Linear Rescaling Research Articles

The added utility of nonlinear methods compared to linear methods in rescaling soil moisture products

In this study, the added utility of nonlinear rescaling methods relative to linear methods in the framework of creating a homogenous soil moisture time series has been explored. The performances of 31 linear and nonlinear rescaling methods are evaluated by rescaling the Land Parameter Retrieval Model (LPRM) soil moisture datasets to station-based watershed average datasets obtained over four United States Department of Agriculture (USDA) Agricultural Research Service (ARS) watersheds. The linear methods include first-order linear regression, multiple linear regression, and multivariate adaptive regression splines (MARS), whereas the nonlinear methods include cumulative distribution function matching (CDF), artificial neural networks (ANN), support vector machines (SVM), Genetic Programming (GEN), and copula methods. MARS, GEN, SVM, ANN, and the copula methods are also implemented to utilize lagged observations to rescale the datasets. The results of a total of 31 different methods show that the nonlinear methods improve the correlation and error statistics of the rescaled product compared to the linear methods. In general, the method that yielded the best results using training data improved the validation correlations, on average, by 0.063, whereas ELMAN ANN and GEN, using lagged observations methods, yielded correlation improvements of 0.052 and 0.048, respectively. The lagged observations improved the correlations when they were incorporated into rescaling equations in linear and nonlinear fashions, with the nonlinear methods (particularly SVM and GEN but not ANN and copula) benefitting from these lagged observations more than the linear methods. The overall results show that a large majority of the similarities between the LPRM and watershed average datasets are due to linear relations; however, nonlinear relations clearly exist, and the use of nonlinear rescaling methods clearly improves the accuracy of the rescaled product.

Quasi-centralized limit order books

A quasi-centralized limit order book (QCLOB) is a limit order book (LOB) in which financial institutions can only access the trading opportunities offered by counterparties with whom they possess sufficient bilateral credit. In this paper, we perform an empirical analysis of a recent, high-quality data set from a large electronic trading platform that utilizes QCLOBs to facilitate trade. We argue that the quote-relative framework often used to study other LOBs is not a sensible reference frame for QCLOBs, so we instead introduce an alternative, trade-relative framework, which we use to study the statistical properties of order flow and LOB state in our data. We also uncover an empirical universality: although the distributions that describe order flow and LOB state vary considerably across days, a simple, linear rescaling causes them to collapse onto a single curve. Motivated by this finding, we propose a semi-parametric model of order flow and LOB state for a single trading day. Our model provides similar performance to that of parametric curve-fitting techniques but is simpler to compute and faster to implement.

Reducing multiplicative bias of satellite soil moisture retrievals

Soil moisture is a principal component of the Earth's climate and hydrological systems that is difficult to monitor and model due to high variability, uncertainty in land surface characterization and uncertainty in soil moisture forcing. Satellite soil moisture retrievals and brightness temperature observations, such as those available from the Soil Moisture and Ocean Salinity (SMOS) mission, can be a valuable source of information for data assimilation and merging with other satellite retrieval datasets. To correct for biases in these data sets, bias correction methods such as cumulative distribution function (CDF) matching, linear rescaling and copulas are used to map satellite soil moisture climatology to that of in situ or model values. This study compared SMOS retrievals to soil moisture observations from the SCAN network for a calibration period of 2010–2011 and validation period of 2012–2013 before and after bias correction. The focus of the study was on the presence and removal of multiplicative bias when comparing SMOS retrievals to SCAN data. Additive bias between SMOS retrievals and SCAN observations was removed by standard bias correction techniques and a new resampling approach was found to reduce multiplicative biases. In addition, the new bias correction technique was found very competitive with the benchmark methods for both the calibration and validation periods.

Early Postnatal Myelin Content Estimate of White Matter via T1w/T2w Ratio.

