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This vignette deals with biplots for separating classes. Topics discussed are
Consider a data matrix \(\mathbf{X}:n \times p\) containing data on \(n\) objects and \(p\) variables. In addition, a vector \(\mathbf{g}:n \times 1\) contains information on class membership of each observation. Let \(G\) indicate the total number of classes. CVA is closely related to linear discriminant anlaysis, in that the \(p\) variables are transformed to \(p\) new variables, called canonical variates, such that the classes are optimally separated in the canonical space. By optimally separated, we mean maximising the between class variance, relative to the within class variance. This can be formulated as follows:
Let \(\mathbf{G}:n \times G\) be an indicator matrix with \(g_{ij} = 0\) unless observation \(i\) belongs to class \(j\) and then \(g_{ij} = 1\). The matrix \(\mathbf{G'G}\) is a diagonal matrix containing the number of observations per class on the diagonal. We can form the matrix of class means \(\bar{\mathbf{X}}:G \times p = (\mathbf{G'G})^{-1} \mathbf{G'X}\). With the usual analysis of variance the total variance can be decomposed into a between class variance and within class variance:
\[ \mathbf{T} = \mathbf{B} + \mathbf{W} \]
\[ \mathbf{X'X} = \mathbf{\bar{\mathbf{X}}'C \bar{\mathbf{X}}} + \mathbf{X' [I - G(G'G)^{-1}C(G'G)^{-1}G'] X} \]
The default choice for the centring matrix \(\mathbf{C = G'G}\) leads to the simplification
\[ \mathbf{X'X} = \mathbf{\bar{\mathbf{X}}'G'G \bar{\mathbf{X}}} + \mathbf{X' [I - G(G'G)^{-1}G'] X}. \]
Other options are \(\mathbf{C = I}\) and \(\mathbf{C} = (\mathbf{I}_G - \frac{1}{G}\mathbf{11'})\). To find the canonical variates we want to maximise the ratio
\[ \frac{\mathbf{m'Bm}}{\mathbf{m'Wm}} \]
subject to \(\mathbf{m'Wm} = 1\). It can be shown that this leads to the following equivalent eigen equations:
\[ \mathbf{W}^{-1}\mathbf{BM} = \mathbf{M \Lambda} \tag{1} \]
\[ \mathbf{BM} = \mathbf{WM \Lambda} \]
\[ (\mathbf{W}^{-\frac{1}{2}} \mathbf{B} \mathbf{W}^{-\frac{1}{2}}) \mathbf{M} = (\mathbf{W}^{-\frac{1}{2}} \mathbf{M}) \mathbf{\Lambda} \]
with \(\mathbf{M'BM}= \mathbf{\Lambda}\) and \(\mathbf{M'WM}= \mathbf{I}\).
Since the matrix \(\mathbf{W}^{-\frac{1}{2}} \mathbf{B} \mathbf{W}^{-\frac{1}{2}}\) is symmetric and positive semi-definite the eigenvalues in the matrix \(\mathbf{\Lambda}\) are positive and ordered. The rank of \(\mathbf{B} = min(p, G-1)\) so that only the first \(rank(\mathbf{B})\) eigenvalues are non-zero. We form the canonical variates with the transformation
\[ \bar{\mathbf{Y}} = \bar{\mathbf{X}}\mathbf{M}. \]
To construct a 2D biplot, we plot the first two canonical variates \(\bar{\mathbf{Z}} = \bar{\mathbf{X}}\mathbf{MJ}_2\) where \(\mathbf{J}_2' = \begin{bmatrix} \mathbf{I}_2 & \mathbf{0} \end{bmatrix}\). We add the individual sample points with the same transformation
\[ \mathbf{Z} = \mathbf{X}\mathbf{MJ}_2 \] where \[ \mathbf{J}_2 = \begin{bmatrix} \mathbf{I}_2\\ \mathbf{0} \end{bmatrix}. \] Interpolation of a new sample \(\mathbf{x}^*:p \times 1\) follows as \({\mathbf{z}^*}':2 \times 1 ={\mathbf{x}^*}' \mathbf{MJ}_2\). Using the inverse transformation \(\mathbf{x}' = \mathbf{y}'\mathbf{M}^{-1}\), all the points that will predict \(\mu\) for variable \(j\) will have the form
\[ \mu = \mathbf{y}'\mathbf{M}^{-1} \mathbf{e}_j \]
where \(\mathbf{e}_j\) is a vector of zeros with a one in position \(j\). All the points in the 2D biplot that predict the value \(\mu\) will have
\[ \mu = \begin{bmatrix} z_1 & z_2 & 0 & \dots & 0\end{bmatrix}\mathbf{M}^{-1} \mathbf{e}_j \]
defining the prediction line as
\[ \mu = \mathbf{z}_{\mu}' \mathbf{J}_2 \mathbf{M}^{-1} \mathbf{e}_j. \]
Writing \(\mathbf{h}_{(j)} = \mathbf{J}_2 \mathbf{M}^{-1} \mathbf{e}_j\) the construction of biplot axes is similar to the discussion in the biplotEZ vignette for PCA biplots. The direction of the axis is given by \(\mathbf{h}_{(j)}\). To find the intersection of the prediction line with \(\mathbf{h}_{(j)}\) we note that \[ \mathbf{z}'_{(\mu)}\mathbf{h}_{(j)} = \| \mathbf{z}_{(\mu)} \|^2 \| \mathbf{h}_{(j)} \|^2 cos(\mathbf{z}_{(\mu)},\mathbf{h}_{(j)}) = \| \mathbf{p} \|^2 \| \mathbf{h}_{(j)} \|^2 \] where \(\mathbf{p}\) is the length of the orthogonal projection of \(\mathbf{z}_{(\mu)}\) on \(\mathbf{h}_{(j)}\).
