The geometry of nonlinear least squares with applications to sloppy models and optimization

Parameter estimation by nonlinear least squares minimization is a common problem with an elegant geometric interpretation: the possible parameter values of a model induce a manifold in the space of data predictions. The minimization problem is then to find the point on the manifold closest to the data. We show that the model manifolds of a large class of models, known as sloppy models, have many universal features; they are characterized by a geometric series of widths, extrinsic curvatures, and parameter-effects curvatures. A number of common difficulties in optimizing least squares problems are due to this common structure. First, algorithms tend to run into the boundaries of the model manifold, causing parameters to diverge or become unphysical. We introduce the model graph as an extension of the model manifold to remedy this problem. We argue that appropriate priors can remove the boundaries and improve convergence rates. We show that typical fits will have many evaporated parameters. Second, bare model parameters are usually ill-suited to describing model behavior; cost contours in parameter space tend to form hierarchies of plateaus and canyons. Geometrically, we understand this inconvenient parametrization as an extremely skewed coordinate basis and show that it induces a large parameter-effects curvature on the manifold. Using coordinates based on geodesic motion, these narrow canyons are transformed in many cases into a single quadratic, isotropic basin. We interpret the modified Gauss-Newton and Levenberg-Marquardt fitting algorithms as an Euler approximation to geodesic motion in these natural coordinates on the model manifold and the model graph respectively. By adding a geodesic acceleration adjustment to these algorithms, we alleviate the difficulties from parameter-effects curvature, improving both efficiency and success rates at finding good fits.Comment: 40 pages, 29 Figure

Controlling magnetic order and quantum disorder in molecule-based magnets

We investigate the structural and magnetic properties of two molecule-based magnets synthesized from the same starting components. Their different structural motifs promote contrasting exchange pathways and consequently lead to markedly different magnetic ground states. Through examination of their structural and magnetic properties we show that [Cu(pyz)(H2O)(gly)2](ClO4)2 may be considered a quasi-one-dimensional quantum Heisenberg antiferromagnet whereas the related compound [Cu(pyz)(gly)](ClO4), which is formed from dimers of antiferromagnetically interacting Cu2+ spins, remains disordered down to at least 0.03 K in zero field but shows a field-temperature phase diagram reminiscent of that seen in materials showing a Bose-Einstein condensation of magnons

Anthropogenic modifications to fire regimes in the wider Serengeti‐Mara ecosystem

Fire is a key driver in savannah systems and widely used as a land management tool. Intensifying human land uses are leading to rapid changes in the fire regimes, with consequences for ecosystem functioning and composition. We undertake a novel analysis describing spatial patterns in the fire regime of the Serengeti‐Mara ecosystem, document multidecadal temporal changes and investigate the factors underlying these patterns. We used MODIS active fire and burned area products from 2001 to 2014 to identify individual fires; summarizing four characteristics for each detected fire: size, ignition date, time since last fire and radiative power. Using satellite imagery, we estimated the rate of change in the density of livestock bomas as a proxy for livestock density. We used these metrics to model drivers of variation in the four fire characteristics, as well as total number of fires and total area burned. Fires in the Serengeti‐Mara show high spatial variability—with number of fires and ignition date mirroring mean annual precipitation. The short‐term effect of rainfall decreases fire size and intensity but cumulative rainfall over several years leads to increased standing grass biomass and fuel loads, and, therefore, in larger and hotter fires. Our study reveals dramatic changes over time, with a reduction in total number of fires and total area burned, to the point where some areas now experience virtually no fire. We suggest that increasing livestock numbers are driving this decline, presumably by inhibiting fire spread. These temporal patterns are part of a global decline in total area burned, especially in savannahs, and we caution that ecosystem functioning may have been compromised. Land managers and policy formulators need to factor in rapid fire regime modifications to achieve management objectives and maintain the ecological function of savannah ecosystems

Higher order commutator estimates and local existence for the non-resistive MHD equations and related models

This paper establishes the local-in-time existence and uniqueness of strong solutions in Hs for s > n/2 to the viscous, non-resistive magnetohydrodynamics (MHD) equations in Rn, n = 2, 3, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato & Ponce (Comm. Pure Appl. Math. 41(7), 891–907, 1988)

Deprived children or deprived neighbourhoods? A public health approach to the investigation of links between deprivation and injury risk with specific reference to child road safety in Devon County, UK

