Determining the number of factors in high-dimensional generalized latent factor models

As a generalization of the classical linear factor model, generalized latent factor models are useful for analysing multivariate data of different types, including binary choices and counts. This paper proposes an information criterion to determine the number of factors in generalized latent factor models. The consistency of the proposed information criterion is established under a high-dimensional setting, where both the sample size and the number of manifest variables grow to infinity, and data may have many missing values. An error bound is established for the parameter estimates, which plays an important role in establishing the consistency of the proposed information criterion. This error bound improves several existing results and may be of independent theoretical interest. We evaluate the proposed method by a simulation study and an application to Eysenck’s personality questionnaire.

The Truth behind the Zeros: A New Approach to Principal Component Analysis of the Neuropsychiatric Inventory

Psychiatric syndromes in dementia are often derived from the Neuropsychiatric Inventory (NPI) using principal component analysis (PCA). The validity of this statistical approach can be questioned, since the excessive proportion of zeros and skewness of NPI items may distort the estimated relations between the items. We propose a novel version of PCA, ZIBP-PCA, where a zero-inflated bivariate Poisson (ZIBP) distribution models the pairwise covariance between the NPI items. We compared the performance of the method to classical PCA under zero-inflation using simulations, and in two dementia-cohorts (N = 830, N = 1349). Simulations showed that component loadings from PCA were biased due to zero-inflation, while the loadings of ZIBP-PCA remained unaffected. ZIBP-PCA obtained a simpler component structure of "psychosis," "mood" and "agitation" in both dementia-cohorts, compared to PCA. The principal components from ZIBP-PCA had component loadings as follows: First, the component interpreted as "psychosis" was loaded by the items delusions and hallucinations. Second, the "mood" component was loaded by depression and anxiety. Finally, the "agitation" component was loaded by irritability and aggression. In conclusion, PCA is not equipped to handle zero-inflation. Using the NPI, PCA fails to identify components with a valid interpretation, while ZIBP-PCA estimates simple and interpretable components to characterize the psychopathology of dementia.

Steerable ePCA: Rotationally Invariant Exponential Family PCA

In photon-limited imaging, the pixel intensities are affected by photon count noise. Many applications require an accurate estimation of the covariance of the underlying 2-D clean images. For example, in X-ray free electron laser (XFEL) single molecule imaging, the covariance matrix of 2-D diffraction images is used to reconstruct the 3-D molecular structure. Accurate estimation of the covariance from low-photon-count images must take into account that pixel intensities are Poisson distributed, hence the classical sample covariance estimator is highly biased. Moreover, in single molecule imaging, including in-plane rotated copies of all images could further improve the accuracy of covariance estimation. In this paper we introduce an efficient and accurate algorithm for covariance matrix estimation of count noise 2-D images, including their uniform planar rotations and possibly reflections. Our procedure, steerable ePCA, combines in a novel way two recently introduced innovations. The first is a methodology for principal component analysis (PCA) for Poisson distributions, and more generally, exponential family distributions, called ePCA. The second is steerable PCA, a fast and accurate procedure for including all planar rotations when performing PCA. The resulting principal components are invariant to the rotation and reflection of the input images. We demonstrate the efficiency and accuracy of steerable ePCA in numerical experiments involving simulated XFEL datasets and rotated face images from Yale Face Database B.