an estimate of the state of the system as an average of the system's predicted state and of the new measurement using a weighted average. The purpose of the weights is that values with better (i.e., smaller) estimated uncertainty are "trusted" more. The weights are calculated from the covariance, a measure of the estimated uncertainty of the prediction of the system's state. The result of the weighted average is a new state estimate that lies between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated at every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that Kalman filter works recursively and requires only the last "best guess", rather than the entire history, of a system's state to calculate a new state. The measurements' certainty-grading and current-state estimate are important considerations. It is common to discuss the filter's response in terms of the Kalman filter's gain. The Kalman gain is the weight given to the measurements and current-state estimate, and can be "tuned" to achieve a particular performance. With a high gain, the filter places more weight on the most recent measurements, and thus conforms to them more responsively. With a low gain, the filter conforms to the model predictions more closely. At the extremes, a high gain (close to one) will result in a more jumpy estimated trajectory, while a low gain (close to zero) will smooth out noise but decrease the responsiveness. When performing the actual calculations for the filter (as discussed below), the state estimate and covariances are coded into matrices because of the multiple dimensions involved in a single set of calculations. This allows for a representation of linear relationships between different state variables (such as position, velocity, and acceleration) in any of the transition models or covariances.
Kalman filter : As an example application, consider the problem of determining the precise location of a truck. The truck can be equipped with a GPS unit that provides an estimate of the position within a few meters. The GPS estimate is likely to be noisy; readings 'jump around' rapidly, though remaining within a few meters of the real position. In addition, since the truck is expected to follow the laws of physics, its position can also be estimated by integrating its velocity over time, determined by keeping track of wheel revolutions and the angle of the steering wheel. This is a technique known as dead reckoning. Typically, the dead reckoning will provide a very smooth estimate of the truck's position, but it will drift over time as small errors accumulate. For this example, the Kalman filter can be thought of as operating in two distinct phases: predict and update. In the prediction phase, the truck's old position will be modified according to the physical laws of motion (the dynamic or "state transition" model). Not only will a new position estimate be calculated, but also a new covariance will be calculated as well. Perhaps the covariance is proportional to the speed of the truck because we are more uncertain about the accuracy of the dead reckoning position estimate at high speeds but very certain about the position estimate at low speeds. Next, in the update phase, a measurement of the truck's position is taken from the GPS unit. Along with this measurement comes some amount of uncertainty, and its covariance relative to that of the prediction from the previous phase determines how much the new measurement will affect the updated prediction. Ideally, as the dead reckoning estimates tend to drift away from the real position, the GPS measurement should pull the position estimate back toward the real position but not disturb it to the point of becoming noisy and rapidly jumping.
Kalman filter : The Kalman filter is an efficient recursive filter estimating 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 systems engineering. Together with the linear-quadratic regulator (LQR), the Kalman filter solves the linear–quadratic–Gaussian control problem (LQG). The Kalman filter, the linear-quadratic regulator, and the linear–quadratic–Gaussian controller are solutions to what arguably are the most fundamental problems of control theory. In most applications, the internal state is much larger (has 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. For the Dempster–Shafer theory, each state equation or observation is considered a special case of a linear belief function and the Kalman filtering is a special case of combining linear belief functions on a join-tree or Markov tree. Additional methods include belief filtering which use Bayes or evidential updates to the state equations. A wide variety of Kalman filters exists by now: Kalman's original formulation - now termed the "simple" Kalman filter, the Kalman–Bucy filter, Schmidt's "extended" filter, the information filter, and a variety of "square-root" filters that were developed by Bierman, Thornton, and many others. Perhaps the most commonly used type of very simple Kalman filter is the phase-locked loop, which is now ubiquitous in radios, especially frequency modulation (FM) radios, television sets, satellite communications receivers, outer space communications systems, and nearly any other electronic communications equipment.
Kalman filter : Kalman filtering is based on linear dynamic systems discretized in the time domain. They are modeled on a Markov chain built on linear operators perturbed by errors that may include Gaussian noise. The state of the target system refers to the ground truth (yet hidden) system configuration of interest, which is represented as a vector of real numbers. At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from the controls on the system if they are known. Then, another linear operator mixed with more noise generates the measurable outputs (i.e., observation) from the true ("hidden") state. The Kalman filter may be regarded as analogous to the hidden Markov model, with the difference that the hidden state variables have values in a continuous space as opposed to a discrete state space as for the hidden Markov model. There is a strong analogy between the equations of a 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 following framework. This means specifying the matrices, for each time-step k , following: F k _ , the state-transition model; H k _ , the observation model; Q k _ , the covariance of the process noise; R k _ , the covariance of the observation noise; and sometimes B k _ , the control-input model as described below; if B k _ is included, then there is also u k _ , the control vector, representing the controlling input into control-input model. As seen below, it is common in many applications that the matrices F , H , Q , R , and B are constant across time, in which case their k index may be dropped. The Kalman filter model assumes the true state at time k is evolved from the state at k − 1 according to x k = F k x k − 1 + B k u k + w k _=\mathbf _\mathbf _+\mathbf _\mathbf _+\mathbf _ where F k _ is the state transition model which is applied to the previous state xk−1; B k _ is the control-input model which is applied to the control vector u k _ ; w k _ is the process noise, which is assumed to be drawn from a zero mean multivariate normal distribution, N , with covariance, Q k _ : w k ∼ N ( 0 , Q k ) _\sim \left(0,\mathbf _\right) . If Q is independent of time, one may, following Roweis and Ghahramani,: 307 write w ∙ _ instead of w k _ to emphasize that the noise has no explicit knowledge of time. At time k an observation (or measurement) z k _ of the true state x k _ is made according to z k = H k x k + v k _=\mathbf _\mathbf _+\mathbf _ where H k _ is the observation model, which maps the true state space into the observed space and v k _ is the observation noise, which is assumed to be zero mean Gaussian white noise with covariance R k _ : v k ∼ N ( 0 , R k ) _\sim \left(0,\mathbf _\right) . Analogously to the situation for w k _ , one may write v ∙ _ instead of v k _ if R is independent of time. The initial state, and the noise vectors at each step _,\mathbf _,\dots ,\mathbf _,\mathbf _,\dots ,\mathbf _\ are all assumed to be mutually independent. Many real-time dynamic systems do not exactly conform to this model. In fact, unmodeled dynamics can seriously degrade the filter performance, even when it was supposed to work with unknown stochastic signals as inputs. The reason for this is that the effect of unmodeled dynamics depends on the input, and, therefore, can bring the estimation algorithm to instability (it diverges). On the other hand, independent white noise signals will not make the algorithm diverge. The problem of distinguishing between measurement noise and unmodeled dynamics is a difficult one and is treated as a problem of control theory using robust control.
Kalman filter : The Kalman filter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. In what follows, the notation x ^ n ∣ m _ represents the estimate of x at time n given observations up to and including at time m ≤ n. The state of the filter is represented by two variables: x ^ k ∣ k _ , the a posteriori state estimate mean at time k given observations up to and including at time k; P k ∣ k _ , the a posteriori estimate covariance matrix (a measure of the estimated accuracy of the state estimate). The algorithm structure of the Kalman filter resembles that of Alpha beta filter. 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 innovation (the pre-fit residual), i.e. the difference between the current a priori prediction and the current observation information, is multiplied by the optimal Kalman gain and combined with the previous state estimate to refine the state estimate. This improved estimate based on the current observation 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 procedures performed. Likewise, if multiple independent observations are available at the same time, multiple update procedures may be performed (typically with different observation matrices Hk).
Kalman filter : Consider a truck on frictionless, straight rails. Initially, the truck is stationary at position 0, but it is buffeted this way and that by random uncontrolled forces. We measure the position of the truck every Δt seconds, but these measurements are imprecise; we want to maintain a model of the truck's position and velocity. We show here how we derive the model from which we create our Kalman filter. Since F , H , R , Q ,\mathbf ,\mathbf ,\mathbf are constant, their time indices are dropped. The position and velocity of the truck are described by the linear state space x k = [ x x ˙ ] _=x\\\end where x ˙ is the velocity, that is, the derivative of position with respect to time. We assume that between the (k − 1) and k timestep, uncontrolled forces cause a constant acceleration of ak that is normally distributed with mean 0 and standard deviation σa. From Newton's laws of motion we conclude that x k = F x k − 1 + G a k _=\mathbf \mathbf _+\mathbf a_ (there is no B u u term since there are no known control inputs. Instead, ak is the effect of an unknown input and G applies that effect to the state vector) where F = [ 1 Δ t 0 1 ] G = [ 1 2 Δ t 2 Δ t ] \mathbf &=1&\Delta t\\0&1\end\\[4pt]\mathbf &=^\\[6pt]\Delta t\end\end so that x k = F x k − 1 + w k _=\mathbf \mathbf _+\mathbf _ where w k ∼ N ( 0 , Q ) Q = G G T σ a 2 = [ 1 4 Δ t 4 1 2 Δ t 3 1 2 Δ t 3 Δ t 2 ] σ a 2 . \mathbf _&\sim N(0,\mathbf )\\\mathbf &=\mathbf \mathbf ^\sigma _^=^&^\\[6pt]^&^\end\sigma _^.\end The matrix Q is not full rank (it is of rank one if Δ t ≠ 0 ). Hence, the distribution N ( 0 , Q ) ) is not absolutely continuous and has no probability density function. Another way to express this, avoiding explicit degenerate distributions is given by w k ∼ G ⋅ N ( 0 , σ a 2 ) . _\sim \mathbf \cdot N\left(0,\sigma _^\right). At each time phase, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise vk is also distributed normally, with mean 0 and standard deviation σz. z k = H x k + v k _=\mathbf _+\mathbf _ where H = [ 1 0 ] =1&0\end and R = E [ v k v k T ] = [ σ z 2 ] =\mathrm \left[\mathbf _\mathbf _^\right]=\sigma _^\end We know the initial starting state of the truck with perfect precision, so we initialize x ^ 0 ∣ 0 = [ 0 0 ] _=0\\0\end and to tell the filter that we know the exact position and velocity, we give it a zero covariance matrix: P 0 ∣ 0 = [ 0 0 0 0 ] _=0&0\\0&0\end If the initial position and velocity are not known perfectly, the covariance matrix should be initialized with suitable variances on its diagonal: P 0 ∣ 0 = [ σ x 2 0 0 σ x ˙ 2 ] _=\sigma _^&0\\0&\sigma _^\end The filter will then prefer the information from the first measurements over the information already in the model.
Kalman filter : For simplicity, assume that the control input u k = 0 _=\mathbf . Then the Kalman filter may be written: x ^ k ∣ k = F k x ^ k − 1 ∣ k − 1 + K k [ z k − H k F k x ^ k − 1 ∣ k − 1 ] . _=\mathbf _ _+\mathbf _[\mathbf _-\mathbf _\mathbf _ _]. A similar equation holds if we include a non-zero control input. Gain matrices K k _ evolve independently of the measurements z k _ . From above, the four equations needed for updating the Kalman gain are as follows: P k ∣ k − 1 = F k P k − 1 ∣ k − 1 F k T + Q k , S k = H k P k ∣ k − 1 H k T + R k , K k = P k ∣ k − 1 H k T S k − 1 , P k \| k = ( I − K k H k ) P k \| k − 1 . \mathbf _&=\mathbf _\mathbf _\mathbf _^+\mathbf _,\\\mathbf _&=\mathbf _\mathbf _\mathbf _^+\mathbf _,\\\mathbf _&=\mathbf _\mathbf _^\mathbf _^,\\\mathbf _&=\left(\mathbf -\mathbf _\mathbf _\right)\mathbf _.\end Since the gain matrices depend only on the model, and not the measurements, they may be computed offline. Convergence of the gain matrices K k _ to an asymptotic matrix K ∞ _ applies for conditions established in Walrand and Dimakis. Simulations establish the number of steps to convergence. For the moving truck example described above, with Δ t = 1 . and σ a 2 = σ z 2 = σ x 2 = σ x ˙ 2 = 1 ^=\sigma _^=\sigma _^=\sigma _^=1 , simulation shows convergence in 10 iterations. Using the asymptotic gain, and assuming H k _ and F k _ are independent of k , the Kalman filter becomes a linear time-invariant filter: x ^ k = F x ^ k − 1 + K ∞ [ z k − H F x ^ k − 1 ] . _=\mathbf _+\mathbf _[\mathbf _-\mathbf \mathbf _]. The asymptotic gain K ∞ _ , if it exists, can be computed by first solving the following discrete Riccati equation for the asymptotic state covariance P ∞ _ : P ∞ = F ( P ∞ − P ∞ H T ( H P ∞ H T + R ) − 1 H P ∞ ) F T + Q . _=\mathbf \left(\mathbf _-\mathbf _\mathbf ^\left(\mathbf \mathbf _\mathbf ^+\mathbf \right)^\mathbf \mathbf _\right)\mathbf ^+\mathbf . The asymptotic gain is then computed as before. K ∞ = P ∞ H T ( R + H P ∞ H T ) − 1 . _=\mathbf _\mathbf ^\left(\mathbf +\mathbf \mathbf _\mathbf ^\right)^. Additionally, a form of the asymptotic Kalman filter more commonly used in control theory is given by x ^ k + 1 = F x ^ k + B u k + K ¯ ∞ [ z k − H x ^ k ] , _=\mathbf _+\mathbf \mathbf _+\mathbf _[\mathbf _-\mathbf _], where K ¯ ∞ = F P ∞ H T ( R + H P ∞ H T ) − 1 . _=\mathbf \mathbf _\mathbf ^\left(\mathbf +\mathbf \mathbf _\mathbf ^\right)^. This leads to an estimator of the form x ^ k + 1 = ( F − K ¯ ∞ H ) x ^ k + B u k + K ¯ ∞ z k , _=(\mathbf - _\mathbf ) _+\mathbf \mathbf _+\mathbf _\mathbf _,
Kalman filter : The Kalman filter can be derived as a generalized least squares method operating on previous data.
