Uncertainty in ellipse fitting using a flatbed scanner: development and experimental verification

In the field of dimensional metrology, the use of optical measuring machines requires the handling of a large number of measurement points, or scanning points, taken from the image of the measurand. The presence of correlation between these measurement points has a significant influence on the uncertainty of the result. The aim of this work is the development of an estimation procedure for the uncertainty of measurement in a geometrically elliptical shape, taking into account the correlation between the scanning points. These points are obtained from an image produced using a commercial flat bed scanner. The characteristic parameters of the ellipse (coordinates of the center, semi-axes and the angle of the semi-major axis with regard to the horizontal) are determined using a least squares fit and orthogonal distance regression. The uncertainty is estimated using the information from the auto-correlation function of the residuals and is propagated through the fitting algorithm according to the rules described in Evaluation of Measurement Data—Supplement 2 to the ‘Guide to the Expression of Uncertainty in Measurement’—Extension to any number of output quantities. By introducing the concept of cut-off length, it can be observed how it is possible to take into account the presence of the correlation in the estimation of uncertainty in a very simple way while avoiding underestimation.

Uncertainty Estimation for Performance Evaluation of a Confocal Microscope as Metrology Equipment

Both in industry and research, the quality control of micrometric manufactured parts is based on the measurement of parameters whose traceability is sometimes difficult to guarantee. In some of these parts, the confocal microscopy shows great aptitudes to characterize a measurand qualitatively and quantitatively. The confocal microscopy allows the acquisition of 2D and 3D images that are easily manipulated. Nowadays, this equipment is manufactured by many different brands, each of them claiming a resolution probably not in accord to their real performance. The Laser Center (Technical University of Madrid) has a confocal microscope to verify the dimensions of the micro mechanizing in their own research projects. The present study pretends to confirm that the magnitudes obtained are true and reliable. To achieve this, a methodology for confocal microscope calibration is proposed, as well as an experimental phase for dimensionally valuing the equipment by 4 different standard positions, with its seven magnifications and the six objective lenses that the equipment currently has, in the x–y and z axis. From the results the uncertainty will be estimated along with an effect analysis of the different magnifications in each of the objective lenses.

Lidar uncertainty in complex terrain development of a bias correction methodology

