Asymptotic and Bootstrap Tests for the Dimension of the Non-Gaussian Subspace

Nordhausen, Klaus; Oja, Hannu; Tyler, David E.; Virta, Joni

doi:10.1109/lsp.2017.2696880

Cited by 38 publications

(71 citation statements)

References 35 publications

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“…The development very closely mimics that of Nordhausen et al (). By Eaton and Tyler (), we have under H 0 k that

n \cdot s_{k}^{2} false(\overset{Σ}{true} false(x_{i} false) false) = n \cdot s^{2} false({\overset{Σ}{true}}_{k} false(x_{i} false) false) + O_{p} false(n^{- 1 false/ 2} false)

, where

s^{2} false({\overset{Σ}{true}}_{k} false(x_{i} false) false)

is the variance of the eigenvalues of

{\overset{Σ}{true}}_{k} false(x_{i} false)

, the lower right r × r block of

\overset{Σ}{true} false(x_{i} false)

.…”

Section: Asymptotic Null Distributionsupporting

confidence: 80%

“…A similar analysis under a noiseless elliptical model was conducted in Nordhausen, Oja, and Tyler (), and the current work improves upon their results by discarding both of the aforementioned assumptions, noiselessness and ellipticity (although; Nordhausen et al, , experimented also with a noisy model using simulations). Similar studies include also Luo and Li () and Nordhausen, Oja, Tyler, and Virta () which provide tools to test for the number of non‐Gaussian components in an ICA model with internal noise. These techniques were extended in Matilainen, Nordhausen, and Virta (), Nordhausen and Virta (), and Virta and Nordhausen () to the second‐order source separation model to test for white noise.…”

Section: Introductionsupporting

confidence: 74%

“…This block equals

\overset{Σ}{true} false(𝛆_{k i} false)

, where ε ki contains the last r components of the noise vectors

𝛆_{i}^{*}

. By the central limit theorem,

\sqrt{n} \cdot vec false(\overset{Σ}{true} false(𝛆_{k i} false) - σ^{2} I_{r} false)

has a limiting multivariate normal distribution, and following the proof of Nordhausen et al, (Theorem 1), we have under H 0 k that

\frac{n r \cdot s_{k}^{2} (\hat{boldΣ} ({boldx}_{i}))}{2 σ^{4} β} ⇝ χ_{\frac{1}{2} (r - 1) (r + 2)}^{2}

for a specific constant β . The limiting distribution stays the same if σ 2 and β are replaced by their consistent estimates.…”

Section: Asymptotic Null Distributionmentioning

confidence: 72%

“…For σ 2 , such an estimate is given by the mean of the r smallest eigenvalues of

\overset{Σ}{true} false(x_{i} false)

. As in Nordhausen et al (), an estimate for β is given by

\hat{β} = \frac{1}{r (r + 2)} \frac{1}{n} \sum_{i = 1}^{n} {[{({\hat{𝛆}}_{k i} - {\bar{𝛆}}_{k})}^{⊤} \hat{boldΣ} {({\hat{𝛆}}_{k i})}^{- 1} ({\hat{𝛆}}_{k i} - {\bar{𝛆}}_{k})]}^{2},

where

{\overset{𝛆}{true}}_{k i} = \overset{R}{true}^{⊤} x_{i}

and

\overset{R}{true}

contains the last r eigenvectors of

\overset{Σ}{true} false(x_{i} false)

. If the assumption of normally distributed noise ε is made, then the population value β = 1 can be substituted instead.…”

Section: Asymptotic Null Distributionmentioning

confidence: 99%

“…As in Nordhausen et al () and Virta and Nordhausen (), asymptotic tests for the different null hypotheses can be combined to yield an estimator for the signal dimension d . That is, for all k = 0,…, p − 1, let c k , n be a deterministic sequence such that c k , n → ∞ and c k , n = o ( n ).…”

Section: Asymptotic Null Distributionmentioning

confidence: 99%

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Estimating the number of signals using principal component analysis

Virta

Nordhausen²

2019

Stat

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In this work, we develop inferential tools for determining the correct number of principal components under a general noisy latent variable model, which includes as a special case, for example, the noisy independent component model. The problem is approached using hypothesis testing, and we provide both a large‐sample test and several resampling‐based alternatives. Simulations and an application to sound data reveal that both types of approaches keep the desired levels and have good power.

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“…The development very closely mimics that of Nordhausen et al (). By Eaton and Tyler (), we have under H 0 k that

n \cdot s_{k}^{2} false(\overset{Σ}{true} false(x_{i} false) false) = n \cdot s^{2} false({\overset{Σ}{true}}_{k} false(x_{i} false) false) + O_{p} false(n^{- 1 false/ 2} false)

, where

s^{2} false({\overset{Σ}{true}}_{k} false(x_{i} false) false)

is the variance of the eigenvalues of

{\overset{Σ}{true}}_{k} false(x_{i} false)

, the lower right r × r block of

\overset{Σ}{true} false(x_{i} false)

.…”

Section: Asymptotic Null Distributionsupporting

confidence: 80%

Section: Introductionsupporting

confidence: 74%

“…This block equals

\overset{Σ}{true} false(𝛆_{k i} false)

