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| | Andrew Simcox is the name his parents gave him and he completely loves this title. For years she's been operating as a journey agent. For years he's been residing in Alaska and he doesn't plan on altering it. To play lacross is the thing I love most of all.<br><br>Here is my blog: real psychics ([http://www.ideocongo.com/profile.php?u=Ha9496 ideocongo.com]) |
| '''Tikhonov regularization''', named for [[Andrey Nikolayevich Tikhonov|Andrey Tikhonov]], is the most commonly used method of [[regularization (mathematics)|regularization]] of [[ill-posed problem]]s. In [[statistics]], the method is known as '''ridge regression''', and, with multiple independent discoveries, it is also variously known as the '''Tikhonov–Miller method''', the '''Phillips–Twomey method''', the '''constrained linear inversion''' method, and the method of '''linear regularization'''. It is related to the [[Levenberg–Marquardt algorithm]] for [[non-linear least squares|non-linear least-squares]] problems.
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| When the following problem is not [[Well-posed problem|well posed]] (either because of non-existence or non-uniqueness of <math>x</math>)
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| : <math>A\mathbf{x}=\mathbf{b},</math>
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| then the standard approach is known as [[ordinary least squares]] and seeks to minimize the [[Residual (numerical analysis)|residual]]
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| : <math>\|A\mathbf{x}-\mathbf{b}\|^2 </math>
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| where <math>\left \| \cdot \right \|</math> is the [[Norm (mathematics)#Euclidean norm|Euclidean norm]]. This may be due to the system being [[Overdetermined system|overdetermined]] or [[Underdetermined system|underdetermined]] (<math>A</math> may be [[ill-conditioned]] or [[Singular matrix|singular]]). In the latter case this is no better than the original problem. In order to give preference to a particular solution with desirable properties, the regularization term is included in this minimization:
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| : <math>\|A\mathbf{x}-\mathbf{b}\|^2+ \|\Gamma \mathbf{x}\|^2</math>
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| for some suitably chosen '''Tikhonov matrix''', <math>\Gamma </math>. In many cases, this matrix is chosen as the [[identity matrix]] <math>\Gamma= I </math>, giving preference to solutions with smaller [[Norm (mathematics)|norms]]. In other cases, [[lowpass]] operators (e.g., a [[difference operator]] or a weighted [[discrete fourier transform|Fourier operator]]) may be used to enforce smoothness if the underlying vector is believed to be mostly continuous.
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| This regularization improves the conditioning of the problem, thus enabling a direct numerical solution. An explicit solution, denoted by <math>\hat{x}</math>, is given by:
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| : <math>\hat{x} = (A^{T}A+ \Gamma^{T} \Gamma )^{-1}A^{T}\mathbf{b}</math>
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| The effect of regularization may be varied via the scale of matrix <math>\Gamma</math>. For <math>\Gamma = 0</math> this reduces to the unregularized least squares solution provided that (A<sup>T</sup>A)<sup>−1</sup> exists.
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| ==History==
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| Tikhonov regularization has been invented independently in many different contexts.
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| It became widely known from its application to integral equations from the work of
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| [[Andrey Nikolayevich Tikhonov|Tikhonov]] and David L. Phillips.
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| Some authors use the term '''Tikhonov–Phillips regularization'''.
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| The finite dimensional case was expounded by Arthur E. Hoerl, who took a statistical approach, and by Manus Foster, who interpreted this method as a [[Norbert Wiener|Wiener]]–[[Andrey Nikolaevich Kolmogorov|Kolmogorov]] filter. Following Hoerl, it is known in the statistical literature as '''ridge regression'''.
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| ==Generalized Tikhonov regularization==
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| For general multivariate normal distributions for <math>x</math> and the data error, one can apply a transformation of the variables to reduce to the case above. Equivalently, one can seek an <math>x</math> to minimize
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| :<math>\|Ax-b\|_P^2 + \|x-x_0\|_Q^2\,</math>
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| where we have used <math>\left \| x \right \|_Q^2</math> to stand for the weighted norm <math>x^T Q x</math> (compare with the [[Mahalanobis distance]]). In the Bayesian interpretation <math>P</math> is the inverse [[covariance matrix]] of <math>b</math>, <math>x_0</math> is the [[expected value]] of <math>x</math>, and <math>Q</math> is the inverse covariance matrix of <math>x</math>. The Tikhonov matrix is then given as a factorization of the matrix <math> Q = \Gamma^T \Gamma </math> (e.g. the [[Cholesky factorization]]), and is considered a [[White noise#Whitening a random vector|whitening filter]].
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| This generalized problem has an optimal solution <math>x^*</math> which can be solved explicitly using the formula
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| : <math>x^* = (A^T PA + Q)^{-1} (A^T Pb+Qx_0).\,</math>
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| or equivalently
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| : <math>x^* = x_0 + (A^T PA + Q)^{-1} (A^T P(b-Ax_0)).\,</math> | |
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| ==Regularization in Hilbert space==
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| Typically discrete linear ill-conditioned problems result from discretization of [[integral equation]]s, and one can formulate a Tikhonov regularization in the original infinite dimensional context. In the above we can interpret <math>A</math> as a [[compact operator]] on [[Hilbert space]]s, and <math>x</math> and <math>b</math> as elements in the domain and range of <math>A</math>. The operator <math>A^* A + \Gamma^T \Gamma </math> is then a [[Hermitian adjoint|self-adjoint]] bounded invertible operator.
