GGE Biplot Analysis

A Graphical Tool for Breeders, Geneticists, and Agronomists

Weikai Yan, Manjit S. Kang

• Publisher: CRC Press
• Year: 2002
• eBook ISBN: 9781000611120
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Research data is expensive and precious, yet it is seldom fully utilized due to our ability of comprehension. Graphical display is desirable, if not absolutely necessary, for fully understanding large data sets with complex interconnectedness and interactions. The newly developed GGE biplot methodology is a superior approach to the graphical analys

Section II

GGE Biplot and Multi-Environment Trial Data Analysis

3 Theory of Biplot

SUMMARY

This chapter introduces the theory of biplot through examining the process of matrix multiplication. A rank-two matrix results from multiplying one matrix with two columns with another matrix with two rows. A biplot displays a rank-two matrix through plotting its two component matrices in a scatter diagram. A biplot not only displays the two component matrices but also the product matrix, though implicitly. The elements of the product matrix are recovered through the inner-product property of the biplot. Moreover, a biplot allows visualization of the product matrix from all perspectives. These include:

• Ranking of the rows in each column

• Ranking of the columns in each row

• Comparison of two rows for all columns

• Comparison of two columns for all rows

• Identification of the largest value (row) within each column

• Identification of the largest value (column) within each row

• Visualization of the interrelationships among rows

• Visualization of the interrelationships among columns

These form the basis for visualizing two-way datasets. If the mathematics looks complicated, one may just skip this chapter. Nevertheless, it will not hinder one’s ability to understand and apply the genotype and genotype-by-interaction (GGE) biplot methodology to one’s data.

3.1 THE CONCEPT OF BIPLOT

3.1.1 MATRIX MULTIPLICATION

Two matrices can only be multiplied if the number of columns in the first matrix located on the left side of the multiplication sign equals the number of rows in the second matrix located on the right side of the multiplication sign. The product matrix assumes the number of rows of the left matrix and the number of columns of the right matrix. Thus the left matrix may be conveniently referred to as the row matrix and the right matrix the column matrix.

Assume that we have a row matrix:

$G=\left[\begin{array}{cc}\hfill 4& \hfill 3\\ \hfill -3& \hfill 3\\ \hfill 1& \hfill -3\\ \hfill 4& \hfill 0\end{array}\right]$

and a column matrix:

$E=\left[\begin{array}{ccc}\hfill 2& \hfill -3& \hfill 3\\ \hfill 4& \hfill 1& \hfill -2\end{array}\right].$

Since G has two columns and E has two rows, they can be multiplied; and since G has four rows and E has three columns, the product matrix Y will have four rows and three columns:

$Y=G\times E=\left[\begin{array}{cc}\hfill 4& \hfill 3\\ \hfill -3& \hfill 3\\ \hfill 1& \hfill -3\\ \hfill 4& \hfill 0\end{array}\right]\left[\begin{array}{ccc}\hfill 2& \hfill -3& \hfill 3\\ \hfill 4& \hfill 1& \hfill -2\end{array}\right]=\left[\begin{array}{ccc}\hfill 20& \hfill -9& \hfill 6\\ \hfill 6& \hfill 12& \hfill -15\\ \hfill -10& \hfill -6& \hfill 9\\ \hfill 8& \hfill -12& \hfill 12\end{array}\right].$

The rule of matrix multiplication is that an element in Y is the sum of the product between each element of a row in the row matrix (G) and each element of a column in the column matrix (E). Thus, the 4 × 3 = 12 elements in Y are calculated as follows:

$\begin{array}{l}\text{Y11}=\text{G11 × E11}+\text{G12}\times \text{E21 =}\left(4\right)\left(2\right)+\left(3\right)\left(4\right)=20\\ \text{Y12}=\text{G11 × E12}+\text{G12}\times \text{E22 =}\left(4\right)\left(-3\right)+\left(3\right)\left(1\right)=-9\\ \text{Y13}=\text{G11 × E13}+\text{G12}\times \text{E23 =}\left(4\right)\left(3\right)+\left(3\right)\left(-2\right)=6\\ \\ \text{Y21}=\text{G21 × E11}+\text{G22}\times \text{E21 =}\left(-3\right)\left(2\right)+\left(3\right)\left(4\right)=6\\ \text{Y22}=\text{G21 × E12}+\text{G22}\times \text{E22 =}\left(-3\right)\left(-3\right)+\left(3\right)\left(1\right)=12\\ \text{Y23}=\text{G21 × E13}+\text{G22}\times \text{E23 =}\left(-3\right)\left(3\right)+\left(3\right)\left(-2\right)=-15\\ \\ \text{Y31}=\text{G31 × E11}+\text{G32}\times \text{E21 =}\left(1\right)\left(2\right)+\left(-3\right)\left(4\right)=-10\\ \text{Y32}=\text{G31 × E12}+\text{G32}\times \text{E22 =}\left(1\right)\left(-3\right)+\left(-3\right)\left(1\right)=-6\\ \text{Y33}=\text{G31 × E13}+\text{G32}\times \text{E23 =}\left(1\right)\left(3\right)+\left(-3\right)\left(-2\right)=9\\ \\ \text{Y41}=\text{G41 × E11}+\text{G42}\times \text{E21 =}\left(4\right)\left(2\right)+\left(0\right)\left(4\right)=8\\ \text{Y42}=\text{G41 × E12}+\text{G42}\times \text{E22 =}\left(4\right)\left(-3\right)+\left(0\right)\left(1\right)=-12\\ \text{Y43}=\text{G41 × E13}+\text{G42}\times \text{E23 =}\left(4\right)\left(3\right)+\left(0\right)\left(-2\right)=12\end{array}$

Y is called a rank-two matrix, since it results from multiplication of G, which has two columns, and E, which has two rows. A rank-two matrix can be displayed by a two-dimensional biplot (Gabriel, 1971).

3.1.2 PRESENTING THE MATRIX MULTIPLICATION PROCESS IN A BIPLOT

Now, we present matrices G and E in the form of a table (Table 3.1), in which the two columns of G are named as x and y, and the two rows of E are also named x and y. Likewise, we present matrix Y in the form of a table (Table 3.2), which indicates the row and column origin of each element. For example, Y11 is summation of the product between the first row of matrix G (G1) and the first column of matrix E (E1), and Y32 is the product between the third row of matrix G (G3) and the second column of matrix E (E2), etc.

If the x values are plotted against the y values in Table 3.1, a two-dimensional scatter diagram is generated, in which each row of G is represented by a point. Similarly, each column of E is represented by a point (Figure 3.1). This scatter diagram is called a biplot because it displays both the row and column matrices. The biplot not only presents the two original matrices, G and E, but it also displays their product, matrix Y. We show how Y is displayed next.

TABLE 3.1
The Matrices G and E to be Multiplied

Names	x	y
	Row matrix (G)
G1	4	3
G2	−3	3
G3	1	−3
G4	4	0
	Column matrix (E)
E1	2	4
E2	−3	1
E3	3	−2

TABLE 3.2
Elements of the Product Matrix Y

	Columns
Rows	E1	E2	E3
G1	20	−9	6
G2	6	12	−15
G3	−10	−6	9
G4	8	−12	12

FIGURE 3.1 A biplot that displays both matrix G, which ha...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Foreword
• Preface
• Table of Contents
• SECTION I Genotype-by-Environment Interaction and Stability Analysis
• SECTION II GGE Biplot and Multi-Environment Trial Data Analysis
• SECTION III GGE Biplot Software and Applications to Other Types of Two-Way Data
• References
• Index