Constrained total least squares

Abstract

The Total Least Squares (TLS) method is a generalized least square technique to solve an overdetermined system of equations <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ax\simeqb</tex> . The TLS solution differs from the usual Least Square (LS) in that it tries to compensate for arbitrary noise present in both <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> . In certain problems the noise perturbations of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> are linear functions of a common "noise source" vector. In this case we obtain a generalization of the TLS criterion called the Constrained Total Least Squares (CTLS) method by taking into account the linear dependence of the noise terms in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> . If the noise columns of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> are linearly related then the CTLS solution is obtained in terms of the largest eigenvalue and corresponding eigenvector of a certain matrix. The CTLS technique can be applied to problems like Maximum Likelihood Signal Parameter Estimation, Frequency Estimation of Sinusoids in white or colored noise by Linear Prediction and others.