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In many dimension reduction problems in statistics and machine learning, such as principal component analysis, canonical correlation analysis, independent component analysis, and sufficient dimension reduction, it is important to determine the dimension of the reduced predictor, which often amounts to estimating the rank of a matrix. This problem is called order determination. In this paper, we propose a novel and highly effective order-determination method based on the idea of predictor augmentation. We show that, if we augment the predictor by an artificially generated random vector, then the part of the eigenvectors of the matrix induced by the augmentation display a pattern that reveals information about the order to be determined. This information, when combined with the information provided by the eigenvalues of the matrix, greatly enhances the accuracy of order determination.

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Abstract. We consider the distribution of the orbits of the number 1 under the β-transformations T β as β varies. Mainly, the size of the set of β > 1 for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. That is, the dimension of the following set, for infinitely many n ∈ N is determined, where x 0 is a given point in [0, 1] and {ℓ n } n≥1 is a sequence of integers tending to infinity as n → ∞. For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of β with a common prefix in the expansion of 1) in the parameter space { β ∈ R : β > 1 }.