Classical XY model

Template:Regression bar Tikhonov regularization, named for Andrey Tikhonov, is the most commonly used method of regularization of ill-posed problems. 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 problems.

When the following problem is not well posed (either because of non-existence or non-uniqueness of $x$)

$A\mathbf {x} =\mathbf {b} ,$

then the standard approach is known as ordinary least squares and seeks to minimize the residual

$\|A\mathbf {x} -\mathbf {b} \|^{2}$

where $\left\|\cdot \right\|$ is the Euclidean norm. This may be due to the system being overdetermined or underdetermined ($A$ may be ill-conditioned or 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:

$\|A\mathbf {x} -\mathbf {b} \|^{2}+\|\Gamma \mathbf {x} \|^{2}$

for some suitably chosen Tikhonov matrix, $\Gamma$. In many cases, this matrix is chosen as the identity matrix $\Gamma =I$, giving preference to solutions with smaller norms. In other cases, lowpass operators (e.g., a difference operator or a weighted Fourier operator) may be used to enforce smoothness if the underlying vector is believed to be mostly continuous. This regularization improves the conditioning of the problem, thus enabling a direct numerical solution. An explicit solution, denoted by ${\hat {x}}$, is given by:

${\hat {x}}=(A^{T}A+\Gamma ^{T}\Gamma )^{-1}A^{T}\mathbf {b}$

The effect of regularization may be varied via the scale of matrix $\Gamma$. For $\Gamma =0$ this reduces to the unregularized least squares solution provided that (A^TA)⁻¹ exists.

History

Tikhonov regularization has been invented independently in many different contexts. It became widely known from its application to integral equations from the work of Tikhonov and David L. Phillips. Some authors use the term Tikhonov–Phillips regularization. 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 Wiener–Kolmogorov filter. Following Hoerl, it is known in the statistical literature as ridge regression.

Generalized Tikhonov regularization

For general multivariate normal distributions for $x$ and the data error, one can apply a transformation of the variables to reduce to the case above. Equivalently, one can seek an $x$ to minimize

$\|Ax-b\|_{P}^{2}+\|x-x_{0}\|_{Q}^{2}\,$

where we have used $\left\|x\right\|_{Q}^{2}$ to stand for the weighted norm $x^{T}Qx$ (compare with the Mahalanobis distance). In the Bayesian interpretation $P$ is the inverse covariance matrix of $b$, $x_{0}$ is the expected value of $x$, and $Q$ is the inverse covariance matrix of $x$. The Tikhonov matrix is then given as a factorization of the matrix $Q=\Gamma ^{T}\Gamma$ (e.g. the Cholesky factorization), and is considered a whitening filter.

This generalized problem has an optimal solution $x^{*}$ which can be solved explicitly using the formula

$x^{*}=(A^{T}PA+Q)^{-1}(A^{T}Pb+Qx_{0}).\,$

or equivalently

$x^{*}=x_{0}+(A^{T}PA+Q)^{-1}(A^{T}P(b-Ax_{0})).\,$

Regularization in Hilbert space

Typically discrete linear ill-conditioned problems result from discretization of integral equations, and one can formulate a Tikhonov regularization in the original infinite dimensional context. In the above we can interpret $A$ as a compact operator on Hilbert spaces, and $x$ and $b$ as elements in the domain and range of $A$. The operator $A^{*}A+\Gamma ^{T}\Gamma$ is then a self-adjoint bounded invertible operator.

Relation to probabilistic formulation

The probabilistic formulation of an inverse problem introduces (when all uncertainties are Gaussian) a covariance matrix $C_{M}$ representing the a priori uncertainties on the model parameters, and a covariance matrix $C_{D}$ representing the uncertainties on the observed parameters (see, for instance, Tarantola, 2005 [1]). In the special case when these two matrices are diagonal and isotropic, $C_{M}=\sigma _{M}^{2}I$ and $C_{D}=\sigma _{D}^{2}I$, and, in this case, the equations of inverse theory reduce to the equations above, with $\alpha ={\sigma _{D}}/{\sigma _{M}}$.

Bayesian interpretation

Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix $\Gamma$ seems rather arbitrary, the process can be justified from a 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 $x$ 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 $\sigma _{x}$. The data are also subject to errors, and the errors in $b$ are also assumed to be independent with zero mean and standard deviation $\sigma _{b}$. Under these assumptions the Tikhonov-regularized solution is the most probable solution given the data and the a priori distribution of $x$, according to Bayes' theorem.

If the assumption of normality is replaced by assumptions of homoskedasticity and uncorrelatedness of errors, and if one still assumes zero mean, then the Gauss–Markov theorem entails that the solution is the minimal unbiased estimate.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

See also

• LASSO estimator is another regularization method in statistics.

References

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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.
• Tikhonov A.N., Leonov A.S., Yagola A.G., 1998, Nonlinear Ill-Posed Problems, V. 1, V. 2, Chapman and Hall.
• Hansen, P.C., 1998, Rank-deficient and Discrete ill-posed problems, SIAM
• Hoerl AE, 1962, Application of ridge analysis to regression problems, Chemical Engineering Progress, 1958, 54–59.
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Tarantola A, 2005, Inverse Problem Theory (free PDF version), Society for Industrial and Applied Mathematics, ISBN 0-89871-572-5
• Wahba, G, 1990, Spline Models for Observational Data, Society for Industrial and Applied Mathematics