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- W57041892 abstract "We consider the multivariate linear regression model with p explanatory variables X and q ≥ 1 response variables Y. Moreover we assume that the regressors are multicollinear. This situation often occurs in the calibration of chemometrical data, where the X-variables correspond with spectra that are measured at many frequencies. It is well known that the classical least squares estimator has a large variance in the presence of multicollinearity. Moreover it can not be computed when p > n since the X t X matrix then becomes singular. Therefore many biased estimators have been proposed. A very appealing method is principal components regression (PCR) since it is easy to understand and to compute. PCR first constructs a new set of uncorrelated explanatory variables, which are called the principal components. They correspond to the eigenvectors of the sample covariance matrix of the X-variables. The response variables are then regressed on these components using the (multivariate) least squares estimator. Both stages of this procedure are however very sensitive to the presence of outliers in the data. We present a robust principal components regression method which also consists of two steps. First a robust principal components analysis is applied to the X-variables (Hubert and Rousseeuw, 2002), yielding a smaller set of k orthogonal regressors. We then regress the response variables Y on these regressors using the multivariate MCD-regression method (Rousseeuw et al. 2000). To select the number of regressors, we propose a robust R 2. It is roughly defined as the proportion of the robust variance of Y which is explained by the robust fit. Moreover we propose several diagnostic plots, which allow us to visualize and to distinguish the possible outliers in the data." @default.
- W57041892 created "2016-06-24" @default.
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- W57041892 date "2002-01-01" @default.
- W57041892 modified "2023-10-17" @default.
- W57041892 title "Robust Principal Components Regression" @default.
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- W57041892 doi "https://doi.org/10.1007/978-3-642-57489-4_79" @default.
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