Methods for estimating uncertainty in PMF solutions : Examples with ambient air and water quality data and guidance on reporting PMF results

The new version of EPA's positive matrix factorization (EPA PMF) software, 5.0, includes three error estimation (EE) methods for analyzing factor analytic solutions: classical bootstrap (BS), displacement of factor elements (DISP), and bootstrap enhanced by displacement (BS-DISP). These methods capture the uncertainty of PMF analyses due to randomerrors and rotational ambiguity. To demonstrate the utility of the EEmethods, results are presented for three data sets: (1) speciated PM2.5 data froma chemical speciation network (CSN) site in Sacramento, California (2003-2009); (2) trace metal, ammonia, and other species inwater quality samples taken at an inline storage system (ISS) in Milwaukee, Wisconsin (2006); and (3) an organic aerosol data set from high- resolution aerosolmass spectrometer (HR-AMS) measurements in Las Vegas, Nevada (January 2008). We present an interpretation of EE diagnostics for these data sets, results fromsensitivity tests of EE diagnostics using additional and fewer factors, and recommendations for reporting PMF results. BS-DISP and BS are found useful in understanding the uncertainty of factor profiles; they also suggest if the data are over-fitted by specifying toomany factors. DISP diagnosticswere consistently robust, indicating its use for understanding rotational uncertainty and as a first step in assessing a solution's viability. The uncertainty of each factor's identifying species is shown to be a useful gauge for evaluating multiple solutions, e.g., with a different number of factors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer reviewe

Existence and uniqueness of the maximum likelihood estimator for the two-parameter negative binomial distribution

Given a sample with mean x and second moment s2, Anscombe in 1950 conjectured that the maximum likelihood equations for the two-parameter negative binomial distribution have a unique solution if and only if s2 > x. We give a proof of his conjecture.Maximum likelihood estimator negative binomial distribution Newton's method

Delayed creation of entanglement in superconducting qubits interacting with a microwave field

We explore the role played by the intrinsic decoherence in superconducting charge qubits in the presence of a microwave field applied as a magnetic flux. We study how the delayed creation of entanglement, which is opposite to the sudden death of entanglement, can be induced. We compute the time evolution of the population inversion, total correlation and entanglement, taking into account the junction mixed state and dissipation of the cavity field. We show that although decoherence destroys the correlation of the junction and field, information of the initial state may be obtained via quasi-probability distribution functions. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2010