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For the least significant bit (LSB) repalcement steganography, an improved cover estimation method is proposed by combining the weighted-stego(WS) residuals and the maximum a posteriori probability (MAP) cover estimator. First, the WS residuals are used to perform preliminary payload location. Next, the preliminary location result is applied to the improved MAP cover estimation method as a priori information so that a more accurate cover estimation algorithm will be obtained. Finally, the residual is computed to derive the final payload location result. Previous experiments have proved that the proposed method can improve the accuracy of payload location for the LSB replacement for either the spatial domain BMP image or the JPEG decompression image. It has a good robustness for different embedding rates.

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Covariance operators are fundamental concepts and modelling tools for many functional data analysis methods, such as functional principal component analysis. However, the empirical (or estimated) covariance operator becomes too costly to compute when the functional dataset gets big. This paper studies a randomized algorithm for covariance operator estimation. The algorithm works by sampling and rescaling observations from the large functional data collection to form a sketch of much smaller size and performs computation on the sketch to obtain the subsampled empirical covariance operator. The proposed algorithm is theoretically justified via nonasymptotic bounds between the subsampled and the full‐sample empirical covariance operator in terms of the Hilbert‐Schmidt norm and the operator norm. It is shown that the optimal sampling probability that minimizes the expected squared Hilbert‐Schmidt norm of the subsampling error is determined by the norm of each function. Simulated and real data examples are used to illustrate the effectiveness of the proposed algorithm.

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