Kalman filter : The Kalman filtering equations provide an estimate of the state x ^ k ∣ k _ and its error covariance P k ∣ k _ recursively. The estimate and its quality depend on the system parameters and the noise statistics fed as inputs to the estimator. This section analyzes the effect of uncertainties in the statistical inputs to the filter. In the absence of reliable statistics or the true values of noise covariance matrices Q k _ and R k _ , the expression P k ∣ k = ( I − K k H k ) P k ∣ k − 1 ( I − K k H k ) T + K k R k K k T _=\left(\mathbf -\mathbf _\mathbf _\right)\mathbf _\left(\mathbf -\mathbf _\mathbf _\right)^+\mathbf _\mathbf _\mathbf _^ no longer provides the actual error covariance. In other words, P k ∣ k ≠ E [ ( x k − x ^ k ∣ k ) ( x k − x ^ k ∣ k ) T ] _\neq E\left[\left(\mathbf _- _\right)\left(\mathbf _- _\right)^\right] . In most real-time applications, the covariance matrices that are used in designing the Kalman filter are different from the actual (true) noise covariances matrices. This sensitivity analysis describes the behavior of the estimation error covariance when the noise covariances as well as the system matrices F k _ and H k _ that are fed as inputs to the filter are incorrect. Thus, the sensitivity analysis describes the robustness (or sensitivity) of the estimator to misspecified statistical and parametric inputs to the estimator. This discussion is limited to the error sensitivity analysis for the case of statistical uncertainties. Here the actual noise covariances are denoted by Q k a _^ and R k a _^ respectively, whereas the design values used in the estimator are Q k _ and R k _ respectively. The actual error covariance is denoted by P k ∣ k a _^ and P k ∣ k _ as computed by the Kalman filter is referred to as the Riccati variable. When Q k ≡ Q k a _\equiv \mathbf _^ and R k ≡ R k a _\equiv \mathbf _^ , this means that P k ∣ k = P k ∣ k a _=\mathbf _^ . While computing the actual error covariance using P k ∣ k a = E [ ( x k − x ^ k ∣ k ) ( x k − x ^ k ∣ k ) T ] _^=E\left[\left(\mathbf _- _\right)\left(\mathbf _- _\right)^\right] , substituting for x ^ k ∣ k _ and using the fact that E [ w k w k T ] = Q k a _\mathbf _^\right]=\mathbf _^ and E [ v k v k T ] = R k a _\mathbf _^\right]=\mathbf _^ , results in the following recursive equations for P k ∣ k a _^ : P k ∣ k − 1 a = F k P k − 1 ∣ k − 1 a F k T + Q k a _^=\mathbf _\mathbf _^\mathbf _^+\mathbf _^ and P k ∣ k a = ( I − K k H k ) P k ∣ k − 1 a ( I − K k H k ) T + K k R k a K k T _^=\left(\mathbf -\mathbf _\mathbf _\right)\mathbf _^\left(\mathbf -\mathbf _\mathbf _\right)^+\mathbf _\mathbf _^\mathbf _^ While computing P k ∣ k _ , by design the filter implicitly assumes that E [ w k w k T ] = Q k _\mathbf _^\right]=\mathbf _ and E [ v k v k T ] = R k _\mathbf _^\right]=\mathbf _ . The recursive expressions for P k ∣ k a _^ and P k ∣ k _ are identical except for the presence of Q k a _^ and R k a _^ in place of the design values Q k _ and R k _ respectively. Researches have been done to analyze Kalman filter system's robustness.
Kalman filter : One problem with the Kalman filter is its numerical stability. If the process noise covariance Qk is small, round-off error often causes a small positive eigenvalue of the state covariance matrix P to be computed as a negative number. This renders the numerical representation of P indefinite, while its true form is positive-definite. Positive definite matrices have the property that they have a factorization into the product of a non-singular, lower-triangular matrix S and its transpose : P = S·ST . The factor S can be computed efficiently using the Cholesky factorization algorithm. This product form of the covariance matrix P is guaranteed to be symmetric, and for all 1 <= k <= n, the k-th diagonal element Pkk is equal to the euclidean norm of the k-th row of S, which is necessarily positive. An equivalent form, which avoids many of the square root operations involved in the Cholesky factorization algorithm, yet preserves the desirable numerical properties, is the U-D decomposition form, P = U·D·UT, where U is a unit triangular matrix (with unit diagonal), and D is a diagonal matrix. Between the two, the U-D factorization uses the same amount of storage, and somewhat less computation, and is the most commonly used triangular factorization. (Early literature on the relative efficiency is somewhat misleading, as it assumed that square roots were much more time-consuming than divisions,: 69 while on 21st-century computers they are only slightly more expensive.) Efficient algorithms for the Kalman prediction and update steps in the factored form were developed by G. J. Bierman and C. L. Thornton. The L·D·LT decomposition of the innovation covariance matrix Sk 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·LT 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 well-conditioned. The pivoting algorithm must retain any portion of the innovation covariance matrix directly corresponding to observed state-variables Hk·xk\|k-1 that are associated with auxiliary observations in yk. The l·d·lt square-root filter requires orthogonalization of the observation vector. This may be done with the inverse square-root of the covariance matrix for the auxiliary variables using Method 2 in Higham (2002, p. 263).
Kalman filter : The Kalman filter is efficient for sequential data processing on central processing units (CPUs), but in its original form it is inefficient on parallel architectures such as graphics processing units (GPUs). It is however possible to express the filter-update routine in terms of an associative operator using the formulation in Särkkä and García-Fernández (2021). The filter solution can then be retrieved by the use of a prefix sum algorithm which can be efficiently implemented on GPU. This reduces the computational complexity from O ( N ) in the number of time steps to O ( log ⁡ ( N ) ) .
Kalman filter : The Kalman filter can be presented as one of the simplest dynamic Bayesian networks. The Kalman filter calculates estimates of the true values of states recursively over time using incoming measurements and a mathematical process model. Similarly, recursive Bayesian estimation calculates estimates of an unknown probability density function (PDF) recursively over time using incoming measurements and a mathematical process model. 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. p ( x k ∣ x 0 , … , x k − 1 ) = p ( x k ∣ x k − 1 ) _\mid \mathbf _,\dots ,\mathbf _)=p(\mathbf _\mid \mathbf _) Similarly, the measurement at the k-th timestep is dependent only upon the current state and is conditionally independent of all other states given the current state. p ( z k ∣ x 0 , … , x k ) = p ( z k ∣ x k ) _\mid \mathbf _,\dots ,\mathbf _)=p(\mathbf _\mid \mathbf _) Using these assumptions the probability distribution over all states of the hidden Markov model can be written simply as: p ( x 0 , … , x k , z 1 , … , z k ) = p ( x 0 ) ∏ i = 1 k p ( z i ∣ x i ) p ( x i ∣ x i − 1 ) _,\dots ,\mathbf _,\mathbf _,\dots ,\mathbf _\right)=p\left(\mathbf _\right)\prod _^p\left(\mathbf _\mid \mathbf _\right)p\left(\mathbf _\mid \mathbf _\right) However, when a 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 results in the predict and update phases of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is the sum (integral) of the products of the probability distribution associated with the transition from the (k − 1)-th timestep to the k-th and the probability distribution associated with the previous state, over all possible x k − 1 . p ( x k ∣ Z k − 1 ) = ∫ p ( x k ∣ x k − 1 ) p ( x k − 1 ∣ Z k − 1 ) d x k − 1 _\mid \mathbf _\right)=\int p\left(\mathbf _\mid \mathbf _\right)p\left(\mathbf _\mid \mathbf _\right)\,d\mathbf _ The measurement set up to time t is Z t = _=\left\ _,\dots ,\mathbf _\right\ The probability distribution of the update is proportional to the product of the measurement likelihood and the predicted state. p ( x k ∣ Z k ) = p ( z k ∣ x k ) p ( x k ∣ Z k − 1 ) p ( z k ∣ Z k − 1 ) _\mid \mathbf _\right)= _\mid \mathbf _\right)p\left(\mathbf _\mid \mathbf _\right) _\mid \mathbf _\right) The denominator p ( z k ∣ Z k − 1 ) = ∫ p ( z k ∣ x k ) p ( x k ∣ Z k − 1 ) d x k _\mid \mathbf _\right)=\int p\left(\mathbf _\mid \mathbf _\right)p\left(\mathbf _\mid \mathbf _\right)\,d\mathbf _ is a normalization term. The remaining probability density functions are p ( x k ∣ x k − 1 ) = N ( F k x k − 1 , Q k ) p ( z k ∣ x k ) = N ( H k x k , R k ) p ( x k − 1 ∣ Z k − 1 ) = N ( x ^ k − 1 , P k − 1 ) p\left(\mathbf _\mid \mathbf _\right)&=\left(\mathbf _\mathbf _,\mathbf _\right)\\p\left(\mathbf _\mid \mathbf _\right)&=\left(\mathbf _\mathbf _,\mathbf _\right)\\p\left(\mathbf _\mid \mathbf _\right)&=\left( _,\mathbf _\right)\end The PDF at the previous timestep is assumed inductively to be the estimated state and covariance. This is justified because, as an optimal estimator, the Kalman filter makes best use of the measurements, therefore the PDF for x k _ given the measurements Z k _ is the Kalman filter estimate.