Since \(\mathbf{p}\) is along \(\mathbf{h}_{(j)}\) we can write \(\mathbf{p} = c\mathbf{h}_{(j)}\) and all points on the prediction line \(\mu = \mathbf{z}'_{\mu}\mathbf{h}_{(j)}\) project on the same point \(c_{\mu}\mathbf{h}_{(j)}\). We solve for \(c_{\mu}\) from \[ \mu = \mathbf{z}'_{\mu}\mathbf{h}_{(j)}=\| \mathbf{p} \|^2 \| \mathbf{h}_{(j)} \|^2 = \| c_{\mu}\mathbf{h}_{(j)} \|^2 \| \mathbf{h}_{(j)} \|^2 \]
\[ c_{\mu} = \frac{\mu}{\mathbf{h}_{(j)}'\mathbf{h}_{(j)}}. \] If we select ‘nice’ scale markers \(\tau_{1}, \tau_{2}, \cdots \tau_{k}\) for variable \(j\), then \(\tau_{h}-\bar{x}_j = \mu_{h}\) and positions of these scale markers on \(\mathbf{h}_{(j)}\) are given by \(p_{\mu_{1}}, p_{\mu_{2}}, \cdots p_{\mu_{k}}\) with \[ p_{\mu_h} = c_{\mu_h}\mathbf{h}_{(j)} = \frac{\mu_h}{\mathbf{h}_{(j)}'\mathbf{h}_{(j)}}\mathbf{h}_{(j)} \]
\[ = \frac{\mu_h}{\mathbf{e}_{(j)}' \mathbf{M'}^{-1} \mathbf{J} \mathbf{M}^{-1} \mathbf{e}_{(j)}}\ \mathbf{J}_2 \mathbf{M}^{-1} \mathbf{e}_{(j)} \]
with \[ \mathbf{J} = \begin{bmatrix} \mathbf{I}_2 & \mathbf{0}\\ \mathbf{0} & \mathbf{0} \end{bmatrix}. \]
CVA()
To obtain a CVA biplot of the state.x77
data set,
optimally separating the classes according to state.region
we call
Fitting \(\alpha\)-bags to the classes makes it easier to compare class overlap and separation. For a detailed discussion on \(\alpha\)-bags, see the biplotEZ vignette.
biplot(state.x77) |> CVA(classes = state.region) |> alpha.bags() |>
legend.type (bags = TRUE) |>
plot()
#> Computing 0.95 -bag for Northeast
#> Computing 0.95 -bag for South
#> Computing 0.95 -bag for North Central
#> Computing 0.95 -bag for West
means()
This function controls the aesthetics of the class means in the
biplot. The function accepts as first argument an object of class
biplot
where the aesthetics should be applied. Let us first
construct a CVA biplot of the state.x77
data with samples
optimally separated according to state.division
.
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
legend.type(means = TRUE) |>
plot()
Instead of adding a legend, we can choose to label the class means. Furthermore, the colour of each class mean defaults to the colour of the samples. We wish to select a different colour and plotting character for the class means.
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
means(label = TRUE, col = "olivedrab", pch = 15) |>
plot()
If we choose to only show the class means for the central states, the
argument which
is used either indicating the number(s) in
the sequence of levels (which = 4:7
), or as shown below,
the levels themselves:
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
means (which = c("West North Central", "West South Central", "East South Central",
"East North Central"), label = TRUE) |>
plot()
The size of the labels is controlled with label.cex
which can be specified either as a single value (for all class means) or
a vector indicating size values for each individual sample. The colour
of the labels defaults to the colour(s) of the class means. However,
individual label colours can be spesified with label.col
,
similar to label.cex
as either a single value of a vector
of length equal to the number of classes.
