engleski

The influence of the Huber estimator tuning constant on the performance of the iteratively reweighted least squares method

In the estimation of the model parameters from observed data, the outliers can have a strong impact. The classical least-squares (LS) estimation method minimizes the squared residuals and it has the highest asymptotic efficiency in the case of normally distributed errors, but it is unsuccessful in fitting a correct model to the data corrupted with outliers or heavy-tailed noise. To reduce the influence of outliers, the iteratively-reweighted least-squares (IRWLS) method [1] can be used. It iteratively minimizes the weighted least squares, thus emphasizing the influence of valid samples and deweighting the outliers. The weights are calculated in each iteration as a function of errors from the previous iteration taking into account the statistical properties of residuals, their dispersion, and central tendency. The main problem is to separate the valid samples from outliers, i.e., to determine the threshold δ which is calculated from the statistical analysis of residuals. Instead of using the classical estimators of the sample mean and sample variance for the statistical analysis of errors, the robust measures of central tendency and dispersion should be used due to their resistance to outliers. However, the prerequisite for the application of robust statistics is at least 50% of valid samples in the observed data, which is the breakdown point of a robust estimator. The first step in the calculation of weights is to determine the threshold δ that is obtained by the multiplication of the robust measure of error dispersion and the tuning factor k. Some often used robust measures of error dispersion are median absolute deviation and interquartile range. The samples with an error larger than the threshold δ are deweighted to reduce their contribution to the final estimate, but they are still included in the estimation process to avoid the loss of useful information. In this paper, the weights are calculated using the Huber estimator [2], which belongs to the group of M-estimators and combines the L1 and L2 norms. The L2 norm is applied to valid samples while the L1 norm is applied to the samples with errors larger than the threshold δ.

iteratively-reweighted least-squares ; Huber loss ; two-dimensional Gaussian profile estimation

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