Improving the condition number of estimated covariance matrices

High dimensional error covariance matrices and their inverses are used to weight the contribution of observation and background information in data assimilation procedures. As observation error covariance matrices are often obtained by sampling methods, estimates are often degenerate or ill-conditioned, making it impossible to invert an observation error covariance matrix without the use of techniques to reduce its condition number. In this paper we present new theory for two existing methods that can be used to ‘recondition’ any covariance matrix: ridge regression, and the minimum eigenvalue method. We compare these methods with multiplicative variance inflation, which cannot alter the condition number of a matrix, but is often used to account for neglected correlation information. We investigate the impact of reconditioning on variances and correlations of a general covariance matrix in both a theoretical and practical setting. Improved theoretical understanding provides guidance to users regarding method selection, and choice of target condition number. The new theory shows that, for the same target condition number, both methods increase variances compared to the original matrix, with larger increases for ridge regression than the minimum eigenvalue method. We prove that the ridge regression method strictly decreases the absolute value of off-diagonal correlations. Theoretical comparison of the impact of reconditioning and multiplicative variance inflation on the data assimilation objective function shows that variance inflation alters information across all scales uniformly, whereas reconditioning has a larger effect on scales corresponding to smaller eigenvalues. We then consider two examples: a general correlation function, and an observation error covariance matrix arising from interchannel correlations. The minimum eigenvalue method results in smaller overall changes to the correlation matrix than ridge regression, but can increase off-diagonal correlations. Data assimilation experiments reveal that reconditioning corrects spurious noise in the analysis but underestimates the true signal compared to multiplicative variance inflation

Spectral structure and decompositions of optical states, and their applications

We discuss the spectral structure and decomposition of multi-photon states. Ordinarily `multi-photon states' and `Fock states' are regarded as synonymous. However, when the spectral degrees of freedom are included this is not the case, and the class of `multi-photon' states is much broader than the class of `Fock' states. We discuss the criteria for a state to be considered a Fock state. We then address the decomposition of general multi-photon states into bases of orthogonal eigenmodes, building on existing multi-mode theory, and introduce an occupation number representation that provides an elegant description of such states that in many situations simplifies calculations. Finally we apply this technique to several example situations, which are highly relevant for state of the art experiments. These include Hong-Ou-Mandel interference, spectral filtering, finite bandwidth photo-detection, homodyne detection and the conditional preparation of Schr\"odinger Kitten and Fock states. Our techniques allow for very simple descriptions of each of these examples.Comment: 12 page

Simulation of Yield and Environmental Impacts of Wheat after Rice in Bangladesh and Australia

CERES-wheat and SWAGMAN Destiny models, respectively, were used to estimate the optimum time of sowing, and trade-off between yield and net recharge of the watertable, for wheat grown after rice in northern Bangladesh and southern NSW, Australia. Simulated wheat yields in Bangladesh, for sowings from Sept to Jan, with two supplemental irrigations, ranged from 0.4 to 4.6 t/ha. November-sown crops yielded more than the earlier- or later-sown crops due to reduced water and heat stress during grain filling. In Australia, simulated yields of non-irrigated wheat were always greater for April than June sowings due to less water deficit at the end of the season. With an initial shallow (0.5 m), fresh (1 dS/m) watertable, simulated yields usually exceeded 3 t/ha, and declined as watertable salinity increased. Nonirrigated wheat almost always lowered the watertable. Frequent irrigation increased simulated yields to 5-6 t/ha, regardless of initial conditions and sowing date, but this was at the cost of decreased discharge or increased recharge leading to rising watertables

A Nutrient-Regulated Cyclic Diguanylate Phosphodiesterase Controls Clostridium difficile Biofilm and Toxin Production during Stationary Phase

ABSTRACT The signaling molecule cyclic diguanylate (c-di-GMP) mediates physiological adaptation to extracellular stimuli in a wide range of bacteria. The complex metabolic pathways governing c-di-GMP synthesis and degradation are highly regulated, but the specific cues that impact c-di-GMP signaling are largely unknown. In the intestinal pathogen Clostridium difficile , c-di-GMP inhibits flagellar motility and toxin production and promotes pilus-dependent biofilm formation, but no specific biological functions have been ascribed to any of the individual c-di-GMP synthases or phosphodiesterases (PDEs). Here, we report the functional and biochemical characterization of a c-di-GMP PDE, PdcA, 1 of 37 confirmed or putative c-di-GMP metabolism proteins in C. difficile 630. Our studies reveal that pdcA transcription is controlled by the nutrient-regulated transcriptional regulator CodY and accordingly increases during stationary phase. In addition, PdcA PDE activity is allosterically regulated by GTP, further linking c-di-GMP levels to nutrient availability. Mutation of pdcA increased biofilm formation and reduced toxin biosynthesis without affecting swimming motility or global intracellular c-di-GMP. Analysis of the transcriptional response to pdcA mutation indicates that PdcA-dependent phenotypes manifest during stationary phase, consistent with regulation by CodY. These results demonstrate that inactivation of this single PDE gene is sufficient to impact multiple c-di-GMP-dependent phenotypes, including the production of major virulence factors, and suggest a link between c-di-GMP signaling and nutrient availability