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
means (col = "olivedrab", pch = 15, cex = 1.5,
label = TRUE, label.col = c("blue","green","gold","cyan","magenta",
"black","red","grey","purple")) |>
plot()
We can also make use of the functionality of the ggrepel
package to place the labels.
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
samples (label = "ggrepel", label.cex=0.65) |>
means (label = "ggrepel", label.cex=0.8) |> plot()
classify()
Classification regions can be added to the CVA biplot with the
function classify()
. The argument
classify.regions
must be set equal to TRUE
to
render the regions in the plot. Other arguments such as
col
, opacity
and borders
allows
to change the aesthetics of the regions.
biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
classify(classify.regions = TRUE,opacity = 0.2) |>
plot()
#> [[1]]
#> [,1] [,2]
#> [1,] -0.67657397 0.6265532938
#> [2,] -0.51718973 0.4035482285
#> [3,] -0.51718973 0.4035482285
#> [4,] -0.46620135 0.3322070130
#> [5,] -0.46620135 0.3322070130
#> [6,] -0.35689824 0.1792737857
#> [7,] -0.35689824 0.1792737857
#> [8,] -0.35689824 0.1792737857
#> [9,] -0.35689824 0.1792737857
#> [10,] -0.34732509 0.1658793547
#> [11,] -0.34732509 0.1658793547
#> [12,] -0.33728853 0.1518365385
#> [13,] -0.33728853 0.1518365385
#> [14,] -0.33454505 0.1479979592
#> [15,] -0.33454505 0.1479979592
#> [16,] -0.32050157 0.1283487876
#> [17,] -0.32050157 0.1283487876
#> [18,] -0.30163236 0.1019476277
#> [19,] -0.30163236 0.1019476277
#> [20,] -0.23730033 0.0119364253
#> [21,] -0.23730033 0.0119364253
#> [22,] -0.22896788 0.0002779352
#> [23,] -0.22896788 0.0002779352
#> [24,] -0.20810213 -0.0289167114
#> [25,] -0.20810213 -0.0289167114
#> [26,] -0.19588594 -0.0460091924
#> [27,] -0.19588594 -0.0460091924
#> [28,] -0.18127394 -0.0664538056
#> [29,] -0.18127394 -0.0664538056
#> [30,] -0.16972825 -0.0826081564
#> [31,] -0.16972825 -0.0826081564
#> [32,] -0.16920367 -0.0833421322
#> [33,] -0.16920367 -0.0833421322
#> [34,] -0.16920367 -0.0833421322
#> [35,] -0.16920367 -0.0833421322
#> [36,] -0.11912749 -0.1534070312
#> [37,] -0.11912749 -0.1534070312
#> [38,] -0.10924433 -0.1672352114
#> [39,] -0.10924433 -0.1672352114
#> [40,] -0.10718222 -0.1701204463
#> [41,] -0.10718222 -0.1701204463
#> [42,] -0.08096737 -0.2067993775
#> [43,] -0.08096737 -0.2067993775
#> [44,] -0.08087036 -0.2069351135
#> [45,] -0.08087036 -0.2069351135
#> [46,] 0.03579740 -0.3701727113
#> [47,] 0.03579740 -0.3701727113
#> [48,] 0.08231319 -0.4352560403
#> [49,] 0.08231319 -0.4352560403
#> [50,] 0.08505992 -0.4390991632
#> [51,] 0.08505992 -0.4390991632
#> [52,] 0.21172646 -0.6163267259
#> [53,] 0.21172646 -0.6163267259
#> [54,] 0.21231045 -0.6171438178
#>
#> [[2]]
#> [,1] [,2]
#> [1,] -0.346175039 -0.617143818
#> [2,] -0.194080378 -0.332658832
#> [3,] -0.194080378 -0.332658832
#> [4,] -0.168265117 -0.284372758
#> [5,] -0.168265117 -0.284372758
#> [6,] -0.128467559 -0.209933539
#> [7,] -0.128467559 -0.209933539
#> [8,] -0.115211563 -0.185138902
#> [9,] -0.115211563 -0.185138902
#> [10,] -0.110336767 -0.176020855
#> [11,] -0.110336767 -0.176020855
#> [12,] -0.107182222 -0.170120446
#> [13,] -0.107182222 -0.170120446