Kalman filter : Related to the recursive Bayesian interpretation described above, the Kalman filter can be viewed as a generative model, i.e., a process for generating a stream of random observations z = (z0, z1, z2, ...). Specifically, the process is Sample a hidden state x 0 _ from the Gaussian prior distribution p ( x 0 ) = N ( x ^ 0 ∣ 0 , P 0 ∣ 0 ) _\right)=\left( _,\mathbf _\right) . Sample an observation z 0 _ from the observation model p ( z 0 ∣ x 0 ) = N ( H 0 x 0 , R 0 ) _\mid \mathbf _\right)=\left(\mathbf _\mathbf _,\mathbf _\right) . For k = 1 , 2 , 3 , … , do Sample the next hidden state x k _ from the transition model p ( x k ∣ x k − 1 ) = N ( F k x k − 1 + B k u k , Q k ) . _\mid \mathbf _\right)=\left(\mathbf _\mathbf _+\mathbf _\mathbf _,\mathbf _\right). Sample an observation z k _ from the observation model p ( z k ∣ x k ) = N ( H k x k , R k ) . _\mid \mathbf _\right)=\left(\mathbf _\mathbf _,\mathbf _\right). This process has identical structure to the hidden Markov model, except that the discrete state and observations are replaced with continuous variables sampled from Gaussian distributions. In some applications, it is useful to compute the probability that a Kalman filter with a given set of parameters (prior distribution, transition and observation models, and control inputs) would generate a particular observed signal. This probability is known as the marginal likelihood because it integrates over ("marginalizes out") the values of the hidden state variables, so it can be computed using only the observed signal. The marginal likelihood can be useful to evaluate different parameter choices, or to compare the Kalman filter against other models using Bayesian model comparison. It is straightforward to compute the marginal likelihood as a side effect of the recursive filtering computation. By the chain rule, the likelihood can be factored as the product of the probability of each observation given previous observations, p ( z ) = ∏ k = 0 T p ( z k ∣ z k − 1 , … , z 0 ) )=\prod _^p\left(\mathbf _\mid \mathbf _,\ldots ,\mathbf _\right) , and because the Kalman filter describes a Markov process, all relevant information from previous observations is contained in the current state estimate x ^ k ∣ k − 1 , P k ∣ k − 1 . _,\mathbf _. Thus the marginal likelihood is given by p ( z ) = ∏ k = 0 T ∫ p ( z k ∣ x k ) p ( x k ∣ z k − 1 , … , z 0 ) d x k = ∏ k = 0 T ∫ N ( z k ; H k x k , R k ) N ( x k ; x ^ k ∣ k − 1 , P k ∣ k − 1 ) d x k = ∏ k = 0 T N ( z k ; H k x ^ k ∣ k − 1 , R k + H k P k ∣ k − 1 H k T ) = ∏ k = 0 T N ( z k ; H k x ^ k ∣ k − 1 , S k ) , p(\mathbf )&=\prod _^\int p\left(\mathbf _\mid \mathbf _\right)p\left(\mathbf _\mid \mathbf _,\ldots ,\mathbf _\right)d\mathbf _\\&=\prod _^\int \left(\mathbf _;\mathbf _\mathbf _,\mathbf _\right)\left(\mathbf _; _,\mathbf _\right)d\mathbf _\\&=\prod _^\left(\mathbf _;\mathbf _ _,\mathbf _+\mathbf _\mathbf _\mathbf _^\right)\\&=\prod _^\left(\mathbf _;\mathbf _ _,\mathbf _\right),\end i.e., a product of Gaussian densities, each corresponding to the density of one observation zk under the current filtering distribution H k x ^ k ∣ k − 1 , S k _ _,\mathbf _ . This can easily be computed as a simple recursive update; however, to avoid numeric underflow, in a practical implementation it is usually desirable to compute the log marginal likelihood ℓ = log ⁡ p ( z ) ) instead. Adopting the convention ℓ ( − 1 ) = 0 =0 , this can be done via the recursive update rule ℓ ( k ) = ℓ ( k − 1 ) − 1 2 ( y ~ k T S k − 1 y ~ k + log ⁡ \| S k \| + d y log ⁡ 2 π ) , =\ell ^-\left( _^\mathbf _^ _+\log \left\|\mathbf _\right\|+d_\log 2\pi \right), where d y is the dimension of the measurement vector. An important application where such a (log) likelihood of the observations (given the filter parameters) is used is multi-target tracking. For example, consider an object tracking scenario where a stream of observations is the input, however, it is unknown how many objects are in the scene (or, the number of objects is known but is greater than one). For such a scenario, it can be unknown apriori which observations/measurements were generated by which object. A multiple hypothesis tracker (MHT) typically will form different track association hypotheses, where each hypothesis can be considered as a Kalman filter (for the linear Gaussian case) with a specific set of parameters associated with the hypothesized object. Thus, it is important to compute the likelihood of the observations for the different hypotheses under consideration, such that the most-likely one can be found.
Kalman filter : In cases where the dimension of the observation vector y is bigger than the dimension of the state space vector x, the information filter can avoid the inversion of a bigger matrix in the Kalman gain calculation at the price of inverting a smaller matrix in the prediction step, thus saving computing time. Additionally, the information filter allows for system information initialization according to I 1 \| 0 = P 1 \| 0 − 1 = 0 =P_^=0 , which would not be possible for the regular Kalman filter. In the information filter, or inverse covariance filter, the estimated covariance and estimated state are replaced by the information matrix and information vector respectively. These are defined as: Y k ∣ k = P k ∣ k − 1 y ^ k ∣ k = P k ∣ k − 1 x ^ k ∣ k \mathbf _&=\mathbf _^\\ _&=\mathbf _^ _\end Similarly the predicted covariance and state have equivalent information forms, defined as: Y k ∣ k − 1 = P k ∣ k − 1 − 1 y ^ k ∣ k − 1 = P k ∣ k − 1 − 1 x ^ k ∣ k − 1 \mathbf _&=\mathbf _^\\ _&=\mathbf _^ _\end and the measurement covariance and measurement vector, which are defined as: I k = H k T R k − 1 H k i k = H k T R k − 1 z k \mathbf _&=\mathbf _^\mathbf _^\mathbf _\\\mathbf _&=\mathbf _^\mathbf _^\mathbf _\end The information update now becomes a trivial sum. Y k ∣ k = Y k ∣ k − 1 + I k y ^ k ∣ k = y ^ k ∣ k − 1 + i k \mathbf _&=\mathbf _+\mathbf _\\ _&= _+\mathbf _\end The main advantage of the information filter is that N measurements can be filtered at each time step simply by summing their information matrices and vectors. Y k ∣ k = Y k ∣ k − 1 + ∑ j = 1 N I k , j y ^ k ∣ k = y ^ k ∣ k − 1 + ∑ j = 1 N i k , j \mathbf _&=\mathbf _+\sum _^\mathbf _\\ _&= _+\sum _^\mathbf _\end 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. M k = [ F k − 1 ] T Y k − 1 ∣ k − 1 F k − 1 C k = M k [ M k + Q k − 1 ] − 1 L k = I − C k Y k ∣ k − 1 = L k M k + C k Q k − 1 C k T y ^ k ∣ k − 1 = L k [ F k − 1 ] T y ^ k − 1 ∣ k − 1 \mathbf _&=\left[\mathbf _^\right]^\mathbf _\mathbf _^\\\mathbf _&=\mathbf _\left[\mathbf _+\mathbf _^\right]^\\\mathbf _&=\mathbf -\mathbf _\\\mathbf _&=\mathbf _\mathbf _+\mathbf _\mathbf _^\mathbf _^\\ _&=\mathbf _\left[\mathbf _^\right]^ _\end
Kalman filter : The optimal fixed-lag smoother provides the optimal estimate of x ^ k − N ∣ k _ for a given fixed-lag N using the measurements from z 1 _ to z k _ . It can be derived using the previous theory via an augmented state, and the main equation of the filter is the following: [ x ^ t ∣ t x ^ t − 1 ∣ t ⋮ x ^ t − N + 1 ∣ t ] = [ I 0 ⋮ 0 ] x ^ t ∣ t − 1 + [ 0 … 0 I 0 ⋮ ⋮ ⋱ ⋮ 0 … I ] [ x ^ t − 1 ∣ t − 1 x ^ t − 2 ∣ t − 1 ⋮ x ^ t − N + 1 ∣ t − 1 ] + [ K ( 0 ) K ( 1 ) ⋮ K ( N − 1 ) ] y t ∣ t − 1 _\\ _\\\vdots \\ _\\\end=\mathbf \\0\\\vdots \\0\\\end _+0&\ldots &0\\\mathbf &0&\vdots \\\vdots &\ddots &\vdots \\0&\ldots &\mathbf \\\end _\\ _\\\vdots \\ _\\\end+\mathbf ^\\\mathbf ^\\\vdots \\\mathbf ^\\\end\mathbf _ where: x ^ t ∣ t − 1 _ is estimated via a standard Kalman filter; y t ∣ t − 1 = z t − H x ^ t ∣ t − 1 _=\mathbf _-\mathbf _ is the innovation produced considering the estimate of the standard Kalman filter; the various x ^ t − i ∣ t _ with i = 1 , … , N − 1 are new variables; i.e., they do not appear in the standard Kalman filter; the gains are computed via the following scheme: K ( i + 1 ) = P ( i ) H T [ H P H T + R ] − 1 ^=\mathbf ^\mathbf ^\left[\mathbf \mathbf \mathbf ^+\mathbf \right]^ and P ( i ) = P [ ( F − K H ) T ] i ^=\mathbf \left[\left(\mathbf -\mathbf \mathbf \right)^\right]^ where P and K are the prediction error covariance and the gains of the standard Kalman filter (i.e., P t ∣ t − 1 _ ). If the estimation error covariance is defined so that P i := E [ ( x t − i − x ^ t − i ∣ t ) ∗ ( x t − i − x ^ t − i ∣ t ) ∣ z 1 … z t ] , _:=E\left[\left(\mathbf _- _\right)^\left(\mathbf _- _\right)\mid z_\ldots z_\right], then we have that the improvement on the estimation of x t − i _ is given by: P − P i = ∑ j = 0 i [ P ( j ) H T ( H P H T + R ) − 1 H ( P ( i ) ) T ] -\mathbf _=\sum _^\left[\mathbf ^\mathbf ^\left(\mathbf \mathbf \mathbf ^+\mathbf \right)^\mathbf \left(\mathbf ^\right)^\right]
Kalman filter : The optimal fixed-interval smoother provides the optimal estimate of x ^ k ∣ n _ ( k < n ) using the measurements from a fixed interval z 1 _ to z n _ . This is also called "Kalman Smoothing". There are several smoothing algorithms in common use.
Kalman filter : Pioneering research on the perception of sounds at different frequencies was conducted by Fletcher and Munson in the 1930s. Their work led to a standard way of weighting measured sound levels within investigations of industrial noise and hearing loss. Frequency weightings have since been used within filter and controller designs to manage performance within bands of interest. Typically, a frequency shaping function is used to weight the average power of the error spectral density in a specified frequency band. Let y − y ^ - denote the output estimation error exhibited by a conventional Kalman filter. Also, let W denote a causal frequency weighting transfer function. The optimum solution which minimizes the variance of W ( y − y ^ ) \left(\mathbf - \right) arises by simply constructing W − 1 y ^ ^ . The design of W remains an open question. One way of proceeding is to identify a system which generates the estimation error and setting W equal to the inverse of that system. This procedure may be iterated to obtain mean-square error improvement at the cost of increased filter order. The same technique can be applied to smoothers.
Kalman filter : The basic Kalman filter is limited to a linear assumption. More complex systems, however, can be nonlinear. The nonlinearity can be associated either with the process model or with the observation model or with both. The most common variants of Kalman filters for non-linear systems are the Extended Kalman Filter and Unscented Kalman filter. The suitability of which filter to use depends on the non-linearity indices of the process and observation model.
Kalman filter : Adaptive Kalman filters allow to adapt for process dynamics which are not modeled in the process model F ( t ) (t) , which happens for example in the context of a maneuvering target when a constant velocity (reduced order) Kalman filter is employed for tracking.
Kalman filter : Kalman–Bucy filtering (named for Richard Snowden Bucy) is a continuous time version of Kalman filtering. It is based on the state space model d d t x ( t ) = F ( t ) x ( t ) + B ( t ) u ( t ) + w ( t ) z ( t ) = H ( t ) x ( t ) + v ( t ) \mathbf (t)&=\mathbf (t)\mathbf (t)+\mathbf (t)\mathbf (t)+\mathbf (t)\\\mathbf (t)&=\mathbf (t)\mathbf (t)+\mathbf (t)\end where Q ( t ) (t) and R ( t ) (t) represent the intensities of the two white noise terms w ( t ) (t) and v ( t ) (t) , respectively. The filter consists of two differential equations, one for the state estimate and one for the covariance: d d t x ^ ( t ) = F ( t ) x ^ ( t ) + B ( t ) u ( t ) + K ( t ) ( z ( t ) − H ( t ) x ^ ( t ) ) d d t P ( t ) = F ( t ) P ( t ) + P ( t ) F T ( t ) + Q ( t ) − K ( t ) R ( t ) K T ( t ) (t)&=\mathbf (t) (t)+\mathbf (t)\mathbf (t)+\mathbf (t)\left(\mathbf (t)-\mathbf (t) (t)\right)\\\mathbf (t)&=\mathbf (t)\mathbf (t)+\mathbf (t)\mathbf ^(t)+\mathbf (t)-\mathbf (t)\mathbf (t)\mathbf ^(t)\end where the Kalman gain is given by K ( t ) = P ( t ) H T ( t ) R − 1 ( t ) (t)=\mathbf (t)\mathbf ^(t)\mathbf ^(t) Note that in this expression for K ( t ) (t) the covariance of the observation noise R ( t ) (t) represents at the same time the covariance of the prediction error (or innovation) y ~ ( t ) = z ( t ) − H ( t ) x ^ ( t ) (t)=\mathbf (t)-\mathbf (t) (t) ; these covariances are equal only in the case of continuous time. The distinction between the prediction and update steps of discrete-time Kalman filtering does not exist in continuous time. The second differential equation, for the covariance, is an example of a Riccati equation. Nonlinear generalizations to Kalman–Bucy filters include continuous time extended Kalman filter.