To develop and evaluate a novel processing framework for the relative quantification of myelin content in cerebral white matter (WM) regions from brain MRI data via a computed ratio of T1 to T2 weighted intensity values. We employed high resolution (1mm3 isotropic) T1 and T2 weighted MRI from 46 (28 male, 18 female) neonate subjects (typically developing controls) scanned on a Siemens Tim Trio 3T at UC Irvine. We developed a novel, yet relatively straightforward image processing framework for WM myelin content estimation based on earlier work by Glasser et al. We first co-register the structural MRI data to correct for motion. Then, background areas are masked out via a joint T1w and T2 foreground mask computed. Raw T1w/T2w-ratios images are computed next. For purpose of calibration across subjects, we first coarsely segment the fat-rich facial regions via an atlas co-registration. Linear intensity rescaling based on median T1w/T2w-ratio values in those facial regions yields calibrated T1w/T2w-ratio images. Mean values in lobar regions are evaluated using standard statistical analysis to investigate their interaction with age at scan. Several lobes have strongly positive significant interactions of age at scan with the computed T1w/T2w-ratio. Most regions do not show sex effects. A few regions show no measurable effects of change in myelin content change within the first few weeks of postnatal development, such as cingulate and CC areas, which we attribute to sample size and measurement variability. We developed and evaluated a novel way to estimate white matter myelin content for use in studies of brain white matter development.

Reproducibility in urine peptidome profiling using MALDI-TOF

MALDI-TOF profiling of low molecular weight peptides (peptidome) usage is limited due to the lack of reproducibility from the confounding inferences of sample preparation, data acquisition, and processing. We applied MALDI-TOF analysis to profile urine peptidome with the aims to: (i) compare centrifugal ultrafiltration and dialysis pretreatments, (ii) determine whether using signal LOD (sLOD), together with data normalization, may reduce MALDI-TOF variability. We also investigated the influence of peaks detection on reproducibility. Dialysis allowed to obtain better MALDI-TOF spectra than ultrafiltration. Within the 1000-4000 m/z range, we identified 120 and 129 peaks in intra- and interassay studies, respectively. To estimate the sLOD, serial dilution of pooled urines up to 1/256 were analyzed in triplicate. Six data normalization strategies were investigated-the mean, median, internal standard, relative intensity, TIC, and linear rescaling normalization. Normalization methods alone performed poorly in reducing features variability while when combined to sLOD adjustment showed an overall reduction in features CVs. Applying a feedback signal processing approach, after median normalization and sLOD adjustment, CVs were reduced from 103 to 26% and 113 to 25% for the intra- and interassay, respectively, and spectra became more comparable in terms of data dispersion.

Quasi-Centralized Limit Order Books

A quasi-centralized limit order book (QCLOB) is a limit order book (LOB) in which financial institutions can only access the trading opportunities offered by counterparties with whom they possess sufficient bilateral credit. We perform an empirical analysis of a recent, high-quality data set from a large electronic trading platform that utilizes QCLOBs to facilitate trade. We find many significant differences between our results and those widely reported for other LOBs. We also uncover a remarkable empirical universality: although the distributions describing order flow and market state vary considerably across days, a simple, linear rescaling causes them to collapse onto a single curve. Motivated by this finding, we propose a semi-parametric model of order flow and market state in a QCLOB on a single trading day. Our model provides similar performance to that of parametric curve-fitting techniques, while being simpler to compute and faster to implement.

Analysis of macular OCT images using deformable registration.

Optical coherence tomography (OCT) of the macula has become increasingly important in the investigation of retinal pathology. However, deformable image registration, which is used for aligning subjects for pairwise comparisons, population averaging, and atlas label transfer, has not been well-developed and demonstrated on OCT images. In this paper, we present a deformable image registration approach designed specifically for macular OCT images. The approach begins with an initial translation to align the fovea of each subject, followed by a linear rescaling to align the top and bottom retinal boundaries. Finally, the layers within the retina are aligned by a deformable registration using one-dimensional radial basis functions. The algorithm was validated using manual delineations of retinal layers in OCT images from a cohort consisting of healthy controls and patients diagnosed with multiple sclerosis (MS). We show that the algorithm overcomes the shortcomings of existing generic registration methods, which cannot be readily applied to OCT images. A successful deformable image registration algorithm for macular OCT opens up a variety of population based analysis techniques that are regularly used in other imaging modalities, such as spatial normalization, statistical atlas creation, and voxel based morphometry. Examples of these applications are provided to demonstrate the potential benefits such techniques can have on our understanding of retinal disease. In particular, included is a pilot study of localized volumetric changes between healthy controls and MS patients using the proposed registration algorithm.