#> [14,] -0.047777235 -0.059006573
#> [15,] -0.047777235 -0.059006573
#> [16,] -0.042467626 -0.049075232
#> [17,] -0.042467626 -0.049075232
#> [18,] -0.039453583 -0.043437625
#> [19,] -0.039453583 -0.043437625
#> [20,] -0.019914041 -0.006889948
#> [21,] -0.019914041 -0.006889948
#> [22,] -0.013415304 0.005265593
#> [23,] -0.013415304 0.005265593
#> [24,] -0.008636955 0.014203241
#> [25,] -0.008636955 0.014203241
#> [26,] -0.002413403 0.025844066
#> [27,] -0.002413403 0.025844066
#> [28,] 0.001256994 0.032709347
#> [29,] 0.001256994 0.032709347
#> [30,] 0.005293778 0.040259938
#> [31,] 0.005293778 0.040259938
#> [32,] 0.022746390 0.072904122
#> [33,] 0.022746390 0.072904122
#> [34,] 0.031886641 0.090000475
#> [35,] 0.031886641 0.090000475
#> [36,] 0.033004443 0.092091265
#> [37,] 0.033004443 0.092091265
#> [38,] 0.046517674 0.117367045
#> [39,] 0.046517674 0.117367045
#> [40,] 0.062295437 0.146878512
#> [41,] 0.062295437 0.146878512
#> [42,] 0.073186309 0.167249311
#> [43,] 0.073186309 0.167249311
#> [44,] 0.085076236 0.189488787
#> [45,] 0.085076236 0.189488787
#> [46,] 0.099790112 0.217010311
#> [47,] 0.099790112 0.217010311
#> [48,] 0.130965985 0.275323125
#> [49,] 0.130965985 0.275323125
#> [50,] 0.213327502 0.429375968
#> [51,] 0.213327502 0.429375968
#> [52,] 0.214827323 0.432181305
#> [53,] 0.214827323 0.432181305
#> [54,] 0.215424980 0.433299189
#> [55,] 0.215424980 0.433299189
#> [56,] 0.430659075 0.835883134
#>
#> [[3]]
#> [,1] [,2]
#> [1,] -0.618406164 -0.61714382
#> [2,] -0.405228050 -0.34858650
#> [3,] -0.405228050 -0.34858650
#> [4,] -0.369819727 -0.30397983
#> [5,] -0.369819727 -0.30397983
#> [6,] -0.304380313 -0.22154063
#> [7,] -0.304380313 -0.22154063
#> [8,] -0.224637133 -0.12108184
#> [9,] -0.224637133 -0.12108184
#> [10,] -0.221911419 -0.11764804
#> [11,] -0.221911419 -0.11764804
#> [12,] -0.199499371 -0.08941382
#> [13,] -0.199499371 -0.08941382
#> [14,] -0.181273945 -0.06645381
#> [15,] -0.181273945 -0.06645381
#> [16,] -0.121765704 0.00851343
#> [17,] -0.121765704 0.00851343
#> [18,] -0.099197431 0.03694447
#> [19,] -0.099197431 0.03694447
#> [20,] -0.090151675 0.04834012
#> [21,] -0.090151675 0.04834012
#> [22,] -0.089504293 0.04915568
#> [23,] -0.089504293 0.04915568
#> [24,] -0.084646079 0.05527596
#> [25,] -0.084646079 0.05527596
#> [26,] -0.081006463 0.05986107
#> [27,] -0.081006463 0.05986107
#> [28,] -0.078402226 0.06314183
#> [29,] -0.078402226 0.06314183
#> [30,] -0.055454998 0.09205027
#> [31,] -0.055454998 0.09205027
#> [32,] -0.046641188 0.10315372
#> [33,] -0.046641188 0.10315372
#> [34,] -0.027600268 0.12714107
#> [35,] -0.027600268 0.12714107
#> [36,] -0.022262264 0.13386578
#> [37,] -0.022262264 0.13386578
#> [38,] -0.011840931 0.14699436
#> [39,] -0.011840931 0.14699436
#> [40,] -0.008124176 0.15167665
#> [41,] -0.008124176 0.15167665
#> [42,] 0.032462978 0.20280749
#> [43,] 0.032462978 0.20280749
#> [44,] 0.032661934 0.20305813
#> [45,] 0.032661934 0.20305813
#> [46,] 0.032792400 0.20322249
#> [47,] 0.032792400 0.20322249
#> [48,] 0.047609001 0.22188813
#> [49,] 0.047609001 0.22188813
#> [50,] 0.060679714 0.23835435
#> [51,] 0.060679714 0.23835435
#> [52,] 0.213722347 0.43115425
#> [53,] 0.213722347 0.43115425
