Statistical Pattern Recognition - A Review

# The choice of a criterion function we also need to

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the choice of a criterion function, we also need to determine the appropriate dimensionality of the reduced feature space. The answer to this question is embedded in the notion of the intrinsic dimensionality of data. Intrinsic dimensionality essentially determines whether the given d -dimensional patterns can be described adequately in a subspace of dimensionality less than d . For example, d -dimensional patterns along a reasonably smooth curve have an intrinsic dimensionality of one, irrespective of the value of d . Note that the intrinsic dimensionality is not the same as the linear dimensionality which is a global property of the data involving the number of significant eigenvalues of the covariance matrix of the data. While several algorithms are available to estimate the intrinsic dimension- ality [81], they do not indicate how a subspace of the identified dimensionality can be easily identified. We now briefly discuss some of the commonly used methods for feature extraction and feature selection. 4.1 Feature Extraction Feature extraction methods determine an appropriate sub- space of dimensionality m (either in a linear or a nonlinear way) in the original feature space of dimensionality d ( m ´ d ). Linear transforms, such as principal component analysis, factor analysis, linear discriminant analysis, and projection pursuit have been widely used in pattern recognition for feature extraction and dimensionality reduction. The best known linear feature extractor is the principal component analysis (PCA) or Karhunen-Loe `ve expansion, that computes the m largest eigenvectors of the d ² d covariance matrix of the n d -dimensional patterns. The linear transformation is defined as Y XH; 7 where X is the given n ² d pattern matrix, Y is the derived n ² m pattern matrix, and H is the d ² m matrix of linear transformation whose columns are the eigenvectors. Since PCA uses the most expressive features (eigenvectors with the largest eigenvalues), it effectively approximates the data by a linear subspace using the mean squared error criterion. Other methods, like projection pursuit [53] and independent component analysis (ICA) [31], [11], [24], [96] are more appropriate for non-Gaussian distributions since they do not rely on the second-order property of the data. ICA has been successfully used for blind-source separation [78]; extracting linear feature combinations that define independent sources. This demixing is possible if at most one of the sources has a Gaussian distribution. Whereas PCA is an unsupervised linear feature extrac- tion method, discriminant analysis uses the category information associated with each pattern for (linearly) extracting the most discriminatory features. In discriminant analysis, interclass separation is emphasized by replacing the total covariance matrix in PCA by a general separability measure like the Fisher criterion, which results in finding the eigenvectors of S ³ 1 w S b (the product of the inverse of the within-class scatter matrix, S w , and the between-class 12

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