DOCSLIB.ORG

Biplots for Exploring Multidimensional Reality

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Biplots for Exploring Multidimensional Reality Graphics and visualisation in practice: Biplots for exploring multidimensional reality Sugnet Gardner & Niël J le Roux Department of Statistics and Actuarial Science, University of Stellenbosch Private Bag X1, Matieland, 7602, South Africa [email protected] 1. Introduction In considering the traditional biplot of Gabriel (1971) as the multivariate analogue of a scatterplot, Gower & Hand (1996) provide a unified methodology for representing multivariate data graphically. In this paper several extensions and novel applications of this philosophy in exploring multidimensional reality are demonstrated. When the above biplotology (biplot methodology) is applied to a multidimensional scaling application, the graphical display of the sample points is enhanced by adding information about the variables. The flexibility of this method facilitates incorporating both continuous and categorical measurements, representing large data sets making it suitable for datamining applications, as well as extending the mere representation of data to an exploratory analysis in itself by the application of several novel ideas. In this paper focus will not be on the underlying theoretical development of these ideas, discussed in Gardner (2001), but rather on illustrating the extensions through practical examples. 2. Multidimensional scatterplot with acceptance regions The principal component analysis (PCA) biplot is the most basic multidimensional extension of a scatterplot. PCA is based on the singular value decomposition of the covariance matrix of the data. A complete discussion can be found in Gower & Hand (1996). However, a few core concepts need to be explained. The principal axes resulting from the PCA are used only as scaffolding for plotting purposes and are not shown in the graphical representation. Instead, biplot axes are constructed representing the original variables measured. The terms interpolation and prediction are used for ‘moving’ from the original p-dimensional space to the biplot space (usually 2 dimensional) and back, respectively. In general p > 2, therefore the biplot display will be a best fit (according to some criteria) approximation of the original data matrix. Contrary to the scatterplot, a biplot needs two sets of biplot axes, interpolation axes for interpolating new samples onto the display and prediction axes for inferring the values of the original variables for a point in the biplot. Specific to the PCA biplot is the optimal representation of multidimensional variation. The quality index (QI) biplot optimally represents the variation in 66 32.5 18 5.6 monthly mean values of 15 quality measurements from a manufacturing 1.8 Jul00 process. By interpolating a multivariate target onto the biplot the 1.2 64 5.4 1.7 ‘distance’ from each month to the target is graphically represented. The 16 2.5 Mar01 1.1 14.2 20 prediction biplot axes suggest on which variables a particular month did 31 3 not conform to the target. A QI is calculated from the monthly values Aug00 A2 Apr0079 TARGET 20.5 A5 Jun001.5 45 49 resulting in index values where 0 – 50 is considered poor, 50 – 80 B5 22 30 D6 26 27 Feb01 50 21 29 D7 43 Jan01 20.5 satisfactory while 80 – 100 is superior quality. To associate the index Feb00 1.4 Dec00 A1 May00 12 Sep00 4.5 values with the biplot display, a grid is superimposed on the biplot space. Mar00 Jan00 56 29.5 Nov00 14.5 4.6 Oct00 0.8 For the midpoint of each grid cell the values of the original variables are C5 E5 C4 D4C7 C8 C6 A4 A3 predicted for calculating a QI value. By colour-coding the cells, acceptance regions are constructed on the biplot. 3. Classification of an unknown entity The canonical variate analysis (CVA) biplot is used to optimally separate classes in data. This biplot, used as a graphical representation in a linear discriminant analysis (LDA) context is also discussed by Gower & Hand (1996). Interpolating the sample points onto the CVA biplot of the class means allows for visual appraisal of the separation or overlap among classes. Three species wood of the genus Ocotea are used to demonstrate the use of a CVA biplot. Two of these, O.bullata (stinkwood) and O.porosa (imbuia), were traditionally used in the manufacturing of Old Cape furniture produced in the Cape region of South Africa during 1652 – 1900. The correct identification of the type of wood is important since furniture made from stinkwood has a high prestige value. For the classification of the type of wood of a chair to be auctioned, 500 a CVA biplot optimally separating the three species, is constructed. 