#> [54,] 0.215424980 0.43329919
#> [55,] 0.215424980 0.43329919
#> [56,] 0.230005490 0.45166741
#> [57,] 0.230005490 0.45166741
#> [58,] 0.534992085 0.83588313
#>
#> [[4]]
#> [,1] [,2]
#> [1,] -0.676573973 -0.42876538
#> [2,] -0.469412084 -0.28772730
#> [3,] -0.469412084 -0.28772730
#> [4,] -0.379472366 -0.22649536
#> [5,] -0.379472366 -0.22649536
#> [6,] -0.269401084 -0.15155763
#> [7,] -0.269401084 -0.15155763
#> [8,] -0.224637133 -0.12108184
#> [9,] -0.224637133 -0.12108184
#> [10,] -0.220709826 -0.11840809
#> [11,] -0.220709826 -0.11840809
#> [12,] -0.171392123 -0.08483206
#> [13,] -0.171392123 -0.08483206
#> [14,] -0.169203666 -0.08334213
#> [15,] -0.169203666 -0.08334213
#> [16,] -0.099735793 -0.03604765
#> [17,] -0.099735793 -0.03604765
#> [18,] -0.092302824 -0.03098720
#> [19,] -0.092302824 -0.03098720
#> [20,] -0.079436253 -0.02222750
#> [21,] -0.079436253 -0.02222750
#> [22,] -0.063355181 -0.01127933
#> [23,] -0.063355181 -0.01127933
#> [24,] -0.025163154 0.01472222
#> [25,] -0.025163154 0.01472222
#> [26,] -0.019341098 0.01868594
#> [27,] -0.019341098 0.01868594
#> [28,] -0.006334286 0.02754112
#> [29,] -0.006334286 0.02754112
#> [30,] -0.003126520 0.02972500
#> [31,] -0.003126520 0.02972500
#> [32,] 0.001256994 0.03270935
#> [33,] 0.001256994 0.03270935
#> [34,] 0.037287563 0.05723935
#> [35,] 0.037287563 0.05723935
#> [36,] 0.048283238 0.06472533
#> [37,] 0.048283238 0.06472533
#> [38,] 0.049421667 0.06550038
#> [39,] 0.049421667 0.06550038
#> [40,] 0.074840492 0.08280580
#> [41,] 0.074840492 0.08280580
#> [42,] 0.091822081 0.09436705
#> [43,] 0.091822081 0.09436705
#> [44,] 0.127641770 0.11875348
#> [45,] 0.127641770 0.11875348
#> [46,] 0.147334391 0.13216043
#> [47,] 0.147334391 0.13216043
#> [48,] 0.191189546 0.16201750
#> [49,] 0.191189546 0.16201750
#> [50,] 0.349266341 0.26963791
#> [51,] 0.349266341 0.26963791
#> [52,] 0.776452979 0.56047124
#>
#> [[5]]
#> [,1] [,2]
#> [1,] -0.01903954 -0.61714382
#> [2,] 0.03579740 -0.37017271
#> [3,] 0.03579740 -0.37017271
#> [4,] 0.05132942 -0.30022060
#> [5,] 0.05132942 -0.30022060
#> [6,] 0.05322565 -0.29168049
#> [7,] 0.05322565 -0.29168049
#> [8,] 0.05470615 -0.28501268
#> [9,] 0.05470615 -0.28501268
#> [10,] 0.07434792 -0.19655135
#> [11,] 0.07434792 -0.19655135
#> [12,] 0.08632527 -0.14260853
#> [13,] 0.08632527 -0.14260853
#> [14,] 0.10367564 -0.06446706
#> [15,] 0.10367564 -0.06446706
#> [16,] 0.10895243 -0.04070176
#> [17,] 0.10895243 -0.04070176
#> [18,] 0.11010240 -0.03552264
#> [19,] 0.11010240 -0.03552264
#> [20,] 0.11092197 -0.03183149
#> [21,] 0.11092197 -0.03183149
#> [22,] 0.11199799 -0.02698538
#> [23,] 0.11199799 -0.02698538
#> [24,] 0.11266837 -0.02396615
#> [25,] 0.11266837 -0.02396615
#> [26,] 0.11555689 -0.01095704
#> [27,] 0.11555689 -0.01095704
#> [28,] 0.13416930 0.07286835
#> [29,] 0.13416930 0.07286835
#> [30,] 0.14159789 0.10632473
#> [31,] 0.14159789 0.10632473
#> [32,] 0.14462426 0.11995474
#> [33,] 0.14462426 0.11995474
#> [34,] 0.14733439 0.13216043
#> [35,] 0.14733439 0.13216043
#> [36,] 0.15598731 0.17113088
#> [37,] 0.15598731 0.17113088
#> [38,] 0.16943647 0.23170238
#> [39,] 0.16943647 0.23170238
#> [40,] 0.19256919 0.33588603
