Biplot AMMI graphic representation of specific combining ... - SBMP

Crop Breeding and Applied Biotechnology, v. 2, n. 2, p. 161-170, 2002163only the few first terms (one, two or three) alreadydescribe a high proportion of this sum of squares. Thisallows the description of the SCA using a parsimoniousmodel (in the sense of using few degrees of freedom)and with good capacity to explain the original variabilitypresent in the matrix. Therefore, the analysis allowsa interpretation of the SS SCAby an approximation ofrank n of the SCA matrix. As n is possibly smallerthan p (e.g. n =1, 2 or 3) the analysis results in aninformative model which can be representedgraphically, since a complex p-dimensional space isrepresented in a bi or tridimensional graph.In the context of the AMMI method, the objective ofthe analysis is to describe the interaction by a reducednumber of axes. It is expected that, successively, eachaxis will capture a smaller portion of the patternunderlying to the interaction (general law whichgoverns the phenomena) and more noise present inthe data. Thus, a simpler model would discard a highproportion of noise which has no agronomic interestand can hamper good predictions, retaining mainlythe portion rich in deterministic information (Gauchand Zobel, 1988; Gauch, 1990). This separation inpattern and noise effects may also be expressed interms of sum of squares:p222SS SCA = ∑ λ = ( ∑ λ ) + ( ∑ λ )nkkkk= 1 k= 1 k= n+1= (SS SCA_pattern) + (SS SCA_noise).The biplot graphic displayAn important property of the SVD approximation isits allowance for the representation of effects of eachrow and of each column in a single graph calledbiplot. The name does not come from the spacedimension of representation, but from the fact that itdisplays simultaneously the rows and the columns ofthe matrix.To understand the mathematical origin of a biplotconsider the SVD n-approximation for the B matrixof rank p>n: ˆn ~~ ~ ~B( fxm)= ∑ k= 1 λkukv′k= USV′; where U( fxn)has~in its columns only the n first u (fx1)vectors; V ′( nxm)has~in its rows the n first v’ (1xm)vectors; and S( n)is thediagonal matrix with the first singular values,λ 1,λ 2,...,λ n(Duarte and Vencovsky, 1999). Theexpression may further be written as:Bˆ =~~ 1/ 2 ~ 1/ 2 ~( U S )( S V ′); or simply Bˆ =~ ~( G )( H ′), where~G( fxn)=~~ 2~( U S1/ ) and H ′( nxm)= ~ 1/ 2 ~( S V ′). Thus, ~ G matrixwill have f vectors called row markers, each one withn elements, the coordinates of the females (F) forpeach singular axis selected. Similarly, ~ H ′ will have mcolumn markers, each also with n elements,corresponding to the male (M) coordinates for thesame n axes. Given that these axes are orthogonal,the f females and m males can be plotted in a singlecartesian system of n perpendicular axes. In practice,similar to the scatter plots in PCA, the rank of theapproximate matrix (n) to be displayed should be one,two or three, which would allow a graphicrepresentation in one, two or three dimensions,respectively.A property of biplot representation is that any elementof the approximate matrix is exactly the inner productof the vectors (markers) corresponding to its row andto its column (Gabriel, 1971). In this case, the productbetween the coordinates of the i th female and the j thmale yields the expected SCA value for that hybridcombination ( ŝAMMIij ). Clearly, this value does notcorrespond to that in the diallel table because thebiplot represents the SVD n-approximation of theoriginal B matrix. However, it is expected that thisprediction, rich in pattern, is closer to the real SCA(s ij) than the values ŝ ij of the diallel table, which arepotentially rich in noise (Gauch, 1990). In this sense,a male/female pair with coordinates of highmagnitude and the same signal has high SCA (positiveinteraction), characterizing a favorable combinationfor hybridization. On the other hand, a combinationwhose coordinates have opposite signs and negativeSCA is unfavorable. Thus the intergroup genotypedistances (F/M) have inverse interpretation in relationto the distance visualized in the graph, that is, pointsclose together do not indicate genotypic similarity butrather affinity for hybrid combination.On the other hand, inspecting only the female (ormale) markers, it is possible to assess the intragroupdivergences. In this case, close points indicate greatergenotypic similarity. In complete diallels, the femalesand males are the same genotypes and, therefore, suchinspection would enable the assessment of thedivergence among all the parents. Thus for genotypeclustering, it would be enough to plot only the femalemarkers (or the males) as in PCA display. However,in this case the opportunity to graphically predict theSCA magnitude and to identify the most promisinghybrid combinations would be lost.APPLICATIONTo illustrate the use of the biplot graphic display inSCA description, Griffing (1956) method IV results2002, Brazilian Society of Plant Breeding