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Asymptotic performance of PCA for high-dimensional heteroscedastic data.

ABSTRACT: Principal Component Analysis (PCA) is a classical method for reducing the dimensionality of data by projecting them onto a subspace that captures most of their variation. Effective use of PCA in modern applications requires understanding its performance for data that are both high-dimensional and heteroscedastic. This paper analyzes the statistical performance of PCA in this setting, i.e., for high-dimensional data drawn from a low-dimensional subspace and degraded by heteroscedastic noise. We provide simplified expressions for the asymptotic PCA recovery of the underlying subspace, subspace amplitudes and subspace coefficients; the expressions enable both easy and efficient calculation and reasoning about the performance of PCA. We exploit the structure of these expressions to show that, for a fixed average noise variance, the asymptotic recovery of PCA for heteroscedastic data is always worse than that for homoscedastic data (i.e., for noise variances that are equal across samples). Hence, while average noise variance is often a practically convenient measure for the overall quality of data, it gives an overly optimistic estimate of the performance of PCA for heteroscedastic data.

SUBMITTER: Hong D

PROVIDER: S-EPMC6377200 | biostudies-literature | 2018 Sep

REPOSITORIES: biostudies-literature

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Asymptotic performance of PCA for high-dimensional heteroscedastic data.

Hong David D Balzano Laura L Fessler Jeffrey A JA

Journal of multivariate analysis 20180619

Principal Component Analysis (PCA) is a classical method for reducing the dimensionality of data by projecting them onto a subspace that captures most of their variation. Effective use of PCA in modern applications requires understanding its performance for data that are both high-dimensional and heteroscedastic. This paper analyzes the statistical performance of PCA in this setting, i.e., for high-dimensional data drawn from a low-dimensional subspace and degraded by heteroscedastic noise. We p ...[more]

PMID: 30778267

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Project description:Objective/methodsNeuroimaging research has predominantly focused on exploring how cortical or subcortical brain abnormalities are related to language dysfunction in patients with neurological disease through the use of single modality imaging. Still, limited knowledge exists on how various MRI measures relate to each other and to patients' language performance. In this study, we explored the relationship between measures of regional cortical thickness, gray-white matter contrast (GWMC), white matter diffusivity [mean diffusivity (MD) and fractional anisotropy (FA)] and the relative contributions of these MRI measures to predicting language function across patients with temporal lobe epilepsy (TLE) and healthy controls. T1- and diffusion-weighted MRI data were collected from 56 healthy controls and 52 patients with TLE. By focusing on frontotemporal regions implicated in language function, we reduced each domain of MRI data to its principal component (PC) and quantified the correlations among these PCs and the ability of these PCs to explain the variation in vocabulary, naming and fluency. We followed up our significant findings by assessing the predictive power of the implicated PCs with respect to language impairment in our sample.ResultsWe found significant positive associations between PCs representing cortical thickness, GWMC and FA that appeared to be partially mediated by changes in total brain volume. We also found a significant association between reduced FA and increased MD after controlling for confounding factors (e.g., age, field strength, total brain volume). Reduced FA was significantly associated with reductions in visual naming while increased MD was associated with reductions in auditory naming scores, even after controlling for the variability explained by reductions in hippocampal volumes. Inclusion of FA and MD PCs in predictive models of language impairment resulted in significant improvements in sensitivity and specificity of the predictions.ConclusionsQuantitative MRI measures from T1 and diffusion-weighted scans are unlikely to represent perfectly orthogonal vectors of disease in individuals with epilepsy. On the contrary, they exhibit highly intercorrelated PCs in their factor structures, which is consistent with an underlying pathological process that affects both the cortical and the subcortical structures simultaneously. In addition to hippocampal volume, the PCs of diffusion weighted measures (FA and MD) increase the sensitivity and specificity for determining naming impairment in patients with TLE. These findings underline the importance of combining multimodal imaging measures to better predict language performance in TLE that could extend to other patients with prominent language impairments.

Dataset Information

Asymptotic performance of PCA for high-dimensional heteroscedastic data.

Publications

Asymptotic performance of PCA for high-dimensional heteroscedastic data.

Similar Datasets

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