80 Through the transformation from the original to the canonical (biplot) Obul 400 2000 1600 1200120 space, Mahalanobis distance in the original space becomes Pythagorean 800 Oken distance in the biplot space. A sample is therefore classified to its nearest Opor class mean in the biplot space. By colour coding grid cells according to 160 300 the nearest class mean, classification regions can be constructed. Interpolating the measurements of the chair onto the biplot, the chair is 200 clearly classified as manufactured from imbuia. 4. Exploring classes in data In an allometric study investigating morphological differences between tortoises of the species Homopus Areolatus from different regions in South Africa, the CVA biplot proves to be a useful exploratory tool when the very few observations available severely limited formal statistical analysis. -0.15 The Western Cape Nature Conservation Board (WCNCB) 0.12 -0.1 suspected that tortoises from the Karoo region have a flatter shell profile. 0.08 -0.05 PCA is used to decompose the data matrix into a size and shape + ‘error’ CH component. Utilising only the length, width and height of the shells, a -0.1 0 PCA biplot after the size is removed, clearly displays the Karoo tortoises separate from the other data points. Using the prediction biplot axes -0.08 0.1 (calibrated in log-transformed units) enough evidence of a flatter shell -0.12 0.15 profile is thus provided to support an application for funding a more in- CW CL depth study. Nelspruit 0 5. Application of bagplots in exploring classes 10 Ratio 20 2.5 30 When 16828 observations are plotted in a CVA biplot to explore 2 0 40 20 differences between the shape of export lemons from different cultivation 50 40 1.5 60 8060 areas, the spread of observations is concealed by overplotting. The 100 1 120 70 bagplot (Rousseeuw, Ruts & Tukey, 1999), as a form of a two- 140 80 160 0.5 dimensional boxplot is useful as a graphical summary of bivariate data Length 180 90 points. Suppressing the plotting of the sample points and superimposing 100 Diameter a bagplot onto the biplot allows for visual comparison of classes via the Vaalharts 0 prediction biplot axes. Since certain requirements are set for export 10 20 standards on two of the three variables, the biplots can be fitted with Ratio 2.5 30 acceptance regions. Comparing each region’s bagplot with the 2 0 40 20 50 40 acceptance region, clearly indicates its suitability as an export lemon 1.5 60 8060 100 producing region. 1 120 70 140 80 160 0.5 Length 180 90 100 Diameter 6. Introduction of the α-bag Although superimposing bagplots onto the biplot scaffolding is very useful as summary of the cloud of points, each bagplot has to be plotted separately. The ‘bag’ of the bagplot is constructed based on the concept of halfspace location depth such that it contains the inner 50% of the data points. Using the algorithm of Rousseeuw, Ruts & Tukey (1999), the 50% cut-off is replaced by a value α ranging between 0 and 1. Typically a value of 0.9 or 0.95 will be useful for enclosing a class of observations, excluding the 10% or 5% of the observations at the extremes. The living standards and development survey was conducted among approximately 9000 South African households preceding the -2 4 ExpDec 14 1994 elections to provide policy makers with information regarding 0 12 15 MEdY 14 10 2 literacy in order to best address racial inequalities. Furnishing the CVA 12 8 6 10 4 6 biplot with an 80% bag for each of four race groups, the differences and White 8 Black Coloured8 6 4 6 4 8 2 overlap can be assessed. The α-bag also provides for a quantitative Indian 2 0 10 16 0 measure of the overlap between groups: the smallest α for which there is 1012 14 overlap. This measure provides a reference for evaluating progress in a 16 12 18 follow-up study. EduYrs TotScore Age 7. Exploring overlap between classes A sample of Middle Stone Age artefacts excavated from the caves at Klasies River on the southern coast of South Africa is explored in a 0 CVA biplot. The construction of α-bags in the biplot allows for 10 70 exploring differences and overlap between artefacts classified as Bladelength 20 0 -5 40 20 10 belonging to the sub-stage layers MSA I, MSA II upper and MSA II 0 80 60 5 40 lower. There is a considerable amount of overlap between the two MSA Bladethickn 20 20 0 8010 II sub-stages with some difference on blade width. The MSA I sub-stage Ratio 15 30 20 40 100 90 differs from the MSA II sub-stages on blade- and platform thickness as 40 120 MSA I Platfthickn MSA II L well as the ratio length:platform thickness. The shape and overlap of the 50 50 140 MSA II U α-bags further highlight that the MSA II artefacts overlap with the MSA PlatfwidthPlatfangle Bladewidth I artefacts, but that a substantial proportion of the MSA I artefacts are distinctly different from MSA II.