A Nutrient-Regulated Cyclic Diguanylate Phosphodiesterase Controls Clostridium difficile Biofilm and Toxin Production During Stationary Phase

The signaling molecule cyclic diguanylate (c-di-GMP) mediates physiological adaptation to extracellular stimuli in a wide range of bacteria. The complex metabolic pathways governing c-di-GMP synthesis and degradation are highly regulated, but the specific cues that impact c-di-GMP signaling are largely unknown. In the intestinal pathogen Clostridium difficile, c-di-GMP inhibits flagellar motility and toxin production and promotes pilus-dependent biofilm formation, but no specific biological functions have been ascribed to any of the individual c-di-GMP synthases or phosphodiesterases (PDEs). Here, we report the functional and biochemical characterization of a c-di-GMP PDE, PdcA, 1 of 37 confirmed or putative c-di-GMP metabolism proteins in C. difficile 630. Our studies reveal that pdcA transcription is controlled by the nutrient-regulated transcriptional regulator CodY and accordingly increases during stationary phase. In addition, PdcA PDE activity is allosterically regulated by GTP, further linking c-di-GMP levels to nutrient availability. Mutation of pdcA increased biofilm formation and reduced toxin biosynthesis without affecting swimming motility or global intracellular c-di-GMP. Analysis of the transcriptional response to pdcA mutation indicates that PdcA-dependent phenotypes manifest during stationary phase, consistent with regulation by CodY. These results demonstrate that inactivation of this single PDE gene is sufficient to impact multiple c-di-GMP-dependent phenotypes, including the production of major virulence factors, and suggest a link between c-di-GMP signaling and nutrient availability

Myosin X and its motorless isoform differentially modulate dendritic spine development by regulating trafficking and retention of vasodilator-stimulated phosphoprotein

Myosin X (Myo10) is an unconventional myosin with two known isoforms: full-length (FL)-Myo10 that has motor activity, and a recently identified brain-expressed isoform, headless (Hdl)-Myo10, which lacks most of the motor domain. FL-Myo10 is involved in the regulation of filopodia formation in non-neuronal cells; however, the biological function of Hdl-Myo10 remains largely unknown. Here, we show that FL- and Hdl-Myo10 have important, but distinct, roles in the development of dendritic spines and synapses in hippocampal neurons. FL-Myo10 induces formation of dendritic filopodia and modulates filopodia dynamics by trafficking the actin-binding protein vasodilator-stimulated phosphoprotein (VASP) to the tips of filopodia. By contrast, Hdl-Myo10 acts on dendritic spines to enhance spine and synaptic density as well as spine head expansion by increasing the retention of VASP in spines. Thus, this study demonstrates a novel biological function for Hdl-Myo10 and an important new role for both Myo10 isoforms in the development of dendritic spines and synapses

Myo10 in brain: developmental regulation, identification of a headless isoform and dynamics in neurons

Although Myo10 (myosin-X) is an unconventional myosin associated with filopodia, little is known about its isoforms and roles in the nervous system. We report here that, in addition to full-length Myo10, brain expresses a shorter form of Myo10 that lacks a myosin head domain. This ;headless' Myo10 is thus unable to function as a molecular motor, but is otherwise identical to full-length Myo10 and, like it, contains three pleckstrin homology (PH) domains, a myosin-tail homology 4 (MyTH4) domain, and a band-4.1/ezrin/radixin/moesin (FERM) domain. Immunoblotting demonstrates that both full-length and headless Myo10 exhibit dramatic developmental regulation in mouse brain. Immunofluorescence with an antibody that detects both isoforms demonstrates that Myo10 is expressed in neurons, such as Purkinje cells, as well as non-neuronal cells, such as astrocytes and ependymal cells. CAD cells, a neuronal cell line, express both full-length and headless Myo10, and this endogenous Myo10 is present in cell bodies, neurites, growth cones and the tips of filopodia. To investigate the dynamics of the two forms of Myo10 in neurons, CAD cells were transfected with GFP constructs corresponding to full-length or headless Myo10. Only full-length Myo10 localizes to filopodial tips and undergoes intrafilopodial motility, demonstrating that the motor domain is necessary for these activities. Live cell imaging also reveals that full-length Myo10 localizes to the tips of neuronal filopodia as they explore and interact with their surroundings, suggesting that this myosin has a role in neuronal actin dynamics