BACKGROUND: Worldwide, injuries from road traffic collisions are a rapidly growing problem in terms of morbidity and mortality. The UK has amongst the worst records in Europe with regard to child pedestrian safety. A traditional view holds that resources should be directed towards training child pedestrians. In order to reduce socio-economic differentials in child pedestrian casualty rates it is suggested that these should be directed at deprived children. This paper seeks to question whether analysis of extant routinely collected data supports this view. METHODS: Routine administrative data on road collisions has been used. A deprivation measure has been assigned to the location where a collision was reported, and the home postcode of the casualty. Aggregate data was analysed using a number of epidemiological models, concentrating on the Generalised Linear Mixed Model. RESULTS: This study confirms evidence suggesting a link between increasing deprivation and increasing casualty involvement of child pedestrians. However, suggestions are made that it may be necessary to control for the urban nature of an area where collisions occur. More importantly, the question is raised as to whether the casualty rate is more closely associated with deprivation measures of the ward in which the collision occurred than with the deprivation measures of the home address of the child. CONCLUSION: Conclusions have to be drawn with great caution. Limitations in the utility of the officially collected data are apparent, but the implication is that the deprivation measures of the area around the collision is a more important determinant of socio-economic differentials in casualty rates than the deprivation measures of the casualties' home location. Whilst this result must be treated with caution, if confirmed by individual level case-controlled studies this would have a strong implication for the most appropriate interventions

A national-scale assessment of climate change impacts on species: assessing the balance of risks and opportunities for multiple taxa

It is important for conservationists to be able to assess the risks that climate change poses to species, in order to inform decision making. Using standardised and repeatable methods, we present a national-scale assessment of the risks of range loss and opportunities for range expansion, that climate change could pose for over 3,000 plants and animals that occur in England. A basic risk assessment that compared projected future changes in potential range with recently observed changes classified 21% of species as being at high risk and 6% at medium risk of range loss under a B1 climate change scenario. A greater number of species were classified as having a medium (16%) or high (38%) opportunity to potentially expand their distribution. A more comprehensive assessment, incorporating additional ecological information, including potentially confounding and exacerbating factors, was applied to 402 species, of which 35 % were at risk of range loss and 42 % may expand their range extent. This study covers a temperate region with a significant proportion of species at their poleward range limit. The balance of risks and opportunities from climate change may be different elsewhere. The outcome of both risk assessments varied between taxonomic groups, with bryophytes and vascular plants containing the greatest proportion of species at risk from climate change. Upland habitats contained more species at risk than other habitats. Whilst the overall pattern was clear, confidence was generally low for individual assessments, with the exception of well-studied taxa such as birds. In response to climate change, nature conservation needs to plan for changing species distributions and increasing uncertainty of the future

Deployment of contact-based ultrasonic thickness measurements using over-actuated UAVs

Unmanned Aerial Vehicles (UAVs) are increasingly being utilized for the structural health assessment of on and off-shore structures. Visual inspection is the usual methodology for acquiring data from these structures, but there is often a need for contact based structural measurements, for example to assess local thickness on corroding structures. Conventional UAV platform dynamics have not traditionally allowed for such contact measurements. The limited dynamic control afforded by fixed plane rotor UAVs means that forward thrust (to apply contact forces for surface transduction) is only possible by tilting the whole platform, thus taking the UAV into a non-stationary state and limiting positional accuracy. An over-actuated UAV platform (with fully vectored thrust capability) may provide the required contact force for such thickness measurements whilst maintaining stable hovering next to the structure. The authors herein present a contact based ultrasonic thickness measurement technique, whereby an ultrasonic wheel probe deployed from a UAV was used to make single point and scanned measurements across a surface to provide a set of local thickness measurements. A 5 MHz, dry-coupled, dual-element, ultrasonic wheel probe is used to measure the thickness of an aluminum sample plate with thicknesses of 8.2 mm, 4.5 mm and 3.2 mm, and a precision stepped calibration block with size from 31.5 mm to 17.5 mm in steps of 1 mm, then steps of 0.1 mm down to 16.5 mm over a total length of 500 mm. The thickness resolution obtainable from the ultrasonic wheel probe was typically 0.1 mm, and the positional accuracy attained from the over-actuated deployment platform was 16.6 mm when performing single point measurements