Natural Metric Research Articles

Site preferences and participation of waterbird recreationists: Using choice modelling to inform habitat management

Recreationists utilizing wetland birds for hunting or birdwatching contribute significantly to local economies and conservation efforts. The waterfowl management community, through the 2012 North American Waterfowl Management Plan (NAWMP), has explicitly recognized the need to increase recruitment and retention of wetland bird recreationists to help halt the national decline in social support for conservation. Achieving this goal requires that waterfowl and wetland managers consider how recreation site characteristics and management impact participation and support for wetland conservation. The outdoor recreation management and natural resource economics fields have extensive experience with similar problems and frequently use recreational site choice models to link attributes of recreational areas to use by recreationists. The waterfowl management community has several publicly available, long-term data sets that can be used in site choice models. We introduce U.S. Fish and Wildlife Service waterfowl harvest survey data, U.S. Geological Survey band encounter data, and Cornell Lab of Ornithology eBird data and use those data to summarize travel characteristics of wetland bird enthusiasts. We used harvest survey and eBird data in choice models to predict and compare the impacts of three seemingly similar proposed habitat acquisitions on use by recreationists in the state of Georgia; the proposal that had the greatest increase in predicted trips would result in twice as many additional hunting trips and >10 times more additional birdwatching trips than the proposal that generated the fewest additional trips. This case study demonstrates the potential of these and similar data and analytical methods for incorporating recreation participation and site preferences into habitat planning and delivery under the NAWMP. We encourage the outdoor recreation management and economics communities and the waterfowl management community to build partnerships and cooperative projects to improve our understanding of the relationships between wetland bird users and habitat conservation. Management implications•Results from our site choice models can provide a natural and intuitive metric that can be used to evaluate the relative merits of alternative management actions in achieving social objectives. An improvement over current proposals evaluated on a qualitative description of public access might be achieved through using the choice modeling approach to obtain quantitative estimates of the expected increase in public use resulting from a proposal.•Future research efforts that utilize finer resolution recreation site choice information would allow for a better understanding of recreationists’ decision processes and could provide managers with valuable insights into the benefits of quality improvements at state parks, managed areas, and other management units of interest.•Researchers should also work with state agencies to identify sources of finer resolution site characteristic data, such as the distribution and availability of public land, recreational regulations, and presence of infrastructure.•Researchers should explore interdisciplinary collaborations with ecological modelers to model expected sightings or expected harvest as a function of spatially-explicit environmental and landscape characteristics. Such a model could then be linked to a recreation site choice model, allowing the tracing of wetland restoration/enhancement benefits through a change in the biological system, and ultimately the change in trip locations, total trips, and/or economic benefits.•We encourage the natural resource management and economics professionals to build partnerships and cooperative projects with the NAWMP community to improve our understanding of the preferences, participation, and motivation of wetland bird users and their relationships to habitat conservation and management.

Spherical Process Models for Global Spatial Statistics

Statistical models used in geophysical, environmental, and climate science applications must reflect the curvature of the spatial domain in global data. Over the past few decades, statisticians have developed covariance models that capture the spatial and temporal behavior of these global data sets. Though the geodesic distance is the most natural metric for measuring distance on the surface of a sphere, mathematical limitations have compelled statisticians to use the chordal distance to compute the covariance matrix in many applications instead, which may cause physically unrealistic distortions. Therefore, covariance functions directly defined on a sphere using the geodesic distance are needed. We discuss the issues that arise when dealing with spherical data sets on a global scale and provide references to recent literature. We review the current approaches to building process models on spheres, including the differential operator, the stochastic partial differential equation, the kernel convolution, and the deformation approaches. We illustrate realizations obtained from Gaussian processes with different covariance structures and the use of isotropic and nonstationary covariance models through deformations and geographical indicators for global surface temperature data. To assess the suitability of each method, we compare their log-likelihood values and prediction scores, and we end with a discussion of related research problems.