Load more

Recommended publications
  • A Generalized Linear Model for Principal Component Analysis of Binary Data
    A Generalized Linear Model for Principal Component Analysis of Binary Data Andrew I. Schein Lawrence K. Saul Lyle H. Ungar Department of Computer and Information Science University of Pennsylvania Moore School Building 200 South 33rd Street Philadelphia, PA 19104-6389 {ais,lsaul,ungar}@cis.upenn.edu Abstract they are not generally appropriate for other data types. Recently, Collins et al.[5] derived generalized criteria We investigate a generalized linear model for for dimensionality reduction by appealing to proper- dimensionality reduction of binary data. The ties of distributions in the exponential family. In their model is related to principal component anal- framework, the conventional PCA of real-valued data ysis (PCA) in the same way that logistic re- emerges naturally from assuming a Gaussian distribu- gression is related to linear regression. Thus tion over a set of observations, while generalized ver- we refer to the model as logistic PCA. In this sions of PCA for binary and nonnegative data emerge paper, we derive an alternating least squares respectively by substituting the Bernoulli and Pois- method to estimate the basis vectors and gen- son distributions for the Gaussian. For binary data, eralized linear coefficients of the logistic PCA the generalized model's relationship to PCA is anal- model. The resulting updates have a simple ogous to the relationship between logistic and linear closed form and are guaranteed at each iter- regression[12]. In particular, the model exploits the ation to improve the model's likelihood. We log-odds as the natural parameter of the Bernoulli dis- evaluate the performance of logistic PCA|as tribution and the logistic function as its canonical link.
    [Show full text]
  • Exploratory and Confirmatory Factor Analysis Course Outline Part 1
    Course Outline Exploratory and Confirmatory Factor Analysis 1 Principal components analysis Part 1: Overview, PCA and Biplots FA vs. PCA Least squares fit to a data matrix Biplots Michael Friendly 2 Basic Ideas of Factor Analysis Parsimony– common variance small number of factors. Linear regression on common factors→ Partial linear independence Psychology 6140 Common vs. unique variance 3 The Common Factor Model Factoring methods: Principal factors, Unweighted Least Squares, Maximum λ1 X1 z1 likelihood ξ λ2 Factor rotation X2 z2 4 Confirmatory Factor Analysis Development of CFA models Applications of CFA PCA and Factor Analysis: Overview & Goals Why do Factor Analysis? Part 1: Outline Why do “Factor Analysis”? 1 PCA and Factor Analysis: Overview & Goals Why do Factor Analysis? Data Reduction: Replace a large number of variables with a smaller Two modes of Factor Analysis number which reflect most of the original data [PCA rather than FA] Brief history of Factor Analysis Example: In a study of the reactions of cancer patients to radiotherapy, measurements were made on 10 different reaction variables. Because it 2 Principal components analysis was difficult to interpret all 10 variables together, PCA was used to find Artificial PCA example simpler measure(s) of patient response to treatment that contained most of the information in data. 3 PCA: details Test and Scale Construction: Develop tests and scales which are “pure” 4 PCA: Example measures of some construct. Example: In developing a test of English as a Second Language, 5 Biplots investigators calculate correlations among the item scores, and use FA to Low-D views based on PCA construct subscales.
