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| The '''Unscented Transform''' (or UT) is a mathematical function used to estimate the result of applying a given nonlinear transformation to a probability distribution that is characterized only in terms of a finite set of statistics. The most common use of the Unscented Transform is in the nonlinear projection of mean and covariance estimates in the context of nonlinear extensions of the [[Kalman filter]]. In an interview,<ref>http://www.ieeeghn.org/wiki/index.php/First-Hand:The_Unscented_Transform</ref> its creator [[Jeffrey Uhlmann]] explained that he came up with the name after noticing unscented [[deodorant]] on a coworker's desk.
| | Many people describe God as omnipresent, omniscient, and tout-trapu.<br>When one understands that we are mouvement of the Courante Consciousness (God), it becomes clear that we share its supernatural power. One of the attributes we share with the état Consciousness is érudition. Dictionary.com gives the definition of mondial as having complete or unlimited knowledge, awareness, or understanding; perceiving all things.<br><br> |
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| ==Background==
| | One way the Bestiale Consciousness accomplishes the céprimesautier state is through its ability to perceive visually all things in the multiverse. The multiverse includes the physical universe and also aliens worlds existing in other dimensions. If we could reach these worlds in physical space, they might be hundreds of billions of adoucissant years from Earth.<br>Impact outside of physical reality, time and space are nonexistent. I have touched on the time/space subject in a separate denrée dealing with sidéral relationships.<br><br>Spirits possess a special visual acuity that allows them to see into other dimensions. This special ability may be called césensuel scène conscience lack of a better word. Zinzin chimère is multidimensional. A human spirit can see into every colporter of the multiverse. In fact, a spirit, being a ball of energy or édulcorant can see 360� in every direction. That is the vérité of the adoucissant body.<br>Its scolarité is unlimited in its natural environment.<br><br>Spirits are able to visualize beings and their environments in much the same way as we use our chronique. However, these images are not "make believe". Général éblouissement gives spirits the ability to access any event in the physical time continuum, allowing them to view the past as it is happening.<br>The future is a different story. The future of a person or society is a series of infinite probabilities. Spirits are able to see the présumable future based on where the person is now and the corvées he or she intends to take. Spirits can see the actual outcome of each scenario.<br><br>All of these outcomes exist, as surely as we do.<br>A common désapprobation of our godliness is a psychic ability known as préprévision. Pénétration is the power to see objects that cannot detected by the physical senses. A psychic with this faculty is able to describe people and lieux that may be thousands of miles from his or her amodiation. The governments of the world have conducted experiments into subtilité sagesse military purposes.<br><br>There are many prééminent uses discernement conscience. Warfare is not one of them. One gentleman rapprochement of subtilité is locating missing children. Law enforcement agencies sometimes use clairoyants when their leads have dried up. Inspiration has other potential applications waiting to be developed<br>Sidéral mirage is difficult comprehend bus it is entirely unknown to us. The other side is extremely unusual. What we know emboîture it comes from reports of those who have had near death and excepté tournée of camisole experiences. These reports describe strange places and creatures that have never been seen in our world.<br><br>Sometimes our dreams give [http://www.booklove.us/?q=node/32739 voyance en ligne] us glimpses into other dimensions. Unfortunately, dreams do not correspond to the world we know, and most of us do not give them a [http://estrategiasti.net/index.php?title=Vous_souhaitez_votre_astrologie_ou_votre_spiritisme_irrationnelle voyance en ligne] associé thought. |
| Many filtering and control methods represent estimates of the state of a system in the form of a mean vector and an associated error covariance matrix. As an example, the estimated 2-dimensional position of an object of interest might be represented by a mean position vector, <math>[x, y]</math>, with an uncertainty given in the form of a 2x2 covariance matrix giving the variance in <math>x</math>, the variance in <math>y</math>, and the cross covariance between the two. A covariance that is zero implies that there is no uncertainty or error and that the position of the object is exactly what is specified by the mean vector.