Colour preference, naturalness, vividness and colour quality metrics, Part 5: A colour preference experiment at 2000 lx in a real room

Subjects rated the colour preference of a colourful object scene illuminated to 2000 lx by 28 different spectra from a four-channel LED light engine at different object saturation levels and different white points (3200 K, 4200 K, 5000 K and 5600 K). Mean colour preference ratings had a maximum at a moderate saturation level. The lowest correlated colour temperature (3200 K) exhibited lower preference ratings than the three higher correlated colour temperatures. The influence of saturation level was modelled by existing colour rendition measures and by the measure colour quality combining a colour fidelity index and an overall object colour object saturation measure taken from Part 4 of this work.

Metric ultraproducts of classical groups

Simple non-discrete metric ultraproducts of classical groups are geodesic spaces with respect to a natural metric.

Facilitating Adoption of Internet Technologies and Services with Externalities via Cost Subsidization

This article models the temporal adoption dynamics of an abstracted Internet technology or service, where the instantaneous net value of the service perceived by each (current or potential) user/customer incorporates three key features: (i) user service affinity heterogeneity, (ii) a network externality, and (iii) a subscription cost. Internet technologies and services with network externalities face a “chicken-and-egg” adoption problem in that the service requires an established customer base to attract new customers. In this article, we study cost subsidization as a means to “reach the knee,” at which point the externality drives rapid service adoption, and thereby change the equilibrium service fractional adoption level from an initial near-zero level to a final near-one level (full adoption). We present three simple subsidy models and evaluate them under two natural performance metrics: (i) the duration required for the subsidized service to reach a given target adoption level and (ii) the aggregate cost of the subsidy born by the service provide. First, we present a “two-target adoption subsidy” that subsidizes the cost to keep the fraction of users with positive net utility at a (constant) target level until the actual adoption target is reached. Second, we study a special case of the above where the target ensures all users have positive net utility, corresponding to a “quickest adoption” subsidy (QAS). Third, we introduce an approximation of QAS that only requires the service provider adjust the subsidy level a prescribed number of times. Fourth, we study equilibria and their stability under uniformly and normally distributed user service affinities, highlighting the unstable equilibrium in each case as the natural target adoption level for the provider. Finally, we provide a fictional case study to illustrate the application of the results in a (hopefully) realistic scenario, along with a brief discussion of the limitations of the model and analysis.

Matrix geometric means based on shuffled inductive sequences

We propose a new deterministic sequence converging to the least squares mean of N symmetric positive definite (SPD) matrices. By “least squares mean”, we refer to the Riemannian barycenter with respect to the natural metric (also known as the trace metric or affine-invariant metric) on the set Pn of n×n SPD matrices. In some papers, this mean is also referred to as the Karcher mean. The sequence we propose belongs to the family of inductive sequences, obtained by letting a point take successive steps towards the data points, with step lengths progressively diminishing to zero. We use the word “inductive” since each point of those sequences can be seen as the inductive mean of a well-chosen set of points containing multiple replications of the data points. We show that visiting the data points in a cyclic order usually results in a slow convergence, and remedy this weakness by reordering repeatedly the data points using a shuffling algorithm. We prove the convergence of the resulting inductive sequence to the least squares mean of the data in the general framework of NPC spaces (complete metric spaces with non-positive curvature), which includes the set Pn. We also illustrate numerically on Pn that some points of this sequence are fairly accurate estimates of the least squares mean, while being considerably cheaper to compute, and perform well as initializers for state-of-the-art algorithms. To reduce further the computation time, we finally exploit inductive sequences to produce a family of parallelizable algorithms for estimating the least squares mean.

Colour preference, naturalness, vividness and colour quality metrics, Part 4: Experiments with still life arrangements at different correlated colour temperatures

Subjective colour preference, naturalness and vividness assessments of two different colourful still life arrangements viewed in a real room were analysed and modelled with the aid of the Rf colour fidelity metric combined with a chroma difference metric. Coloured objects were illuminated by a four-channel LED light engine with 36 different spectra at four correlated colour temperatures and nine object oversaturation levels. Results imply a significant dependence of the subjective judgments on correlated colour temperature.