Kalman filter : Most physical systems are represented as continuous-time models while discrete-time measurements are made frequently for state estimation via a digital processor. Therefore, the system model and measurement model are given by x ˙ ( t ) = F ( t ) x ( t ) + B ( t ) u ( t ) + w ( t ) , w ( t ) ∼ N ( 0 , Q ( t ) ) z k = H k x k + v k , v k ∼ N ( 0 , R k ) (t)&=\mathbf (t)\mathbf (t)+\mathbf (t)\mathbf (t)+\mathbf (t),&\mathbf (t)&\sim N\left(\mathbf ,\mathbf (t)\right)\\\mathbf _&=\mathbf _\mathbf _+\mathbf _,&\mathbf _&\sim N(\mathbf ,\mathbf _)\end where x k = x ( t k ) _=\mathbf (t_) .
Kalman filter : The traditional Kalman filter has also been employed for the recovery of sparse, possibly dynamic, signals from noisy observations. Recent works utilize notions from the theory of compressed sensing/sampling, such as the restricted isometry property and related probabilistic recovery arguments, for sequentially estimating the sparse state in intrinsically low-dimensional systems.
Kalman filter : Since linear Gaussian state-space models lead to Gaussian processes, Kalman filters can be viewed as sequential solvers for Gaussian process regression.
Kalman filter : A New Approach to Linear Filtering and Prediction Problems, by R. E. Kalman, 1960 Kalman and Bayesian Filters in Python. Open source Kalman filtering textbook. How a Kalman filter works, in pictures. Illuminates the Kalman filter with pictures and colors Kalman–Bucy Filter, a derivation of the Kalman–Bucy Filter MIT Video Lecture on the Kalman filter on YouTube Kalman filter in Javascript. Open source Kalman filter library for node.js and the web browser. An Introduction to the Kalman Filter Archived 2021-02-24 at the Wayback Machine, SIGGRAPH 2001 Course, Greg Welch and Gary Bishop Kalman Filter webpage, with many links Kalman Filter Explained Simply, Step-by-Step Tutorial of the Kalman Filter with Equations "Kalman filters used in Weather models" (PDF). SIAM News. 36 (8). October 2003. Archived from the original (PDF) on 2011-05-17. Retrieved 2007-01-27. Haseltine, Eric L.; Rawlings, James B. (2005). "Critical Evaluation of Extended Kalman Filtering and Moving-Horizon Estimation". Industrial & Engineering Chemistry Research. 44 (8): 2451. doi:10.1021/ie034308l. Gerald J. Bierman's Estimation Subroutine Library: Corresponds to the code in the research monograph "Factorization Methods for Discrete Sequential Estimation" originally published by Academic Press in 1977. Republished by Dover. Matlab Toolbox implementing parts of Gerald J. Bierman's Estimation Subroutine Library: UD / UDU' and LD / LDL' factorization with associated time and measurement updates making up the Kalman filter. Matlab Toolbox of Kalman Filtering applied to Simultaneous Localization and Mapping: Vehicle moving in 1D, 2D and 3D The Kalman Filter in Reproducing Kernel Hilbert Spaces A comprehensive introduction. Matlab code to estimate Cox–Ingersoll–Ross interest rate model with Kalman Filter Archived 2014-02-09 at the Wayback Machine: Corresponds to the paper "estimating and testing exponential-affine term structure models by kalman filter" published by Review of Quantitative Finance and Accounting in 1999. Online demo of the Kalman Filter. Demonstration of Kalman Filter (and other data assimilation methods) using twin experiments. kalman-filter.com. Insights into the use of Kalman Filters in different domains. Botella, Guillermo; Martín h., José Antonio; Santos, Matilde; Meyer-Baese, Uwe (2011). "FPGA-Based Multimodal Embedded Sensor System Integrating Low- and Mid-Level Vision". Sensors. 11 (12): 1251–1259. Bibcode:2011Senso..11.8164B. doi:10.3390/s110808164. PMC 3231703. PMID 22164069. Examples and how-to on using Kalman Filters with MATLAB Archived 2024-02-29 at the Wayback Machine A Tutorial on Filtering and Estimation Explaining Filtering (Estimation) in One Hour, Ten Minutes, One Minute, and One Sentence by Yu-Chi Ho Simo Särkkä (2013). "Bayesian Filtering and Smoothing". Cambridge University Press. Full text available on author's webpage https://users.aalto.fi/~ssarkka/.
Kruskal count : The Kruskal count (also known as Kruskal's principle, Dynkin–Kruskal count, Dynkin's counting trick, Dynkin's card trick, coupling card trick or shift coupling) is a probabilistic concept originally demonstrated by the Russian mathematician Evgenii Borisovich Dynkin in the 1950s or 1960s discussing coupling effects and rediscovered as a card trick by the American mathematician Martin David Kruskal in the early 1970s as a side-product while working on another problem. It was published by Kruskal's friend Martin Gardner and magician Karl Fulves in 1975. This is related to a similar trick published by magician Alexander F. Kraus in 1957 as Sum total and later called Kraus principle. Besides uses as a card trick, the underlying phenomenon has applications in cryptography, code breaking, software tamper protection, code self-synchronization, control-flow resynchronization, design of variable-length codes and variable-length instruction sets, web navigation, object alignment, and others.
Kruskal count : The trick is performed with cards, but is more a magical-looking effect than a conventional magic trick. The magician has no access to the cards, which are manipulated by members of the audience. Thus sleight of hand is not possible. Rather the effect is based on the mathematical fact that the output of a Markov chain, under certain conditions, is typically independent of the input. A simplified version using the hands of a clock is as follows. A volunteer picks a number from one to twelve and does not reveal it to the magician. The volunteer is instructed to start from 12 on the clock and move clockwise by a number of spaces equal to the number of letters that the chosen number has when spelled out. This is then repeated, moving by the number of letters in the new number. The output after three or more moves does not depend on the initially chosen number and therefore the magician can predict it.
Kruskal count : Coupling (probability) Discrete logarithm Equifinality Ergodic theory Geometric distribution Overlapping instructions Pollard's kangaroo algorithm Random walk Self-synchronizing code
Kruskal count : Dynkin [Ды́нкин], Evgenii Borisovich [Евге́ний Бори́сович]; Uspenskii [Успе́нский], Vladimir Andreyevich [Влади́мир Андре́евич] (1963). Written at University of Moscow, Moscow, Russia. Putnam, Alfred L.; Wirszup, Izaak (eds.). Random Walks (Mathematical Conversations Part 3). Survey of Recent East European Mathematical Literature. Vol. 3. Translated by Whaland, Jr., Norman D.; Titelbaum, Olga A. (1 ed.). Boston, Massachusetts, US: The University of Chicago / D. C. Heath and Company. LCCN 63-19838. Retrieved 2023-09-03. (1+9+80+9+1 pages) [8] (NB. This is a translation of the first Russian edition published as "Математические беседы: Задачи о многоцветной раскраске / Задачи из теории чисел / Случайные блуждания"[9] by GTTI (ГТТИ) in March 1952 as Number 6 in Library of the Mathematics Circle (Библиотека математического кружка). It is based on seminars held at the School Mathematics Circle in 1945/1946 and 1946/1947 at Moscow State University.) Dynkin [Ды́нкин], Evgenii Borisovich [Евге́ний Бори́сович] (1965) [1963-03-10, 1962-03-31]. Written at University of Moscow, Moscow, Russia. Markov Processes-I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Vol. I (121). Translated by Fabius, Jaap [at Wikidata]; Greenberg, Vida Lazarus [at Wikidata]; Maitra, Ashok Prasad [at Wikidata]; Majone, Giandomenico (1 ed.). New York, US / Berlin, Germany: Springer-Verlag (Academic Press, Inc.). doi:10.1007/978-3-662-00031-1. ISBN 978-3-662-00033-5. ISSN 0072-7830. LCCN 64-24812. S2CID 251691119. Title-No. 5104. Retrieved 2023-09-02. [10] (xii+365+1 pages); Dynkin, Evgenii Borisovich (1965) [1963-03-10, 1962-03-31]. Written at University of Moscow, Moscow, Russia. Markov Processes-II. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Vol. II (122). Translated by Fabius, Jaap [at Wikidata]; Greenberg, Vida Lazarus [at Wikidata]; Maitra, Ashok Prasad [at Wikidata]; Majone, Giandomenico (1 ed.). New York, US / Berlin, Germany: Springer-Verlag. doi:10.1007/978-3-662-25360-1. ISBN 978-3-662-23320-7. ISSN 0072-7830. LCCN 64-24812. Title-No. 5105. Retrieved 2023-09-02. (viii+274+2 pages) (NB. This was originally published in Russian as "Markovskie prot︠s︡essy" (Марковские процессы) by Fizmatgiz (Физматгиз) in 1963 and translated to English with the assistance of the author.) Dynkin [Ды́нкин], Evgenii Borisovich [Евге́ний Бори́сович]; Yushkevish [Юшкевич], Aleksandr Adol'fovich [Александр Адольфович] [in German] (1969) [1966-01-22]. Written at University of Moscow, Moscow, Russia. Markov Processes: Theorems and Problems (PDF). Translated by Wood, James S. (1 ed.). New York, US: Plenum Press / Plenum Publishing Corporation. LCCN 69-12529. Archived (PDF) from the original on 2023-09-06. Retrieved 2023-09-03. (x+237 pages) (NB. 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(18 pages) Kijima, Shuji; Montenegro, Ravi [at Wikidata] (2015-03-15) [2015-03-30/2015-04-01]. Written at Gaithersburg, Maryland, US. Katz, Jonathan (ed.). Collision of Random Walks and a Refined Analysis of Attacks on the Discrete Logarithm Problem (PDF). Proceedings of the 18th IACR International Conference on Practice and Theory in Public-Key Cryptography. Lecture Notes in Computer Science. Berlin & Heidelberg, Germany: International Association for Cryptologic Research / Springer Science+Business Media. pp. 127–149. doi:10.1007/978-3-662-46447-2_6. ISBN 978-3-662-46446-5. LNCS 9020. Archived (PDF) from the original on 2023-09-03. Retrieved 2023-09-03. (23 pages) Jose, Harish (2016-06-14) [2016-06-02]. "PDCA and the Roads to Rome: Can a lean purist and a Six Sigma purist reach the same answer to a problem?". Lean. Archived from the original on 2023-09-07. Retrieved 2023-09-07. [12][13] Lamprecht, Daniel; Dimitrov, Dimitar; Helic, Denis; Strohmaier, Markus (2016-08-17). "Evaluating and Improving Navigability of Wikipedia: A Comparative Study of Eight Language Editions". Proceedings of the 12th International Symposium on Open Collaboration (PDF). OpenSym, Berlin, Germany: Association for Computing Machinery. pp. 1–10. doi:10.1145/2957792.2957813. ISBN 978-1-4503-4451-7. S2CID 13244770. Archived (PDF) from the original on 2023-09-04. Retrieved 2021-03-17. Jämthagen, Christopher (November 2016). On Offensive and Defensive Methods in Software Security (PDF) (Thesis). Lund, Sweden: Department of Electrical and Information Technology, Lund University. p. 96. ISBN 978-91-7623-942-1. ISSN 1654-790X. Archived (PDF) from the original on 2023-08-26. Retrieved 2023-08-26. (1+xvii+1+152 pages) Mannam, Pragna; Volkov, Jr., Alexander; Paolini, Robert; Chirikjian, Gregory Scott; Mason, Matthew Thomas (2019-02-06) [2018-12-04]. "Sensorless Pose Determination Using Randomized Action Sequences". Entropy. 21 (2). Basel, Switzerland: Multidisciplinary Digital Publishing Institute: 154. arXiv:1812.01195. Bibcode:2019Entrp..21..154M. doi:10.3390/e21020154. ISSN 1099-4300. PMC 7514636. PMID 33266870. S2CID 54444590. Article 154. p. 2: [...] The phenomenon, while also reminiscent of contraction mapping, is similar to an interesting card trick called the Kruskal Count [...] so we have dubbed the phenomenon as "Kruskal effect". [...] (13 pages) Blackburn, Simon Robert; Esfahani, Navid Nasr; Kreher, Donald Lawson; Stinson, Douglas "Doug" Robert (2023-08-22) [2022-11-18]. "Constructions and bounds for codes with restricted overlaps". IEEE Transactions on Information Theory. arXiv:2211.10309. (17 pages) (NB. This source does not mention Dynkin or Kruskal specifically.)
