Biplot of Anderson's iris data set

Spectramap biplot of Anderson's iris data set

Discriminant Analysis Biplot of Fisher's Iris Data (Greenacre, 2010)

Biplots are a type of exploratory graph used in statistics, a generalization of the simple two-variable scatterplot. A biplot allows information on both samples and variables of a data matrix to be displayed graphically. Samples are displayed as points while variables are displayed either as vectors, linear axes or nonlinear trajectories. In the case of categorical variables, category level points may be used to represent the levels of a categorical variable. A generalised biplot displays information on both continuous and categorical variables.

Introduction and history[link]

The biplot was introduced by Gabriel (1971). Gower and Hand (1996) wrote a monograph on biplots. Yan and Kang (2003) described various methods which can be used in order to visualize and interpret a biplot. The book by Greenacre (2010) is a practical user-oriented guide to biplots, along with scripts in the open-source R programming language, to generate biplots associated with principal component analysis (PCA), multidimensional scaling (MDS), log-ratio analysis (LRA) - also known as spectral mapping^[1][2] - discriminant analysis (DA) and various forms of correspondence analysis: simple correspondence analysis (CA), multiple correspondence analysis (MCA) and canonical correspondence analysis (CCA). The book by Gower, Lubbe and le Roux (2011) aims to popularize biplots as a useful and reliable method for the visualization of multivariate data when researchers want to consider, for example, principal component analysis (PCA), canonical variates analysis (CVA) or various types of correspondence analysis.

Construction[link]

A biplot is constructed by using the singular value decomposition (SVD) to obtain a low-rank approximation to a transformed version of the data matrix X, whose n rows are the samples (also called the cases, or objects), and whose p columns are the variables. The transformed data matrix Y is obtained from the original matrix X by centering and optionally standardizing the columns (the variables). Using the SVD, we can write Y = ∑_k=1,...pd_ku_kv_k^T;, where the u_k are n-dimensional column vectors, the v_k are p-dimensional column vectors, and the d_k are a non-increasing sequence of non-negative scalars. The biplot is formed from two scatterplots that share a common set of axes and have a between-set scalar product interpretation. The first scatterplot is formed from the points (d₁^αu_1i,  d₂^αu_2i), for i = 1,...,n. The second plot is formed from the points (d₁^1-αv_1j, d₂^1-αv_2j), for j = 1,...,p. This is the biplot formed by the dominant two terms of the SVD, which can then be represented in a two-dimensional display. Typical choices of α are 1 (to give a distance interpretation to the row display) and 0 (to give a distance interpretation to the column display), and in some rare cases α=1/2 to obtain a symmetrically scaled biplot (which gives no distance interpretation to the rows or the columns, but only the scalar product interpretation). The set of points depicting the variables can be drawn as arrows from the origin to reinforce the idea that they represent biplot axes onto which the samples can be projected to approximate the original data.

References[link]

Sources[link]

• Gabriel, K.R. (1971). "The biplot graphic display of matrices with application to principal component analysis". Biometrika 58 (3): 453–467. DOI:10.1093/biomet/58.3.453. 
• Gower, J.C., Lubbe, S. and le Roux, N. (due January 2011). Understanding Biplots. Wiley. ISBN 978-0-470-01255-0
• Gower, J.C. and Hand, D.J (1996). Biplots. Chapman & Hall, London, UK. ISBN 0-412-71630-5
• Greenacre, M. (2010). Biplots in Practice. BBVA Foundation, Madrid, Spain. Available for free download ISBN 978-84-923846-8-6, with materials.
• Yan, W. and Kang, M.S. (2003). GGE Biplot Analysis. CRC Press, Boca Raton, FL. ISBN 0-8493-1338-4
• Vicente-Villardón, J.L., Galindo-Villardón, M.P. and Blázquez-Zaballos, A. (2006). Logistic Biplots. In: Multiple Correspondence Analysis and Related Methods. Greenacre, M. and Blasius, J. (Eds) Chapman & Hall/CRC Press. Boca Raton. USA. ISBN 1-58488-628-5
• Demey, J.R., Vicente-Villardón, J.L., Galindo-Villardón, M.P. and Zambrano, A.Y. (2008). Identifying molecular markers associated with classification of genotypes by External Logistic Biplots. Bioinformatics. 24(24):2832-2838

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John Gower shooting the world, a sphere of earth, air, and water (from a manuscript of his works ca. 1400)

John Gower (c. 1330 – October 1408) was an English poet, a contemporary of William Langland and a personal friend of Geoffrey Chaucer. He is remembered primarily for three major works, the Mirroir de l'Omme, Vox Clamantis, and Confessio Amantis, three long poems written in French, Latin, and English respectively, which are united by common moral and political themes.^[1]

Life[link]

Few details are known of Gower's early life. He was probably born into a prominent Yorkshire family which held properties in Kent, Yorkshire, Norfolk and Suffolk. It is thought that he practiced law in or around London.

