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Boxplots are useful displays that convey rough information about the distribution of a variable. Boxplots were designed to be drawn by hand and work best for small datasets, where detailed estimates of tail behavior beyond the quartiles may not be trustworthy. Larger datasets afford more precise estimates of tail behavior, but boxplots do not take advantage of this precision, instead presenting large numbers of extreme, though not unexpected, observations. Letter-value plots address this problem by including more detailed information about the tails using \"letter values,\" an order statistic defined by Tukey. Boxplots display the first two letter values (the median and quartiles); letter-value plots display further letter values so far as they are reliable estimates of their corresponding quantiles. We illustrate letter-value plots with real data that demonstrate their usefulness for large datasets. All graphics are created using the R package lvplot , and code and data are available in the supplementary materials.

Given data in , a Tukey κ-trimmed region is the set of all points that have at least Tukey depth κ w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and interpreted, their practical use in applications has been impeded so far by the lack of efficient computational procedures in dimension p > 2. We construct two novel algorithms to compute a Tukey κ-trimmed region, a naïve one and a more sophisticated one that is much faster than known algorithms. Further, a strict bound on the number of facets of a Tukey region is derived. In a large simulation study the novel fast algorithm is compared with the naïve one, which is slower and by construction exact, yielding in every case the same correct results. Finally, the approach is extended to an algorithm that calculates the innermost Tukey region and its barycenter, the Tukey median. Supplementary materials for this article are available online.

We propose new tools for visualizing large amounts of functional data in the form of smooth curves. The proposed tools include functional versions of the bagplot and boxplot, which make use of the first two robust principal component scores, Tukey's data depth and highest density regions. By-products of our graphical displays are outlier detection methods for functional data. We compare these new outlier detection methods with existing methods for detecting outliers in functional data, and show that our methods are better able to identify outliers. An R-package containing computer code and datasets is available in the online supplements.

A detailed investigation has been conducted to study the shift location of the point of the maximum scour depth for both bridge abutment-collar and pier-collar arrangements. In the present study, an experimental program has been conducted for abutment-collar arrangements and additionally, the data obtained from the literature for the pier-collar arrangements have been revisited and analyzed to complement the framework of this study. For the abutment-collar arrangements, a series of experiments under clear-water conditions were carried out for different abutment lengths with fixed values of collar location and collar width. For pier-collar arrangements, data used from the literature have been involved constant pier diameter with various collar sizes at various elevations. To describe the locations of these maximum scour depths, their coordinates with respect to the location of the abutment or the pier were obtained. Results from this investigation showed that when a collar placed on or below the bed level was used as a countermeasure against scouring, either on an abutment or on a pier, it was observed that the maximum scour depth was routed downstream of the bridge structure.

We propose the bagplot, a bivariate generalization of the univariate boxplot. The key notion is the half space location depth of a point relative to a bivariate dataset, which extends the univariate concept of rank. The \"depth median\" is the deepest location, and it is surrounded by a \"bag\" containing the n/2 observations with largest depth. Magnifying the bag by a factor 3 yields the \"fence\" (which is not plotted). Observations between the bag and the fence are marked by a light gray loop, whereas observations outside the fence are flagged as outliers. The bagplot visualizes the location, spread, correlation, skewness, and tails of the data. It is equivariant for linear transformations, and not limited to elliptical distributions. Software for drawing the bagplot is made available for the S-Plus and MATLAB environments. The bagplot is illustrated on several datasets-for example, in a scatterplot matrix of multivariate data.

A fast nonparametric procedure for classifying functional data is introduced. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional space. The functional data are first mapped into a finite-dimensional location-slope space and then transformed by a multivariate depth function into the DD -plot, which is a subset of the unit square. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification in [ 0 , 1 ] 2 . The resulting classifier has to be cross-validated over a small range of parameters only, which is restricted by a Vapnik–Chervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust, efficiently computable, and has been implemented in an R environment. Applicability of the new approach is demonstrated by simulations as well as by a benchmark study.