El autor ha trabajado como parte del equipo de investigación en mediciones de viento en el Centro Nacional de Energías Renovables (CENER), España, en cooperación con la Universidad Politécnica de Madrid y la Universidad Técnica de Dinamarca. El presente reporte recapitula el trabajo de investigación realizado durante los últimos 4.5 años en el estudio de las fuentes de error de los sistemas de medición remota de viento, basados en la tecnología lidar, enfocado al error causado por los efectos del terreno complejo. Este trabajo corresponde a una tarea del paquete de trabajo dedicado al estudio de sistemas remotos de medición de viento, perteneciente al proyecto de intestigación europeo del 7mo programa marco WAUDIT. Adicionalmente, los datos de viento reales han sido obtenidos durante las campañas de medición en terreno llano y terreno complejo, pertenecientes al también proyecto de intestigación europeo del 7mo programa marco SAFEWIND. El principal objetivo de este trabajo de investigación es determinar los efectos del terreno complejo en el error de medición de la velocidad del viento obtenida con los sistemas de medición remota lidar. Con este conocimiento, es posible proponer una metodología de corrección del error de las mediciones del lidar. Esta metodología está basada en la estimación de las variaciones del campo de viento no uniforme dentro del volumen de medición del lidar. Las variaciones promedio del campo de viento son predichas a partir de los resultados de las simulaciones computacionales de viento RANS, realizadas para el parque experimental de Alaiz. La metodología de corrección es verificada con los resultados de las simulaciones RANS y validadas con las mediciones reales adquiridas en la campaña de medición en terreno complejo. Al inicio de este reporte, el marco teórico describiendo el principio de medición de la tecnología lidar utilizada, es presentado con el fin de familiarizar al lector con los principales conceptos a utilizar a lo largo de este trabajo. Posteriormente, el estado del arte es presentado en donde se describe los avances realizados en el desarrollo de la la tecnología lidar aplicados al sector de la energía eólica. En la parte experimental de este trabajo de investigación se ha estudiado los datos adquiridos durante las dos campañas de medición realizadas. Estas campañas has sido realizadas en terreno llano y complejo, con el fin de complementar los conocimiento adquiridos en casa una de ellas y poder comparar los efectos del terreno en las mediciones de viento realizadas con sistemas remotos lidar. La primer campaña experimental se desarrollo en terreno llano, en el parque de ensayos de aerogeneradores H0vs0re, propiedad de DTU Wind Energy (anteriormente Ris0). La segunda campaña experimental se llevó a cabo en el parque de ensayos de aerogeneradores Alaiz, propiedad de CENER. Exactamente los mismos dos equipos lidar fueron utilizados en estas campañas, haciendo de estos experimentos altamente relevantes en el contexto de evaluación del recurso eólico. Un equipo lidar está basado en tecnología de onda continua, mientras que el otro está basado en tecnología de onda pulsada. La velocidad del viento fue medida, además de con los equipos lidar, con anemómetros de cazoletas, veletas y anemómetros verticales, instalados en mástiles meteorológicos. Los sensores del mástil meteorológico son considerados como las mediciones de referencia en el presente estudio. En primera instancia, se han analizado los promedios diez minútales de las medidas de viento. El objetivo es identificar las principales fuentes de error en las mediciones de los equipos lidar causadas por diferentes condiciones atmosféricas y por el flujo no uniforme de viento causado por el terreno complejo. El error del lidar ha sido estudiado como función de varias propiedades estadísticas del viento, como lo son el ángulo vertical de inclinación, la intensidad de turbulencia, la velocidad vertical, la estabilidad atmosférica y las características del terreno. El propósito es usar este conocimiento con el fin de definir criterios de filtrado de datos. Seguidamente, se propone una metodología para corregir el error del lidar causado por el campo de viento no uniforme, producido por la presencia de terreno complejo. Esta metodología está basada en el análisis matemático inicial sobre el proceso de cálculo de la velocidad de viento por los equipos lidar de onda continua. La metodología de corrección propuesta hace uso de las variaciones de viento calculadas a partir de las simulaciones RANS realizadas para el parque experimental de Alaiz. Una ventaja importante que presenta esta metodología es que las propiedades el campo de viento real, presentes en las mediciones instantáneas del lidar de onda continua, puede dar paso a análisis adicionales como parte del trabajo a futuro. Dentro del marco del proyecto, el trabajo diario se realizó en las instalaciones de CENER, con supervisión cercana de la UPM, incluyendo una estancia de 1.5 