Missing value imputation in a data matrix using the regularised singular value decomposition.

Some statistical analysis techniques may require complete data matrices, but a frequent problem in the construction of databases is the incomplete collection of information for different reasons. One option to tackle the problem is to estimate and impute the missing data. This paper describes a form of imputation that mixes regression with lower rank approximations. To improve the quality of the imputations, a generalisation is proposed that replaces the singular value decomposition (SVD) of the matrix with a regularised SVD in which the regularisation parameter is estimated by cross-validation. To evaluate the performance of the proposal, ten sets of real data from multienvironment trials were used. Missing values were created in each set at four percentages of missing not at random, and three criteria were then considered to investigate the effectiveness of the proposal. The results show that the regularised method proves very competitive when compared to the original method, beating it in several of the considered scenarios. As it is a very general system, its application can be extended to all multivariate data matrices. •The imputation method is modified through the inclusion of a stable and efficient computational algorithm that replaces the classical SVD least squares criterion by a penalised criterion. This penalty produces smoothed eigenvectors and eigenvalues that avoid overfitting problems, improving the performance of the method when the penalty is necessary. The size of the penalty can be determined by minimising one of the following criteria: the prediction errors, the Procrustes similarity statistic or the critical angles between subspaces of principal components.

Estimating the number of principal components via Split-Half Eigenvector Matching (SHEM).

Estimating the number of principal components to retain for dimension reduction is a critical step in many applications of principal component analysis. Common methods may not be optimal, however. The current paper presents an alternative procedure that aims to recover the true number of principal components, in the sense of the number of independent vectors involved in the generation of the data.•Data are split into random halves repeatedly.•For each split, the eigenvectors in one half are compared to those in the other.•The split between high and low similarities is used to estimate the number of principal components. The method is a proof of principle that similarity over split-halves of the data may provide a useful approach to estimating the number of components in dimension reduction, or of similar dimensions in other models.

Hydropower Plants as Dispersal Barriers in Freshwater Species Distribution Models: Using Restrictions through Asymmetrical Dispersal Predictors.

Hydropower plants represent one of the greatest threats for freshwater fish by fragmenting the habitat and avoiding the species dispersal. This type of dispersal barrier is often disregarded when predicting freshwater species distribution due to the complexity in inserting the species dispersal routes, and thus the barriers, into the models. Here, we evaluate the impact of including hydroelectric dams into species distribution models through asymmetrical dispersal predictors on the predicted geographic distribution of freshwater fish species. For this, we used asymmetrical dispersal (i.e., AEM) as predictors for modeling the distribution of 29 native fish species of Tocantins-Araguaia River basin. After that, we included the hydropower power plant (HPP) location into the asymmetrical binary matrix for the AEM construction by removing the connections where the HPP is located, representing the downstream disconnection a dam causes in the fish species dispersal route. Besides having higher predicted accuracy, the models using the HPP information generated more realistic predictions, avoiding overpredictions to areas suitable but limited to the species dispersal due to an anthropic barrier. Furthermore, the predictions including HPPs showed higher loss of species richness and nestedness (i.e., loss of species instead of replacement), especially for the southeastern area which concentrates most planned and built HPPs. Therefore, using dispersal constraints in species distribution models increases the reliability of the predictions by avoiding overpredictions based on premise of complete access by the species to any area that is climatically suitable regardless of dispersal barriers or capacity. In conclusion, in this study, we use a novel method of including dispersal constraints into distribution models through a priori insertion of their location within the asymmetrical dispersal predictors, avoiding a posteriori adjustment of the predicted distribution.

Correlation between diffusion tensor indices and fascicular morphometric parameters of peripheral nerve.