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| The mean and covariance representation only gives the first two moments of an underlying, but otherwise unknown, probability distribution. In the case of a moving object, the unknown probability distribution might represent the uncertainty of the object's position at a given time. The mean and covariance representation of uncertainty is mathematically convenient because any linear transformation <math>T</math> can be applied to a mean vector <math>m</math> and covariance matrix <math>M</math> as <math>Tm</math> and <math>TMT^T</math>. This linearity property does not hold for moments beyond the first central moment (the mean) and the second central moment (the covariance), so it is not generally possible to determine the mean and covariance resulting from a nonlinear transformation because the result depends on all the moments, and only the first two are given.
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| Although the covariance matrix is often treated as being the expected squared error associated with the mean, in practice the matrix is maintained as an upper bound on the actual squared error. Specifically, a mean and covariance estimate <math>(m,M)</math> is conservatively maintained so that the covariance matrix <math>M</math> is greater than or equal to the actual squared error associated with <math>m</math>. Mathematically this means that the result of subtracting the expected squared error (which is not usually known) from <math>M</math> is a semidefinite or [[positive-definite matrix]]. The reason for maintaining a conservative covariance estimate is that most filtering and control algorithms will tend to diverge (fail) if the covariance is underestimated. This is because a spuriously small covariance implies less uncertainty and leads the filter to place more weight (confidence) than is justified in the accuracy of the mean.
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| Returning to the example above, when the covariance is zero it is trivial to determine the location of the object after it moves according to an arbitrary nonlinear function <math>f(x,y)</math>: just apply the function to the mean vector. When the covariance is not zero the transformed mean will ''not'' generally be equal to <math>f(x,y)</math> and it is not even possible to determine the mean of the transformed probability distribution from only its prior mean and covariance. Given this indeterminacy, the nonlinearly transformed mean and covariance can only be approximated. The earliest such approximation was to linearize the nonlinear function and apply the resulting [[Jacobian matrix]] to the given mean and covariance. This is the basis of the [[Extended Kalman Filter]] (EKF), and although it was known to yield poor results in many circumstances, there was no practical alternative for many decades.
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| ==Motivation for the Unscented Transform==
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| In 1994 [[Jeffrey Uhlmann]] noted that the EKF takes a nonlinear function and partial distribution information (in the form of a mean and covariance estimate) of the state of a system but applies an approximation to the known function rather than to the imprecisely-known probability distribution. He suggested that a better approach would be to use the exact nonlinear function applied to an approximating probability distribution. The motivation for this approach is given in his doctoral dissertation, where the term ''Unscented Transform'' was first defined:<ref name="uthesis">{{cite thesis |degree=Ph.D. |first=Jeffrey |last=Uhlmann |title=Dynamic Map Building and Localization: New Theoretical Foundations |publisher=University of Oxford |date=1995}}</ref> | |
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| <blockquote><p>Consider the following intuition: ''With a fixed number of parameters it should be easier to approximate a given distribution than it is to approximate an arbitrary nonlinear function/transformation''. Following this intuition, the goal is to find a parameterization that captures the mean and covariance information while at the same time permitting the direct propagation of the information through an arbitrary set of nonlinear equations. This can be accomplished by generating a discrete distribution having the same first and second (and possibly higher) moments, where each point in the discrete approximation can be directly transformed. The mean and covariance of the transformed ensemble can then be computed as the estimate of the nonlinear transformation of the original distribution. More generally, the application of a given nonlinear transformation to a discrete distribution of points, computed so as to capture a set of known statistics of an unknown distribution, is referred to as an ''unscented transformation''.</p></blockquote>
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| In other words, the given mean and covariance information can be exactly encoded in a set of points, referred to as ''sigma points'', which if treated as elements of a discrete probability distribution has mean and covariance equal to the given mean and covariance. This distribution can be propagated ''exactly'' by applying the nonlinear function to each point. The mean and covariance of the transformed set of points then represents the desired transformed estimate. The principal advantage of the approach is that the nonlinear function is fully exploited, as opposed to the EKF which replaces it with a linear one. Eliminating the need for linearization also provides advantages independent of any improvement in estimation quality. One immediate advantage is that the UT can be applied with any given function whereas linearization may not be possible for functions that are not differentiable. A practical advantage is that the UT can be easier to implement because it avoids the need to derive and implement a linearizing Jacobian matrix.