meses en la universidad. Durante esta estancia, se definió el análisis matemático de las mediciones de viento realizadas por el equipo lidar de onda continua. Adicionalmente, los efectos del campo de viento no uniforme sobre el error de medición del lidar fueron analíticamente definidos, después de asumir algunas simplificaciones. Adicionalmente, durante la etapa inicial de este proyecto se desarrollo una importante trabajo de cooperación con DTU Wind Energy. Gracias a esto, el autor realizó una estancia de 1.5 meses en Dinamarca. Durante esta estancia, el autor realizó una visita a la campaña de medición en terreno llano con el fin de aprender los aspectos básicos del diseño de campañas de medidas experimentales, el estudio del terreno y los alrededores y familiarizarse con la instrumentación del mástil meteorológico, el sistema de adquisición y almacenamiento de datos, así como de el estudio y reporte del análisis de mediciones. ABSTRACT The present report summarizes the research work performed during last 4.5 years of investigation on the sources of lidar bias due to complex terrain. This work corresponds to one task of the remote sensing work package, belonging to the FP7 WAUDIT project. Furthermore, the field data from the wind velocity measurement campaigns of the FP7 SafeWind project have been used in this report. The main objective of this research work is to determine the terrain effects on the lidar bias in the measured wind velocity. With this knowledge, it is possible to propose a lidar bias correction methodology. This methodology is based on an estimation of the wind field variations within the lidar scan volume. The wind field variations are calculated from RANS simulations performed from the Alaiz test site. The methodology is validated against real scale measurements recorded during an eight month measurement campaign at the Alaiz test site. Firstly, the mathematical framework of the lidar sensing principle is introduced and an overview of the state of the art is presented. The experimental part includes the study of two different, but complementary experiments. The first experiment was a measurement campaign performed in flat terrain, at DTU Wind Energy H0vs0re test site, while the second experiment was performed in complex terrain at CENER Alaiz test site. Exactly the same two lidar devices, based on continuous wave and pulsed wave systems, have been used in the two consecutive measurement campaigns, making this a relevant experiment in the context of wind resource assessment. The wind velocity was sensed by the lidars and standard cup anemometry and wind vanes (installed on a met mast). The met mast sensors are considered as the reference wind velocity measurements. The first analysis of the experimental data is dedicated to identify the main sources of lidar bias present in the 10 minute average values. The purpose is to identify the bias magnitude introduced by different atmospheric conditions and by the non-uniform wind flow resultant of the terrain irregularities. The lidar bias as function of several statistical properties of the wind flow like the tilt angle, turbulence intensity, vertical velocity, atmospheric stability and the terrain characteristics have been studied. The aim of this exercise is to use this knowledge in order to define useful lidar bias data filters. Then, a methodology to correct the lidar bias caused by non-uniform wind flow is proposed, based on the initial mathematical analysis of the lidar measurements. The proposed lidar bias correction methodology has been developed focusing on the the continuous wave lidar system. In a last step, the proposed lidar bias correction methodology is validated with the data of the complex terrain measurement campaign. The methodology makes use of the wind field variations obtained from the RANS analysis. The results are presented and discussed. The advantage of this methodology is that the wind field properties at the Alaiz test site can be studied with more detail, based on the instantaneous measurements of the CW lidar. Within the project framework, the daily basis work has been done at CENER, with close guidance and support from the UPM, including an exchange period of 1.5 months. During this exchange period, the mathematical analysis of the lidar sensing of the wind velocity was defined. Furthermore, the effects of non-uniform wind fields on the lidar bias were analytically defined, after making some assumptions for the sake of simplification. Moreover, there has been an important cooperation with DTU Wind Energy, where a secondment period of 1.5 months has been done as well. During the secondment period at DTU Wind Energy, an important introductory learning has taken place. The learned aspects include the design of an experimental measurement campaign in flat terrain, the site assessment study of obstacles and terrain conditions, the data acquisition and processing, as well as the study and reporting of the measurement analysis.

Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelengths of Bragg gratings

A method of discriminating between temperature and strain effects in fibre sensing using a conventionally written, in-fibre Bragg grating is presented. The technique uses wavelength information from the first and second diffraction orders of the grating element to determine the wavelength dependent strain and temperature coefficients, from which independent temperature and strain measurements can be made. The authors present results that validate this matrix inversion technique and quantify the strain and temperature errors which can arise for a given uncertainty in the measurement of the reflected wavelength.

Uses of frontier efficiency methodologies and multi-criteria decision making for performance measurement in the energy sector

Guest editorial Ali Emrouznejad is a Senior Lecturer at the Aston Business School in Birmingham, UK. His areas of research interest include performance measurement and management, efficiency and productivity analysis as well as data mining. He has published widely in various international journals. He is an Associate Editor of IMA Journal of Management Mathematics and Guest Editor to several special issues of journals including Journal of Operational Research Society, Annals of Operations Research, Journal of Medical Systems, and International Journal of Energy Management Sector. He is in the editorial board of several international journals and co-founder of Performance Improvement Management Software. William Ho is a Senior Lecturer at the Aston University Business School. Before joining Aston in 2005, he had worked as a Research Associate in the Department of Industrial and Systems Engineering at the Hong Kong Polytechnic University. His research interests include supply chain management, production and operations management, and operations research. He has published extensively in various international journals like Computers & Operations Research, Engineering Applications of Artificial Intelligence, European Journal of Operational Research, Expert Systems with Applications, International Journal of Production Economics, International Journal of Production Research, Supply Chain Management: An International Journal, and so on. His first authored book was published in 2006. He is an Editorial Board member of the International Journal of Advanced Manufacturing Technology and an Associate Editor of the OR Insight Journal. Currently, he is a Scholar of the Advanced Institute of Management Research. Uses of frontier efficiency methodologies and multi-criteria decision making for performance measurement in the energy sector This special issue aims to focus on holistic, applied research on performance measurement in energy sector management and for publication of relevant applied research to bridge the gap between industry and academia. After a rigorous refereeing process, seven papers were included in this special issue. The volume opens with five data envelopment analysis (DEA)-based papers. Wu et al. apply the DEA-based Malmquist index to evaluate the changes in relative efficiency and the total factor productivity of coal-fired electricity generation of 30 Chinese administrative regions from 1999 to 2007. Factors considered in the model include fuel consumption, labor, capital, sulphur dioxide emissions, and electricity generated. The authors reveal that the east provinces were relatively and technically more efficient, whereas the west provinces had the highest growth rate in the period studied. Ioannis E. Tsolas applies the DEA approach to assess the performance of Greek fossil fuel-fired power stations taking undesirable outputs into consideration, such as carbon dioxide and sulphur dioxide emissions. In addition, the bootstrapping approach is deployed to address the uncertainty surrounding DEA point estimates, and provide bias-corrected estimations and confidence intervals for the point estimates. The author revealed from the sample that the non-lignite-fired stations are on an average more efficient than the lignite-fired stations. Maethee Mekaroonreung and Andrew L. Johnson compare the relative performance of three DEA-based measures, which estimate production frontiers and evaluate the relative efficiency of 113 US petroleum refineries while considering undesirable outputs. Three inputs (capital, energy consumption, and crude oil consumption), two desirable outputs (gasoline and distillate generation), and an undesirable output (toxic release) are considered in the DEA models. The authors discover that refineries in the Rocky Mountain region performed the best, and about 60 percent of oil refineries in the sample could improve their efficiencies further. H. Omrani, A. Azadeh, S. F. Ghaderi, and S. Abdollahzadeh presented an integrated approach, combining DEA, corrected ordinary least squares (COLS), and principal component analysis (PCA) methods, to calculate the relative efficiency scores of 26 Iranian electricity distribution units from 2003 to 2006. Specifically, both DEA and COLS are used to check three internal consistency conditions, whereas PCA is used to verify and validate the final ranking results of either DEA (consistency) or DEA-COLS (non-consistency). Three inputs (network length, transformer capacity, and number of employees) and two outputs (number of customers and total electricity sales) are considered in the model. Virendra Ajodhia applied three DEA-based models to evaluate the relative performance of 20 electricity distribution firms from the UK and the Netherlands. The first model is a traditional DEA model for analyzing cost-only efficiency. The second model includes (inverse) quality by modelling total customer minutes lost as an input data. The third model is based on the idea of using total social costs, including the firm’s private costs and the interruption costs incurred by consumers, as an input. Both energy-delivered and number of consumers are treated as the outputs in the models. After five DEA papers, Stelios Grafakos, Alexandros Flamos, Vlasis Oikonomou, and D. Zevgolis presented a multiple criteria analysis weighting approach to evaluate the energy and climate policy. The proposed approach is akin to the analytic hierarchy process, which consists of pairwise comparisons, consistency verification, and criteria prioritization. In the approach, stakeholders and experts in the energy policy field are incorporated in the evaluation process by providing an interactive mean with verbal, numerical, and visual representation of their preferences. A total of 14 evaluation criteria were considered and classified into four objectives, such as climate change mitigation, energy effectiveness, socioeconomic, and competitiveness and technology. Finally, Borge Hess applied the stochastic frontier analysis approach to analyze the impact of various business strategies, including acquisition, holding structures, and joint ventures, on a firm’s efficiency within a sample of 47 natural gas transmission pipelines in the USA from 1996 to 2005. The author finds that there were no significant changes in the firm’s efficiency by an acquisition, and there is a weak evidence for efficiency improvements caused by the new shareholder. Besides, the author discovers that parent companies appear not to influence a subsidiary’s efficiency positively. In addition, the analysis shows a negative impact of a joint venture on technical efficiency of the pipeline company. To conclude, we are grateful to all the authors for their contribution, and all the reviewers for their constructive comments, which made this special issue possible. We hope that this issue would contribute significantly to performance improvement of the energy sector.

Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelengths of Bragg gratings

A method of discriminating between temperature and strain effects in fibre sensing using a conventionally written, in-fibre Bragg grating is presented. The technique uses wavelength information from the first and second diffraction orders of the grating element to determine the wavelength dependent strain and temperature coefficients, from which independent temperature and strain measurements can be made. The authors present results that validate this matrix inversion technique and quantify the strain and temperature errors which can arise for a given uncertainty in the measurement of the reflected wavelength.

Application of calorimetric method for loss measurement of a SynRM drive system

Synchronous reluctance motors (SynRMs) are gaining in popularity in industrial drives due to their permanent magnet-free, competitive performance, and robust features. This paper studies the power losses in a 90-kW converter-fed SynRM drive by a calorimetric method in comparison of the traditional input-output method. After the converter and the motor were measured simultaneously in separate chambers, the converter was installed inside the large-size chamber next to the motor and the total drive system losses were obtained using one chamber. The uncertainty of both measurement methods is analyzed and discussed.