, where ε ki contains the last r components of the noise vectors

𝛆_{i}^{*}

. By the central limit theorem,

\sqrt{n} \cdot vec false(\overset{Σ}{true} false(𝛆_{k i} false) - σ^{2} I_{r} false)

has a limiting multivariate normal distribution, and following the proof of Nordhausen et al, (Theorem 1), we have under H 0 k that

\frac{n r \cdot s_{k}^{2} (\hat{boldΣ} ({boldx}_{i}))}{2 σ^{4} β} ⇝ χ_{\frac{1}{2} (r - 1) (r + 2)}^{2}

for a specific constant β . The limiting distribution stays the same if σ 2 and β are replaced by their consistent estimates.…”

Section: Asymptotic Null Distributionmentioning

confidence: 72%

“…For σ 2 , such an estimate is given by the mean of the r smallest eigenvalues of

\overset{Σ}{true} false(x_{i} false)

. As in Nordhausen et al (), an estimate for β is given by

\hat{β} = \frac{1}{r (r + 2)} \frac{1}{n} \sum_{i = 1}^{n} {[{({\hat{𝛆}}_{k i} - {\bar{𝛆}}_{k})}^{⊤} \hat{boldΣ} {({\hat{𝛆}}_{k i})}^{- 1} ({\hat{𝛆}}_{k i} - {\bar{𝛆}}_{k})]}^{2},

where

{\overset{𝛆}{true}}_{k i} = \overset{R}{true}^{⊤} x_{i}

and

\overset{R}{true}

contains the last r eigenvectors of

\overset{Σ}{true} false(x_{i} false)

. If the assumption of normally distributed noise ε is made, then the population value β = 1 can be substituted instead.…”

Section: Asymptotic Null Distributionmentioning

confidence: 99%

Section: Asymptotic Null Distributionmentioning

confidence: 99%

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Estimating the number of signals using principal component analysis

Virta

Nordhausen²

2019

Stat

Self Cite

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Optimization and testing in linear non‐Gaussian component analysis

Jin

Risk

Matteson

2019

Statistical Analysis

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Independent component analysis (ICA) decomposes multivariate data into mutually independent components (ICs). The ICA model is subject to a constraint that at most one of these components is Gaussian, which is required for model identifiability. Linear non-Gaussian component analysis (LNGCA) generalizes the ICA model to a linear latent factor model with any number of both non-Gaussian components (signals) and Gaussian components (noise), where observations are linear combinations of independent components. Although the individual Gaussian components are not identifiable, the Gaussian subspace is identifiable. We introduce an estimator along with its optimization approach in which non-Gaussian and Gaussian components are estimated simultaneously, maximizing the discrepancy of each non-Gaussian component from Gaussianity while minimizing the discrepancy of each Gaussian component from Gaussianity. When the number of non-Gaussian components is unknown, we develop a statistical test to determine it based on resampling and the discrepancy of estimated components. Through a variety of simulation studies, we demonstrate the improvements of our estimator over competing estimators, and we illustrate the effectiveness of our test to determine the number of non-Gaussian components. Further, we apply our method to real data examples and show its practical value. KEYWORDSdimension reduction, hypothesis testing, independent component analysis, multivariate analysis, projection pursuit, subspace estimation 1 −1∕2 Y be an uncorrelating matrix. Let Z = HY = (Z 1 , … , Z p ) T ∈ R p be a random vector of uncorrelated observations, such that Σ Z = Cov(Z) = I p , the p × p identity matrix. The ICA model further assumes that the components X 1 , … , X p are mutually independent, in which the number of Gaussian components is at most one. Then the Stat Anal Data Min: The ASA Data Sci Journal. 2019;12:141-156 wileyonlinelibrary.com/sam

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Independent component analysis: A statistical perspective

Nordhausen¹,

Oja

2018

WIREs Computational Stats

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1. z $ N p (0, I p ) ) multivariate normal (Gaussian) model. 2. z spherical, that is, Uz $ z for all orthogonal U ) elliptical model. 3. z has independent components ) independent component model. Both the elliptical model and IC model are extensions of the Gaussian model but in diverse directions. In the case of the Gaussian and elliptical model, A is identifiable only up to a postmultiplication by an orthogonal matrix, and the inference is only for parameters E(x) = b and Cov(x) = AA 0. Classical multivariate statistical inference methods, such as Hotelling's T 2 , multivariate analysis of variance, multivariate regression, and PCA, rely on the assumption of multivariate normality. Gaussianity and ellipticity assumptions mean that no hidden structures or latent variables are allowed in the data analysis.The IC model is a so called BSS model with latent variables in z and latent structures caused by different marginal distributions of the components of z. In the case of the IC model one tries to estimate an unmixing matrix to go back to the non-2 of 23 NORDHAUSEN AND OJA

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Asymptotic and Bootstrap Tests for the Dimension of the Non-Gaussian Subspace

Cited by 38 publications

References 35 publications

Estimating the number of signals using principal component analysis

Estimating the number of signals using principal component analysis

Optimization and testing in linear non‐Gaussian component analysis

Independent component analysis: A statistical perspective

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