Introduction: Diffusion tensor imaging (DTI) is a magnetic resonance imaging (MRI) technique that measures the anisotropy of water diffusion. Clinical magnetic resonance imaging scanners enable visualization of the structural integrity of larger axonal bundles in the central nervous system and smaller structures like peripheral nerves; however, their resolution for the depiction of nerve fascicular morphology is limited. Accordingly, high-field strength MRI and strong magnetic field gradients are needed to depict the fascicular pattern. The study aimed to quantify diffusion tensor indices with high-field strength MRI within different anatomical compartments of the median nerve and determine if they correlate with nerve structure at the fascicular level. Methods: Three-dimensional pulsed gradient spin-echo (PGSE) imaging sequence in 19 different gradient directions and b value 1,150 s/mm2 was performed on a 9.4T wide-bore vertical superconducting magnet. Nine-millimeter-long segments of five median nerve samples were obtained from fresh cadavers and acquired in sixteen 0.625 mm thick slices. Each nerve sample had the fascicles, perineurium, and interfascicular epineurium segmented. The diffusion tensor was calculated from the region-average diffusion-weighted signals for all diffusion gradient directions. Subsequently, correlations between diffusion tensor indices of segmentations and nerve structure at the fascicular level (number of fascicles, fascicular ratio, and cross-sectional area of fascicles or nerve) were assessed. The acquired diffusion tensor imaging data was employed for display with trajectories and diffusion ellipsoids. Results: The nerve fascicles proved to be the most anisotropic nerve compartment with fractional anisotropy 0.44 ± 0.05. In the interfascicular epineurium, the diffusion was more prominent in orthogonal directions with fractional anisotropy 0.13 ± 0.02. Diffusion tensor indices within the fascicles and perineurium differed significantly between the subjects (p < 0.0001); however, there were no differences within the interfascicular epineurium (p ≥ 0.37). There were no correlations between diffusion tensor indices and nerve structure at the fascicular level (p ≥ 0.29). Conclusion: High-field strength MRI enabled the depiction of the anisotropic diffusion within the fascicles and perineurium. Diffusion tensor indices of the peripheral nerve did not correlate with nerve structure at the fascicular level. Future studies should investigate the relationship between diffusion tensor indices at the fascicular level and axon- and myelin-related parameters.

Moisture Detection in Tree Trunks in Semiarid Lands Using Low-Cost Non-Invasive Capacitive Sensors with Statistical Based Anomaly Detection Approach.

This paper focuses on building a non-invasive, low-cost sensor that can be fitted over tree trunks growing in a semiarid land environment. It also proposes a new definition that characterizes tree trunks' water retention capabilities mathematically. The designed sensor measures the variations in capacitance across its probes. It uses amplification and filter stages to smooth the readings, requires little power, and is operational over a 100 kHz frequency. The sensor sends data via a Long Range (LoRa) transceiver through a gateway to a processing unit. Field experiments showed that the system provides accurate readings of the moisture content. As the sensors are non-invasive, they can be fitted to branches and trunks of various sizes without altering the structure of the wood tissue. Results show that the moisture content in tree trunks increases exponentially with respect to the measured capacitance and reflects the distinct differences between different tree types. Data of known healthy trees and unhealthy trees and defective sensor readings have been collected and analysed statistically to show how anomalies in sensor reading baseds on eigenvectors and eigenvalues of the fitted curve coefficient matrix can be detected.

Translation in the cell under fierce competition for shared resources: a mathematical model.

During translation, mRNAs 'compete' for shared resources. Under stress conditions, during viral infection and also in high-throughput heterologous gene expression, these resources may become scarce, e.g. the pool of free ribosomes is starved, and then the competition may have a dramatic effect on the global dynamics of translation in the cell. We model this scenario using a network that includes m ribosome flow models (RFMs) interconnected via a pool of free ribosomes. Each RFM models ribosome flow along an mRNA molecule, and the pool models the shared resource. We assume that the number of mRNAs is large, so many ribosomes are attached to the mRNAs, and the pool is starved. Our analysis shows that adding an mRNA has an intricate effect on the total protein production. The new mRNA produces new proteins, but the other mRNAs produce less proteins, as the pool that feeds these mRNAs now has a smaller abundance of ribosomes. As the number of mRNAs increases, the marginal utility of adding another mRNA diminishes, and the total protein production rate saturates to a limiting value. We demonstrate our approach using an example of insulin protein production in a cell-free system.

An Improved Kinect Recognition Method for Identifying Unsafe Behaviors of Metro Passengers.