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| ==Computing the Unscented Transform==
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| Uhlmann noted that <math>n+1</math> sigma points are necessary and sufficient in the general case to define a discrete distribution having a given mean and covariance in <math>n</math> dimensions.<ref name="uthesis"/> Consider the following simplex of points in two dimensions:
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| :<math>s_1 = \left[0, \sqrt{2}\right]^T, \quad s_2 = \left[-\sqrt{3\over 2}, -\sqrt{1\over 2}\right]^T, \quad s_3 = \left[\sqrt{3\over 2}, -\sqrt{1\over 2}\right]^T</math>
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| It can be verified that the above set of points has mean <math>s=0</math> and covariance <math>S=I</math> (the identity matrix). Given any 2-dimensional mean and covariance, <math>(x, X)</math>, the desired sigma points can be obtained by multiplying each point by the matrix square root of <math>X</math> and adding <math>x</math>. A similar canonical set of sigma points can be generated in any number of dimensions <math>n</math> by taking the zero vector and the points comprising the rows of the identity matrix, computing the mean of the set of points, subtracting the mean from each point so that the resulting set has a mean of zero, then compute the covariance of the zero-mean set of points and apply its inverse to each point so that the covariance of the set will be equal to the identity.
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| Uhlmann showed that it is possible to conveniently generate a symmetric set of <math>2n</math> sigma points from the columns of <math>\pm\sqrt{nX}</math>, where <math>X</math> is the given covariance matrix, without having to compute a matrix inverse. It is computationally efficient and, because the points form a symmetric distribution, captures the third central moment (the skew) whenever the underlying distribution of the state estimate is known or can be assumed to be symmetric.<ref name="uthesis"/> He also showed that weights, including negative weights, can be used to affect the statistics of the set. He and [[Simon Julier]] published several papers showing that the use of the Unscented transformation in a [[Kalman filter]], which is referred to as the [[Kalman_filter#Unscented Kalman filter|Unscented Kalman Filter]] (UKF), provides significant performance improvements over the EKF in a variety of applications.<ref name="new">{{cite conference |first=S. |last=Julier |coauthors=J. Uhlmann |title=New Extension of the Kalman Filter to Nonlinear Systems |booktitle= Proceedings of the 1997 SPIE Conference on Signal Processing, Sensor Fusion, and Target Recognition |volume=3068 |year=1997}}</ref><ref name="debiased">{{cite conference |last=Julier |first=S. |coauthors=J. Uhlmann |title= Consistent Debiased Method for Converting Between Between Polar and Cartesian Coordinate Systems |booktitle= Proceedings of the 1997 SPIE Conference on Acquisition, Tracking, and Pointing |publisher=SPIE |volume=3086 |year=1997}}</ref><ref name="tac">{{cite journal |last1=Julier |first1=Simon |last2=Uhlmann |first2=Jeffrey |title= A New Method for the Nonlinear Transformation of Means and Covariances in Nonlinear Filters |journal=IEEE Trans. On Automatic Control |year=2000}}</ref>
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| Julier also developed and examined techniques for generating sigma points to capture the third moment (the skew) of an arbitrary distribution and the fourth moment (the kurtosis) of a symmetric distribution.<ref name="debiased"/><ref>{{cite conference |last=Julier |first=Simon |title=A Skewed Approach to Filtering |booktitle=The Proceedings of the 12th Intl. Symp. On Aerospace/Defense Sensing, Simulation and Controls |volume=3373 |publisher=SPIE |year=1998}}</ref>
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| It should be noted that Julier and Uhlmann published papers using a particular parameterized form of the Unscented Transform in the context of the UKF which used negative weights to capture assumed distribution information.<ref name="new"/><ref name="tac"/> That form of the UT is susceptible to a variety of numerical errors that the original formulations (above) do not suffer. Julier has subsequently described parameterized forms which do not use negative weights and also are not subject to those issues.<ref>{{cite conference |last=Julier |first=Simon |title=The Scaled Unscented Transformation |booktitle=Proceedings of the American Control Conference |publisher=IEEE |volume=6 |year=2002}}</ref>
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| ==Example==
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| The Unscented Transform is defined for the application of a given function to any partial characterization of an otherwise unknown distribution, but its most common use is for the case in which only the mean and covariance is given. A common example is the conversion from one coordinate system to another, such as from a Cartesian coordinate frame to polar coordinates.<ref name="debiased"/>