    [Show full text]
  • A Resampling Test for Principal Component Analysis of Genotype-By-Environment Interaction
    A RESAMPLING TEST FOR PRINCIPAL COMPONENT ANALYSIS OF GENOTYPE-BY-ENVIRONMENT INTERACTION JOHANNES FORKMAN Abstract. In crop science, genotype-by-environment interaction is of- ten explored using the \genotype main effects and genotype-by-environ- ment interaction effects” (GGE) model. Using this model, a singular value decomposition is performed on the matrix of residuals from a fit of a linear model with main effects of environments. Provided that errors are independent, normally distributed and homoscedastic, the signifi- cance of the multiplicative terms of the GGE model can be tested using resampling methods. The GGE method is closely related to principal component analysis (PCA). The present paper describes i) the GGE model, ii) the simple parametric bootstrap method for testing multi- plicative genotype-by-environment interaction terms, and iii) how this resampling method can also be used for testing principal components in PCA. 1. Introduction Forkman and Piepho (2014) proposed a resampling method for testing in- teraction terms in models for analysis of genotype-by-environment data. The \genotype main effects and genotype-by-environment interaction ef- fects"(GGE) analysis (Yan et al. 2000; Yan and Kang, 2002) is closely related to principal component analysis (PCA). For this reason, the method proposed by Forkman and Piepho (2014), which is called the \simple para- metric bootstrap method", can be used for testing principal components in PCA as well. The proposed resampling method is parametric in the sense that it assumes homoscedastic and normally distributed observations. The method is \simple", because it only involves repeated sampling of standard normal distributed values. Specifically, no parameters need to be estimated.
    [Show full text]
  • GGL Biplot Analysis of Durum Wheat (Triticum Turgidum Spp
    756 Bulgarian Journal of Agricultural Science, 19 (No 4) 2013, 756-765 Agricultural Academy GGL BIPLOT ANALYSIS OF DURUM WHEAT (TRITICUM TURGIDUM SPP. DURUM) YIELD IN MULTI-ENVIRONMENT TRIALS N. SABAGHNIA2*, R. KARIMIZADEH1 and M. MOHAMMADI1 1 Department of Agronomy and Plant Breeding, Faculty of Agriculture, University of Maragheh, Maragheh, Iran 2 Dryland Agricultural Research Institute (DARI), Gachsaran, Iran Abstract SABAGHNIA, N., R. KARIMIZADEH and M. MOHAMMADI, 2013. GGL biplot analysis of durum wheat (Triticum turgidum spp. durum) yield in multi-environment trials. Bulg. J. Agric. Sci., 19: 756-765 Durum wheat (Triticum turgidum spp. durum) breeders have to determine the new genotypes responsive to the environ- mental changes for grain yield. Matching durum wheat genotype selection with its production environment is challenged by the occurrence of significant genotype by environment (GE) interaction in multi-environment trials (MET). This investigation was conducted to evaluate 20 durum wheat genotypes for their stability grown in five different locations across three years using randomized completely block design with 4 replications. According to combined analysis of variance, the main effects of genotypes, locations and years, were significant as well as the interactions effects. The first two principal components of the site regression model accounted for 60.3 % of the total variation. Polygon view of genotype plus genotype by location (GGL) biplot indicated that there were three winning genotypes in three mega-environments for durum wheat in rain-fed conditions. Genotype G14 was the most favorable genotype for location Gachsaran and the most favorable genotype of mega-environment Kouhdasht and Ilam was G12 while G10 was the most favorable genotypes for mega-environment Gonbad and Moghan.
    [Show full text]
  • A Biplot-Based PCA Approach to Study the Relations Between Indoor and Outdoor Air Pollutants Using Case Study Buildings
    buildings Article A Biplot-Based PCA Approach to Study the Relations between Indoor and Outdoor Air Pollutants Using Case Study Buildings He Zhang * and Ravi Srinivasan UrbSys (Urban Building Energy, Sensing, Controls, Big Data Analysis, and Visualization) Laboratory, M.E. Rinker, Sr. School of Construction Management, University of Florida, Gainesville, FL 32603, USA; sravi@ufl.edu * Correspondence: rupta00@ufl.edu Abstract: The 24 h and 14-day relationship between indoor and outdoor PM2.5, PM10, NO2, relative humidity, and temperature were assessed for an elementary school (site 1), a laboratory (site 2), and a residential unit (site 3) in Gainesville city, Florida. The primary aim of this study was to introduce a biplot-based PCA approach to visualize and validate the correlation among indoor and outdoor air quality data. The Spearman coefficients showed a stronger correlation among these target environmental measurements on site 1 and site 2, while it showed a weaker correlation on site 3. The biplot-based PCA regression performed higher dependency for site 1 and site 2 (p < 0.001) when compared to the correlation values and showed a lower dependency for site 3. The results displayed a mismatch between the biplot-based PCA and correlation analysis for site 3. The method utilized in this paper can be implemented in studies and analyzes high volumes of multiple building environmental measurements along with optimized visualization. Keywords: air pollution; indoor air quality; principal component analysis; biplot Citation: Zhang, H.; Srinivasan, R. A Biplot-Based PCA Approach to Study the Relations between Indoor and 1. Introduction Outdoor Air Pollutants Using Case The 2020 Global Health Observatory (GHO) statistics show that indoor and outdoor Buildings 2021 11 Study Buildings.
