Produces a biplot from a set of variates (S.A. Harding).

### Options

`PRINT` = string tokens |
Printed output from the analysis (`singular` , `scores` ); default `*` i.e. no output |
---|---|

`GRAPHICS` = string token |
What sort of graphics to use (`lineprinter` , `highresolution` ); default `high` |

`WINDOW` = scalar |
Window number for the graph; default 3 |

`SCREEN` = string token |
Whether to clear the screen before plotting or to continue plotting on the old screen (`clear` , `keep` ); default `clea` |

`METHOD` = string token |
Type of analysis required (`principalcomponent` , `variate` , `diagnostic` ); default `prin` |

`STANDARDIZE` = string tokens |
Whether to centre the configurations (at the origin), and/or to normalize them (to unit sum of squares) prior to analysis (`centre` , `normalize` ); default `cent` , `norm` |

`LABELS` = factor or text |
Labels to identify the points for the individuals |

`VLABELS` = factor or text |
Labels to identify the points for the variates |

`NDIMENSIONS` = scalar |
Number of dimensions to save with `COORDINATES` and `VCOORDINATES` ; default 2 |

### Parameters

`DATA` = pointers |
Each pointer contains a set of variates to be analysed |
---|---|

`COORDINATES` = matrices |
Used to store the scores for the individuals |

`VCOORDINATES` = matrices |
Used to store the scores for the variates |

### Description

`BIPLOT`

produces a graphical representation of the relationships between data units and variates, as described by Gabriel (1971).

The data for the procedure consist of a set of variates, contained in a pointer specified by the `DATA`

parameter. The data may be centred at the origin and/or normalized before plotting, by setting the `STANDARDIZE`

option. The variates must not contain any missing values, nor should they be restricted. The values of the variates remain unaltered on exit from the procedure. The `METHOD`

option allows the user to select which form of the biplot is to be plotted: principal component, variate, or diagnostic biplot.

Printed output is controlled by the option `PRINT`

with settings: `singular`

to print the singular values, and `scores`

to print the scores. By default, nothing is printed.

The `GRAPHICS`

option controls whether the biplot is plotted in high-resolution or line-printer styles; or setting `GRAPHICS=*`

suppresses the plot. The `WINDOW`

option specifies the window to use for a high-resolution plot (default 3), and the `SCREEN`

option controls whether or not to clear the screen first (default `clear`

).

Results from the analysis can be saved using the parameters `COORDINATES`

and `VCOORDINATES`

. The structures specified for these parameters need not be declared in advance. The number of dimensions that are saved is specified by the `NDIMENSIONS`

option; default 2.

Options: `PRINT`

, `GRAPHICS`

, `WINDOW`

, `SCREEN`

, `METHOD`

, `STANDARDIZE`

, `LABELS`

, `VLABELS`

, `NDIMENSIONS`

.

Parameters: `DATA`

, `COORDINATES`

, `VCOORDINATES`

.

### Method

The biplot (Gabriel, 1971) is a graphical representation of the relationships between *n* individuals and between *p* variates. If these variates are arranged as a matrix *X* (*n* × *p*), the singular value decomposition of *X* ( *X* = *U* *S* *V* ′ ) is used to express the least-squares approximation to *X* in two dimensions in the form *X2* = *A* *B* ′, where *X2* is (*n* × 2); *A* (*n* × 2) and *B* (*p* × 2) are given by the first two columns of ( *U* *S ^{r}* ) and (

*V*

*S*

^{(1-r)}) respectively. The matrices

*A*and

*B*give the coordinates of the row and column markers, and the constant

*r*can be set to either 0, 0.5, or 1 to obtain the form of the biplot requested by the

`METHOD`

option.### Action with `RESTRICT`

The variates must not be restricted.

### References

Gabriel, K.R. (1971). The biplot graphic display of matrices with application to pricipal component analysis. *Biometrika*, 58, 453.

Gower, J.C. and Digby, P.G.N. (1981). Expressing Complex Relationships in Two Dimensions. In: *Interpreting Multivariate Data* (ed. V. Barnett). Wiley, New York.

### See also

Procedures: `DBIPLOT`

, `CABIPLOT`

, `CRBIPLOT`

, `CRTRIPLOT`

.

Commands for: Multivariate and cluster analysis, Graphics.

### Example

CAPTION 'BIPLOT example'; STYLE=meta VARIATE [NVALUES=20] v[1...7] READ v[] 4.0 11.0 4.0 28.0 31.0 17.0 21.0 5.0 11.0 5.0 29.0 30.0 16.0 21.0 7.0 9.0 6.0 25.0 30.0 17.0 23.0 3.0 9.0 5.0 28.0 32.0 12.0 15.0 5.0 15.0 6.0 29.0 34.0 18.0 21.0 3.0 10.0 5.0 23.0 27.0 17.0 20.0 3.0 10.0 7.0 24.0 28.0 18.0 21.0 3.0 13.0 7.0 29.0 34.0 18.0 21.0 3.0 10.0 5.0 26.0 21.0 17.0 28.0 5.0 10.0 6.0 26.0 30.0 16.0 23.0 7.0 9.0 5.0 26.0 30.0 16.0 23.0 4.0 11.0 8.0 27.0 31.0 17.0 22.0 3.0 12.0 6.0 26.0 31.0 18.0 24.0 4.0 11.0 7.0 26.0 31.0 18.0 23.0 6.0 10.0 9.0 28.0 31.0 21.0 27.0 4.0 12.0 9.0 27.0 32.0 16.0 25.0 5.0 12.0 8.0 29.0 33.0 15.0 22.0 4.0 14.0 6.0 23.0 29.0 16.0 19.0 4.0 10.0 6.0 25.0 29.0 19.0 22.0 3.0 15.0 7.0 25.0 29.0 16.0 19.0 : TEXT [VALUES=va,vb,vc,vd,ve,vf,vg] vlabs BIPLOT [PRINT=singular,scores; VLABELS=vlabs] v