In order to solve the problem of the low action recognition accuracy of passengers' unsafe behaviors caused by redundant joints, this study proposes an efficient recognition method based on a Kinect sensor. The method uses the pelvis as the starting point of the vector and high-frequency bone joints as the end point to construct the recognition feature vector. The joint angle difference between actions is obtained by using the cosine law, and the initial test result is converted into action similarity combined with the DTW similarity algorithm. Considering the combination of 3 angle features and 4 joint feature selection methods, 12 combined recognition models are formed. A comparative experiment was carried out to identify five types of unsafe behaviors of metro passengers-punch, one-armed call for help, arms for help, trip forward and trip backwards. The results show that the overall selection of joints has a poor similarity effect and cannot achieve the purpose of recognition. The overall recognition model effect of the local "pelvis divergence method" is higher than that of the local "adjacent joint method", and the local "pelvis divergence method" has the highest recognition result of the maximum angle difference model, and the recognition results of the five unsafe behaviors are 86.9%, 89.2%, 85.5%, 86.7%, and 88.3%, respectively, and the recognition accuracy of this method is 95.7%, indicating the feasibility of the model. The recognition results are more concentrated and more stable, which significantly improves the recognition rate of metro passengers' unsafe behavior.

Asymptotic Theory of Eigenvectors for Random Matrices with Diverging Spikes.

Characterizing the asymptotic distributions of eigenvectors for large random matrices poses important challenges yet can provide useful insights into a range of statistical applications. To this end, in this paper we introduce a general framework of asymptotic theory of eigenvectors (ATE) for large spiked random matrices with diverging spikes and heterogeneous variances, and establish the asymptotic properties of the spiked eigenvectors and eigenvalues for the scenario of the generalized Wigner matrix noise. Under some mild regularity conditions, we provide the asymptotic expansions for the spiked eigenvalues and show that they are asymptotically normal after some normalization. For the spiked eigenvectors, we establish asymptotic expansions for the general linear combination and further show that it is asymptotically normal after some normalization, where the weight vector can be arbitrary. We also provide a more general asymptotic theory for the spiked eigenvectors using the bilinear form. Simulation studies verify the validity of our new theoretical results. Our family of models encompasses many popularly used ones such as the stochastic block models with or without overlapping communities for network analysis and the topic models for text analysis, and our general theory can be exploited for statistical inference in these large-scale applications.

On the rank-deficient canonical correlation technique solved by analytic spectral decomposition.

Regularization is a well-known and used statistical approach covering individual points or limit approximations. In this study, the canonical correlation analysis (CCA) process of the paths is discussed with partial least squares (PLS) as the other boundary covering transformation to a symmetric eigenvalue (or singular value) problem dependent on a parameter. Two regularizations of the original criterion in the parameterization domain are compared, i.e. using projection and by identity matrix. We discuss the existence and uniqueness of the analytic path for eigenvalues and corresponding elements of eigenvectors. Specifically, canonical analysis is applied to an ill-conditioned case of singular within-sets input matrices encompassing tourism accommodation data.

Missing value imputation using least squares techniques in contaminated matrices.

This paper describes strategies to reduce the possible effect of outliers on the quality of imputations produced by a method that uses a mixture of two least squares techniques: regression and lower rank approximation of a matrix. To avoid the influence of discrepant data and maintain the computational speed of the original scheme, pre-processing options were explored before applying the imputation method. The first proposal is to previously use a robust singular value decomposition, the second is to detect outliers and then treat the potential outliers as missing. To evaluate the proposed methods, a cross-validation study was carried out on ten complete matrices of real data from multi-environment trials. The imputations were compared with the original data using three statistics: a measure of goodness of fit, the squared cosine between matrices and the prediction error. The results show that the original method should be replaced by one of the options presented here because outliers can cause low quality imputations or convergence problems.•The imputation algorithm based on Gabriel's cross-validation method uses two least squares techniques that can be affected by the presence of outliers. The inclusion of a robust singular value decomposition allows both to robustify the procedure and to detect outliers and consider them later as missing. These forms of pre-processing ensure that the algorithm performs well on any dataset that has a matrix form with suspected contamination.

A Vital Signs Fast Detection and Extraction Method of UWB Impulse Radar Based on SVD.