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| ==Relation to probabilistic formulation==
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| The probabilistic formulation of an [[inverse problem]] introduces (when all uncertainties are Gaussian) a covariance matrix <math> C_M</math> representing the ''a priori'' uncertainties on the model parameters, and a covariance matrix <math> C_D</math> representing the uncertainties on the observed parameters (see, for instance, Tarantola, 2005 [http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/SIAM/index.html]). In the special case when these two matrices are diagonal and isotropic, <math> C_M = \sigma_M^2 I </math> and <math> C_D = \sigma_D^2 I </math>, and, in this case, the equations of inverse theory reduce to the equations above, with <math> \alpha = {\sigma_D}/{\sigma_M} </math>.
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| ==Bayesian interpretation==
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| Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix <math>\Gamma</math> seems rather arbitrary, the process can be justified from a [[Bayesian probability|Bayesian point of view]]. Note that for an ill-posed problem one must necessarily introduce some additional assumptions in order to get a stable solution. Statistically, the [[prior probability]] distribution of <math>x</math> is sometimes taken to be a [[multivariate normal distribution]]. For simplicity here, the following assumptions are made: the means are zero; their components are independent; the components have the same [[standard deviation]] <math>\sigma _x</math>. The data are also subject to errors, and the errors in <math>b</math> are also assumed to be [[statistical independence|independent]] with zero mean and standard deviation <math>\sigma _b</math>. Under these assumptions the Tikhonov-regularized solution is the [[maximum a posteriori|most probable]] solution given the data and the ''a priori'' distribution of <math>x</math>, according to [[Bayes' theorem]].<ref>{{cite book |author=Vogel, Curtis R. |title=Computational methods for inverse problems |publisher=Society for Industrial and Applied Mathematics |location=Philadelphia |year=2002 |pages= |isbn=0-89871-550-4 |oclc= |doi= |accessdate=}}</ref>
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| If the assumption of [[normal distribution|normality]] is replaced by assumptions of [[homoskedasticity]] and uncorrelatedness of [[errors and residuals in statistics|errors]], and if one still assumes zero mean, then the [[Gauss–Markov theorem]] entails that the solution is the minimal [[Bias of an estimator|unbiased estimate]].{{Citation needed|date=April 2012}}
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| ==See also==
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| * [[Lasso_(statistics)#Lasso_method|LASSO estimator]] is another regularization method in statistics.
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| {{More footnotes|date=April 2012}}
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| ==References==
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| {{Reflist}}
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| * {{Cite journal| last=Tikhonov | first=Andrey Nikolayevich | authorlink=Andrey Nikolayevich Tikhonov | year=1943 | title=Об устойчивости обратных задач | trans_title=On the stability of inverse problems | journal=[[Doklady Akademii Nauk SSSR]] | volume=39 | issue=5 | pages=195–198}}
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| * {{Cite journal| last=Tikhonov | first=A. N. | year=1963 | title=О решении некорректно поставленных задач и методе регуляризации | trans_title=Solution of incorrectly formulated problems and the regularization method | journal=Doklady Akademii Nauk SSSR | volume=151 | pages=501–504}}. Translated in {{Cite journal| journal=Soviet Mathematics | volume=4 | pages=1035–1038}}
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| * {{Cite book| last=Tikhonov | first=A. N. | coauthors=V. Y. Arsenin | year=1977 | title=Solution of Ill-posed Problems | publisher=Winston & Sons | location=Washington | isbn=0-470-99124-0}}
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| * Tikhonov A.N., Goncharsky A.V., Stepanov V.V., Yagola A.G., 1995, ''Numerical Methods for the Solution of Ill-Posed Problems'', Kluwer Academic Publishers.
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| * Tikhonov A.N., Leonov A.S., Yagola A.G., 1998, ''Nonlinear Ill-Posed Problems'', V. 1, V. 2, Chapman and Hall.
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| * Hansen, P.C., 1998, ''Rank-deficient and Discrete ill-posed problems'', SIAM
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| * Hoerl AE, 1962, ''Application of ridge analysis to regression problems'', Chemical Engineering Progress, 1958, 54–59.
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| * {{cite journal | last = Hoerl | first = A.E. | coauthors = R.W. Kennard | year = 1970 | title=Ridge regression: Biased estimation for nonorthogonal problems | journal=Technometrics | volume=12 | jstor=1271436 | issue=1 | pages = 55–67}}
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| * {{cite doi|10.1137/0109031}}
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| * {{cite doi|10.1145/321105.321114}}
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| * {{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 19.5. Linear Regularization Methods | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1006}}
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| * Tarantola A, 2005, ''Inverse Problem Theory'' ([http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/SIAM/index.html free PDF version]), Society for Industrial and Applied Mathematics, ISBN 0-89871-572-5
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| * Wahba, G, 1990, ''Spline Models for Observational Data'', Society for Industrial and Applied Mathematics
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| [[Category:Linear algebra]]
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| [[Category:Estimation theory]]
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| [[Category:Inverse problems]]
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Andrew Simcox is the name his parents gave him and he completely loves this title. For years she's been operating as a journey agent. For years he's been residing in Alaska and he doesn't plan on altering it. To play lacross is the thing I love most of all.
Here is my blog: real psychics (ideocongo.com)