High-Throughput Calculations of Molecular Properties in the MedeA Environment: Accuracy of PM7 in Predicting Vibrational Frequencies, Ideal Gas Entropies, Heat Capacities, and Gibbs Free Energies of Organic Molecules

The atomistic and molecular simulation environment MedeA (MedeA: Materials Exploration and Design Analysis, version 2.14.6; Material Design, Inc.: Angel Fire, NM, 1998–2014; http://www.materialsdesign.com) in its functionalities and graphical user interface has been enhanced to prepare and submit on the order of 1000 simulations on different structures, and to collect and help in the analysis of the results. We illustrate this with the determination of the accuracy of the semiempirical (SE) package MOPAC2012 (Stewart, J. J. P. MOPAC2012; Stewart Computational Chemistry: Colorado Springs, CO, USA, 2012; http://OpenMOPAC.net) with the PM7 method (Stewart, J. J. P. Optimization of parameters for semiempirical methods VI: more modifications to the NDDO approximations and reoptimization of parameters. J. Mol. Model. 2013, 19, 1–32) to compute frequencies of vibration and thermodynamic properties, specifically the zero point energies, ideal gas heat capacity at constant pressure, entropy, and Gibbs free energy, between 200 and 1000 K for 795 organic molecules. The results were compared with experimental data and density functional theory (DFT) values (using B3LYP/TZVP and BP86/TZVP DFT methods). This comparison showed that the PM7 frequencies of vibration above 2500 cm–1 are systematically underestimated. An a posteriori correction using a linear relationship rescaling of the frequencies permitted resetting to zero the average relative deviations with respect to experimental reference values. This frequency correction also removed the bias from the zero point energies, ideal gas heat capacity, and entropy average deviations from the PM7 results. The root-mean-square deviation (RMSD) of PM7 and the DFT heat capacities of 160 organic molecules were equivalent with respect to experimental values, being about 5 %, 2.5 %, and 3 % at 300 K, 600 K, and 1000 K, respectively. The RMSD of PM7, when compared to the DFT values, became 4 %, 2 %, and 1 % for the same temperatures when the analysis was extended to a set of 795 molecules. In the case of the ideal gas entropies, the RMSD of the PM7 relative to DFT values were between 5 % and 4 % between 300 K and 1000 K, respectively. The RMSD of the Gibbs free energies of PM7 were 15 kJ mol–1 and 30 kJ mol–1 at 300 K and 1000 K, respectively. The efficiency of this semiempirical approach was tested on a set of approximately 5800 molecules. This set was processed in about a day, thus demonstrating the scalability of the approach to big data sets.

Content-Aware Video Retargeting Using Object-Preserving Warping

A novel content-aware warping approach is introduced for video retargeting. The key to this technique is adapting videos to fit displays with various aspect ratios and sizes while preserving both visually salient content and temporal coherence. Most previous studies solve this spatiotemporal problem by consistently resizing content in frames. This strategy significantly improves the retargeting results, but does not fully consider object preservation, sometimes causing apparent distortions on visually salient objects. We propose an object-preserving warping scheme with object-based significance estimation to reduce this unpleasant distortion. In the proposed scheme, visually salient objects in 3D space-time space are forced to undergo as-rigid-as-possible warping, while low-significance contents are warped as close as possible to linear rescaling. These strategies enable our method to consistently preserve both the spatial shapes and temporal motions of visually salient objects and avoid overdeformations on low-significance objects, yielding a pleasing motion-aware video retargeting. Qualitative and quantitative analyses, including a user study and experiments on complex videos containing diverse cameras and dynamic motions, show a clear superiority of our method over related video retargeting methods.