The identification of weak vital signs has always been one of the difficulties in the field of life detection. In this paper, a novel vital sign detection and extraction method with high efficiency, high precision, high sensitivity and high signal-to-noise ratio is proposed. Based on the NVA6100 pulse radar system, the radar matrix which contains several radar pulse detection signals is received. According to the characteristics of vital signs and radar matrices, the Singular Value Decomposition (SVD) is adopted to perform signal denoising and decomposition after preprocessing, and the temporal and spatial eigenvectors of each principal component are obtained. Through the energy proportion screening, the Wavelet Transform decomposition and linear trend suppression, relatively pure vital signs in each principal component, are obtained. The human location is detected by the Energy Entropy of spatial eigenvectors, and the respiratory signal and heartbeat signal are restored through a Butterworth Filter and an MTI harmonic canceller. Finally, through an analysis of the performance of the algorithm, it is proved to have the properties of efficiency and accuracy.

Posterior cortical atrophy phenotypic heterogeneity revealed by decoding 18F-FDG-PET.

Posterior cortical atrophy is a neurodegenerative syndrome with a heterogeneous clinical presentation due to variable involvement of the left, right, dorsal and ventral parts of the visual system, as well as inconsistent involvement of other cognitive domains and systems. 18F-fluorodeoxyglucose (FDG)-PET is a sensitive marker for regional brain damage or dysfunction, capable of capturing the pattern of neurodegeneration at the single-participant level. We aimed to leverage these inter-individual differences on FDG-PET imaging to better understand the associations of heterogeneity of posterior cortical atrophy. We identified 91 posterior cortical atrophy participants with FDG-PET data and abstracted demographic, neurologic, neuropsychological and Alzheimer's disease biomarker data. The mean age at reported symptom onset was 59.3 (range: 45-72 years old), with an average disease duration of 4.2 years prior to FDG-PET scan, and a mean education of 15.0 years. Females were more common than males at 1.6:1. After standard preprocessing steps, the FDG-PET scans for the cohort were entered into an unsupervised machine learning algorithm which first creates a high-dimensional space of inter-individual covariance before performing an eigen-decomposition to arrive at a low-dimensional representation. Participant values ('eigenbrains' or latent vectors which represent principle axes of inter-individual variation) were then compared to the clinical and biomarker data. Eight eigenbrains explained over 50% of the inter-individual differences in FDG-PET uptake with left (eigenbrain 1) and right (eigenbrain 2) hemispheric lateralization representing 24% of the variance. Furthermore, eigenbrain-loads mapped onto clinical and neuropsychological data (i.e. aphasia, apraxia and global cognition were associated with the left hemispheric eigenbrain 1 and environmental agnosia and apperceptive prosopagnosia were associated with the right hemispheric eigenbrain 2), suggesting that they captured important axes of normal and abnormal brain function. We used NeuroSynth to characterize the eigenbrains through topic-based decoding, which supported the idea that the eigenbrains map onto a diverse set of cognitive functions. These eigenbrains captured important biological and pathophysiologic data (i.e. limbic predominant eigenbrain 4 patterns being associated with older age of onset compared to frontoparietal eigenbrain 7 patterns being associated with younger age of onset), suggesting that approaches that focus on inter-individual differences may be important to better understand the variability observed within a neurodegenerative syndrome like posterior cortical atrophy.

Beamforming Using Exact Evaluation of Leakage and Ergodic Capacity of MU-MIMO System.

Closed-form evaluation of key performance indicators (KPIs) of telecommunication networks help perform mathematical analysis under several network configurations. This paper deals with a recent mathematical approach of indefinite quadratic forms to propose simple albeit exact closed-form expressions of the expectation of two significant logarithmic functions. These functions formulate KPIs which include the ergodic capacity and leakage rate of multi-user multiple-input multiple-output (MU-MIMO) systems in Rayleigh fading channels. Our closed-form expressions are generic in nature and they characterize several network configurations under statistical channel state information availability. As a demonstrative example of the proposed characterization, the derived expressions are used in the statistical transmit beamformer design in a broadcast MU-MIMO system to portray promising diversity gains using standalone or joint maximization techniques of the ergodic capacity and leakage rate. The results presented are validated by Monte Carlo simulations.

ASYMMETRY HELPS: EIGENVALUE AND EIGENVECTOR ANALYSES OF ASYMMETRICALLY PERTURBED LOW-RANK MATRICES.

