Biplots: Taking Stock John Gower Mathematics Department The Open University Milton Keynes, U.K.
Scope Biplots simultaneously display two kinds of information; typically, the variables (categorical and numerical) and sample units described by a multivariate data matrix or the items labeling the rows and columns of a two-way table. Approximation is important. Biplots are useful for visualizing multidimensional analyses, e.g., principal component analysis, canonical variate analysis, multidimensional scaling, multiplicative interactions and correspondence analysis – and many more.
Ptolemy from Wikipaedia • The first part of the Geographia is a discussion of the data and of the methods he used. Ptolemy put all this information into a grand scheme. Following Marinos, he assigned coordinates to all the places and geographic features he knew, in a grid that spanned the globe. Latitude was measured from the equator, as it is today, but Ptolemy preferred in book 8 to express it as the length of the longest day rather than degrees of arc (the length of the midsummer day increases from 12h to 24h as one goes from the equator to the polar circle). In books 2 through 7, he used degrees and put the meridian of 0 longitude at the most western land he knew, the "Blessed Islands", probably the Cape Verde islands (not the Canary Islands, as long accepted) as suggested by the location of the six dots labeled the "FORTUNATA" islands near the left extreme of the blue sea of Ptolemy's map here reproduced.
• Geography • Main article: Geographia (Ptolemy) • Ptolemy's other main work is his Geographia . This also is a compilation of what was known about the world's geography in the Roman Empire during his time. He relied somewhat on the work of an earlier geographer, Marinos of Tyre, and on gazetteers of the Roman and ancient Persian Empire, but most of his sources beyond the perimeter of the Empire were unreliable.[ citation needed ] • The first part of the Geographia is a discussion of the data and of the methods he used. As with the model of the solar system in the Almagest , Ptolemy put all this information into a grand scheme. Following Marinos, he assigned coordinates to all the places and geographic features he knew, in a grid that spanned the globe. Latitude was measured from the equator, as it is today, but Ptolemy preferred in book 8 to express it as the length of the longest day rather than degrees of arc (the length of the midsummer day increases from 12h to 24h as one goes from the equator to the polar circle). In books 2 through 7, he used degrees and put the meridian of 0 longitude at the most western land he knew, the "Blessed Islands", probably the Cape Verde islands (not the Canary Islands, as long accepted) as suggested by the location of the six dots labelled the "FORTUNATA" islands near the left extreme of the blue sea of Ptolemy's map here reproduced.
Ptolemy and Biplots • Ptolemy’s maps have all the ingrediants of a biplot: (a) Points representing cities (b) Lines representing latitude/longitude coordinates (c) Approximation N.B. Ptolemy seems to have recognised that the world is a globe.
Descartes from Wikipaedia This system was developed in 1637 in two writings by Descartes and independently by Pierre de Fermat, (unpublished). Descartes introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides. In La Géométrie , he further explores the above-mentioned concepts. Some note that the master artists of the Renaissance used a grid, in the form of a wire mesh, as a tool for breaking up the component parts of their subjects they painted. This may have influenced Descartes. Nicole Oresme, a French philosopher of the 14th Century, used constructions similar to Cartesian coordinates.
Types of approximation 1. Through the least-squares properties of the singular-value decomposition. Representing “cases” by any form of 2. MDS and then superimposing the “variables” either by: (i) The regression method (ii) by superimposing nonlinear trajectories (nonlinear biplots)
Singular Value Decomposition • SVD X = U Σ V ' Beltrami (1873), Jordan (1874), Sylvester (1889) ˆ 2 ˆ • Approximation: min where rank = r X X X ˆ X = U Σ JV ' Eckart and Young (1936) • Ruben Gabriel (1971) Biplots PCA and Two way Table, Diagnostic biplots Canonical Biplots • Paul Horst ??
Paul Horst: Willem Heiser • What I recall is that I said Paul Horst had written about direct decomposition of data matrices, rather than defining factor analysis in the usual way, as decomposition of secondary structures such as the correlation matrix. He was a man of numbers, not plots, so I don't believe he actually plotted the coordinates obtained (as least, I couldn't find that back). • • See his book "Factor Analysis of Data Matrices" (Holt, Rinehart & Winston, 1965), and note the "Data Matrices" in the title. In chapter 3 he introduces the singular value decomposition, and in chapter 4 the decomposition of the data matrix as the basic one from which other methods follow. Chapter 22 gives an interesting treatment on homogeneity analysis, 10 years before Gifi started to make a whole system out of it. To my own surprise, it also contains a method to eliminate the horseshoe effect in multiple correspondence analysis!