A fish-based multimetric index for Brazilian savanna streams

Multimetric indices (MMI) have been widely used to assess ecosystem conditions because they are low-cost, employ a rapid field method, and can incorporate various biological metrics at different levels of biological organization. Our objective was to create a fish-based multimetric index applicable to all streams of the Brazilian savanna biome (Cerrado), the second largest biome in Brazil and deemed a global biodiversity hotspot. We evaluated 156 sites in two river basins (Paraná and São Francisco) and selected metrics capable of distinguishing stream-sites across a gradient of anthropogenic disturbances. We employed two different MMI approaches to determine if an MMI based on natural variation-adjusted metrics performed better than one based on unadjusted metrics. In addition, we assessed the performance of the two final MMIs and their sensitivity to anthropogenic pressures at local (LDI), catchment (CDI) and both scales integrated (IDI). Finally, we employed the power of a probability sample survey design to infer headwater stream conditions across a hydrologic region of approximately 47,000km2. Our final MMI for Brazilian savanna streams included six metrics: % common species; % characiform individuals; % loricariid individuals; % trichomycterid individuals; % invertivore species, and % Poecilia reticulata individuals. MMI1 (unadjusted metrics) performed better than MMI2 (natural variation-adjusted metrics) in discriminating least- and most-disturbed sites, but MMI2 distinguished intermediate from most-disturbed sites better than MMI1. Both indices were negatively correlated with the CDI scores; however, only MMI2 was negatively correlated with the IDI scores. We inferred that 709km (9.35%) of streams in the studied hydrologic region were in good condition, 8115km (82.73%) were intermediate, and 641km (7.91%) were in poor condition. We conclude that the MMIs proposed in this study have great potential for widespread application because they integrate data from two of the most important Brazilian river basins included in a biome that represents more than 20% of the country. Furthermore, the metrics retained in the indexes are easy to access with a rapid low-cost field method. However, their feasibility in areas influenced by mining, as well as in different biomes, should be tested.

A tool for signage placement recommendation in hospitals based on wayfinding metrics

Navigating a healthcare facility can prove challenging to both new and existing patients and visitors. Poor or ineffective use of signage within the facility may enhance navigational difficulties. Signage strategies within facility design tend to be produced without consideration of how people typically navigate a space. Thus, strategies that ‘work on paper’ may not, in reality, aid or optimize patient and visitor wayfinding. Existing strategies for determining signage placement may also prove costly in terms of time spent on manual analysis of a facility’s floor space, including the potential for overlooking prime signage locations when analysing large floor plans. This paper presents a tool which aims to aid signage placement strategies by analysing facility design and routes within it, based on natural wayfinding metrics found in existing literature. The tool is designed to enable quick analysis of large designs for analysing multiple routes, highlighting areas where signage placement would aid natural wayfinding. The outputs of the tool are presented as a colour map which overlays the original 3D model design, highlighting the key areas where signage may be appropriate. An example of how the tool can be utilised to aid effective sign strategy is demonstrated on a small healthcare facility design.

Remarks on curvature in the transportation metric

According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the “hyperbolic” toric Kahler–Einstein equation e Φ = detD 2Φ on proper convex cones. We prove a generalization of this theorem by showing that for every Φ solving this equation on a proper convex domain Ω the corresponding metric measure space (D 2Φ, e Φ dx) has a non-positive Bakry–Emery tensor. Modifying the Calabi computations we obtain this result by applying the tensorial maximum principle to the weighted Laplacian of the Bakry–Emery tensor. Our computations are carried out in a generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of the log-concave probability measures we prove a third-order uniform dimension-free apriori estimate in the spirit of the second-order Caffarelli contraction theorem, which has numerous applications in probability theory.