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix M ⋆ ∈ ℝ n × n , yet only a randomly perturbed version M is observed. The noise matrix M - M ⋆ is composed of independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise if, for example, we have two independent samples for each entry of M ⋆ and arrange them in an asymmetric fashion. The aim is to estimate the leading eigenvalue and the leading eigenvector of M ⋆. We demonstrate that the leading eigenvalue of the data matrix M can be O ( n ) times more accurate (up to some log factor) than its (unadjusted) leading singular value of M in eigenvalue estimation. Moreover, the eigen-decomposition approach is fully adaptive to heteroscedasticity of noise, without the need of any prior knowledge about the noise distributions. In a nutshell, this curious phenomenon arises since the statistical asymmetry automatically mitigates the bias of the eigenvalue approach, thus eliminating the need of careful bias correction. Additionally, we develop appealing non-asymptotic eigenvector perturbation bounds; in particular, we are able to bound the perterbation of any linear function of the leading eigenvector of M (e.g. entrywise eigenvector perturbation). We also provide partial theory for the more general rank-r case. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigen-decomposition could sometimes be quite beneficial.

SEIR order parameters and eigenvectors of the three stages of completed COVID-19 epidemics: with an illustration for Thailand January to May 2020.

By end of October 2020, the COVID-19 pandemic has taken a tragic toll of 1150 000 lives and this number is expected to increase. Despite the pandemic is raging in most parts of the world, in a few countries COVID-19 epidemics subsided due to successful implementations of intervention measures. A unifying perspective of the beginnings, middle stages, and endings of such completed COVID-19 epidemics is developed based on the order parameter and eigenvalue concepts of nonlinear physics, in general, and synergetics, in particular. To this end, a standard susceptible-exposed-infected-recovered (SEIR) epidemiological model is used. It is shown that COVID-19 epidemic outbreaks follow a suitably defined SEIR order parameter. Intervention measures switch the eigenvalue of the order parameter from a positive to a negative value, and in doing so, stabilize the COVID-19 disease-free state. The subsiding of COVID-19 epidemics eventually follows the remnant of the order parameter of the infection dynamical system. These considerations are illustrated for the COVID-19 epidemic in Thailand from January to May 2020. The decay of effective contact rates throughout the three epidemic stages is demonstrated. Evidence for the sign-switching of the dominant eigenvalue is given and the order parameter and its stage-3 remnant are identified. The presumed impacts of interventions measures implemented in Thailand are discussed in this context.

Modelling road traffic Noise under heterogeneous traffic conditions using the graph-theoretic approach.

A traffic noise system involves several subsystems like road traffic subsystem, human subsystem, environment subsystem, traffic network subsystem, and urban prosperity subsystem. The study's main aim was to develop road traffic noise models using a graph theory approach involving the parameters related to road traffic subsystem. The road traffic subsystem variables selected for the modeling purposes included vehicular speed, traffic volume, carriageway width, number of heavy vehicles, and number of honking events. The interaction of the selected variables considered in the form of permanent noise function is given in the matrix form. Eigenvalues and corresponding eigenvectors are calculated for removing any human judgmental error. The permanent noise function matrix was then updated using the eigenvectors, which was ultimately utilized for obtaining the permanent noise index. Data regarding the selected variables were collected for three months, and the noise parameters included in the study were equivalent noise level (Leq,1h), maximum noise level (L10,1h), and background noise level (L90,1h). A logarithmic transformation was applied to the permanent noise index and linear regression models were developed for Leq,1h , L90,1h , and L10,1h respectively. The models were validated using the data collected from the same locations for nine months. The models were found to provide satisfactory results, although the results were somewhat overestimated. The method can prove beneficial for estimating future noise levels, given the expected changes in values for the independent variables considered in the study.

Structure of the G-matrix in relation to phenotypic contributions to fitness.

Classical theory in population genetics includes derivation of the stationary distribution of allele frequencies under balance between selection, genetic drift, and mutation. Here we investigate the simplest generalization of these single locus models to quantitative genetics with many loci, assuming simple additive effects on a set of phenotypes and a linear approximation to the fitness function. Genetic effects and pleiotropy are simulated by a prescribed stochastic model. Our goal is to analyze the structure of the G-matrix at stasis when the model is not very close to being neutral. The smallest eigenvalue of the G-matrix is practically zero by Fisher's fundamental theorem for natural selection and the fitness function is approximately a linear function of the corresponding eigenvector. Evolution of genetic trade-offs is closely linked to the fitness function. If a single locus never codes for more than two traits, then additive genetic covariance between two phenotype components always has the opposite sign of the product of their coefficients in the fitness function under no mutation, a pattern that is likely to occur frequently also in more complex models. In our major examples only 1-2 percent of the loci are over-dominant for fitness, but they still account for practically all dominance variance in fitness as well as all contributions to the G-matrix. These analyses show that the structure of the G-matrix reveals important information about the contribution of different traits to fitness.

