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.
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In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods. The three main linear least squares formulations are: Ordinary least squares (OLS) is the most common estimator.
Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences.
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