#> [41,] 0.19256919 0.33588603
#> [42,] 0.21332750 0.42937597
#> [43,] 0.21332750 0.42937597
#> [44,] 0.21372235 0.43115425
#> [45,] 0.21372235 0.43115425
#> [46,] 0.21403896 0.43258018
#> [47,] 0.21403896 0.43258018
#> [48,] 0.30358750 0.83588313
#>
#> [[6]]
#> [,1] [,2]
#> [1,] -0.67657397 -0.2460987
#> [2,] -0.50255891 -0.2346169
#> [3,] -0.50255891 -0.2346169
#> [4,] -0.47171558 -0.2325818
#> [5,] -0.47171558 -0.2325818
#> [6,] -0.37947237 -0.2264954
#> [7,] -0.37947237 -0.2264954
#> [8,] -0.30438031 -0.2215406
#> [9,] -0.30438031 -0.2215406
#> [10,] -0.27208005 -0.2194094
#> [11,] -0.27208005 -0.2194094
#> [12,] -0.19250468 -0.2141588
#> [13,] -0.19250468 -0.2141588
#> [14,] -0.17926981 -0.2132856
#> [15,] -0.17926981 -0.2132856
#> [16,] -0.12846756 -0.2099335
#> [17,] -0.12846756 -0.2099335
#> [18,] -0.11443848 -0.2090079
#> [19,] -0.11443848 -0.2090079
#> [20,] -0.08096737 -0.2067994
#> [21,] -0.08096737 -0.2067994
#> [22,] -0.08082421 -0.2067899
#> [23,] -0.08082421 -0.2067899
#> [24,] 0.02691068 -0.1996814
#> [25,] 0.02691068 -0.1996814
#> [26,] 0.03888149 -0.1988915
#> [27,] 0.03888149 -0.1988915
#> [28,] 0.07434792 -0.1965514
#> [29,] 0.07434792 -0.1965514
#> [30,] 0.12883306 -0.1929563
#> [31,] 0.12883306 -0.1929563
#> [32,] 0.15341907 -0.1913341
#> [33,] 0.15341907 -0.1913341
#> [34,] 0.21189224 -0.1874759
#> [35,] 0.21189224 -0.1874759
#> [36,] 0.26141660 -0.1842082
#> [37,] 0.26141660 -0.1842082
#> [38,] 0.43741547 -0.1725954
#> [39,] 0.43741547 -0.1725954
#> [40,] 0.45359560 -0.1715278
#> [41,] 0.45359560 -0.1715278
#> [42,] 0.52374646 -0.1668991
#> [43,] 0.52374646 -0.1668991
#> [44,] 0.59694004 -0.1620696
#> [45,] 0.59694004 -0.1620696
#> [46,] 0.77645298 -0.1502250
#>
#> [[7]]
#> [,1] [,2]
#> [1,] -0.676573973 0.31763727
#> [2,] -0.356898239 0.17927379
#> [3,] -0.356898239 0.17927379
#> [4,] -0.323308154 0.16473517
#> [5,] -0.323308154 0.16473517
#> [6,] -0.311845155 0.15977371
#> [7,] -0.311845155 0.15977371
#> [8,] -0.265868309 0.13987380
#> [9,] -0.265868309 0.13987380
#> [10,] -0.165187999 0.09629690
#> [11,] -0.165187999 0.09629690
#> [12,] -0.152464511 0.09078986
#> [13,] -0.152464511 0.09078986
#> [14,] -0.149015897 0.08929722
#> [15,] -0.149015897 0.08929722
#> [16,] -0.132935526 0.08233724
#> [17,] -0.132935526 0.08233724
#> [18,] -0.085999077 0.06202199
#> [19,] -0.085999077 0.06202199
#> [20,] -0.083696109 0.06102521
#> [21,] -0.083696109 0.06102521
#> [22,] -0.081006463 0.05986107
#> [23,] -0.081006463 0.05986107
#> [24,] -0.071141513 0.05559128
#> [25,] -0.071141513 0.05559128
#> [26,] -0.067507231 0.05401827
#> [27,] -0.067507231 0.05401827
#> [28,] -0.030854193 0.03815394
#> [29,] -0.030854193 0.03815394
#> [30,] -0.012373001 0.03015482
#> [31,] -0.012373001 0.03015482
#> [32,] -0.006334286 0.02754112
#> [33,] -0.006334286 0.02754112
#> [34,] -0.002413403 0.02584407
#> [35,] -0.002413403 0.02584407
#> [36,] 0.094178922 -0.01596346
#> [37,] 0.094178922 -0.01596346
#> [38,] 0.103145728 -0.01984451
#> [39,] 0.103145728 -0.01984451
#> [40,] 0.112668375 -0.02396615
#> [41,] 0.112668375 -0.02396615
#> [42,] 0.117224127 -0.02593799
#> [43,] 0.117224127 -0.02593799
#> [44,] 0.126734286 -0.03005422
