PCA consistency in high dimension, low sample size context

Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the asymptotic behavior of the Principal Component (PC) directions. HDLSS asymptotics are used to study consistency, strong inconsistency and subspace consistency. We show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others, then the corresponding estimated PC directions are consistent or converge to the appropriate subspace (subspace consistency) and most other PC directions are strongly inconsistent. Broad sets of sufficient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations. In preparation for these results, we show that the geometric representation of HDLSS data holds under general conditions, which includes a $\rho$-mixing condition and a broad range of sphericity measures of the covariance matrix.Comment: Published in at http://dx.doi.org/10.1214/09-AOS709 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Principal arc analysis on direct product manifolds

We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS370 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Third-order Intermodulation Reduction in Mobile Power Amplifiers by the First Stage Bias Control

In this paper, the third order intermodulation distortion (IMD3) of three-stage power amplifier (PA) is analyzed using the Volterra series. The analysis explains how the total IMD3 of the three-stage power amplifier can be reduced by the first-stage bias condition. The three-stage PA, which is fabricated using InGaP/GaAs hetero-junction bipolar transistor (HBT), operates with an optimized first driver stage bias for higher P1dB and good gain flatness. The power amplifier has been designed for 1626.5 MHz~1660.5 MHz satellite mobile communications. With π/4 DQPSK modulation signals, this PA can deliver a highly linear output power of 33 dBm from 3.6V supply voltage. At 33 dBm output power, it shows a gain of 31.9 dB, a power-added efficiency (PAE) of 39.8%, an adjacent channel power ratio (ACPR) of -28.2 dBc at a 31.25 KHz offset frequency

Measurements of Pressure Distributions and Force Coefficients in a Squeeze Film Damper. Part 2: Partially Sealed Configuration

Experimental results from a partially sealed squeeze film damper (SFD) test rig, executing a circular centered orbit are presented and discussed. A serrated piston ring is installed at the damper exit. This device involves a new sealing concept which produces high damping values while allowing for oil flow to cool the damper. In the partially sealed damper, large cavitation regions are observed in the pressure fields at orbit radii epsilon equals 0.5 and epsilon equals 0.8. The cavitated pressure distributions and the corresponding force coefficients are compared with a cavitated bearing solution. The experimental results show the significance of fluid inertia and vapor cavitation in the operation of squeeze film dampers. Squeeze film Reynolds numbers tested reach up to Re equals 50, spanning the range of contemporary applications

Measurements of Pressure Distributions and Force Coefficients in a Squeeze Film Damper. Part 1: Fully Open Ended Configuration

Measurements of pressure distributions and force coefficients were carried out in two types of squeeze film dampers, executing a circular centered orbit, an open-ended configuration, and a partially sealed one, in order to investigate the effect of fluid inertia and cavitation on pressure distributions and force coefficients. Dynamic pressure measurements were carried out for two orbit radii, epsilon 0.5 and 0.8. It was found that the partially sealed configuration was less influenced by fluid inertia than the open ended configuration

Gas Dynamics of the Nickel-56 Decay Heating in Pair-Instability Supernovae

Very massive 140-260 Msun stars can die as highly-energetic pair-instability supernovae (PI SNe) with energies of up to 100 times those of core-collapse SNe that can completely destroy the star, leaving no compact remnant behind. These explosions can synthesize $0.1-30$ Msun of radioactive Ni56, which can cause them to rebrighten at later times when photons due to Ni56 decay diffuse out of the ejecta. However, heat from the decay of such large masses of Ni56 could also drive important dynamical effects deep in the ejecta that are capable of mixing elements and affecting the observational signatures of these events. We have now investigated the dynamical effect of Ni56 heating on PI SN ejecta with high-resolution two-dimensional hydrodynamic simulations performed with the CASTRO code. We find that expansion of the hot Ni56 bubble forms a shell at the base of the silicon layer of the ejecta about 200 days after the explosion but that no hydrodynamical instabilities develop that would mix Ni56 with the Si/O-rich ejecta. However, while the dynamical effects of Ni56 heating may be weak they could affect the observational signatures of some PI SNe by diverting decay energy into internal expansion of the ejecta at the expense of rebrightening at later times.Comment: Accepted to ApJ, 14 page

Some Grüss' Type Inequalities in 2-Inner Product Spaces and Applications for Determinantal Integral Inequalities

Some new Grüss type inequalities in 2-inner product spaces are given. Using this framework, some determinantal integral inequalities for synchronous functions are also derived

Radiation Transport Simulations of Pulsational Pair-Instability Supernovae

Massive stars of helium cores of 35-65 Msun eventually encounter the electron/positron creation instability, and it triggers explosive carbon or oxygen burning that produces several thermonuclear eruptions. The resulting catastrophe collisions of eruptive shells sometimes produce luminous transients with peak luminosity of $10^{43} - 10^{44}$ erg/sec, known as pulsational pair-instability supernovae (PPISNe). Previous 2D simulations of colliding shells show the development of Rayleigh-Taylor (RT) instabilities and mixing. Here we present radiation hydrodynamic PPISNe simulations of a 110 Msun solar-metallicity star that was promising to produce a superluminous transit in the early work. Our comprehensive study contains a suite of one-, two-, and three-dimensional models. We discuss the impact of dimensionality and fluid instabilities on the resulting light curves. The results show the RT mixing found in previous multidimensional hydro studies transforms into a thin and distorted shell due to radiative cooling. Radiation from the wiggly shell peaks at its bolometric light curve of $\sim 2\times10^{43}$ erg/sec, lasting about 150 days and following with a plateau of $\sim 3\times10^{42}$ erg/sec for another two hundred days before it fades away. The total radiation energy emitted from colliding shells is $\sim 1.8 \times 10^{50}$ erg, which is $\sim 27\%$ of the kinetic energy of the major eruption. The dimensional effects also manifest on the physical properties, such as irregularity and thickness of the shell. Our study suggests PPISNe is a promising candidate of luminous SNe, the radiation of which originates from colliding shells with a homogeneous mixing of ejecta.Comment: Submitted to ApJ, 16 pages, comments are welcom