    [Show full text]
  • Logistic Biplot by Conjugate Gradient Algorithms and Iterated SVD
    mathematics Article Logistic Biplot by Conjugate Gradient Algorithms and Iterated SVD Jose Giovany Babativa-Márquez 1,2,* and José Luis Vicente-Villardón 1 1 Department of Statistics, University of Salamanca, 37008 Salamanca, Spain; [email protected] 2 Facultad de Ciencias de la Salud y del Deporte, Fundación Universitaria del Área Andina, Bogotá 1321, Colombia * Correspondence: [email protected] Abstract: Multivariate binary data are increasingly frequent in practice. Although some adaptations of principal component analysis are used to reduce dimensionality for this kind of data, none of them provide a simultaneous representation of rows and columns (biplot). Recently, a technique named logistic biplot (LB) has been developed to represent the rows and columns of a binary data matrix simultaneously, even though the algorithm used to fit the parameters is too computationally demanding to be useful in the presence of sparsity or when the matrix is large. We propose the fitting of an LB model using nonlinear conjugate gradient (CG) or majorization–minimization (MM) algo- rithms, and a cross-validation procedure is introduced to select the hyperparameter that represents the number of dimensions in the model. A Monte Carlo study that considers scenarios with several sparsity levels and different dimensions of the binary data set shows that the procedure based on cross-validation is successful in the selection of the model for all algorithms studied. The comparison of the running times shows that the CG algorithm is more efficient in the presence of sparsity and when the matrix is not very large, while the performance of the MM algorithm is better when the binary matrix is balanced or large.
    [Show full text]
  • Pcatools: Everything Principal Components Analysis
    Package ‘PCAtools’ October 1, 2021 Type Package Title PCAtools: Everything Principal Components Analysis Version 2.5.15 Description Principal Component Analysis (PCA) is a very powerful technique that has wide applica- bility in data science, bioinformatics, and further afield. It was initially developed to anal- yse large volumes of data in order to tease out the differences/relationships between the logi- cal entities being analysed. It extracts the fundamental structure of the data with- out the need to build any model to represent it. This 'summary' of the data is ar- rived at through a process of reduction that can transform the large number of vari- ables into a lesser number that are uncorrelated (i.e. the 'principal compo- nents'), while at the same time being capable of easy interpretation on the original data. PCA- tools provides functions for data exploration via PCA, and allows the user to generate publica- tion-ready figures. PCA is performed via BiocSingular - users can also identify optimal num- ber of principal components via different metrics, such as elbow method and Horn's paral- lel analysis, which has relevance for data reduction in single-cell RNA-seq (scRNA- seq) and high dimensional mass cytometry data. License GPL-3 Depends ggplot2, ggrepel Imports lattice, grDevices, cowplot, methods, reshape2, stats, Matrix, DelayedMatrixStats, DelayedArray, BiocSingular, BiocParallel, Rcpp, dqrng Suggests testthat, scran, BiocGenerics, knitr, Biobase, GEOquery, hgu133a.db, ggplotify, beachmat, RMTstat, ggalt, DESeq2, airway,
    [Show full text]
  • Principal Components Analysis (Pca)
    PRINCIPAL COMPONENTS ANALYSIS (PCA) Steven M. Holand Department of Geology, University of Georgia, Athens, GA 30602-2501 3 December 2019 Introduction Suppose we had measured two variables, length and width, and plotted them as shown below. Both variables have approximately the same variance and they are highly correlated with one another. We could pass a vector through the long axis of the cloud of points and a second vec- tor at right angles to the first, with both vectors passing through the centroid of the data. Once we have made these vectors, we could find the coordinates of all of the data points rela- tive to these two perpendicular vectors and re-plot the data, as shown here (both of these figures are from Swan and Sandilands, 1995). In this new reference frame, note that variance is greater along axis 1 than it is on axis 2. Also note that the spatial relationships of the points are unchanged; this process has merely rotat- ed the data. Finally, note that our new vectors, or axes, are uncorrelated. By performing such a rotation, the new axes might have particular explanations. In this case, axis 1 could be regard- ed as a size measure, with samples on the left having both small length and width and samples on the right having large length and width. Axis 2 could be regarded as a measure of shape, with samples at any axis 1 position (that is, of a given size) having different length to width ratios. PC axes will generally not coincide exactly with any of the original variables.
