The Akaike information criterion (AIC) is a widely used tool for model selection. AIC is derived as an asymptotically unbiased estimator of a function used for ranking candidate models which is a variant of the Kullback-Leibler divergence between the true model and the approximating candidate model. Despite the Kullback-Leibler's computational and theoretical advantages, what can become inconvenient in model selection applications is their lack of symmetry. Simple examples can show that reversing the role of the arguments in the Kullback-Leibler divergence can yield substantially different results. In this paper, three new functions for ranking candidate models are proposed. These functions are constructed by symmetrizing the Kullback-Leibler divergence between the true model and the approximating candidate model. The operations used for symmetrizing are the average, geometric, and harmonic means. It is found that the original AIC criterion is an asymptotically unbiased estimator of these three different functions. Using one of these proposed ranking functions, an example of new bias correction to AIC is derived for univariate linear regression models. A simulation study based on polynomial regression is provided to compare the different proposed ranking functions with AIC and the new derived correction with AICc.

An experiment was conducted to investigate the feasibility of using functional near-infrared spectroscopy (fNIRS) to image cortical activity in the language areas of cochlear implant (CI) users and to explore the association between the activity and their speech understanding ability. Using fNIRS, 15 experienced CI users and 14 normal-hearing participants were imaged while presented with either visual speech or auditory speech. Brain activation was measured from the prefrontal, temporal, and parietal lobe in both hemispheres, including the language-associated regions. In response to visual speech, the activation levels of CI users in an a priori region of interest (ROI)—the left superior temporal gyrus or sulcus—were negatively correlated with auditory speech understanding. This result suggests that increased cross-modal activity in the auditory cortex is predictive of poor auditory speech understanding. In another two ROIs, in which CI users showed significantly different mean activation levels in response to auditory speech compared with normal-hearing listeners, activation levels were significantly negatively correlated with CI users’ auditory speech understanding. These ROIs were located in the right anterior temporal lobe (including a portion of prefrontal lobe) and the left middle superior temporal lobe. In conclusion, fNIRS successfully revealed activation patterns in CI users associated with their auditory speech understanding.

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The Kullback information criterion KIC is a recently developed tool for statistical model selection [I]. KIC serves as an asymptotically unbiased estimator of a variant of the Kullback symmetric divergence, known also as J-divergence. In this paper a bias correction of the Kullback symmetric information criterion is derived for linear models. The conection is of particular use when the sample size is small or when the number of fitted parameters is of moderate to large fraction of the sample sire. For linear regression models, the corrected method called KICc is an exucrly unhiased estimator of a variant of the Kullback symmetric divergence between the true unknown model and the candidate fitted model. Furthermore KICc is found to provide better model order choice than any other asymptotically efficient methods in an application to autoregressive time series models. measures, it functions as a gauge of model disparity, which is arguably more sensitive than either of its individual component. Following the above reasoning, Cavanaugh [ I ] proposed the Kullback information criterion KIC as an asymptotically unbiased estimate of a variant (within a constant) of the J-divergence between the true unknown model and the fitted approximating model. Motivated by the above developments, we propose a bias corrected version of the KIC for linear regression models. The new criterion is shown to outperform classical criteria in a small sample autoregressive modeling. The remainder of this paper is organized as follows. In section 2 we present a short overview of Kullback's directed divergence, AIC, its corrected version AlCc and KIC. In section 3 we introduce the bias corrected version of KIC. Section 4 presents simulation results for autoregressive model selection. We end up by concluding remarks.

In many machine learning and data related applications, it is required to have the knowledge of approximate ranks of large data matrices at hand. In this paper, we present two computationally inexpensive techniques to estimate the approximate ranks of such large matrices. These techniques exploit approximate spectral densities, popular in physics, which are probability density distributions that measure the likelihood of finding eigenvalues of the matrix at a given point on the real line. Integrating the spectral density over an interval gives the eigenvalue count of the matrix in that interval. Therefore the rank can be approximated by integrating the spectral density over a carefully selected interval. Two different approaches are discussed to estimate the approximate rank, one based on Chebyshev polynomials and the other based on the Lanczos algorithm. In order to obtain the appropriate interval, it is necessary to locate a gap between the eigenvalues that correspond to noise and the relevant eigenvalues that contribute to the matrix rank. A method for locating this gap and selecting the interval of integration is proposed based on the plot of the spectral density. Numerical experiments illustrate the performance of these techniques on matrices from typical applications.