On a class of submanifolds in a tangent bundle with a $g$-natural metric

On a class of submanifolds in a tangent bundle with a $g$-natural metric

Computing distances and geodesics between manifold-valued curves in the SRV framework

This paper focuses on the study of open curves in a Riemannian manifold M, and proposes a reparametrization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. to define a Riemannian metric on the space of immersions M'=Imm([0,1],M) by pullback of a natural metric on the tangent bundle TM'. This induces a first-order Sobolev metric on M' and leads to a distance which takes into account the distance between the origins in M and the L2-distance between the SRV representations of the curves. The geodesic equations for this metric are given and exploited to define an exponential map on M'. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of M'. The particular case of curves lying in the hyperbolic half-plane is considered as an example, in the setting of radar signal processing.

Curvature of the Determinant Line Bundle for the Noncommutative Two Torus

We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using an analogue of Kontsevich-Vishik canonical trace, defined on Connes' algebra of classical pseudodifferential symbols for the noncommutative two torus, we compute the curvature form of the determinant line bundle by computing the second variation $\delta_{w}\delta_{\bar{w}}\log\det(\Delta)$.

Metric-space approach to potentials and its relevance to density-functional theory

External potentials play a crucial role in modelling quantum systems, since, for a given inter- particle interaction, they define the system Hamiltonian. We use the metric space approach to quantum mechanics to derive, from the energy conservation law, two natural metrics for potentials. We show that these metrics are well defined for physical potentials, regardless of whether the system is in an eigenstate or if the potential is bounded. In addition, we discuss the gauge freedom of potentials and how to ensure that the metrics preserve physical relevance. Our metrics for potentials, together with the metrics for wavefunctions and densities from [I. D'Amico, J. P. Coe, V. V. Franca, and K. Capelle, Phys. Rev. Lett. 106, 050401 (2011)] paves the way for a comprehensive study of the two fundamental theorems of Density Functional Theory. We explore these by analysing two many- body systems for which the related exact Kohn-Sham systems can be derived. First we consider the information provided by each of the metrics, and we find that the density metric performs best in distinguishing two many-body systems. Next we study for the systems at hand the one-to-one relationships among potentials, ground state wavefunctions, and ground state densities defined by the Hohenberg-Kohn theorem as relationships in metric spaces. We find that, in metric space, these relationships are monotonic and incorporate regions of linearity, at least for the systems considered. Finally, we use the metrics for wavefunctions and potentials in order to assess quantitatively how close the many-body and Kohn-Sham systems are: We show that, at least for the systems analysed, both metrics provide a consistent picture, and for large regions of the parameter space the error in approximating the many-body wavefunction with the Kohn-Sham wavefunction lies under a threshold of 10%.

Connecting natural enemy metrics to biological control activity for aphids in California walnuts

To encourage adoption of conservation biological control, metrics need to be developed that can predict current activity and future potential of biological control. In this study, we evaluated natural enemy metrics to see how well they performed in predicting current and future biological control of Chromaphis juglandicola in California walnut orchards. Metrics based on more direct measures of natural enemy activity, such as percent parasitism and predator–prey ratio, were effective indicators of current biological control activity. Of the natural enemy metrics based on biodiversity, only evenness had a significant relationship with current aphid density, and only in organic orchards. There were two apparent negative relationships between the seasonal change in aphid density and natural enemy metrics, weighted natural enemy units and weighted predator units. June aphid density alone had the strongest influence on the seasonal change in aphid density, however, suggesting that even at low densities walnut aphids show within-year density dependence. Coupled with strong positive relationships between both natural enemy units and predator units and current aphid densities, most likely due to aggregation, this suggests that the apparent predictive power of these two metrics between seasons was an artifact of the density dependence. Overall, the results from this study suggest that for walnut aphid, predator–prey ratio, parasitism and natural enemy evenness (in organic orchards only) can be used to evaluate current biological control activity, but predicting future control potential through the season from natural enemy metrics can be misleading due to the confounding effect of within-year density dependence.