    [Show full text]
  • Recent Outlier Detection Methods with Illustrations Loss Reserving Context
    Recent outlier detection methods with illustrations in loss reserving Benjamin Avanzi, Mark Lavender, Greg Taylor, Bernard Wong School of Risk and Actuarial Studies, UNSW Sydney Insights, 18 September 2017 Recent outlier detection methods with illustrations loss reserving Context Context Reserving Robustness and Outliers Robust Statistical Techniques Robustness criteria Heuristic Tools Robust M-estimation Outlier Detection Techniques Robust Reserving Overview Illustration - Robust Bivariate Chain Ladder Robust N-Dimensional Chain-Ladder Summary and Conclusions References 1/46 Recent outlier detection methods with illustrations loss reserving Context Reserving Context Reserving Robustness and Outliers Robust Statistical Techniques Robustness criteria Heuristic Tools Robust M-estimation Outlier Detection Techniques Robust Reserving Overview Illustration - Robust Bivariate Chain Ladder Robust N-Dimensional Chain-Ladder Summary and Conclusions References 1/46 Recent outlier detection methods with illustrations loss reserving Context Reserving The Reserving Problem i/j 1 2 ··· j ··· I 1 X1;1 X1;2 ··· X1;j ··· X1;J 2 X2;1 X2;2 ··· X2;j ··· . i Xi;1 Xi;2 ··· Xi;j . I XI ;1 Figure: Aggregate claims run-off triangle I Complete the square (or rectangle) I Also - multivariate extensions. 1/46 Recent outlier detection methods with illustrations loss reserving Context Reserving Common Reserving Techniques I Deterministic Chain-Ladder I Stochastic Chain-Ladder (Hachmeister and Stanard, 1975; England and Verrall, 2002) I Mack’s Model (Mack, 1993) I GLMs
    [Show full text]
  • Correlplot: a Collection of Functions for Plotting Correlation Matrices 1
    Correlplot: a collection of functions for plotting correlation matrices version 1.0-2 Jan Graffelman Department of Statistics and Operations Research Universitat Polit`ecnica de Catalunya Avinguda Diagonal 647, 08028 Barcelona, Spain. jan.graff[email protected] October 2013 1 Introduction This documents gives some instructions on how to create plots of correlation matrices in the statistical environment R (R Core Team, 2013) using the pack- age Correlplot. The outline of this guide is as follows: Section 2 explains how to install this package and Section 3 shows how to use the functions of the package for creating pictures like correlograms and biplots for a given correlation matrix. Computation of goodness-of-fit statistics is also illustrated. If you appreciate this software then please cite the following paper in your work: Graffelman, J. (2013) Linear-angle correlation plots: new graphs for revealing correlation structure. Journal of Computational and Graphical Statistics, 22(1) pp. 92-106. (click here to access the paper). 2 Installation The package Correlplot can be installed in R by typing: install.packages("Correlplot") library("Correlplot") This instruction will make, among others, the functions correlogram, linangplot, pco and pfa available. The correlation matrices used in the paper cited above are included in the package, and can be accessed with the data instruction. The instruction data(package="Correlplot") will give a list of all correlation and data matrices available in the package. 3 Plots of a correlation matrix In this section we indicate how to create different plots of a correlation matrix, and how to obtain the goodness-of-fit of the displays.