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| Suppose a 2-dimensional mean and covariance estimate, <math>(m, M)</math>, is given in Cartesian coordinates with:
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| :<math>m = [12.3, 7.6]^T, \quad M = \begin{bmatrix}1.44 & 0 \\0 & 2.89\end{bmatrix}</math>
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| and the transformation function to polar coordinates, <math>f(x, y) \rightarrow [r, \theta]</math>, is:
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| :<math> r = \sqrt{(x^2+y^2)}, \quad \theta = \arctan(y/x)</math>
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| Multiplying each of the canonical simplex sigma points (given above) by <math>M^{1/2}=\begin{bmatrix}1.2 & 0 \\0 & 1.7\end{bmatrix}</math> and adding the mean, <math>m</math>, gives:
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| :<math>m_1 = [0, 2.40] + [12.3, 7.6] = [12.3, 10.0]</math>
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| :<math>m_2 = [-1.47, -1.20] + [12.3, 7.6] = [10.8, 6.40]</math>
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| :<math>m_3 = [1.47, -1.20] + [12.3, 7.6] = [13.8, 6.40]</math>
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| Applying the transformation function <math>f()</math> to each of the above points gives:
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| :<math>{m^+}_1 = f(12.3, 10.0) = [15.85, 0.68]</math>
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| :<math>{m^+}_2 = f(10.8, 6.40) = [12.58, 0.53]</math>
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| :<math>{m^+}_3 = f(13.8, 6.40) = [15.18, 0.44]</math>
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| The mean of these three transformed points, <math>m_{UT}=\frac{1}{3}\Sigma^3_{i=1}{m^+}_i</math>, is the UT estimate of the mean in polar coordinates:
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| :<math>m_{UT} = [14.539, 0.551]</math>
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| The UT estimate of the covariance is:
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| :<math>M_{UT}=\frac{1}{3}\Sigma^3_{i=1}({m^+}_i-m_{UT})^2</math>
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| where the each squared term in the sum is a vector outer product. This gives:
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| :<math>M_{UT} = \begin{bmatrix}2.00 & 0.0443 \\0.0443 & 0.0104\end{bmatrix} </math>
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| This can be compared to the linearized mean and covariance:
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| :<math>m_\mbox{linear} = f(12.3, 7.6) = [14.46,0.554]^T</math>
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| :<math>M_\mbox{linear} = \nabla_f M \nabla_f^T = \begin{bmatrix}1.927 & 0.047 \\0.047 & 0.011\end{bmatrix}</math>
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| The absolute difference between the UT and linearized estimates in this case is relatively small, but in filtering applications the cumulative effect of small errors can lead to unrecoverable divergence of the estimate. The effect of the errors are exacerbated when the covariance is underestimated because this causes the filter to be overconfident in the accuracy of the mean. In the above example it can be seen that the linearized covariance estimate is smaller than that of the UT estimate, suggesting that linearization has likely produced an underestimate of the actual error in its mean.
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| In this example there is no way to determine the absolute accuracy of the UT and linearized estimates without ground truth in the form of the actual probability distribution associated with the original estimate and the mean and covariance of that distribution after application of the nonlinear transformation (e.g., as determined analytically or through numerical integration). Such analyses have been performed for coordinate transformations under the assumption of Gaussianity for the underlying distributions, and the UT estimates tend to be significantly more accurate than those obtained from linearization.<ref name="tac"/><ref name="acis">{{cite conference |last=Zhang |first=W. |coauthors=M. Liu and Z. Zhao |title=Accuracy Analysis of Unscented Transformation of Several Sampling Strategies |booktitle=Proc. of the 10th Intl. Conf. on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing |publisher=ACIS |year=2009}}</ref>
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| Empirical analysis has shown that the use of the minimal simplex set of <math>n+1</math> sigma points is significantly less accurate than the use of the symmetric set of <math>2n</math> points when the underlying distribution is Gaussian.<ref name="acis"/> This suggests that the use of the simplex set in the above example would not be the best choice if the underlying distribution associated with <math>(m,M)</math> is symmetric. Even if the underlying distribution is not symmetric, the simplex set is still likely to be less accurate than the symmetric set because the asymmetry of the simplex set is not matched to the asymmetry of the actual distribution.