Identification of key temperature measurement technologies for the enhancement of product and equipment integrity in the light controlled factory

Thermal effects in uncontrolled factory environments are often the largest source of uncertainty in large volume dimensional metrology. As the standard temperature for metrology of 20°C cannot be achieved practically or economically in many manufacturing facilities, the characterisation and modelling of temperature offers a solution for improving the uncertainty of dimensional measurement and quantifying thermal variability in large assemblies. Technologies that currently exist for temperature measurement in the range of 0-50°C have been presented alongside discussion of these temperature measurement technologies' usefulness for monitoring temperatures in a manufacturing context. Particular aspects of production where the technology could play a role are highlighted as well as practical considerations for deployment. Contact sensors such as platinum resistance thermometers can produce accuracy closest to the desired accuracy given the most challenging measurement conditions calculated to be ∼0.02°C. Non-contact solutions would be most practical in the light controlled factory (LCF) and semi-invasive appear least useful but all technologies can play some role during the initial development of thermal variability models.

Theory of hybrid computational and physical measurement-based integration of thermal and dimensional measurement

In dimensional metrology, often the largest source of uncertainty of measurement is thermal variation. Dimensional measurements are currently scaled linearly, using ambient temperature measurements and coefficients of thermal expansion, to ideal metrology conditions at 20˚C. This scaling is particularly difficult to implement with confidence in large volumes as the temperature is unlikely to be uniform, resulting in thermal gradients. A number of well-established computational methods are used in the design phase of product development for the prediction of thermal and gravitational effects, which could be used to a greater extent in metrology. This paper outlines the theory of how physical measurements of dimension and temperature can be combined more comprehensively throughout the product lifecycle, from design through to the manufacturing phase. The Hybrid Metrology concept is also introduced: an approach to metrology, which promises to improve product and equipment integrity in future manufacturing environments. The Hybrid Metrology System combines various state of the art physical dimensional and temperature measurement techniques with established computational methods to better predict thermal and gravitational effects.

The application of uncertainty evaluating software for the utilisation of machine tool systems for final inspection

As machine tools continue to become increasingly repeatable and accurate, high-precision manufacturers may be tempted to consider how they might utilise machine tools as measurement systems. In this paper, we have explored this paradigm by attempting to repurpose state-of-the-art coordinate measuring machine Uncertainty Evaluating Software (UES) for a machine tool application. We performed live measurements on all the systems in question. Our findings have highlighted some gaps with UES when applied to machine tools, and we have attempted to identify the sources of variation which have led to discrepancies. Implications of this research include requirements to evolve the algorithms within the UES if it is to be adapted for on-machine measurement, improve the robustness of the input parameters, and most importantly, clarify expectations.

Variation and Uncertainty in Evaporation from a Subtropical Estuary: Florida Bay

Variation and uncertainty in estimated evaporation was determined over time and between two locations in Florida Bay, a subtropical estuary. Meteorological data were collected from September 2001 to August 2002 at Rabbit Key and Butternut Key within the Bay. Evaporation was estimated using both vapor flux and energy budget methods. The results were placed into a long-term context using 33 years of temperature and rainfall data collected in south Florida. Evaporation also was estimated from this long-term data using an empirical formula relating evaporation to clear sky solar radiation and air temperature. Evaporation estimates for the 12-mo period ranged from 144 to 175 cm yr21, depending on location and method, with an average of 163 cm yr21 (6 9%). Monthly values ranged from 9.2 to 18.5 cm, with the highest value observed in May, corresponding with the maximum in measured net radiation. Uncertainty estimates derived from measurement errors in the data were as much as 10%, and were large enough to obscure differences in evaporation between the two sites. Differences among all estimates for any month indicate the overall uncertainty in monthly evaporation, and ranged from 9% to 26%. Over a 33-yr period (1970–2002), estimated annual evaporation from Florida Bay ranged from 148 to 181 cm yr21, with an average of 166 cm yr21. Rainfall was consistently lower in Florida Bay than evaporation, with a long-term average of 106 cm yr21. Rainfall considered alone was uncorrelated with evaporation at both monthly and annual time scales; when the seasonal variation in clear sky radiation was also taken into account both net radiation and evaporation were significantly suppressed in months with high rainfall.

