PNNL

Abstract

In statistical mechanics, spectrum estimation is traditionally done by computing Fourier transforms of time-correlation functions[1]-[3] to find the amplitudes of frequency components that fit best the observed data (assuming each component is the only one present). Examples of important time correlation functions include the van Hove, dipole density, and magnetization correlation functions, whose Fourier transforms are related to the structure factor, infrared lineshape, and NMR signal lineshape, respectively.[4], [5] Although the Fourier transform is extremely general, crucial restrictions limiting its applications in statistical mechanics include assumptions of periodic or stationary time series and a linear system, so that data may be projected onto complex sinusoidal basis functions.[6], [7] These and other limitations of the Fourier transform have spurred a recent search for alternatives, but most options are closely related to the Fourier transform,[7]-[9] make similar assumptions, and share similar limitations. Herein, we present an alternative to Fourier-transform-based methods that finds wide use in fields where signal processing,[10] pattern recognition,[11] or multivariate statistical analysis[12] methods are common. The method, known by various names which include principal component analysis (PCA)[10], [12] and the Karhunen-Loève expansion,[10], [11] shares many limitations of Fourier transforms, but is more flexible than Fourier-like methods because it gives data-adaptive basis functions in addition to the amplitudes of frequency components. In fact, PCA is usually used for data compression to find the smallest set of vectors that reproduce a covariance matrix to desired accuracy and is used most frequently in statistical mechanics to compress molecular dynamics trajectories of proteins to a minimal set of vectors which represent only the large amplitude, low frequency motions.[13]-[15] This contribution states the ability of PCA to give optimal spectral density functions in a mean-square sense,[16] gives analytical proof that PCA gives a more accurate spectrum than Fourier analysis for a fixed, finite number of basis vectors, and demonstrates numerically that PCA gives spectra more accurately than fast Fourier transform (FFT) or maximum-entropy algorithms.[17] The potential utility of PCA is illustrated by calculating vibrational frequencies and modes from dynamics trajectories (time series) for water, but PCA is general and equally valid for estimating other spectral densities,[1], [3]-[5] analyzing distributions generated from Monte Carlo simulations[18] or experimental data, and analyzing quantum-mechanical time-correlation functions.[19]