Robust real-time monitoring of high-dimensional data streams has many important real-world applications such as industrial quality control, signal detection, biosurveillance, but unfortunately it is highly non-trivial to develop efficient schemes due to two challenges: (1) the unknown sparse number or subset of affected data streams and (2) the uncertainty of model specification for high-dimensional data. In this article, motivated by the detection of smaller persistent changes in the presence of larger transient outliers, we develop a family of efficient real-time robust detection schemes for high-dimensional data streams through monitoring feature spaces such as PCA or wavelet coefficients when the feature coefficients are from Tukey-Hubers gross error models with outliers. We propose to construct a new local detection statistic for each feature called $L_{alpha}$-CUSUM statistic that can reduce the effect of outliers by using the Box-Cox transformation of the likelihood function, and then raise a global alarm based upon the sum of the soft-thresholding transformation of these local $L_{alpha}$-CUSUM statistics so that to filter out unaffected features. In addition, we propose a new concept called false alarm breakdown point to measure the robustness of online monitoring schemes, and also characterize the breakdown point of our proposed schemes. Asymptotic analysis, extensive numerical simulations and case study of nonlinear profile monitoring are conducted to illustrate the robustness and usefulness of our proposed schemes.
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