#> [45,] 0.126734286 -0.03005422
#> [46,] 0.453595596 -0.17152780
#> [47,] 0.453595596 -0.17152780
#> [48,] 0.776452979 -0.31126838
#>
#> [[8]]
#> [,1] [,2]
#> [1,] -0.676573973 0.20866614
#> [2,] -0.337288528 0.15183654
#> [3,] -0.337288528 0.15183654
#> [4,] -0.329621154 0.15055227
#> [5,] -0.329621154 0.15055227
#> [6,] -0.317723918 0.14855951
#> [7,] -0.317723918 0.14855951
#> [8,] -0.313670632 0.14788059
#> [9,] -0.313670632 0.14788059
#> [10,] -0.289704267 0.14386628
#> [11,] -0.289704267 0.14386628
#> [12,] -0.265868309 0.13987380
#> [13,] -0.265868309 0.13987380
#> [14,] -0.162877946 0.12262313
#> [15,] -0.162877946 0.12262313
#> [16,] -0.134716451 0.11790614
#> [17,] -0.134716451 0.11790614
#> [18,] -0.125035547 0.11628461
#> [19,] -0.125035547 0.11628461
#> [20,] -0.075924615 0.10805863
#> [21,] -0.075924615 0.10805863
#> [22,] -0.075032016 0.10790913
#> [23,] -0.075032016 0.10790913
#> [24,] -0.046641188 0.10315372
#> [25,] -0.046641188 0.10315372
#> [26,] -0.044521893 0.10279874
#> [27,] -0.044521893 0.10279874
#> [28,] -0.032515347 0.10078767
#> [29,] -0.032515347 0.10078767
#> [30,] -0.015665607 0.09796538
#> [31,] -0.015665607 0.09796538
#> [32,] 0.002271677 0.09496092
#> [33,] 0.002271677 0.09496092
#> [34,] 0.014448964 0.09292125
#> [35,] 0.014448964 0.09292125
#> [36,] 0.028606447 0.09054990
#> [37,] 0.028606447 0.09054990
#> [38,] 0.031886641 0.09000048
#> [39,] 0.031886641 0.09000048
#> [40,] 0.074840492 0.08280580
#> [41,] 0.074840492 0.08280580
#> [42,] 0.078581292 0.08217922
#> [43,] 0.078581292 0.08217922
#> [44,] 0.134169301 0.07286835
#> [45,] 0.134169301 0.07286835
#> [46,] 0.189589173 0.06358563
#> [47,] 0.189589173 0.06358563
#> [48,] 0.262673120 0.05134423
#> [49,] 0.262673120 0.05134423
#> [50,] 0.393498449 0.02943126
#> [51,] 0.393498449 0.02943126
#> [52,] 0.776452979 -0.03471282
#>
#> [[9]]
#> [,1] [,2]
#> [1,] -0.676573973 -0.51106498
#> [2,] -0.657020824 -0.49935681
#> [3,] -0.657020824 -0.49935681
#> [4,] -0.405228050 -0.34858650
#> [5,] -0.405228050 -0.34858650
#> [6,] -0.351929240 -0.31667185
#> [7,] -0.351929240 -0.31667185
#> [8,] -0.339104121 -0.30899233
#> [9,] -0.339104121 -0.30899233
#> [10,] -0.190487312 -0.22000247
#> [11,] -0.190487312 -0.22000247
#> [12,] -0.179269811 -0.21328558
#> [13,] -0.179269811 -0.21328558
#> [14,] -0.123503973 -0.17989370
#> [15,] -0.123503973 -0.17989370
#> [16,] -0.109797837 -0.17168664
#> [17,] -0.109797837 -0.17168664
#> [18,] -0.107182222 -0.17012045
#> [19,] -0.107182222 -0.17012045
#> [20,] -0.060230616 -0.14200642
#> [21,] -0.060230616 -0.14200642
#> [22,] 0.003617210 -0.10377516
#> [23,] 0.003617210 -0.10377516
#> [24,] 0.004845522 -0.10303966
#> [25,] 0.004845522 -0.10303966
#> [26,] 0.019640941 -0.09418035
#> [27,] 0.019640941 -0.09418035
#> [28,] 0.100064542 -0.04602372
#> [29,] 0.100064542 -0.04602372
#> [30,] 0.108952433 -0.04070176
#> [31,] 0.108952433 -0.04070176
#> [32,] 0.114515406 -0.03737073
#> [33,] 0.114515406 -0.03737073
#> [34,] 0.121325930 -0.03329267
#> [35,] 0.121325930 -0.03329267
#> [36,] 0.126734286 -0.03005422
#> [37,] 0.126734286 -0.03005422
#> [38,] 0.141851263 -0.02100236
#> [39,] 0.141851263 -0.02100236
#> [40,] 0.223165022 0.02768728
