This vignette deals with biplots based on Multidimensional scaling
(MDS).
Regression biplot
The function regress accepts as arguments an object of
class biplot. The call to the function biplot
should contain the data set that will be used to construct the biplot
axes. The second argument Z contains the coordinates of the
samples in the MDS map. By default linear regression axes will be fitted
to the plot. If \(\mathbf{X}\) denote
the data matrix, the directions of the biplot axes are found by solving
the regression equation
\[
\mathbf{X} = \mathbf{ZH}' + \mathbf{E}.
\] The matrix \(\mathbf{H}'=(\mathbf{Z'Z})^{-1}\mathbf{Z'X}\)
and calibration of the axes proceed analogous to equation (2) in the
biplotEZ vignette. The coordinates of the marker \(/mu\) on biplot axis \(j\) is given as follows
\[
p_{\mu}
= \frac{\mu}{\mathbf{h}_{(j)}'\mathbf{h}_{(j)}}\mathbf{h}_{(j)}.
\]
biplot(rock) |>
regress(Z = MASS::sammon(dist(scale(rock), method="manhattan"))$points) |>
plot()
#> Initial stress : 0.02554
#> stress after 10 iters: 0.01094, magic = 0.500
#> stress after 20 iters: 0.01080, magic = 0.500
#> stress after 30 iters: 0.01078, magic = 0.500
More flexibility is provided by approximating the biplot axes using
B-splines. A calibrated trajectory represented in a matrix \(\mathbf{H}:m \times 2\) snakes through the
samples points \(\mathbf{Z}\) such that
the marker on the trajectory which is closest to a particular sample
gives the predicted value of that sample for the particular variable. To
retain smoothness of the trajectory \(\mathbf{H}\), B-splines are used.
biplot(rock) |>
regress(Z = MASS::sammon(dist(scale(rock), method="manhattan"))$points,
axes = "splines") |>
plot()
#> Initial stress : 0.02554
#> stress after 10 iters: 0.01094, magic = 0.500
#> stress after 20 iters: 0.01080, magic = 0.500
#> stress after 30 iters: 0.01078, magic = 0.500
#> Calculating spline axis for variable 1
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
#> Calculating spline axis for variable 2
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
#> Calculating spline axis for variable 3
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
#> Calculating spline axis for variable 4
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
Principal Coordinate
Analysis (PCO) biplots
Let \(\mathbf{\tilde\Delta}\) be the
\(n \times n\) matrix containing the
values \(-\frac{1}{2}\delta_{ij}^2\)
where \(\delta_{ij}\) represent the
dissimilarity between objects \(i\) and
\(j\). If it is possible to find a set
of coordinates \(\mathbf{Y}:n\times m\)
where typically \(m = n-1\) such that
the Euclidean distances between the rows of \(\mathbf{Y}\) exactly match the
dissimilarities \(\delta_{ij}\), the
dissimilarity is known as a Euclidean embeddable distance metric. J. C. Gower (1982) shows that if the distances
\(\delta_{ij}\) are Euclidean
embeddable, then
\[
(\mathbf{I}-\mathbf{1s}')\mathbf{\tilde\Delta}
(\mathbf{I}-\mathbf{s1}')
\]
is positive semi-definite.
The Euclidean representation of the samples is obtained as \(\mathbf{Y=V\Lambda}^{\frac{1}{2}}\) where
\((\mathbf{I}-\mathbf{1s}')\mathbf{\tilde\Delta}
(\mathbf{I}-\mathbf{s1}') = \mathbf{V \Lambda V'}\).
Since the coordinates in \(\mathbf{Y}\)
is already referred to their principal axes with \(\mathbf{Y'Y=\Lambda}\), the
representation of the samples in a 2D biplot is obtained from the first
two columns of \(\mathbf{Y}\).
In addition to the exact biplot axis representations discussed in
J. C. Gower, Lubbe, and Roux (2011)
approximate axes can be obtained. Linear axes are fitted with the
regression method. The variables in the data matrix \(\mathbf{X}:n \times p\) can be represented
as biplot axes in the PCO biplot with the sample points \(\mathbf{Z=V\Lambda}^{\frac{1}{2}}\mathbf{J}_2\)
according to the regression method discussed in section 2 above.
biplot(rock, scale = TRUE) |> PCO() |> plot()
Using B-splines instead of linear regression provides the user with
more flexibility. This is achieved by setting the argument
axes = "splines".
