28,826 research outputs found
ON THE CONSISTENCY AND ROBUSTNESS PROPERTIES OF LINEAR DISCRIMINANT ANALYSIS
Strong consistency of linear discriminant analysis is established under wide assumptions on the class conditional densities. Robustness to the presence of a mild degree of class dispersion heterogeneity is also analyzed. Results obtained may help to explain analytically the frequent good behavior in applications of linear discrimination techniques.
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
Random projections as regularizers: learning a linear discriminant from fewer observations than dimensions
We prove theoretical guarantees for an averaging-ensemble of randomly projected Fisher linear discriminant classifiers, focusing on the casewhen there are fewer training observations than data dimensions. The specific form and simplicity of this ensemble permits a direct and much more detailed analysis than existing generic tools in previous works. In particular, we are able to derive the exact form of the generalization error of our ensemble, conditional on the training set, and based on this we give theoretical guarantees which directly link the performance of the ensemble to that of the corresponding linear discriminant learned in the full data space. To the best of our knowledge these are the first theoretical results to prove such an explicit link for any classifier and classifier ensemble pair. Furthermore we show that the randomly projected ensemble is equivalent to implementing a sophisticated regularization scheme to the linear discriminant learned in the original data space and this prevents overfitting in conditions of small sample size where pseudo-inverse FLD learned in the data space is provably poor. Our ensemble is learned from a set of randomly projected representations of the original high dimensional data and therefore for this approach data can be collected, stored and processed in such a compressed form. We confirm our theoretical findings with experiments, and demonstrate the utility of our approach on several datasets from the bioinformatics domain and one very high dimensional dataset from the drug discovery domain, both settings in which fewer observations than dimensions are the norm
Discriminant analysis involving count data
A situation giving rise to a violation of the normality assumption in discriminant analysis is that which involves count
observations. For a two-variable case involving count observations, this paper presents a new discriminant analysis approach when
one variable is observed conditional on the other. Two cases involving Poisson-Binomial and Poisson-Poisson distributions were
considered. The derived allocation rules are based on the resulting joint distribution of the two count variables. Applicability of the
suggested allocation rules in discriminant analysis involving count data and its performance in comparison with Fisher linear
discriminant rule was studied under different conditions. Results obtained show promising applicability of the suggested allocation
rules when compared with the Fisher linear discriminant method
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