SEAIR Epidemic spreading model of COVID-19.

We study Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) epidemic spreading model of COVID-19. It captures two important characteristics of the infectiousness of COVID-19: delayed start and its appearance before onset of symptoms, or even with total absence of them. The model is theoretically analyzed in continuous-time compartmental version and discrete-time version on random regular graphs and complex networks. We show analytically that there are relationships between the epidemic thresholds and the equations for the susceptible populations at the endemic equilibrium in all three versions, which hold when the epidemic is weak. We provide theoretical arguments that eigenvector centrality of a node approximately determines its risk to become infected.

Species diffusion in clinopyroxene solid solution in the diopside-anorthite system.

The coupled multicomponent diffusion of the species Ca2Si2O6, CaAl2SiO6 and Mg2Si2O6 was determined in diopside crystals in the diopside/anorthite (Di/An) system at temperatures (T) of 1110-1260 °C and oxygen fugacities (fO 2 ) between 1.0 log unit below and above the fayalite-magnetite-quartz equilibrium (FMQ ± 1). Diffusion couples were prepared by the seed overgrowth technique. Element concentration profiles were measured perpendicular to the rim/core interface by step-scanning profiling with a field emission gun scanning electron microscope (FEG-SEM). The multicomponent diffusion matrix was solved by fitting its eigenvalues (λ) and eigenvectors (v) to the measured concentration profiles. The full diffusion matrix D can be recovered by using the formula D = P Λ P - 1 resulting in the following equation: D Di/An = 1.00 - 0.67 - 0.38 1.00 λ 1 T 0 0 λ 2 T 1.00 - 0.67 - 0.38 1.00 - 1 . The eigenvalues (λ1 and λ2) represent upper limit values and are described by the following Arrhenius-type equations: λ 1 Di/An = 10 - 15.98 ± 1.17 × exp - 114.4 ± 32.8 kJ/mol RT , λ 2 Di/An = 10 - 16.23 ± 1.17 × exp - 114.4 ± 32.8 kJ/mol RT , where λ1 and λ2 are the first and second eigenvalue of the diffusion matrix in m2 s-1 , R is the gas constant and T is the temperature in K. The dominant eigenvalue (λ1) is one quarter order of magnitude larger than the second eigenvalue (λ2). The eigenvectors are constant for all experiments inferring that the entire D matrix can be described with the eigenvalues as the only T-dependent parameter. Additionally, the derived diffusion data and modeling approach were applied to constrain the duration of magmatic processes recorded in zoned clinopyroxene (cpx) phenocrysts from a basaltic, post-plutonic dyke of the Tertiary Adamello batholiths (N-Italy). The results reveal residence times of the overgrown cpx prior to final emplacement in the range of 0.25-1.7 years (lower limit values) testifying that the data and method can be applied to model cpx diffusion profiles in complex natural cpx.

Discrete-time population dynamics on the state space of measures.

If the individual state space of a structured population is given by a metric space S, measures µ on the σ-algebra of Borel subsets T of S offer a modeling tool with a natural interpretation: µ(T) is the number of individuals with structural characteristics in the set T. A discrete-time population model is given by a population turnover map F on the cone of finite nonnegative Borel measures that maps the structural population distribution of a given year to the one of the next year. Under suitable assumptions, F has a first order approximation at the zero measure (the extinction fixed point), which is a positive linear operator on the ordered vector space of real measures and can be interpreted as a basic population turnover operator. For a semelparous population, it can be identified with the next generation operator. A spectral radius can be defined by the usual Gelfand formula.We investigate in how far it serves as a threshold parameter between population extinction and population persistence. The variation norm on the space of measures is too strong to give the basic turnover operator enough compactness that its spectral radius is an eigenvalue associated with a positive eigenmeasure. A suitable alternative is the flat norm (also known as (dual) bounded Lipschitz norm), which, as a trade-off, makes the basic turnover operator only continuous on the cone of nonnegative measures but not on the whole space of real measures.