The Collision Cross Sections of Iodide Salt Cluster Ions in Air via Differential Mobility Analysis-Mass Spectrometry

To date, most collision cross section (CCS) predictions have invoked gas molecule impingement-reemission rules in which specular and elastic scattering of spherical gas molecules from rigid polyatomic surfaces are assumed. Although such predictions have been shown to agree well with CCSs measured in helium bath gas, a number of studies reveal that these predictions do not agree with CCSs for ions in diatomic gases, namely, air and molecular nitrogen. To further examine the validity of specular-elastic versus diffuse-inelastic scattering models, we measured the CCSs of positively charged metal iodide cluster ions of the form [MI]n[M(+)]z, where M = Na, K, Rb, or Cs, n = 1 - 25, and z = 1 - 2. Measurements were made in air via differential mobility analysis mass spectrometry (DMA-MS). The CCSs measured are compared with specular-elastic as well as diffuse-inelastic scattering model predictions with candidate ion structures determined from density functional theory. It is found that predictions from diffuse-inelastic collision models agree well (within 5%) with measurements from sodium iodide cluster ions, while specular-elastic collision model predictions are in better agreement with cesium iodide cluster ion measurements. The agreement with diffuse-inelastic and specular-elastic predictions decreases and increases, respectively, with increasing cation mass. However, even when diffuse-inelastic cluster ion predictions disagree with measurements, the disagreement is of a near-constant factor for all ions, indicating that a simple linear rescaling collapses predictions to measurements. Conversely, rescaling cannot be used to collapse specular-elastic predictions to measurements; hence, although the precise impingement reemission rules remain ambiguous, they are not specular-elastic.

Estimating soil moisture in gullies from adjacent upland measurements through different observation operators

Summary Soil moisture datasets in large gullies are rare due to the difficulty of direct sampling in such landform. This study attempted to estimate spatial soil moisture averages in gullies from measurements of adjacent uplands by using observation operators, based on three-year soil moisture datasets in a gully catchment of the Loess Plateau. Soil moisture datasets in 2010 and 2011 were used for developing observation operators and those in 2012 were used for validation. Several nonlinear and linear methods including cumulative distribution function (CDF) matching method, linear regression (LRG) method, mean relative difference (MRD) method and linear rescaling (LRS) method were used to define observation operators. The results showed observation operators significantly improved the predictions compared to when using spatial averages of uplands as the direct surrogates for gullies. Among different methods, the CDF matching method performed best in estimating soil moisture in gullies followed by the LRG, LRS and MRD methods. Validation analysis showed that the linear observation operators such as LRS, MRD and LRG had better temporal transferability than the nonlinear operators. The MRD observation operators for various layers could successfully transfer in time whereas temporal transferability only succeeds to a limited extent for other observation operators. Furthermore, the MRD, LRG and LRS methods exhibited better vertical transferability than the CDF matching method. However, the transferability of observation operators across the whole root zone layers was not successful.

