349 research outputs found
Dynamic Linear Discriminant Analysis in High Dimensional Space
High-dimensional data that evolve dynamically feature predominantly in the
modern data era. As a partial response to this, recent years have seen
increasing emphasis to address the dimensionality challenge. However, the
non-static nature of these datasets is largely ignored. This paper addresses
both challenges by proposing a novel yet simple dynamic linear programming
discriminant (DLPD) rule for binary classification. Different from the usual
static linear discriminant analysis, the new method is able to capture the
changing distributions of the underlying populations by modeling their means
and covariances as smooth functions of covariates of interest. Under an
approximate sparse condition, we show that the conditional misclassification
rate of the DLPD rule converges to the Bayes risk in probability uniformly over
the range of the variables used for modeling the dynamics, when the
dimensionality is allowed to grow exponentially with the sample size. The
minimax lower bound of the estimation of the Bayes risk is also established,
implying that the misclassification rate of our proposed rule is minimax-rate
optimal. The promising performance of the DLPD rule is illustrated via
extensive simulation studies and the analysis of a breast cancer dataset.Comment: 34 pages; 3 figure
Bootstrapping frequency domain tests in multivariate time series with an application to comparing spectral densities
We propose a general bootstrap procedure to approximate the null distribution of nonparametric frequency domain tests about the spectral density matrix of a multivariate time series. Under a set of easy to verify conditions, we establish asymptotic validity of the proposed bootstrap procedure. We apply a version of this procedure together with a new statistic in order to test the hypothesis that the spectral densities of not necessarily independent time series are equal. The test statistic proposed is based on a L2-distance between the nonparametrically estimated individual spectral densities and an overall, 'pooled' spectral density, the later being obtained using the whole set of m time series considered. The effects of the dependence between the time series on the power behavior of the test are investigated. Some simulations are presented and a real-life data example is discussed. --
Variable-Dependent Partial Dimension Reduction
Sufficient dimension reduction reduces the dimension of a regression model without loss of information by replacing the original predictor with its lower-dimensional linear combinations. Partial (sufficient) dimension reduction arises when the predictors naturally fall into two sets X and W, and pursues a partial dimension reduction of X. Though partial dimension reduction is a very general problem, only very few research results are available when W is continuous. To the best of our knowledge, none can deal with the situation where the reduced lower-dimensional subspace of X varies with W. To address such issue, we in this paper propose a novel variable-dependent partial dimension reduction framework and adapt classical sufficient dimension reduction methods into this general paradigm. The asymptotic consistency of our method is investigated. Extensive numerical studies and real data analysis show that our variable-dependent partial dimension reduction method has superior performance compared to the existing methods
Variable-Dependent Partial Dimension Reduction
Sufficient dimension reduction reduces the dimension of a regression model without loss of information by replacing the original predictor with its lower-dimensional linear combinations. Partial (sufficient) dimension reduction arises when the predictors naturally fall into two sets X and W, and pursues a partial dimension reduction of X. Though partial dimension reduction is a very general problem, only very few research results are available when W is continuous. To the best of our knowledge, none can deal with the situation where the reduced lower-dimensional subspace of X varies with W. To address such issue, we in this paper propose a novel variable-dependent partial dimension reduction framework and adapt classical sufficient dimension reduction methods into this general paradigm. The asymptotic consistency of our method is investigated. Extensive numerical studies and real data analysis show that our variable-dependent partial dimension reduction method has superior performance compared to the existing methods
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