Tukey (1975) introduced the notion of halfspace depth in a data analytic context, as a multivariate analog of rank relative to a finite data set. Here we focus on the depth function of an arbitrary probability distribution on ℝp, and even of a non-probability measure. The halfspace depth of any point θ in ℝp is the smallest measure of a closed halfspace that contains θ. We review the properties of halfspace depth, enriched with some new results. For various measures, uniform as well as non-uniform, we derive an expression for the depth function. We also compute the Tukey median, which is the θ in which the depth function attains its maximal value. Various interesting phenomena occur. For the uniform distribution on a triangle, a square or any regular polygon, the depth function has ridges that correspond to an 'inversion' of depth contours. And for a product of Cauchy distributions, the depth contours are squares. We also consider an application of the depth function to voting theory.

The sub-surface structural analysis to understand the geology and tectonics of an area is always useful to locate the hydrocarbon resources. Oil and gas based energy supplies have become a vital source for Pakistan, which is passing through an era of severe energy crisis. The study area, Buzdar block, in the southern Indus Basin is tectonically an extensional regime and is expected to have a huge hydrocarbon potential. In this study, we did the interpretation of the migrated seismic lines of the 872-SGR-527, 872-SGR-529, 872-SGR-531, 872-SGR-532 of Buzdar block, District TandoAllahyar, Sindh. The lines 872-SGR-529, 872-SGR-531, 872-SGR-532 were oriented W–E whereas the line 872-SGR-527 was oriented NW–SE. The obtained data was analysed and three reflectors were marked named top Khadro Formation, top lower Goru formation and top Chiltan limestone (probable). Through this study faults have been also marked on seismic lines which are normal faults by nature; collectively form horsts and grabens which is the evidence of effect of extensional tectonics in the area. Time contour maps were also generated. After that, time was converted into depth with the help of well velocity from VSP data for lower Goru formation and average velocity for Chiltan limestone (probable) from regression analysis. Finally, depth contour maps were generated which helped to know the basic mechanism of tectonic movement in the area. On the basis of present analysis we propose that a well may be drilled at Lower Goru formation near fault F1 on western side at a depth of 1370 meters and at 1290 meters near fault F4 on eastern side.

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Framed within the Copernicus Climate Change Service (C3S) of the European Commission, the European Centre for Medium-Range Weather Forecasts (ECMWF) is producing an enhanced global dataset for the land component of the fifth generation of European ReAnalysis (ERA5), hereafter referred to as ERA5-Land. Once completed, the period covered will span from 1950 to the present, with continuous updates to support land monitoring applications. ERA5-Land describes the evolution of the water and energy cycles over land in a consistent manner over the production period, which, among others, could be used to analyse trends and anomalies. This is achieved through global high-resolution numerical integrations of the ECMWF land surface model driven by the downscaled meteorological forcing from the ERA5 climate reanalysis, including an elevation correction for the thermodynamic near-surface state. ERA5-Land shares with ERA5 most of the parameterizations that guarantees the use of the state-of-the-art land surface modelling applied to numerical weather prediction (NWP) models. A main advantage of ERA5-Land compared to ERA5 and the older ERA-Interim is the horizontal resolution, which is enhanced globally to 9 km compared to 31 km (ERA5) or 80 km (ERA-Interim), whereas the temporal resolution is hourly as in ERA5. Evaluation against independent in situ observations and global model or satellite-based reference datasets shows the added value of ERA5-Land in the description of the hydrological cycle, in particular with enhanced soil moisture and lake description, and an overall better agreement of river discharge estimations with available observations. However, ERA5-Land snow depth fields present a mixed performance when compared to those of ERA5, depending on geographical location and altitude. The description of the energy cycle shows comparable results with ERA5. Nevertheless, ERA5-Land reduces the global averaged root mean square error of the skin temperature, taking as reference MODIS data, mainly due to the contribution of coastal points where spatial resolution is important. Since January 2020, the ERA5-Land period available has extended from January 1981 to the near present, with a 2- to 3-month delay with respect to real time. The segment prior to 1981 is in production, aiming for a release of the whole dataset in summer/autumn 2021. The high spatial and temporal resolution of ERA5-Land, its extended period, and the consistency of the fields produced makes it a valuable dataset to support hydrological studies, to initialize NWP and climate models, and to support diverse applications dealing with water resource, land, and environmental management. The full ERA5-Land hourly (Muñoz-Sabater, 2019a) and monthly (Muñoz-Sabater, 2019b) averaged datasets presented in this paper are available through the C3S Climate Data Store at https://doi.org/10.24381/cds.e2161bac and https://doi.org/10.24381/cds.68d2bb30, respectively.