A refined index of model performance: a rejoinder

AbstractWillmott et al. [Willmott CJ, Robeson SM, Matsuura K. 2012. A refined index of model performance. International Journal of Climatology, forthcoming. DOI:10.1002/joc.2419.] recently suggest a refined index of model performance (dr) that they purport to be superior to other methods. Their refined index ranges from − 1.0 to 1.0 to resemble a correlation coefficient, but it is merely a linear rescaling of our modified coefficient of efficiency (E1) over the positive portion of the domain of dr. We disagree with Willmott et al. (2012) that dr provides a better interpretation; rather, E1 is more easily interpreted such that a value of E1 = 1.0 indicates a perfect model (no errors) while E1 = 0.0 indicates a model that is no better than the baseline comparison (usually the observed mean). Negative values of E1 (and, for that matter, dr < 0.5) indicate a substantially flawed model as they simply describe a ‘level of inefficacy’ for a model that is worse than the comparison baseline. Moreover, while dr is piecewise continuous, it is not continuous through the second and higher derivatives. We explain why the coefficient of efficiency (E or E2) and its modified form (E1) are superior and preferable to many other statistics, including dr, because of intuitive interpretability and because these indices have a fundamental meaning at zero.We also expand on the discussion begun by Garrick et al. [Garrick M, Cunnane C, Nash JE. 1978. A criterion of efficiency for rainfall‐runoff models. Journal of Hydrology 36: 375‐381.] and continued by Legates and McCabe [Legates DR, McCabe GJ. 1999. Evaluating the use of “goodness‐of‐fit” measures in hydrologic and hydroclimatic model validation. Water Resources Research 35(1): 233‐241.] and Schaefli and Gupta [Schaefli B, Gupta HV. 2007. Do Nash values have value? Hydrological Processes 21: 2075‐2080. DOI: 10.1002/hyp.6825.]. This important discussion focuses on the appropriate baseline comparison to use, and why the observed mean often may be an inadequate choice for model evaluation and development. Copyright © 2012 Royal Meteorological Society

Irrelevant conjunction and the ratio measure or historical skepticism

It is widely believed that one should not become more confident that all swans are white and all lions are brave simply by observing white swans. Irrelevant conjunction or “tacking” of a theory onto another is often thought problematic for Bayesianism, especially given the ratio measure of confirmation considered here. It is recalled that the irrelevant conjunct is not confirmed at all. Using the ratio measure, the irrelevant conjunction is confirmed to the same degree as the relevant conjunct, which, it is argued, is ideal: the irrelevant conjunct is irrelevant. Because the past’s really having been as it now appears to have been is an irrelevant conjunct, present evidence confirms theories about past events only insofar as irrelevant conjunctions are confirmed. Hence the ideal of not confirming irrelevant conjunctions would imply that historical claims are not confirmed. Confirmation measures partially realizing that ideal make the confirmation of historical claims by present evidence depend strongly on the (presumably subjective) degree of belief in the irrelevant conjunct. The unusually good behavior of the ratio measure has a bearing on the problem of measure sensitivity. For non-statistical hypotheses, Bayes’ theorem yields a fractional linear transformation in the prior probability, not a linear rescaling, so even the ratio measure arguably does not aptly measure confirmation in such cases.

Rigidity and mapping class group for abstract tiling spaces

AbstractWe study abstract self-affine tiling actions, which are an intrinsically defined class of minimal expansive actions of ℝdon a compact space. They include the translation actions on the compact spaces associated to aperiodic repetitive tilings or Delone sets in ℝd. In the self-similar case, we show that the existence of a homeomorphism between tiling spaces implies conjugacy of the actions up to a linear rescaling. We also introduce the general linear group of an (abstract) tiling, prove its discreteness, and show that it is naturally isomorphic with the (pointed) mapping class group of the tiling space. To illustrate our theory, we compute the mapping class group for a five-fold symmetric Penrose tiling.

A Decomposition of the Rotation of Circle

The paper concerns the dynamics-related properties of the rotation map of a circle (rotation of the plane). A self-similar structure of orbits of the rotation map is established. That is, a possibility of decomposition of orbits of a given rotation map into a finite set of orbits of other such maps is proved—it is shown that every orbit of iterates of the rotation of circle on irrational angle, after linear re-scaling of its argument can be represented as a finite set of such orbits situated on another circles. A pointwise self-similarity of classical trigonometric system is established and an application to Fourier expansion, which emphasizes a possibility of shifting of signals with respect to time, is presented. The free mechanical motion is also considered. A special dynamical spectrum of frequencies or speeds, associated with a given uniform circular or rectilinear motion, is defined. We prove that an appropriate fragmentation of time axis yields a decomposition of a given orbit of the free continuous-time motion into a set of such orbits propagating in new time and such decomposition is consistent with the decomposition of the per time unit discrete motion. Particularly, our theorems assert that due to a piecewise-linear transform of spatial and time variables the rectilinear rays change their direction.

