Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Total least squares
Résumé
In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models.
The total least squares approximation of the data is generically equivalent to the best, in the Frobenius norm, low-rank approximation of the data matrix.
In the least squares method of data modeling, the objective function, S,
is minimized, where r is the vector of residuals and W is a weighting matrix. In linear least squares the model contains equations which are linear in the parameters appearing in the parameter vector , so the residuals are given by
There are m observations in y and n parameters in β with m>n. X is a m×n matrix whose elements are either constants or functions of the independent variables, x. The weight matrix W is, ideally, the inverse of the variance-covariance matrix of the observations y. The independent variables are assumed to be error-free. The parameter estimates are found by setting the gradient equations to zero, which results in the normal equations
Now, suppose that both x and y are observed subject to error, with variance-covariance matrices and respectively. In this case the objective function can be written as
where and are the residuals in x and y respectively. Clearly these residuals cannot be independent of each other, but they must be constrained by some kind of relationship. Writing the model function as , the constraints are expressed by m condition equations.
Thus, the problem is to minimize the objective function subject to the m constraints. It is solved by the use of Lagrange multipliers. After some algebraic manipulations, the result is obtained.
or alternatively
where M is the variance-covariance matrix relative to both independent and dependent variables.
When the data errors are uncorrelated, all matrices M and W are diagonal.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Proximité ontologique