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Abstract
In this paper, we present the Cumulative Median Estimation (CUMed) algorithm for robust sufficient dimension reduction. Compared with non-robust competitors, this algorithm performs better when there are outliers present in the data and comparably when outliers are not present. This is demonstrated in simulated and real data experiments.
Highlights
Sufficient Dimension Reduction (SDR) is a framework of supervised dimension reduction algorithms that have been proposed mainly for regression and classification settings.The main objective is to reduce the dimension of the p-dimensional predictor vector X without losing information on the conditional distribution of Y | X
As Cumulative Median Estimation (CUMed) is the robust version of Cumulative Mean Estimation (CUME), it has all the advantages CUME has with the additional advantage that it is robust to the presence of outliers
Sliced Inverse Regression (SIR) and CUME as well as Sliced Inverse Median (SIME), which is the robust version of SIR proposed by [8] and, to our case, it uses the L1 median
Summary
Sufficient Dimension Reduction (SDR) is a framework of supervised dimension reduction algorithms that have been proposed mainly for regression and classification settings. The space spanned by such β is called the Central Dimension Reduction. The first approach to the SDR framework was Sliced Inverse Regression (SIR) introduced by [3] and which used inverse means to achieve efficient feature extraction. To avoid this, [7] proposed the Cumulative Mean Estimation (CUME), which uses the first moment and removes the necessity to tune the number of slices. A number of methods were introduced for robust sufficient dimension reduction. [8] proposed Robust SIR by using the L1 median to achieve sufficient dimension reduction and [9] proposed the use of Tukey and Oja medians to robustify SIR. [10] proposed Sliced Inverse Median Difference (SIMeD), the robust version of SIMD using the L1 median. The new method is called Cumulative Median estimation (CUMed).
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