Designing for User-facing Uncertainty in Everyday Sensing and Prediction

Thesis (Ph.D.)--University of Washington, 2016-07

Metal measurement in aquatic environments by passive sampling methods: Lessons learning from an in situ intercomparison exercise

Passive sampling devices (PS) are widely used for pollutant monitoring in water, but estimation of measurement uncertainties by PS has seldom been undertaken. The aim of this work was to identify key parameters governing PS measurements of metals and their dispersion. We report the results of an in situ intercomparison exercise on diffusive gradient in thin films (DGT) in surface waters. Interlaboratory uncertainties of time-weighted average (TWA) concentrations were satisfactory (from 28% to 112%) given the number of participating laboratories (10) and ultra-trace metal concentrations involved. Data dispersion of TWA concentrations was mainly explained by uncertainties generated during DGT handling and analytical procedure steps. We highlight that DGT handling is critical for metals such as Cd, Cr and Zn, implying that DGT assembly/dismantling should be performed in very clean conditions. Using a unique dataset, we demonstrated that DGT markedly lowered the LOQ in comparison to spot sampling and stressed the need for accurate data calculation.

Worldwide trends in blood pressure from 1975 to 2015: a pooled analysis of 1479 population-based measurement studies with 19·1 million participants

Background: Raised blood pressure is an important risk factor for cardiovascular diseases and chronic kidney disease. We estimated worldwide trends in mean systolic and mean diastolic blood pressure, and the prevalence of, and number of people with, raised blood pressure, defined as systolic blood pressure of 140 mm Hg or higher or diastolic blood pressure of 90 mm Hg or higher. Methods: For this analysis, we pooled national, subnational, or community population-based studies that had measured blood pressure in adults aged 18 years and older. We used a Bayesian hierarchical model to estimate trends from 1975 to 2015 in mean systolic and mean diastolic blood pressure, and the prevalence of raised blood pressure for 200 countries. We calculated the contributions of changes in prevalence versus population growth and ageing to the increase in the number of adults with raised blood pressure. Findings: We pooled 1479 studies that had measured the blood pressures of 19·1 million adults. Global age-standardised mean systolic blood pressure in 2015 was 127·0 mm Hg (95% credible interval 125·7–128·3) in men and 122·3 mm Hg (121·0–123·6) in women; age-standardised mean diastolic blood pressure was 78·7 mm Hg (77·9–79·5) for men and 76·7 mm Hg (75·9–77·6) for women. Global age-standardised prevalence of raised blood pressure was 24·1% (21·4–27·1) in men and 20·1% (17·8–22·5) in women in 2015. Mean systolic and mean diastolic blood pressure decreased substantially from 1975 to 2015 in high-income western and Asia Pacific countries, moving these countries from having some of the highest worldwide blood pressure in 1975 to the lowest in 2015. Mean blood pressure also decreased in women in central and eastern Europe, Latin America and the Caribbean, and, more recently, central Asia, Middle East, and north Africa, but the estimated trends in these super-regions had larger uncertainty than in high-income super-regions. By contrast, mean blood pressure might have increased in east and southeast Asia, south Asia, Oceania, and sub-Saharan Africa. In 2015, central and eastern Europe, sub-Saharan Africa, and south Asia had the highest blood pressure levels. Prevalence of raised blood pressure decreased in high-income and some middle-income countries; it remained unchanged elsewhere. The number of adults with raised blood pressure increased from 594 million in 1975 to 1·13 billion in 2015, with the increase largely in low-income and middle-income countries. The global increase in the number of adults with raised blood pressure is a net effect of increase due to population growth and ageing, and decrease due to declining age-specific prevalence. Interpretation: During the past four decades, the highest worldwide blood pressure levels have shifted from high-income countries to low-income countries in south Asia and sub-Saharan Africa due to opposite trends, while blood pressure has been persistently high in central and eastern Europe.

Uncertainty evaluation of multivariate quantities: A case study on electrical impedance

The paper discusses the evaluation of the uncertainty of a multivariate quantity using the Law of Propagation of Uncertainty defined in the Guide to the Expression of Uncertainty in Measurement (GUM) and a Monte Carlo method according to the GUM’s Supplement 2. The quantity analysed is the electrical impedance, which is not a scalar but a complex quantity. The used measuring method allows the evaluation of the impedance and of its uncertainty in different ways and the corresponding results are presented, compared and discussed. For comparison purposes, results of the impedance uncertainty obtained using the NIST Uncertainty Machine are also presented.