On volumes of classical supermanifolds

We consider the volumes of classical supermanifolds (such as the supersphere, complex projective superspace, Stiefel and Grassmann supermanifolds) with respect to natural metrics or symplectic structures. We show that the formulae for the volumes of these supermanifolds can be obtained from the formulae for the volumes of the corresponding ordinary manifolds (under some universal normalization of the volume) by analytic continuation with respect to parameters. The volumes of nontrivial supermanifolds may be identically equal to zero. In the 1970s Berezin showed that the total Haar measure of the unitary supergroup vanishes except in the cases and , when the supergroup is the ordinary unitary group or . Some time ago Witten conjectured that the Liouville volume of a compact even symplectic supermanifold is always equal to zero (except for ordinary manifolds). We give counterexamples to this conjecture, present a simple explanation of Berezin's theorem, and generalize this theorem to the Stiefel supermanifold . We mention a connection with recent work of Mkrtchyan and Veselov on universal formulae in Lie algebra theory. Bibliography: 32 titles.

Colour preference, naturalness, vividness and colour quality metrics, Part 3: Experiments with makeup products and analysis of the complete warm white dataset

Light source colour quality is a complex phenomenon with several visual and cognitive components. Subjective colour preference, naturalness and vividness assessments of reddish cosmetic products in a viewing booth lit by light sources with a correlated colour temperature of 3200 K and to 550 lx were analysed and modelled by a new combined colour quality formula – a linear combination of a colour fidelity index and a measure of chroma change. Results of the warm white light sources described in all three parts of this work were merged. They were modelled by optimising the parameters of the same formula. The formula supports multi-LED light sources with moderately accentuated local spectral maxima to enhance object chroma over a moderate range.

RDF graph alignment with bisimulation

We investigate the problem of aligning two RDF databases, an essential problem in understanding the evolution of ontologies. Our approaches address three fundamental challenges: 1) the use of "blank" (null) names, 2) ontology changes in which different names are used to identify the same entity, and 3) small changes in the data values as well as small changes in the graph structure of the RDF database. We propose approaches inspired by the classical notion of graph bisimulation and extend them to capture the natural metrics of edit distance on the data values and the graph structure. We evaluate our methods on three evolving curated data sets. Overall, our results show that the proposed methods perform well and are scalable.

Riemannian geometry of the space of volume preserving immersions

Given a compact manifold M and a Riemannian manifold N of bounded geometry, we consider the manifold Imm(M,N) of immersions from M to N and its subset Immμ(M,N) of those immersions with the property that the volume-form of the pull-back metric equals μ. We first show that the non-minimal elements of Immμ(M,N) form a splitting submanifold. On this submanifold we consider the Levi-Civita connection for various natural Sobolev metrics, we write down the geodesic equation for which we show local well-posedness in many cases. The question is a natural generalization of the corresponding well-posedness question for the group of volume-preserving diffeomorphisms, which is of importance in fluid mechanics.

Using a Prediction Error Criterion for Model Selection in Forecasting Option Prices

The common way to measure the performance of a volatility prediction model is to assess its ability to predict future volatility. However, as volatility is unobservable, there is no natural metric for measuring the accuracy of any particular model. Noh et al. (1994) assessed the performance of a volatility prediction model by devising trading rules to trade options on a daily basis and using forecasts of option prices obtained by the Black & Scholes (BS) option pricing formula. (An option is a security that gives its owner the right, not the obligation, to buy or sell an asset at a fixed price within a specified period of time, subject to certain conditions. The BS formula amounts to buying (selling) an option when its price forecast for tomorrow is higher (lower) than today’s market settlement price.)In this paper, adopting Noh et al.’s (1994) idea, we assess the performance of a number of Autoregressive Conditional Heteroscedasticity (ARCH) models. For, each trading day, the ARCH model, selected on the basis of the prediction error criterion (PEC) introduced by Xekalaki et al. (2003) and suggested by Degiannakis and Xekalaki (1999) in the context of ARCH models, is used to forecast volatility. According to this criterion, the ARCH model with the lowest sum of squared standardized one step ahead prediction errors is selected for forecasting future volatility. A comparative study is made in order to examine which ARCH volatility estimation method yields the highest profits and whether there is any gain in using the PEC model selection algorithm for speculating with financial derivatives. Among a set of model selection algorithms, even marginally, the PEC algorithm appears to achieve the highest rate of return.