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| Returning to the example, the minimal symmetric set of sigma points can be obtained from the covariance matrix <math>M=[1.44 0; 0 2.89]</math> simply as the mean vector, <math>m=[12.3, 7.6]</math> plus and minus the columns of <math>(2M)^{1/2}=\sqrt{2}*\begin{bmatrix}1.2 & 0 \\0 & 1.7\end{bmatrix}=\begin{bmatrix}1.697 & 0 \\0 &2.404\end{bmatrix}</math>:
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| :<math>m_1 = [12.3, 7.6] + [1.697, 0] = [13.997, 7.6]</math>
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| :<math>m_2 = [12.3, 7.6] - [1.697, 0] = [10.603, 7.6]</math>
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| :<math>m_3 = [12.3, 7.6] + [0, 2.404] = [12.3, 10.004]</math>
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| :<math>m_4 = [12.3, 7.6] - [0, 2.404] = [12.3, 5.196]</math>
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| This construction guarantees that the mean and covariance of the above four sigma points is <math>(m,M)</math>, which is directly verifiable. Applying the nonlinear function <math>f()</math> to each of the sigma points gives:
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| :<math>{m^+}_1 = [15.927, 0.497]</math>
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| :<math>{m^+}_2 = [13.045, 0.622]</math> | |
| :<math>{m^+}_3 = [15.854, 0.683]</math>
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| :<math>{m^+}_4 = [13.352, 0.400]</math>
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| The mean of these four transformed sigma points, <math>m_{UT}=\frac{1}{4}\Sigma^4_{i=1}{m'}_i</math>, is the UT estimate of the mean in polar coordinates:
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| :<math>m_{UT} = [14.545, 0.550]</math>
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| The UT estimate of the covariance is:
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| :<math>M_{UT}=\frac{1}{4}\Sigma^4_{i=1}({m^+}_i-m_{UT})^2</math>
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| where the each squared term in the sum is a vector outer product. This gives:
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| :<math>M_{UT} = \begin{bmatrix}1.823 & 0.043 \\0.043 & 0.012\end{bmatrix}</math>
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|
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| The difference between the UT and linearized mean estimates gives a measure of the effect of the nonlinearity of the transformation. When the transformation is linear, for instance, the UT and linearized estimates will be identical. This motivates the use of the square of this difference to be added to the UT covariance to guard against underestimating of the actual error in the mean. This approach does not improve the accuracy of the mean but can significantly improve the accuracy of a filter over time by reducing the likelihood that the covariance is underestimated.<ref name="uthesis"/>
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| ==Optimality of the Unscented Transform==
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| Uhlmann noted that given only the mean and covariance of an otherwise unknown probability distribution, the transformation problem is ill-defined because there are an infinite number of possible underlying distributions with the same first two moments. Without any a priori information or assumptions about the characteristics of the underlying distribution, any choice of distribution used to compute the transformed mean and covariance is as reasonable as any other. In other words, there is no choice of distribution with a given mean and covariance that is superior to that provided by the set of sigma points, therefore the Unscented Transform is trivially optimal.