    [Show full text]
  • Download from the Resource and Environment Data Cloud Platform (
    sustainability Article Quantitative Assessment for the Dynamics of the Main Ecosystem Services and their Interactions in the Northwestern Arid Area, China Tingting Kang 1,2, Shuai Yang 1,2, Jingyi Bu 1,2 , Jiahao Chen 1,2 and Yanchun Gao 1,* 1 Key Laboratory of Water Cycle and Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, P.O. Box 9719, Beijing 100101, China; [email protected] (T.K.); [email protected] (S.Y.); [email protected] (J.B.); [email protected] (J.C.) 2 University of the Chinese Academy of Sciences, Beijing 100049, China * Correspondence: [email protected] Received: 21 November 2019; Accepted: 13 January 2020; Published: 21 January 2020 Abstract: Evaluating changes in the spatial–temporal patterns of ecosystem services (ESs) and their interrelationships provide an opportunity to understand the links among them, as well as to inform ecosystem-service-based decision making and governance. To research the development trajectory of ecosystem services over time and space in the northwestern arid area of China, three main ecosystem services (carbon sequestration, soil retention, and sand fixation) and their relationships were analyzed using Pearson’s coefficient and a time-lagged cross-correlation analysis based on the mountain–oasis–desert zonal scale. The results of this study were as follows: (1) The carbon sequestration of most subregions improved, and that of the oasis continuously increased from 1990 to 2015. Sand fixation decreased in the whole region except for in the Alxa Plateau–Hexi Corridor area, which experienced an “increase–decrease” trend before 2000 and after 2000; (2) the synergies and trade-off relationships of the ES pairs varied with time and space, but carbon sequestration and soil retention in the most arid area had a synergistic relationship, and most oases retained their synergies between carbon sequestration and sand fixation over time.
    [Show full text]
  • Of Typicality and Predictive Distributions in Discriminant Function Analysis Lyle W
    Wayne State University Human Biology Open Access Pre-Prints WSU Press 8-22-2018 Of Typicality and Predictive Distributions in Discriminant Function Analysis Lyle W. Konigsberg Department of Anthropology, University of Illinois at Urbana–Champaign, [email protected] Susan R. Frankenberg Department of Anthropology, University of Illinois at Urbana–Champaign Recommended Citation Konigsberg, Lyle W. and Frankenberg, Susan R., "Of Typicality and Predictive Distributions in Discriminant Function Analysis" (2018). Human Biology Open Access Pre-Prints. 130. https://digitalcommons.wayne.edu/humbiol_preprints/130 This Open Access Article is brought to you for free and open access by the WSU Press at DigitalCommons@WayneState. It has been accepted for inclusion in Human Biology Open Access Pre-Prints by an authorized administrator of DigitalCommons@WayneState. Of Typicality and Predictive Distributions in Discriminant Function Analysis Lyle W. Konigsberg1* and Susan R. Frankenberg1 1Department of Anthropology, University of Illinois at Urbana–Champaign, Urbana, Illinois, USA. *Correspondence to: Lyle W. Konigsberg, Department of Anthropology, University of Illinois at Urbana–Champaign, 607 S. Mathews Ave, Urbana, IL 61801 USA. E-mail: [email protected]. Short Title: Typicality and Predictive Distributions in Discriminant Functions KEY WORDS: ADMIXTURE, POSTERIOR PROBABILITY, BAYESIAN ANALYSIS, OUTLIERS, TUKEY DEPTH Pre-print version. Visit http://digitalcommons.wayne.edu/humbiol/ after publication to acquire the final version. Abstract While discriminant function analysis is an inherently Bayesian method, researchers attempting to estimate ancestry in human skeletal samples often follow discriminant function analysis with the calculation of frequentist-based typicalities for assigning group membership. Such an approach is problematic in that it fails to account for admixture and for variation in why individuals may be classified as outliers, or non-members of particular groups.
    [Show full text]