biplot(rock, scale = TRUE) |> PCO(axes = "splines") |> plot()
#> Calculating spline axis for variable 1
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
#> attr(,"scaled:scale")
#> area peri shape perm
#> 3.725992e-04 6.984893e-04 1.197656e+01 2.284053e-03
#> Calculating spline axis for variable 2
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
#> attr(,"scaled:scale")
#> area peri shape perm
#> 3.725992e-04 6.984893e-04 1.197656e+01 2.284053e-03
#> Calculating spline axis for variable 3
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
#> attr(,"scaled:scale")
#> area peri shape perm
#> 3.725992e-04 6.984893e-04 1.197656e+01 2.284053e-03
#> Calculating spline axis for variable 4
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
#> attr(,"scaled:scale")
#> area peri shape perm
#> 3.725992e-04 6.984893e-04 1.197656e+01 2.284053e-03
By default the distance metric used for the analysis is Euclidean
distance. The user can also specify the distance matrix
Dmat as an \(n \times n\)
matrix of a dist object. As illustration of a metric that
is Euclidean embeddable, the Clark’s distance
\[
\delta_{ij}^2 =
\sum_{k=1}^{p}\left(\frac{x_{ik}-x_{jk}}{x_{ik}+x_{jk}}\right)^2
\]
as defined by John C. Gower and Ngouenet
(2005) is calculated below and used for constructing a PCO
biplot. Note that the metric is only defined for strictly positive
values, so that the data is scaled to values between \(1\) and \(2\).
Clark.dist <- function(X)
{
n <- nrow(X)
p <- ncol(X)
Dmat <- matrix (0, nrow=n, ncol=n)
for (i in 1:(n-1))
for (j in (i+1):n)
Dmat[i,j] <- sum(((X[i,] - X[j,])/(X[i,] + X[j,]))^2)
sqrt(Dmat + t(Dmat))
}
my.data <- scale(rock, center=apply(rock,2,min), scale=diff(apply(rock,2,range)))+1
biplot(rock) |> PCO(Dmat = Clark.dist (rock), axes = "splines") |> plot()
#> Calculating spline axis for variable 1
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
#> Calculating spline axis for variable 2
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
#> Calculating spline axis for variable 3
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
#> Calculating spline axis for variable 4
#> area peri shape perm
#> 1 4990 2791.900 0.0903296 6.3
#> 2 7002 3892.600 0.1486220 6.3
#> 3 7558 3930.660 0.1833120 6.3
#> 4 7352 3869.320 0.1170630 6.3
#> 5 7943 3948.540 0.1224170 17.1
#> 6 7979 4010.150 0.1670450 17.1
#> 7 9333 4345.750 0.1896510 17.1
#> 8 8209 4344.750 0.1641270 17.1
#> 9 8393 3682.040 0.2036540 119.0
#> 10 6425 3098.650 0.1623940 119.0
#> 11 9364 4480.050 0.1509440 119.0
#> 12 8624 3986.240 0.1481410 119.0
#> 13 10651 4036.540 0.2285950 82.4
#> 14 8868 3518.040 0.2316230 82.4
#> 15 9417 3999.370 0.1725670 82.4
#> 16 8874 3629.070 0.1534810 82.4
#> 17 10962 4608.660 0.2043140 58.6
#> 18 10743 4787.620 0.2627270 58.6
#> 19 11878 4864.220 0.2000710 58.6
#> 20 9867 4479.410 0.1448100 58.6
#> 21 7838 3428.740 0.1138520 142.0
#> 22 11876 4353.140 0.2910290 142.0
#> 23 12212 4697.650 0.2400770 142.0
#> 24 8233 3518.440 0.1618650 142.0
#> 25 6360 1977.390 0.2808870 740.0
#> 26 4193 1379.350 0.1794550 740.0
#> 27 7416 1916.240 0.1918020 740.0
#> 28 5246 1585.420 0.1330830 740.0
#> 29 6509 1851.210 0.2252140 890.0
#> 30 4895 1239.660 0.3412730 890.0
#> 31 6775 1728.140 0.3116460 890.0
#> 32 7894 1461.060 0.2760160 890.0
#> 33 5980 1426.760 0.1976530 950.0
#> 34 5318 990.388 0.3266350 950.0
#> 35 7392 1350.760 0.1541920 950.0
#> 36 7894 1461.060 0.2760160 950.0
#> 37 3469 1376.700 0.1769690 100.0
#> 38 1468 476.322 0.4387120 100.0
#> 39 3524 1189.460 0.1635860 100.0
#> 40 5267 1644.960 0.2538320 100.0
#> 41 5048 941.543 0.3286410 1300.0
#> 42 1016 308.642 0.2300810 1300.0
#> 43 5605 1145.690 0.4641250 1300.0
#> 44 8793 2280.490 0.4204770 1300.0
#> 45 3475 1174.110 0.2007440 580.0
#> 46 1651 597.808 0.2626510 580.0
#> 47 5514 1455.880 0.1824530 580.0
#> 48 9718 1485.580 0.2004470 580.0
#> attr(,"scaled:center")
#> area peri shape perm
#> -7187.7291667 -2682.2119375 -0.2181104 -415.4500000
Alternatively, the user can specify a function which computes a
distance matrix or dist object. The Manhattan distance is
not Euclidean embeddable, but the square root of the distance is. The
function sqrtManhattan is included as an example of a
function computing a Euclidean embeddable dist object.
sqrtManhattan
#> function (X)
#> {
#> sqrt(stats::dist(X, method = "manhattan"))
#> }
#> <bytecode: 0x000002271155f128>
#> <environment: namespace:biplotEZ>
biplot(rock, scaled = TRUE) |> PCO(dist.func = sqrtManhattan) |> plot()