Analysis and Presentation • Biplots are not a method of analysis – merely a way of presenting (graphically) the results of a variety of multidimensional methods of analysis. • Although I shall not be discussing methods of analysis I note that the main differences between methods is often one of the initial transformation of the data X .
Initial transformations Centre and scale PCA 1 ( I N ) XS 2 log X Remove “main effects” I P X I Q , I P log X I Q Biadditive 1 1 Pearson Residuals R XC CA 2 2 1 1 1 1 R XC or R XC 2 2 Row/Col chi-square distance CA 2 Within-group dispersion spectral decomposition CVA W V V 1 canonical var iables X XV Dispersion FA X X Dissimilarity MDS X X Constrained regression Rank,CANOCO Y XB Y XB XB XB 0 0
Types of “Scale” (a) (f) 1 2 3 4 5 (b) small medium big 1 2 3 4 5 (c) 1 2 3 4 5 6 (d) (g) 1 2 7 6 3 5 4 (e) small medium big
Tools of interpretation • The inner-product • Distance • Area • Angle • Orthogonal Projection • Nearness • Convex regions • Content
How to represent inner-products • Two sets of points (or vectors) a i b j cos( θ ij ) • One set of points plus calibrated axes • Areas a i b j cos( θ ij ) = a i b j sin( θ ij +½ π ) i.e rotate one set of points through 90 degrees. Published Biplots need clear indications of what choice has been made (Cartouche).
PCA biplot: Process Quality Control A4 C6 C8 C7 D4 32.5 66 66 A3 1.8 18 5.6 1.2 Jul00 32 32 64 1.7 1.15 5.4 2.5 2.5 31.5 16 Mar01 1.1 1.1 62 62 20 1.6 5.2 3 14.2 31 1.05 60 1 14 3.5 30.5 5 14.3 14.3 Apr00 Aug00 Target 1.5 20.5 79 49 49 0.95 D7 22 22 Jun00 A5 B5 A5 B5 26 21.5 21.5 27 28 29 21 21 30 45 45 20.5 20.5 D6 79.2 Feb01 4 58 58 44 30 30 50 0.9 43 43 4.8 14.4 A2 Dec00 1.4 12 Jan01 May00 A1 21 Sep00 Feb00 0.85 4.5 29.5 Jan00 Mar00 Nov00 56 Oct00 4.6 14.5 14.5 0.8 C5 E5 C4 Poor quality Satisfactory quality Good quality
Orthogonal Parallel Translation of Axes 6 5 4 3 2 (a,b) 6 1 5 0 4 3 2 1 O 0
PCA biplot of 4 variables and 23 cases RGF SLF 6 0 6 5 4 0.1 r 5 c 0 q p 3 j k 2 m g u v i h t 4 2 n d 0.2 6 f e 4 s w 8 1 b SPR a 0 3 0.3 -1 -2 PLF
Orthogonal Parallel Shift and Rotation PLF -4 -3 0.4 0.4 2 -2 0.3 -1 3 0 a b 0.2 1 f 4 e d 2 h i g 0.1 n m s w r p 3 c k 5 t q j u v 0 2 4 6 8 10 4 SPR 0 0 5 6 6 -0.1 RGF SLF
Σ -Scaling Plots based on the inner-product given by SVD X = U Σ V ' give Row-points p A r =U and Column-points q B r = V with α + β = 1,usually. Choices are: (i) α = 1, β = 0 or β = 1, α =0 Preserves I-P; suitable for showing approximations to row or column distances (ii) α = 1, β = 1 I-P not preserved, suitable for showing simultaneous row and column distances (iii) α = ½, β = ½ Preserves I-P; suitable for showing symmetric ˆ approximations to X
λ -Scaling When plotting points given by the rows of p A r and q B r one set may have much greater dispersion than the other. This can be remedied as follows: first observe that AB ' = ( A λ )( B '/ λ ) leaves the inner product unchanged. This simple fact may be used to improve the look of the display. One way of choosing λ is to arrange that the average squared distance of the points in A λ and B / λ are the same. This requires: 2 2 A / p B / q 4 p B / q A
General Scaling • Note that λ could be replaced by any nonsingular matrix L but this does not seem to be helpful, except perhaps when L is diagonal.
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