While in London, he became closely associated with the nobility of his day. He was apparently personally acquainted with Richard II: in the prologue of the first edition of the Confessio Amantis, he tells how the king, chancing to meet him on the Thames (probably circa 1385), invited him aboard the royal barge, and that their conversation then resulted in a commission for the work that would become the Confessio Amantis. Later in life his allegiance switched to the future Henry IV, to whom later editions of the Confessio Amantis were dedicated. Much of this is based on circumstantial rather than documentary evidence, and the history of revisions of the Confessio Amantis, including the different dedications, is yet to be fully understood.

The tomb of John Gower in Southwark Cathedral. For more information click on the picture

Gower's friendship with Chaucer is also well documented. When Chaucer was sent as a diplomat to Italy in 1378, Gower was one of the men to whom he gave power of attorney over his affairs in England. The two poets also paid one another compliments in their verse: Chaucer dedicated his Troilus and Criseyde in part to "moral Gower", and Gower reciprocated by placing a speech in praise of Chaucer in the mouth of Venus at the end of the Confessio Amantis.

At some point during the early 1370s, he took up residence in rooms provided by the Priory of St Mary Overie (now Southwark Cathedral). In 1398, while living here, he married, probably for the second time: his wife, Agnes Groundolf, was to survive him. In his last years, and possibly as early as 1400, he became blind.

After his death in 1408, Gower was interred in an ostentatious tomb in the Priory church (now Southwark Cathedral), which remains today.

Works[link]

Gower's verse is by turns religious, political, historical, and moral—though he has been narrowly defined as "moral Gower" ever since Chaucer graced him with the epithet. His primary mode is allegory, although he shies away from sustained abstractions in favour of the plain style of the raconteur.

His earliest works were probably ballades in Anglo-Norman French, some of which may have later been included in his work the Cinkante Ballades. The first work which has survived is in the same language, however: it is the Speculum Meditantis, also known by the French title Mirour de l'Omme, a poem of just under 30,000 lines, containing a dense exposition of religion and morality.

Gower's second major work, the Vox Clamantis, was written in Latin: it takes as its subject the state of England, and incorporates commentary on the Peasants' Revolt that occurred during the composition of the poem. Gower takes the side of the aristocracy, and appears to have admired the techniques Richard II used to suppress the revolt.

His third work is the Confessio Amantis, a 30,000-line poem in octosyllabic English couplets, which makes use of the structure of a Christian confession (presented allegorically as a confession of sins against Love) as a narrative frame within which a multitude of individual tales are told. Like his previous works, the theme is very much morality, even where the stories themselves have a tendency to describe rather immoral behaviour. One scholar asserts that Confessio Amantis "almost exclusively" made Gower's "poetic reputation."^[2]

In later years Gower wrote a number of minor works in all three languages: the Cinkante Ballades, a series of French ballades on romantic subjects, and several poems addressed to the new Henry IV—in return for which he was granted a pension, in the form of an annual allowance of wine.

Gower's poetry has had a mixed critical reception. In the 15th century, he was generally regarded alongside Chaucer as the father of English poetry. Over the years, however, his reputation declined, largely on account of a perceived didacticism and dullness. During the 20th century he has received more recognition, notably by C. S. Lewis in The Allegory of Love (1936). However, he has not obtained the same following or critical acceptance as other major poets of the period.

List of works[link]

Wikisource has original works written by or about: John Gower

• Mirour de l'Omme, or Speculum Hominis, or Speculum Meditantis (French, c.1376–1379)
• Vox Clamantis (Latin, c.1377–1381)
• Confessio Amantis (English, c.1386–1393)
• Traité (French, 1397)
• Cinkante Balades (French, 1399-1400)
• Cronica Tripertita (Latin, c.1400)
• In praise of peace (English, c.1400)

See also[link]

• Pericles, Prince of Tyre, a play by Shakespeare, based on Gower's work and featuring Gower as the Chorus.
• Henry IV Part II includes Gower as a messenger
• In Henry V, Gower is the ideal English soldier

References[link]

• Macaulay, G. C. (1908) "John Gower," in Ward, A. W., and Waller, A. R., eds. The Cambridge History of English Literature, vol. II. The End of the Middle Ages, chapter VI. Cambridge University Press
• Urban, M. (ed.) (2009) John Gower, Manuscripts, Readers, Contexts, Turnhout: Brepols ISBN 978-2-503-52470-2
• Watt, Diane (2003) Amoral Gower. University of Minnesota Press
• Yeager, R. F. (ed.) (2007) On John Gower: Essays at the Millennium. (Studies in Medieval Culture, XLVI) Kalamazoo: Medieval Institute Publications, Pp. x, 241

External links[link]

Wikiquote has a collection of quotations related to: John Gower

• Complete works at Archive.org
• The International John Gower Society
• The Gower Project http://www.gowerproject.org/
• CrowdSourcing Confessio Amantis http://confessioamantis.org
• Works by John Gower at Project Gutenberg
• Luminarium: John Gower Life, works, essays
• Excerpt from Confessio Amantis - Harvard Chaucer Pages
• Volume 1 of the Confessio Amantis (containing the Prologue, Book 1, and Book 8); Volumes 2 (Books 2-4) and 3 (Books 5-7) can be accessed at the Complete Catalogue of TEAMS texts
• John Gower at the Catholic Encyclopedia