Kruskal count : Humble, Steve "Dr. Maths" (2010). "Dr. Maths Randomness Show". YouTube (Video). Alchemist Cafe, Dublin, Ireland. Retrieved 2023-09-05. [23:40] "Mathematical Card Trick Source". Close-Up Magic. GeniiForum. 2015–2017. Archived from the original on 2023-09-04. Retrieved 2023-09-05. Behr, Denis, ed. (2023). "Kruskal Principle". Conjuring Archive. Archived from the original on 2023-09-10. Retrieved 2023-09-10.
Mark V. Shaney : Mark V. Shaney is a synthetic Usenet user whose postings in the net.singles newsgroups were generated by Markov chain techniques, based on text from other postings. The username is a play on the words "Markov chain". Many readers were fooled into thinking that the quirky, sometimes uncannily topical posts were written by a real person. The system was designed by Rob Pike with coding by Bruce Ellis. Don P. Mitchell wrote the Markov chain code, initially demonstrating it to Pike and Ellis using the Tao Te Ching as a basis. They chose to apply it to the net.singles netnews group. The program is fairly simple. It ingests the sample text (the Tao Te Ching, or the posts of a Usenet group) and creates a massive list of every sequence of three successive words (triplet) which occurs in the text. It then chooses two words at random, and looks for a word which follows those two in one of the triplets in its massive list. If there is more than one, it picks at random (identical triplets count separately, so a sequence which occurs twice is twice as likely to be picked as one which only occurs once). It then adds that word to the generated text. Then, in the same way, it picks a triplet that starts with the second and third words in the generated text, and that gives a fourth word. It adds the fourth word, then repeats with the third and fourth words, and so on. This algorithm is called a third-order Markov chain (because it uses sequences of three words).
Mark V. Shaney : A classic example, from 1984, originally sent as a mail message, later posted to net.singles is reproduced here: >From mvs Fri Nov 16 17:11 EST 1984 remote from alice It looks like Reagan is going to say? Ummm... Oh yes, I was looking for. I'm so glad I remembered it. Yeah, what I have wondered if I had committed a crime. Don't eat with your assessment of Reagon and Mondale. Up your nose with a guy from a firm that specifically researches the teen-age market. As a friend of mine would say, "It really doesn't matter"... It looks like Reagan is holding back the arms of the American eating public have changed dramatically, and it got pretty boring after about 300 games. People, having a much larger number of varieties, and are very different from what one can find in Chinatowns across the country (things like pork buns, steamed dumplings, etc.) They can be cheap, being sold for around 30 to 75 cents apiece (depending on size), are generally not greasy, can be adequately explained by stupidity. Singles have felt insecure since we came down from the Conservative world at large. But Chuqui is the way it happened and the prices are VERY reasonable. Can anyone think of myself as a third sex. Yes, I am expected to have. People often get used to me knowing these things and then a cover is placed over all of them. Along the side of the $$ are spent by (or at least for ) the girls. You can't settle the issue. It seems I've forgotten what it is, but I don't. I know about violence against women, and I really doubt they will ever join together into a large number of jokes. It showed Adam, just after being created. He has a modem and an autodial routine. He calls my number 1440 times a day. So I will conclude by saying that I can well understand that she might soon have the time, it makes sense, again, to get the gist of my argument, I was in that (though it's a Republican administration). _-_-_-_-Mark Other quotations from Mark's Usenet posts are: "I spent an interesting evening recently with a grain of salt." (Alternatively reported as "While at a conference a few weeks back, I spent an interesting evening with a grain of salt.") "I hope that there are sour apples in every bushel." (see also sour grapes)
Mark V. Shaney : In The Usenet Handbook Mark Harrison writes that after September 1981, students joined Usenet en masse, "creating the USENET we know today: endless dumb questions, endless idiots posing as savants, and (of course) endless victims for practical jokes." In December, Rob Pike created the netnews group net.suicide as prank, "a forum for bad jokes". Some users thought it was a legitimate forum, some discussed "riding motorcycles without helmets". At first, most posters were "real people", but soon "characters" began posting. Pike created a "vicious" character named Bimmler. At its peak, net.suicide had ten frequent posters; nine were "known to be characters." But ultimately, Pike deleted the newsgroup because it was too much work to maintain; Bimmler messages were created "by hand". The "obvious alternative" was software, running on a Bell Labs computer created by Bruce Ellis, based on the Markov code by Don Mitchell, which became the online character Mark V. Shaney. Kernighan and Pike listed Mark V. Shaney in the acknowledgements in The Practice of Programming, noting its roots in Mitchell's markov, which, adapted as shaney, was used for "humorous deconstructionist activities" in the 1980s. Dewdney pointed out "perhaps Mark V. Shaney's magnum opus: a 20-page commentary on the deconstructionist philosophy of Jean Baudrillard" directed by Pike, with assistance from Henry S. Baird and Catherine Richards, to be distributed by email. The piece was based on Jean Baudrillard's "The Precession of Simulacra", published in Simulacra and Simulation (1981).
Mark V. Shaney : The program was discussed by A. K. Dewdney in the Scientific American "Computer Recreations" column in 1989, by Penn Jillette in his PC Computing column in 1991, and in several books, including the Usenet Handbook, Bots: the Origin of New Species, Hippo Eats Dwarf: A Field Guide to Hoaxes and Other B.S., and non-computer-related journals such as Texas Studies in Literature and Language. Dewdney wrote about the program's output, "The overall impression is not unlike what remains in the brain of an inattentive student after a late-night study session. Indeed, after reading the output of Mark V. Shaney, I find ordinary writing almost equally strange and incomprehensible!" He noted the reactions of newsgroup users, who have "shuddered at Mark V. Shaney's reflections, some with rage and others with laughter:" The opinions of the new net.singles correspondent drew mixed reviews. Serious users of the bulletin board's services sensed satire. Outraged, they urged that someone "pull the plug" on Mark V. Shaney's monstrous rantings. Others inquired almost admiringly whether the program was a secret artificial intelligence project that was being tested in a human conversational environment. A few may even have thought that Mark V. Shaney was a real person, a tortured schizophrenic desperately seeking a like-minded companion. Concluding, Dewdney wrote, "If the purpose of computer prose is to fool people into thinking that it was written by a sane person, Mark V. Shaney probably falls short." A 2012 article in Observer compared Mark V. Shaney's "strangely beautiful" postings to the Horse_ebooks account on Twitter and music reviews at Pitchfork, saying that "this mash-up of gibberish and human sentiment" is what "made Mark V. Shaney so endlessly fascinating".
Mark V. Shaney : Turing test Dissociated press On the Internet, nobody knows you're a dog Parody generator
Mark V. Shaney : FAQ for the Plan 9 operating system by Mark V. Shaney Unofficial biography "Mark V. Shaney at Your Service" online version by Yisong Yue. "Mark V. Shaney in Common Lisp" at Racine Systems. Every Mark V. Shaney post at Google Groups Usenet archive. "Sable Debutante's Journal", a Mark V. Shaney clone at LiveJournal markovtxt.c, markov.c C source code at Bell Labs
Markov chain : In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, finance, information theory, physics, signal processing, and speech processing. The adjectives Markovian and Markov are used to describe something that is related to a Markov process.
Markov chain : Andrey Markov studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906. Markov Processes in continuous time were discovered long before his work in the early 20th century in the form of the Poisson process. Markov was interested in studying an extension of independent random sequences, motivated by a disagreement with Pavel Nekrasov who claimed independence was necessary for the weak law of large numbers to hold. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption, which had been commonly regarded as a requirement for such mathematical laws to hold. Markov later used Markov chains to study the distribution of vowels in Eugene Onegin, written by Alexander Pushkin, and proved a central limit theorem for such chains. In 1912 Henri Poincaré studied Markov chains on finite groups with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest in 1907, and a branching process, introduced by Francis Galton and Henry William Watson in 1873, preceding the work of Markov. After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by Irénée-Jules Bienaymé. Starting in 1928, Maurice Fréchet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains. Andrey Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as Norbert Wiener's work on Einstein's model of Brownian movement. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes. Independent of Kolmogorov's work, Sydney Chapman derived in a 1928 paper an equation, now called the Chapman–Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement. The differential equations are now called the Kolmogorov equations or the Kolmogorov–Chapman equations. Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in 1930s, and then later Eugene Dynkin, starting in the 1950s.
Markov chain : Mark V. Shaney is a third-order Markov chain program, and a Markov text generator. It ingests the sample text (the Tao Te Ching, or the posts of a Usenet group) and creates a massive list of every sequence of three successive words (triplet) which occurs in the text. It then chooses two words at random, and looks for a word which follows those two in one of the triplets in its massive list. If there is more than one, it picks at random (identical triplets count separately, so a sequence which occurs twice is twice as likely to be picked as one which only occurs once). It then adds that word to the generated text. Then, in the same way, it picks a triplet that starts with the second and third words in the generated text, and that gives a fourth word. It adds the fourth word, then repeats with the third and fourth words, and so on. Random walks based on integers and the gambler's ruin problem are examples of Markov processes. Some variations of these processes were studied hundreds of years earlier in the context of independent variables. Two important examples of Markov processes are the Wiener process, also known as the Brownian motion process, and the Poisson process, which are considered the most important and central stochastic processes in the theory of stochastic processes. These two processes are Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time. A famous Markov chain is the so-called "drunkard's walk", a random walk on the number line where, at each step, the position may change by +1 or −1 with equal probability. From any position there are two possible transitions, to the next or previous integer. The transition probabilities depend only on the current position, not on the manner in which the position was reached. For example, the transition probabilities from 5 to 4 and 5 to 6 are both 0.5, and all other transition probabilities from 5 are 0. These probabilities are independent of whether the system was previously in 4 or 6. A series of independent states (for example, a series of coin flips) satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next state depends on the current one.
Markov chain : Two states are said to communicate with each other if both are reachable from one another by a sequence of transitions that have positive probability. This is an equivalence relation which yields a set of communicating classes. A class is closed if the probability of leaving the class is zero. A Markov chain is irreducible if there is one communicating class, the state space. A state i has period k if k is the greatest common divisor of the number of transitions by which i can be reached, starting from i. That is: k = gcd =i\mid X_=i)>0\ The state is periodic if k > 1 ; otherwise k = 1 and the state is aperiodic. A state i is said to be transient if, starting from i, there is a non-zero probability that the chain will never return to i. It is called recurrent (or persistent) otherwise. For a recurrent state i, the mean hitting time is defined as: M i = E [ T i ] = ∑ n = 1 ∞ n ⋅ f i i ( n ) . =E[T_]=\sum _^n\cdot f_^. State i is positive recurrent if M i is finite and null recurrent otherwise. Periodicity, transience, recurrence and positive and null recurrence are class properties — that is, if one state has the property then all states in its communicating class have the property. A state i is called absorbing if there are no outgoing transitions from the state.
Markov chain : Markov chains have been employed in a wide range of topics across the natural and social sciences, and in technological applications. They have been used for forecasting in several areas: for example, price trends, wind power, stochastic terrorism, and solar irradiance. The Markov chain forecasting models utilize a variety of settings, from discretizing the time series, to hidden Markov models combined with wavelets, and the Markov chain mixture distribution model (MCM).
Markov chain : == External links ==
Markov chain central limit theorem : In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theory, but the quantity in the role taken by the variance in the classic CLT has a more complicated definition. See also the general form of Bienaymé's identity.