Efficiency analysis to incorporate interval-scale data

We develop an approach to efficiency analysis to enable us to incorporate interval-scale data in addition to ratio-scale data. Our approach introduces a measure of inefficiency and identifies efficient units as is done in Data Envelopment Analysis. The basic idea in our approach is to find the “best” hyperplane separating the units that are better and worse than each unit. “Best” is defined in such a way that the number of not-better units is maximal. The efficiency measure is defined as a proportion of not-better units to all units. The results are invariant under a strictly increasing linear re-scaling of any input- or output-variables. Thus zeroes or negative values do not cause problems for the analysis. The approach is used to analyze the data of the research evaluation exercise recently carried out at the University of Joensuu, Finland.

Local E11 and the gauging of the trombone symmetry

In any dimension, the positive level generators of the very extended Kac–Moody algebra E11 with completely antisymmetric spacetime indices are associated with the form fields of the corresponding maximal supergravity. We consider the local E11 algebra, that is the algebra obtained by enlarging these generators of E11 in such a way that the global E11 symmetries are promoted to gauge symmetries. These are the gauge symmetries of the corresponding massless maximal supergravity. We show the existence of a new type of deformation of the local E11 algebra, which corresponds to the gauging of the symmetry under rescaling of the fields. In particular, we show how the gauged IIA theory of Howe, Lambert and West is obtained from an 11-dimensional group element that only depends on the 11th coordinate via a linear rescaling. We then show how this results in ten dimensions in a deformed local E11 algebra of a new type.

Education

Take a random polygon, find the the midpoints of its sides, and construct a new polygon from them. Repeat ad infinitum. What happens? This simple geometric iteration is at the heart of this issue's Education section offering, “From Random Polygon to Ellipse: An Eigenanalysis” by Adam N. Elmachtoub and Charles F. Van Loan. A cursory numerical experiment (although the traditionalist may wish to try this with a compass and straightedge) shows that the polygon shrinks to a point which the astute observer will realize is the centroid of the original vertices. If we now zoom in on the sequence of shrinking polygons, most easily accomplished by rescaling at each iteration, we see even more order. The rescaled vertices approach a binary oscillation between two polygons, both of which can be inscribed in the same ellipse. While students posed with these questions may think they are pursuing a geometry problem, the not-too-secret agenda is to introduce them to some fundamental ideas of numerical linear algebra, namely iteration, rescaling, and convergence. The authors guide us to write the problem as an iterative algorithm whose properties can be analyzed via eigenvalues and eigenvectors and whose convergence is akin to the power method for finding the eigenvalue of largest magnitude of a matrix. One might consider threading this example through a course in numerical analysis or numerical linear algebra, having the students explore the behavior at the beginning of the course (a great programming exercise), and eventually leading them through the analysis after introducing iterative methods for computing eigenvalues and eigenvectors. The problem opens the door to many generalizations, including exploring what happens for different geometric constructions and asking for geometric interpretations for other iterative linear algebra algorithms. The midpoint and rescaling construction is also simple enough to show to freshmen or even high school students; who knows, perhaps a few of them may be drawn to the study of mathematics by the allure of binary blinking ellipses?

Microdynamics for Nash maps

We investigate a family of maps that arises from a model in economics and game theory. It has some features similar to renormalization and some similar to intermittency. In a one-parameter family of maps in dimension 2, when the parameter goes to 0, the maps converge to the identity. Nevertheless, after a linear rescaling of both space and time, we get maps with attracting invariant closed curves. As the parameter goes to 0, those curves converge in a strong sense to a certain circle. We call those phenomena microdynamics. The model can be also understood as a family of discrete time approximations to a Brown-von Neumann differential equation.

Some identities for the moments of random variables having a linear rescaling the Student's t distribution

Taking advantage of Nadarajah's representation of modified Chebyshev polynomials by means of the random variable that has the Student's t distribution [S. Nadarajah, On modified Chebyshev polynomials, J. Math. Anal. Appl. 334 (2007) 1492–1494] the scope of this paper is the generation of the identities for the moments of linear rescaling random variables that have the Student's t distribution.