Quadratic discriminant functions with constraints on the covariance matrices: Some asymptotic results

Flury, Bernhard; Schmid, Martin J.

doi:10.1016/0047-259x(92)90025-b

Cited by 11 publications

(19 citation statements)

References 7 publications

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“…While significantly slower than the median time of 0.0001 seconds to compute the unbiased covariance matrix estimators, less than a second is added to computation time by the use of CPC discrimination rather than ordinary QDA. Flury and Schmid (1992) have shown that the asymptotic variances of the discriminant function coefficients are the same for ordinary QDA and CPC discrimination for k = 2 groups when the CPC model holds. In particular, if λ 1j − λ 1h = λ 2h − λ 2j for all (j, h) pairs of the eigenvalues from two population covariance matrices, where λ ij indicates the j th eigenvalue of the i th population covariance matrix, CPC discrimination and ordinary QDA should perform about equally well.…”

Section: Discriminant Analysis Under the Cpc Modelmentioning

confidence: 93%

“…This hypothesis have been investigated by Schmid (1987), Flury (1988), Flury and Schmid (1992), Flury et al (1994) and Bianco et al (2008). For CPC discrimination, the unbiased covariance matrix estimators in (5) and (6) are simply replaced by covariance matrix estimators under the CPC model.…”

Section: Discriminant Analysis Under the Cpc Modelmentioning

confidence: 99%

“…They suggested that both the proportional and CPC models can perform well in classification applications where the number of variables and/or number of groups are large relative to the sample sizes, and that these models offer a compromise between the assumption of equal covariance matrices on the one extreme and the assumption of unrelated covariance matrices on the other. For sparse data, theoretically incorrect but more parsimonious models can outperform theoretically correct models in discriminant analysis applications, because the discriminant func-2 tion coefficients for a more parsimonious model will generally be less variable (Flury and Schmid, 1992). Flury et al (1994) followed up these investigations with a simulation study to determine the misclassification error rates for the different covariance matrix estimators when plugged into the quadratic discriminant function.…”

Section: Introductionmentioning

confidence: 99%

“…Flury and Schmid (1992) derived asymptotic results for discrimination coefficients under homogeneous, proportional, CPC and unrelated covariance matrix models, respectively. They showed that, although CPC discrimination can improve on the misclassification rate of ordinary quadratic discrimination in some situations, such improvement is usually not substantial.…”

Section: Introductionmentioning

confidence: 99%

“…This method is briefly discussed in Section 3, pointing out a relation to CPC discrimination based on the regularised CPC covariance matrix estimator proposed by Pepler et al (2015). Bianco et al (2008) used partial influence functions to derive the asymptotic variances of the discriminant function coefficients under the same models considered by Flury and Schmid (1992),…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Discriminant analysis under the common principal components model

Pepler

Uys

Nel

2017

Communications in Statistics - Simulation and Computation

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For two or more populations of which the covariance matrices have a common set of eigenvectors, but different sets of eigenvalues, the common principal components (CPC) model is appropriate. Pepler et al. (2015) proposed a regularised CPC covariance matrix estimator and showed that this estimator outperforms the unbiased and pooled estimators in situations where the CPC model is applicable. This paper extends their work to the context of discriminant analysis for two groups, by plugging the regularised CPC estimator into the ordinary quadratic discriminant function.Monte Carlo simulation results show that CPC discriminant analysis offers significant improvements in misclassification error rates in certain situations, and at worst performs similar to ordinary quadratic and linear discriminant analysis. Based on these results, CPC discriminant analysis is recommended for situations where the sample size is small compared to the number of variables, in particular for cases where there is uncertainty about the population covariance matrix structures.

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Section: Discriminant Analysis Under the Cpc Modelmentioning

confidence: 93%

Section: Discriminant Analysis Under the Cpc Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Discriminant analysis under the common principal components model

Pepler

Uys

Nel

2017

Communications in Statistics - Simulation and Computation

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A discriminant analysis extension to mixed models

Tomasko¹,

Helms

Snapinn³

1999

Statist. Med.

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Discriminant analysis is commonly used to classify an observation into one of two (or more) populations on the basis of correlated measurements. Classical discriminant analysis approaches require complete data for all observations. Our extension enables the use of all available longitudinal data, regardless of completeness. Traditionally a linear discriminant function assumes a common unstructured covariance matrix for both populations, which may be taken from a multivariate model. Here, we can model the correlated measurements and use a structured covariance in the discriminant function. We illustrate cases in which the estimated covariance structure is either compound symmetric, heterogeneous compound symmetric or heterogeneous autoregressive. Thus a structured covariance is incorporated into the discrimination process in contrast to standard discriminant analysis methodology. Simulations are performed to obtain a true measure of the effect of structure on the error rate. In addition, the usual multivariate expected value structure is altered. The impact on the discrimination process is contrasted when using the multivariate and random-effects covariance structures and expected values. The random-effects covariance structure leads to an improvement in the error rate in small samples. To illustrate the procedure we consider repeated measurements data from a clinical trial comparing two active treatments; the goal is to determine if the treatment could be unblinded based on repeated anxiety score measurements.

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Common Principal Components for Dependent Random Vectors

Neuenschwander

Flury

2000

Journal of Multivariate Analysis

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Let the kp-variate random vector X be partitioned into k subvectors X i of dimension p each, and let the covariance matrix 9 of X be partitioned analogously into submatrices 9 ij . The common principal component (CPC) model for dependent random vectors assumes the existence of an orthogonal p by p matrix ; such that ; t 9 ij ; is diagonal for all (i, j ). After a formal definition of the model, normal theory maximum likelihood estimators are obtained. The asymptotic theory for the estimated orthogonal matrix is derived by a new technique of choosing proper subsets of functionally independent parameters. Academic PressAMS 1991 subject classifications: 62H25, 62F10, 62F12.

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Quadratic discriminant functions with constraints on the covariance matrices: Some asymptotic results

Cited by 11 publications

References 7 publications

Discriminant analysis under the common principal components model

Discriminant analysis under the common principal components model

A discriminant analysis extension to mixed models

Common Principal Components for Dependent Random Vectors

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