Markov chain central limit theorem : Suppose that: the sequence X 1 , X 2 , X 3 , … ,X_,X_,\ldots of random elements of some set is a Markov chain that has a stationary probability distribution; and the initial distribution of the process, i.e. the distribution of X 1 , is the stationary distribution, so that X 1 , X 2 , X 3 , … ,X_,X_,\ldots are identically distributed. In the classic central limit theorem these random variables would be assumed to be independent, but here we have only the weaker assumption that the process has the Markov property; and g is some (measurable) real-valued function for which var ⁡ ( g ( X 1 ) ) < + ∞ . (g(X_))<+\infty . Now let μ = E ⁡ ( g ( X 1 ) ) , μ ^ n = 1 n ∑ k = 1 n g ( X k ) σ 2 := lim n → ∞ var ⁡ ( n μ ^ n ) = lim n → ∞ n var ⁡ ( μ ^ n ) = var ⁡ ( g ( X 1 ) ) + 2 ∑ k = 1 ∞ cov ⁡ ( g ( X 1 ) , g ( X 1 + k ) ) . \mu &=\operatorname (g(X_)),\\_&=\sum _^g(X_)\\\sigma ^&:=\lim _\operatorname (_)=\lim _n\operatorname (_)=\operatorname (g(X_))+2\sum _^\operatorname (g(X_),g(X_)).\end Then as n → ∞ , we have n ( μ ^ n − μ ) → D Normal ( 0 , σ 2 ) , (_-\mu )\ \ (0,\sigma ^), where the decorated arrow indicates convergence in distribution.
Markov chain central limit theorem : The Markov chain central limit theorem can be guaranteed for functionals of general state space Markov chains under certain conditions. In particular, this can be done with a focus on Monte Carlo settings. An example of the application in a MCMC (Markov Chain Monte Carlo) setting is the following: Consider a simple hard spheres model on a grid. Suppose X = × ⊆ Z 2 \\times \\\subseteq Z^ . A proper configuration on X consists of coloring each point either black or white in such a way that no two adjacent points are white. Let χ denote the set of all proper configurations on X , N χ ( n 1 , n 2 ) (n_,n_) be the total number of proper configurations and π be the uniform distribution on χ so that each proper configuration is equally likely. Suppose our goal is to calculate the typical number of white points in a proper configuration; that is, if W ( x ) is the number of white points in x ∈ χ then we want the value of E π W = ∑ x ∈ χ W ( x ) N χ ( n 1 , n 2 ) W=\sum _n_,n_ If n 1 and n 2 are even moderately large then we will have to resort to an approximation to E π W W . Consider the following Markov chain on χ . Fix p ∈ ( 0 , 1 ) and set X 1 = x 1 =x_ where x 1 ∈ χ \in \chi is an arbitrary proper configuration. Randomly choose a point ( x , y ) ∈ X and independently draw U ∼ U n i f o r m ( 0 , 1 ) (0,1) . If u ≤ p and all of the adjacent points are black then color ( x , y ) white leaving all other points alone. Otherwise, color ( x , y ) black and leave all other points alone. Call the resulting configuration X 1 . Continuing in this fashion yields a Harris ergodic Markov chain ,X_,X_,\ldots \ having π as its invariant distribution. It is now a simple matter to estimate E π W W with w n ¯ = ∑ i = 1 n W ( X i ) / n =\sum _^W(X_)/n . Also, since χ is finite (albeit potentially large) it is well known that X will converge exponentially fast to π which implies that a CLT holds for w n ¯ .
Markov chain central limit theorem : Not taking into account the additional terms in the variance which stem from correlations (e.g. serial correlations in markov chain monte carlo simulations) can result in the problem of pseudoreplication when computing e.g. the confidence intervals for the sample mean.
Markov chain central limit theorem : Gordin, M. I. and Lifšic, B. A. (1978). "Central limit theorem for stationary Markov processes." Soviet Mathematics, Doklady, 19, 392–394. (English translation of Russian original). Geyer, Charles J. (2011). "Introduction to MCMC." In Handbook of Markov Chain Monte Carlo, edited by S. P. Brooks, A. E. Gelman, G. L. Jones, and X. L. Meng. Chapman & Hall/CRC, Boca Raton, pp. 3–48.
Markov chain geostatistics : Markov chain geostatistics uses Markov chain spatial models, simulation algorithms and associated spatial correlation measures (e.g., transiogram) based on the Markov chain random field theory, which extends a single Markov chain into a multi-dimensional random field for geostatistical modeling. A Markov chain random field is still a single spatial Markov chain. The spatial Markov chain moves or jumps in a space and decides its state at any unobserved location through interactions with its nearest known neighbors in different directions. The data interaction process can be well explained as a local sequential Bayesian updating process within a neighborhood. Because single-step transition probability matrices are difficult to estimate from sparse sample data and are impractical in representing the complex spatial heterogeneity of states, the transiogram, which is defined as a transition probability function over the distance lag, is proposed as the accompanying spatial measure of Markov chain random fields.
Markov chain geostatistics : Li, W. 2007. Markov chain random fields for estimation of categorical variables. Math. Geol., 39(3): 321–335. Li, W. et al. 2015. Bayesian Markov chain random field cosimulation for improving land cover classification accuracy. Math. Geosci., 47(2): 123–148. Li, W., and C. Zhang. 2019. Markov chain random fields in the perspective of spatial Bayesian networks and optimal neighborhoods for simulation of categorical fields. Computational Geosciences, 23(5): 1087-1106. http://gisweb.grove.ad.uconn.edu/weidong/Markov_chain_spatial_statistics.htm
Markov chain Monte Carlo : In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that is, the Markov chain's equilibrium distribution matches the target distribution. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Markov chain Monte Carlo methods are used to study probability distributions that are too complex or too highly dimensional to study with analytic techniques alone. Various algorithms exist for constructing such Markov chains, including the Metropolis–Hastings algorithm.
Markov chain Monte Carlo : MCMC methods are primarily used for calculating numerical approximations of multi-dimensional integrals, for example in Bayesian statistics, computational physics, computational biology and computational linguistics. In Bayesian statistics, Markov chain Monte Carlo methods are typically used to calculate moments and credible intervals of posterior probability distributions. The use of MCMC methods makes it possible to compute large hierarchical models that require integrations over hundreds to thousands of unknown parameters. In rare event sampling, they are also used for generating samples that gradually populate the rare failure region.
Markov chain Monte Carlo : Markov chain Monte Carlo methods create samples from a continuous random variable, with probability density proportional to a known function. These samples can be used to evaluate an integral over that variable, as its expected value or variance. Practically, an ensemble of chains is generally developed, starting from a set of points arbitrarily chosen and sufficiently distant from each other. These chains are stochastic processes of "walkers" which move around randomly according to an algorithm that looks for places with a reasonably high contribution to the integral to move into next, assigning them higher probabilities. Random walk Monte Carlo methods are a kind of random simulation or Monte Carlo method. However, whereas the random samples of the integrand used in a conventional Monte Carlo integration are statistically independent, those used in MCMC are autocorrelated. Correlations of samples introduces the need to use the Markov chain central limit theorem when estimating the error of mean values. These algorithms create Markov chains such that they have an equilibrium distribution which is proportional to the function given.
Markov chain Monte Carlo : While MCMC methods were created to address multi-dimensional problems better than generic Monte Carlo algorithms, when the number of dimensions rises they too tend to suffer the curse of dimensionality: regions of higher probability tend to stretch and get lost in an increasing volume of space that contributes little to the integral. One way to address this problem could be shortening the steps of the walker, so that it does not continuously try to exit the highest probability region, though this way the process would be highly autocorrelated and expensive (i.e. many steps would be required for an accurate result). More sophisticated methods such as Hamiltonian Monte Carlo and the Wang and Landau algorithm use various ways of reducing this autocorrelation, while managing to keep the process in the regions that give a higher contribution to the integral. These algorithms usually rely on a more complicated theory and are harder to implement, but they usually converge faster.
Markov chain Monte Carlo : Usually it is not hard to construct a Markov chain with the desired properties. The more difficult problem is to determine how many steps are needed to converge to the stationary distribution within an acceptable error. A good chain will have rapid mixing: the stationary distribution is reached quickly starting from an arbitrary position. A standard empirical method to assess convergence is to run several independent simulated Markov chains and check that the ratio of inter-chain to intra-chain variances for all the parameters sampled is close to 1. Typically, Markov chain Monte Carlo sampling can only approximate the target distribution, as there is always some residual effect of the starting position. More sophisticated Markov chain Monte Carlo-based algorithms such as coupling from the past can produce exact samples, at the cost of additional computation and an unbounded (though finite in expectation) running time. Many random walk Monte Carlo methods move around the equilibrium distribution in relatively small steps, with no tendency for the steps to proceed in the same direction. These methods are easy to implement and analyze, but unfortunately it can take a long time for the walker to explore all of the space. The walker will often double back and cover ground already covered. Further consideration of convergence is at Markov chain central limit theorem. See for a discussion of the theory related to convergence and stationarity of the Metropolis–Hastings algorithm.
Markov chain Monte Carlo : Several software programs provide MCMC sampling capabilities, for example: ParaMonte parallel Monte Carlo software available in multiple programming languages including C, C++, Fortran, MATLAB, and Python. Packages that use dialects of the BUGS model language: WinBUGS / OpenBUGS/ MultiBUGS JAGS MCSim Julia language with packages like Turing.jl DynamicHMC.jl AffineInvariantMCMC.jl Gen.jl and the ones in StanJulia repository. Python (programming language) with the packages: Blackjax. emcee, NumPyro PyMC R (programming language) with the packages adaptMCMC, atmcmc, BRugs, mcmc, MCMCpack, ramcmc, rjags, rstan, etc. Stan TensorFlow Probability (probabilistic programming library built on TensorFlow) Korali high-performance framework for Bayesian UQ, optimization, and reinforcement learning. MacMCMC — Full-featured application (freeware) for MacOS, with advanced functionality, available at causaScientia
Markov chain Monte Carlo : Coupling from the past Integrated nested Laplace approximations Markov chain central limit theorem Metropolis-adjusted Langevin algorithm
Markov partition : A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift. The appellation 'Markov' is appropriate because the resulting dynamics of the system obeys the Markov property. The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm determinants and the like.
Markov partition : Let ( M , φ ) be a discrete dynamical system. A basic method of studying its dynamics is to find a symbolic representation: a faithful encoding of the points of M by sequences of symbols such that the map φ becomes the shift map. Suppose that M has been divided into a number of pieces E 1 , E 2 , … , E r ,E_,\ldots ,E_ which are thought to be as small and localized, with virtually no overlaps. The behavior of a point x under the iterates of φ can be tracked by recording, for each n , the part E i which contains φ n ( x ) (x) . This results in an infinite sequence on the alphabet which encodes the point. In general, this encoding may be imprecise (the same sequence may represent many different points) and the set of sequences which arise in this way may be difficult to describe. Under certain conditions, which are made explicit in the rigorous definition of a Markov partition, the assignment of the sequence to a point of M becomes an almost one-to-one map whose image is a symbolic dynamical system of a special kind called a shift of finite type. In this case, the symbolic representation is a powerful tool for investigating the properties of the dynamical system ( M , φ ) .
Markov partition : A Markov partition is a finite cover of the invariant set of the manifold by a set of curvilinear rectangles ,E_,\ldots ,E_\ such that For any pair of points x , y ∈ E i , that W s ( x ) ∩ W u ( y ) ∈ E i (x)\cap W_(y)\in E_ Int ⁡ E i ∩ Int ⁡ E j = ∅ E_\cap \operatorname E_=\emptyset for i ≠ j If x ∈ Int ⁡ E i E_ and φ ( x ) ∈ Int ⁡ E j E_ , then φ [ W u ( x ) ∩ E i ] ⊃ W u ( φ x ) ∩ E j (x)\cap E_\right]\supset W_(\varphi x)\cap E_ φ [ W s ( x ) ∩ E i ] ⊂ W s ( φ x ) ∩ E j (x)\cap E_\right]\subset W_(\varphi x)\cap E_ Here, W u ( x ) (x) and W s ( x ) (x) are the unstable and stable manifolds of x, respectively, and Int ⁡ E i E_ simply denotes the interior of E i . These last two conditions can be understood as a statement of the Markov property for the symbolic dynamics; that is, the movement of a trajectory from one open cover to the next is determined only by the most recent cover, and not the history of the system. It is this property of the covering that merits the 'Markov' appellation. The resulting dynamics is that of a Markov shift; that this is indeed the case is due to theorems by Yakov Sinai (1968) and Rufus Bowen (1975), thus putting symbolic dynamics on a firm footing. Variants of the definition are found, corresponding to conditions on the geometry of the pieces E i .