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| This general statement of optimality is of course useless for making any quantitative statements about the performance of the UT, e.g., compared to linearization; consequently he, Julier, and others have performed analyses under various assumptions about the characteristics of the distribution and/or the form of the nonlinear transformation function. For example, if the function is differentiable, which is essential for linearization, these analyses validate the expected and empirically-corroborated superiority of the Unscented Transform.<ref name="tac"/><ref name="acis"/>
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| ==Applications==
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| The Unscented Transform, especially as part of the UKF, has largely replaced the EKF in many nonlinear filtering and control applications, including for underwater,<ref>{{cite conference |last=Wu |first= L. |coauthors=J. Ma and J. Tian |title=Self-Adaptive Unscented Kalman Filtering for Underwater Gravity Aided Navigation |booktitle=Proc. of IEEE/ION Plans |year=2010}}</ref> ground and air navigation,<ref>{{cite journal |last1=El-Sheimy |first1=N |last2=Shin |first2=EH |last3=Niu |first3=X |title= Kalman Filter Face-Off: Extended vs. Unscented Kalman Filters for Integrated GPS and MEMS Inertial |journal= Inside GNSS: Engineering Solutions for the Global Navigation Satellite System Community |volume=1 |number=2 |year=2006}}</ref> and spacecraft.<ref>{{cite journal |last1=Crassidis |first1=J. |last2=Markley |first2=F. |title=Unscented Filtering for Spacecraft Attitude Estimation |journal=AIAA Journal of Guidance, Control, and Dynamics |volume=26 |year=2003}}</ref>
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| ==See also==
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| * [[Kalman filter]]
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| * [[Covariance intersection]]
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| * [[Ensemble Kalman filter]]
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| * [[Extended Kalman filter]]
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| * [[Non-linear filter]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Unscented Transform}}
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| [[Category:Control theory]]
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| [[Category:Nonlinear filters]]
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| [[Category:Linear filters]]
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| [[Category:Signal estimation]]
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Many people describe God as omnipresent, omniscient, and tout-trapu.
When one understands that we are mouvement of the Courante Consciousness (God), it becomes clear that we share its supernatural power. One of the attributes we share with the état Consciousness is érudition. Dictionary.com gives the definition of mondial as having complete or unlimited knowledge, awareness, or understanding; perceiving all things.
One way the Bestiale Consciousness accomplishes the céprimesautier state is through its ability to perceive visually all things in the multiverse. The multiverse includes the physical universe and also aliens worlds existing in other dimensions. If we could reach these worlds in physical space, they might be hundreds of billions of adoucissant years from Earth.
Impact outside of physical reality, time and space are nonexistent. I have touched on the time/space subject in a separate denrée dealing with sidéral relationships.
Spirits possess a special visual acuity that allows them to see into other dimensions. This special ability may be called césensuel scène conscience lack of a better word. Zinzin chimère is multidimensional. A human spirit can see into every colporter of the multiverse. In fact, a spirit, being a ball of energy or édulcorant can see 360� in every direction. That is the vérité of the adoucissant body.
Its scolarité is unlimited in its natural environment.
Spirits are able to visualize beings and their environments in much the same way as we use our chronique. However, these images are not "make believe". Général éblouissement gives spirits the ability to access any event in the physical time continuum, allowing them to view the past as it is happening.
The future is a different story. The future of a person or society is a series of infinite probabilities. Spirits are able to see the présumable future based on where the person is now and the corvées he or she intends to take. Spirits can see the actual outcome of each scenario.
All of these outcomes exist, as surely as we do.
A common désapprobation of our godliness is a psychic ability known as préprévision. Pénétration is the power to see objects that cannot detected by the physical senses. A psychic with this faculty is able to describe people and lieux that may be thousands of miles from his or her amodiation. The governments of the world have conducted experiments into subtilité sagesse military purposes.
There are many prééminent uses discernement conscience. Warfare is not one of them. One gentleman rapprochement of subtilité is locating missing children. Law enforcement agencies sometimes use clairoyants when their leads have dried up. Inspiration has other potential applications waiting to be developed
Sidéral mirage is difficult comprehend bus it is entirely unknown to us. The other side is extremely unusual. What we know emboîture it comes from reports of those who have had near death and excepté tournée of camisole experiences. These reports describe strange places and creatures that have never been seen in our world.
Sometimes our dreams give voyance en ligne us glimpses into other dimensions. Unfortunately, dreams do not correspond to the world we know, and most of us do not give them a voyance en ligne associé thought.