Markov partition : Markov partitions have been constructed in several situations. Anosov diffeomorphisms of the torus. Dynamical billiards, in which case the covering is countable. Markov partitions make homoclinic and heteroclinic orbits particularly easy to describe. The system ( [ 0 , 1 ) , x ↦ 2 x m o d 1 ) has the Markov partition E 0 = ( 0 , 1 / 2 ) , E 1 = ( 1 / 2 , 1 ) =(0,1/2),E_=(1/2,1) , and in this case the symbolic representation of a real number in [ 0 , 1 ) is its binary expansion. For example: x ∈ E 0 , T x ∈ E 1 , T 2 x ∈ E 1 , T 3 x ∈ E 1 , T 4 x ∈ E 0 ⇒ x = ( 0.01110... ) 2 ,Tx\in E_,T^x\in E_,T^x\in E_,T^x\in E_\Rightarrow x=(0.01110...)_ . The assignment of points of [ 0 , 1 ) to their sequences in the Markov partition is well defined except on the dyadic rationals - morally speaking, this is because ( 0.01111 … ) 2 = ( 0.10000 … ) 2 =(0.10000\dots )_ , in the same way as 1 = 0.999 … in decimal expansions.
Markov partition : Lind, Douglas; Marcus, Brian (1995). An introduction to symbolic dynamics and coding. Cambridge University Press. ISBN 978-0-521-55124-3. Zbl 1106.37301. Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 978-3-540-44141-0. Zbl 1014.11015.
Markov property : In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process, which means that its future evolution is independent of its history. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the Ising model. A discrete-time stochastic process satisfying the Markov property is known as a Markov chain.
Markov property : A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past. A process with this property is said to be Markov or Markovian and known as a Markov process. Two famous classes of Markov process are the Markov chain and Brownian motion. Note that there is a subtle, often overlooked and very important point that is often missed in the plain English statement of the definition: the statespace of the process is constant through time. The conditional description involves a fixed "bandwidth". For example, without this restriction we could augment any process to one which includes the complete history from a given initial condition and it would be made to be Markovian. But the state space would be of increasing dimensionality over time and does not meet the definition.
Markov property : Let ( Ω , F , P ) ,P) be a probability space with a filtration ( F s , s ∈ I ) _,\ s\in I) , for some (totally ordered) index set I ; and let ( S , S ) ) be a measurable space. A ( S , S ) ) -valued stochastic process X = t ∈ I :\Omega \to S\_ adapted to the filtration is said to possess the Markov property if, for each A ∈ S and each s , t ∈ I with s < t , P ( X t ∈ A ∣ F s ) = P ( X t ∈ A ∣ X s ) . \in A\mid _)=P(X_\in A\mid X_). In the case where S is a discrete set with the discrete sigma algebra and I = N , this can be reformulated as follows: P ( X n + 1 = x n + 1 ∣ X n = x n , … , X 1 = x 1 ) = P ( X n + 1 = x n + 1 ∣ X n = x n ) for all n ∈ N . =x_\mid X_=x_,\dots ,X_=x_)=P(X_=x_\mid X_=x_)n\in \mathbb .
Markov property : Alternatively, the Markov property can be formulated as follows. E ⁡ [ f ( X t ) ∣ F s ] = E ⁡ [ f ( X t ) ∣ σ ( X s ) ] [f(X_)\mid _]=\operatorname [f(X_)\mid \sigma (X_)] for all t ≥ s ≥ 0 and f : S → R bounded and measurable.
Markov property : Suppose that X = ( X t : t ≥ 0 ) :t\geq 0) is a stochastic process on a probability space ( Ω , F , P ) ,P) with natural filtration t ≥ 0 _\_ . Then for any stopping time τ on Ω , we can define F τ = ∩ A ∈ F t _=\:\forall t\geq 0,\\cap A\in _\ . Then X is said to have the strong Markov property if, for each stopping time τ , conditional on the event , we have that for each t ≥ 0 , X τ + t is independent of F τ _ given X τ . The strong Markov property implies the ordinary Markov property since by taking the stopping time τ = t , the ordinary Markov property can be deduced.
Markov property : In the fields of predictive modelling and probabilistic forecasting, the Markov property is considered desirable since it may enable the reasoning and resolution of the problem that otherwise would not be possible to be resolved because of its intractability. Such a model is known as a Markov model.
Markov property : Assume that an urn contains two red balls and one green ball. One ball was drawn yesterday, one ball was drawn today, and the final ball will be drawn tomorrow. All of the draws are "without replacement". Suppose you know that today's ball was red, but you have no information about yesterday's ball. The chance that tomorrow's ball will be red is 1/2. That's because the only two remaining outcomes for this random experiment are: On the other hand, if you know that both today and yesterday's balls were red, then you are guaranteed to get a green ball tomorrow. This discrepancy shows that the probability distribution for tomorrow's color depends not only on the present value, but is also affected by information about the past. This stochastic process of observed colors doesn't have the Markov property. Using the same experiment above, if sampling "without replacement" is changed to sampling "with replacement," the process of observed colors will have the Markov property. An application of the Markov property in a generalized form is in Markov chain Monte Carlo computations in the context of Bayesian statistics.
Markov property : Causal Markov condition Chapman–Kolmogorov equation Hysteresis Markov blanket Markov chain Markov decision process Markov model == References ==
Markov switching multifractal : In financial econometrics (the application of statistical methods to economic data), the Markov-switching multifractal (MSM) is a model of asset returns developed by Laurent E. Calvet and Adlai J. Fisher that incorporates stochastic volatility components of heterogeneous durations. MSM captures the outliers, log-memory-like volatility persistence and power variation of financial returns. In currency and equity series, MSM compares favorably with standard volatility models such as GARCH(1,1) and FIGARCH both in- and out-of-sample. MSM is used by practitioners in the financial industry for different types of forecasts.
Markov switching multifractal : The MSM model can be specified in both discrete time and continuous time.
Markov switching multifractal : When M has a discrete distribution, the Markov state vector M t takes finitely many values m 1 , . . . , m d ∈ R + k ¯ ,...,m^\in R_^ . For instance, there are d = 2 k ¯ possible states in binomial MSM. The Markov dynamics are characterized by the transition matrix A = ( a i , j ) 1 ≤ i , j ≤ d )_ with components a i , j = P ( M t + 1 = m j \| M t = m i ) =P\left(M_=m^\|M_=m^\right) . Conditional on the volatility state, the return r t has Gaussian density f ( r t \| M t = m i ) = 1 2 π σ 2 ( m i ) exp ⁡ [ − ( r t − μ ) 2 2 σ 2 ( m i ) ] . \|M_=m^)=(m^)\exp \left[--\mu )^(m^)\right].
Markov switching multifractal : Given r 1 , … , r t ,\dots ,r_ , the conditional distribution of the latent state vector at date t + n is given by: Π ^ t , n = Π t A n . _=\Pi _A^.\, MSM often provides better volatility forecasts than some of the best traditional models both in and out of sample. Calvet and Fisher report considerable gains in exchange rate volatility forecasts at horizons of 10 to 50 days as compared with GARCH(1,1), Markov-Switching GARCH, and Fractionally Integrated GARCH. Lux obtains similar results using linear predictions.
Markov switching multifractal : MSM is a stochastic volatility model with arbitrarily many frequencies. MSM builds on the convenience of regime-switching models, which were advanced in economics and finance by James D. Hamilton. MSM is closely related to the Multifractal Model of Asset Returns. MSM improves on the MMAR's combinatorial construction by randomizing arrival times, guaranteeing a strictly stationary process. MSM provides a pure regime-switching formulation of multifractal measures, which were pioneered by Benoit Mandelbrot.
Markov switching multifractal : Brownian motion Rogemar Mamon Markov chain Multifractal model of asset returns Multifractal Stochastic volatility
Markov switching multifractal : Financial Time Series, Multifractals and Hidden Markov Models
MegaHAL : MegaHAL is a computer conversation simulator, or "chatterbot", created by Jason Hutchens.
MegaHAL : In 1996, Jason Hutchens entered the Loebner Prize Contest with HeX, a chatterbot based on ELIZA. HeX won the competition that year and took the $2000 prize for having the highest overall score. In 1998, Hutchens again entered the Loebner Prize Contest with his new program, MegaHAL. MegaHAL made its debut in the 1998 Loebner Prize Contest. Like many chatterbots, the intent is for MegaHAL to appear as a human fluent in a natural language. As a user types sentences into MegaHAL, MegaHAL will respond with sentences that are sometimes coherent and at other times complete gibberish. MegaHAL learns as the conversation progresses, remembering new words and sentence structures. It will even learn new ways to substitute words or phrases for other words or phrases. Many would consider conversation simulators like MegaHAL to be a primitive form of artificial intelligence. However, MegaHAL doesn't understand the conversation or even the sentence structure. It generates its conversation based on sequential and mathematical relationships. In the world of conversation simulators, MegaHAL is based on relatively old technology and could be considered primitive. However, its popularity has grown due to its humorous nature; it has been known to respond with twisted or nonsensical statements that are often amusing.
MegaHAL : MegaHal is based at least in part on a so-called "hidden Markov Model", so that the first thing that Megahal does when it "trains" on a script or text is to build a database of text fragments encompassing every possible subset of perhaps 4, 5, or even 6 consecutive words, so that for example - if MegaHal trains on the Declaration of Independence, then MegaHal will build a database containing text fragments such as "When in the course", "in the course of", "the course of human", "course of human events", "of human events, one", "human events, one people", and so on. Then if Megahal is fed another text, such has "Superman, Yes! It's Superman - he can change the course of mighty rivers, bend steel with his bare hands - and who disguised at Clark Kent …" IT MIGHT induce Megahal to apparently bemuse itself to proffer whether Superman can change the course of human events, or something else altogether - such as some rambling about "when in the course of mighty rivers", and so on. Thus likewise - if a phrase like "the White house said" comes up a lot in some text; then Megahal's ability to switch randomly between different contexts which otherwise share some similarity can result at times in some surprising lucidity, or else it might otherwise seem quite bizarre.
MegaHAL : There are some sentences that MegaHAL generated: CHESS IS A FUN SPORT, WHEN PLAYED WITH SHOT GUNS. and COWS FLY LIKE CLOUDS BUT THEY ARE NEVER COMPLETELY SUCCESSFUL.
MegaHAL : MegaHAL is distributed under the Unlicense. Its source code can be downloaded from the Github repository.
MegaHAL : Loebner prize ELIZA
MegaHAL : Hutchens, Jason L.; Alder, Michael D. (1998), "Introducing MegaHAL" (PDF), NeMLaP3 / CoNLL98 Workshop on Human-Computer Conversation, ACL (271): 274
MegaHAL : Github repository
Models of DNA evolution : A number of different Markov models of DNA sequence evolution have been proposed. These substitution models differ in terms of the parameters used to describe the rates at which one nucleotide replaces another during evolution. These models are frequently used in molecular phylogenetic analyses. In particular, they are used during the calculation of likelihood of a tree (in Bayesian and maximum likelihood approaches to tree estimation) and they are used to estimate the evolutionary distance between sequences from the observed differences between the sequences.
Models of DNA evolution : These models are phenomenological descriptions of the evolution of DNA as a string of four discrete states. These Markov models do not explicitly depict the mechanism of mutation nor the action of natural selection. Rather they describe the relative rates of different changes. For example, mutational biases and purifying selection favoring conservative changes are probably both responsible for the relatively high rate of transitions compared to transversions in evolving sequences. However, the Kimura (K80) model described below only attempts to capture the effect of both forces in a parameter that reflects the relative rate of transitions to transversions. Evolutionary analyses of sequences are conducted on a wide variety of time scales. Thus, it is convenient to express these models in terms of the instantaneous rates of change between different states (the Q matrices below). If we are given a starting (ancestral) state at one position, the model's Q matrix and a branch length expressing the expected number of changes to have occurred since the ancestor, then we can derive the probability of the descendant sequence having each of the four states. The mathematical details of this transformation from rate-matrix to probability matrix are described in the mathematics of substitution models section of the substitution model page. By expressing models in terms of the instantaneous rates of change we can avoid estimating a large numbers of parameters for each branch on a phylogenetic tree (or each comparison if the analysis involves many pairwise sequence comparisons). The models described on this page describe the evolution of a single site within a set of sequences. They are often used for analyzing the evolution of an entire locus by making the simplifying assumption that different sites evolve independently and are identically distributed. This assumption may be justifiable if the sites can be assumed to be evolving neutrally. If the primary effect of natural selection on the evolution of the sequences is to constrain some sites, then models of among-site rate-heterogeneity can be used. This approach allows one to estimate only one matrix of relative rates of substitution, and another set of parameters describing the variance in the total rate of substitution across sites.
Models of DNA evolution : Molecular evolution Molecular clock UPGMA
Models of DNA evolution : DAWG: DNA Assembly With Gaps — free software for simulating sequence evolution
MRF optimization via dual decomposition : In dual decomposition a problem is broken into smaller subproblems and a solution to the relaxed problem is found. This method can be employed for MRF optimization. Dual decomposition is applied to markov logic programs as an inference technique.
MRF optimization via dual decomposition : Discrete MRF Optimization (inference) is very important in Machine Learning and Computer vision, which is realized on CUDA graphical processing units. Consider a graph G = ( V , E ) with nodes V and Edges E . The goal is to assign a label l p to each p ∈ V so that the MRF Energy is minimized: (1) min Σ p ∈ V θ p ( l p ) + Σ p q ∈ ε θ p q ( l p ) ( l q ) \theta _(l_)+\Sigma _\theta _(l_)(l_) Major MRF Optimization methods are based on Graph cuts or Message passing. They rely on the following integer linear programming formulation (2) min x E ( θ , x ) = θ . x = ∑ p ∈ V θ p . x p + ∑ p q ∈ ε θ p q . x p q E(\theta ,x)=\theta .x=\sum _\theta _.x_+\sum _\theta _.x_ In many applications, the MRF-variables are -variables that satisfy: x p ( l ) = 1 (l)=1 ⇔ label l is assigned to p , while x p q ( l , l ′ ) = 1 (l,l^)=1 , labels l , l ′ are assigned to p , q .
MRF optimization via dual decomposition : The main idea behind decomposition is surprisingly simple: decompose your original complex problem into smaller solvable subproblems, extract a solution by cleverly combining the solutions from these subproblems. A sample problem to decompose: min x Σ i f i ( x ) \Sigma _f^(x) where x ∈ C In this problem, separately minimizing every single f i ( x ) (x) over x is easy; but minimizing their sum is a complex problem. So the problem needs to get decomposed using auxiliary variables \ and the problem will be as follows: min , x Σ i f i ( x i ) \,x\Sigma _f^(x^) where x i ∈ C , x i = x \in C,x^=x Now we can relax the constraints by multipliers \ which gives us the following Lagrangian dual function: g ( ) = min , x Σ i f i ( x i ) + Σ i λ i . ( x i − x ) = min , x Σ i [ f i ( x i ) + λ i . x i ] − ( Σ i λ i ) x \)=\min _\in C\,x\Sigma _f^(x^)+\Sigma _\lambda ^.(x^-x)=\min _\in C\,x\Sigma _[f^(x^)+\lambda ^.x^]-(\Sigma _\lambda ^)x Now we eliminate x from the dual function by minimizing over x and dual function becomes: g ( ) = min Σ i [ f i ( x i ) + λ i . x i ] \)=\min _\in C\\Sigma _[f^(x^)+\lambda ^.x^] We can set up a Lagrangian dual problem: (3) max ∈ Λ g ( λ i ) = Σ i g i ( x i ) , \\in \Lambda g()=\Sigma _g^(x^), The Master problem (4) g i ( x i ) = m i n x i f i ( x i ) + λ i . x i (x^)=min_f^(x^)+\lambda ^.x^ where x i ∈ C \in C The Slave problems
MRF optimization via dual decomposition : The original MRF optimization problem is NP-hard and we need to transform it into something easier. τ is a set of sub-trees of graph G where its trees cover all nodes and edges of the main graph. And MRFs defined for every tree T in τ will be smaller. The vector of MRF parameters is θ T and the vector of MRF variables is x T , these two are just smaller in comparison with original MRF vectors θ , x . For all vectors θ T we'll have the following: (5) ∑ T ∈ τ ( p ) θ p T = θ p , ∑ T ∈ τ ( p q ) θ p q T = θ p q . \theta _^=\theta _,\sum _\theta _^=\theta _. Where τ ( p ) and τ ( p q ) denote all trees of τ than contain node p and edge p q respectively. We simply can write: (6) E ( θ , x ) = ∑ T ∈ τ E ( θ T , x T ) E(\theta ^,x^) And our constraints will be: (7) x T ∈ χ T , x T = x \| T , ∀ T ∈ τ \in \chi ^,x^=x_,\forall T\in \tau Our original MRF problem will become: (8) min , x Σ T ∈ τ E ( θ T , x T ) \,x\Sigma _E(\theta ^,x^) where x T ∈ χ T , ∀ T ∈ τ \in \chi ^,\forall T\in \tau and x T ∈ x \| T , ∀ T ∈ τ \in x_,\forall T\in \tau And we'll have the dual problem we were seeking: (9) max ∈ Λ g ( ) = ∑ T ∈ τ g T ( λ T ) , \\in \Lambda g(\\)=\sum _g^(\lambda ^), The Master problem where each function g T ( . ) (.) is defined as: (10) g T ( λ T ) = min x T E ( θ T + λ T , x T ) (\lambda ^)=\min _E(\theta ^+\lambda ^,x^) where x T ∈ χ T \in \chi ^ The Slave problems
MRF optimization via dual decomposition : Theorem 1. Lagrangian relaxation (9) is equivalent to the LP relaxation of (2). min , x \,x\^=s_,x^\in (\chi ^)\ Theorem 2. If the sequence of multipliers \ satisfies α t ≥ 0 , lim t → ∞ α t = 0 , ∑ t = 0 ∞ α t = ∞ \geq 0,\lim _\alpha _=0,\sum _^\alpha _=\infty then the algorithm converges to the optimal solution of (9). Theorem 3. The distance of the current solution \ to the optimal solution ^\ , which decreases at every iteration. Theorem 4. Any solution obtained by the method satisfies the WTA (weak tree agreement) condition. Theorem 5. For binary MRFs with sub-modular energies, the method computes a globally optimal solution. == References ==
Multiple sequence alignment : Multiple sequence alignment (MSA) is the process or the result of sequence alignment of three or more biological sequences, generally protein, DNA, or RNA. These alignments are used to infer evolutionary relationships via phylogenetic analysis and can highlight homologous features between sequences. Alignments highlight mutation events such as point mutations (single amino acid or nucleotide changes), insertion mutations and deletion mutations, and alignments are used to assess sequence conservation and infer the presence and activity of protein domains, tertiary structures, secondary structures, and individual amino acids or nucleotides. Multiple sequence alignments require more sophisticated methodologies than pairwise alignments, as they are more computationally complex. Most multiple sequence alignment programs use heuristic methods rather than global optimization because identifying the optimal alignment between more than a few sequences of moderate length is prohibitively computationally expensive. However, heuristic methods generally cannot guarantee high-quality solutions and have been shown to fail to yield near-optimal solutions on benchmark test cases.
Multiple sequence alignment : Given m sequences S i , i = 1 , ⋯ , m similar to the form below: S := S_=(S_,S_,\ldots ,S_)\\S_=(S_,S_,\cdots ,S_)\\\,\,\,\,\,\,\,\,\,\,\vdots \\S_=(S_,S_,\ldots ,S_)\end A multiple sequence alignment is taken of this set of sequences S by inserting any amount of gaps needed into each of the S i sequences of S until the modified sequences, S i ′ , all conform to length L ≥ max \mid i=1,\ldots ,m\ and no values in the sequences of S of the same column consists of only gaps. The mathematical form of an MSA of the above sequence set is shown below: S ′ := S'_=(S'_,S'_,\ldots ,S'_)\\S'_=(S'_,S'_,\ldots ,S'_)\\\,\,\,\,\,\,\,\,\,\,\vdots \\S'_=(S'_,S'_,\ldots ,S'_)\end To return from each particular sequence S i ′ to S i , remove all gaps.
Multiple sequence alignment : A general approach when calculating multiple sequence alignments is to use graphs to identify all of the different alignments. When finding alignments via graph, a complete alignment is created in a weighted graph that contains a set of vertices and a set of edges. Each of the graph edges has a weight based on a certain heuristic that helps to score each alignment or subset of the original graph.
Multiple sequence alignment : There are various alignment methods used within multiple sequence to maximize scores and correctness of alignments. Each is usually based on a certain heuristic with an insight into the evolutionary process. Most try to replicate evolution to get the most realistic alignment possible to best predict relations between sequences.
Multiple sequence alignment : The necessary use of heuristics for multiple alignment means that for an arbitrary set of proteins, there is always a good chance that an alignment will contain errors. For example, an evaluation of several leading alignment programs using the BAliBase benchmark found that at least 24% of all pairs of aligned amino acids were incorrectly aligned. These errors can arise because of unique insertions into one or more regions of sequences, or through some more complex evolutionary process leading to proteins that do not align easily by sequence alone. As the number of sequence and their divergence increases many more errors will be made simply because of the heuristic nature of MSA algorithms. Multiple sequence alignment viewers enable alignments to be visually reviewed, often by inspecting the quality of alignment for annotated functional sites on two or more sequences. Many also enable the alignment to be edited to correct these (usually minor) errors, in order to obtain an optimal 'curated' alignment suitable for use in phylogenetic analysis or comparative modeling. However, as the number of sequences increases and especially in genome-wide studies that involve many MSAs it is impossible to manually curate all alignments. Furthermore, manual curation is subjective. And finally, even the best expert cannot confidently align the more ambiguous cases of highly diverged sequences. In such cases it is common practice to use automatic procedures to exclude unreliably aligned regions from the MSA. For the purpose of phylogeny reconstruction (see below) the Gblocks program is widely used to remove alignment blocks suspect of low quality, according to various cutoffs on the number of gapped sequences in alignment columns. However, these criteria may excessively filter out regions with insertion/deletion events that may still be aligned reliably, and these regions might be desirable for other purposes such as detection of positive selection. A few alignment algorithms output site-specific scores that allow the selection of high-confidence regions. Such a service was first offered by the SOAP program, which tests the robustness of each column to perturbation in the parameters of the popular alignment program CLUSTALW. The T-Coffee program uses a library of alignments in the construction of the final MSA, and its output MSA is colored according to confidence scores that reflect the agreement between different alignments in the library regarding each aligned residue. Its extension, Transitive Consistency Score (TCS), uses T-Coffee libraries of pairwise alignments to evaluate any third party MSA. Pairwise projections can be produced using fast or slow methods, thus allowing a trade-off between speed and accuracy. Another alignment program that can output an MSA with confidence scores is FSA, which uses a statistical model that allows calculation of the uncertainty in the alignment. The HoT (Heads-Or-Tails) score can be used as a measure of site-specific alignment uncertainty due to the existence of multiple co-optimal solutions. The GUIDANCE program calculates a similar site-specific confidence measure based on the robustness of the alignment to uncertainty in the guide tree that is used in progressive alignment programs. An alternative, more statistically justified approach to assess alignment uncertainty is the use of probabilistic evolutionary models for joint estimation of phylogeny and alignment. A Bayesian approach allows calculation of posterior probabilities of estimated phylogeny and alignment, which is a measure of the confidence in these estimates. In this case, a posterior probability can be calculated for each site in the alignment. Such an approach was implemented in the program BAli-Phy. There are free programs available for visualization of multiple sequence alignments, for example Jalview and UGENE.
Multiple sequence alignment : Multiple sequence alignments can be used to create a phylogenetic tree. This is made possible by two reasons. The first is because functional domains that are known in annotated sequences can be used for alignment in non-annotated sequences. The other is that conserved regions known to be functionally important can be found. This makes it possible for multiple sequence alignments to be used to analyze and find evolutionary relationships through homology between sequences. Point mutations and insertion or deletion events (called indels) can be detected. Multiple sequence alignments can also be used to identify functionally important sites, such as binding sites, active sites, or sites corresponding to other key functions, by locating conserved domains. When looking at multiple sequence alignments, it is useful to consider different aspects of the sequences when comparing sequences. These aspects include identity, similarity, and homology. Identity means that the sequences have identical residues at their respective positions. On the other hand, similarity has to do with the sequences being compared having similar residues quantitatively. For example, in terms of nucleotide sequences, pyrimidines are considered similar to each other, as are purines. Similarity ultimately leads to homology, in that the more similar sequences are, the closer they are to being homologous. This similarity in sequences can then go on to help find common ancestry.

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