#> [41,] 0.223165022 0.02768728
#> [42,] 0.227191661 0.03009838
#> [43,] 0.227191661 0.03009838
#> [44,] 0.262673120 0.05134423
#> [45,] 0.262673120 0.05134423
#> [46,] 0.370134930 0.11569099
#> [47,] 0.370134930 0.11569099
#> [48,] 0.750914303 0.34369683
#> [49,] 0.750914303 0.34369683
#> [50,] 0.776452979 0.35898907
#>
#> [[10]]
#> [,1] [,2]
#> [1,] -0.676573973 -0.349757473
#> [2,] -0.546214401 -0.275193890
#> [3,] -0.546214401 -0.275193890
#> [4,] -0.471715578 -0.232581759
#> [5,] -0.471715578 -0.232581759
#> [6,] -0.265942908 -0.114883094
#> [7,] -0.265942908 -0.114883094
#> [8,] -0.245116012 -0.102970443
#> [9,] -0.245116012 -0.102970443
#> [10,] -0.233235575 -0.096175024
#> [11,] -0.233235575 -0.096175024
#> [12,] -0.181273945 -0.066453806
#> [13,] -0.181273945 -0.066453806
#> [14,] -0.139963058 -0.042824642
#> [15,] -0.139963058 -0.042824642
#> [16,] -0.096773716 -0.018121031
#> [17,] -0.096773716 -0.018121031
#> [18,] -0.090238181 -0.014382810
#> [19,] -0.090238181 -0.014382810
#> [20,] -0.069756023 -0.002667344
#> [21,] -0.069756023 -0.002667344
#> [22,] -0.026894088 0.021848995
#> [23,] -0.026894088 0.021848995
#> [24,] -0.012373001 0.030154824
#> [25,] -0.012373001 0.030154824
#> [26,] -0.004237262 0.034808336
#> [27,] -0.004237262 0.034808336
#> [28,] -0.004077380 0.034899786
#> [29,] -0.004077380 0.034899786
#> [30,] -0.003976802 0.034957315
#> [31,] -0.003976802 0.034957315
#> [32,] 0.005293778 0.040259938
#> [33,] 0.005293778 0.040259938
#> [34,] 0.036486887 0.058101896
#> [35,] 0.036486887 0.058101896
#> [36,] 0.048224618 0.064815689
#> [37,] 0.048224618 0.064815689
#> [38,] 0.049421667 0.065500382
#> [39,] 0.049421667 0.065500382
#> [40,] 0.075327560 0.080318138
#> [41,] 0.075327560 0.080318138
#> [42,] 0.078581292 0.082179220
#> [43,] 0.078581292 0.082179220
#> [44,] 0.096095931 0.092197313
#> [45,] 0.096095931 0.092197313
#> [46,] 0.133462080 0.113570150
#> [47,] 0.133462080 0.113570150
#> [48,] 0.144624265 0.119954740
#> [49,] 0.144624265 0.119954740
#> [50,] 0.195437568 0.149019135
#> [51,] 0.195437568 0.149019135
#> [52,] 0.209758232 0.157210324
#> [53,] 0.209758232 0.157210324
#> [54,] 0.439649532 0.288704459
#> [55,] 0.439649532 0.288704459
#> [56,] 0.776452979 0.481350625
#>
#> [[11]]
#> [,1] [,2]
#> [1,] -0.676573973 -0.185026253
#> [2,] -0.386704730 -0.126932394
#> [3,] -0.386704730 -0.126932394
#> [4,] -0.250415784 -0.099618179
#> [5,] -0.250415784 -0.099618179
#> [6,] -0.242663952 -0.098064603
#> [7,] -0.242663952 -0.098064603
#> [8,] -0.233235575 -0.096175024
#> [9,] -0.233235575 -0.096175024
#> [10,] -0.199499371 -0.089413816
#> [11,] -0.199499371 -0.089413816
#> [12,] -0.170466793 -0.083595280
#> [13,] -0.170466793 -0.083595280
#> [14,] -0.169203666 -0.083342132
#> [15,] -0.169203666 -0.083342132
#> [16,] -0.169203666 -0.083342132
#> [17,] -0.169203666 -0.083342132
#> [18,] -0.105438846 -0.070562768
#> [19,] -0.105438846 -0.070562768
#> [20,] -0.097013822 -0.068874276
#> [21,] -0.097013822 -0.068874276
#> [22,] -0.047777235 -0.059006573
#> [23,] -0.047777235 -0.059006573
#> [24,] -0.032898450 -0.056024656
#> [25,] -0.032898450 -0.056024656
#> [26,] -0.030461323 -0.055536222
#> [27,] -0.030461323 -0.055536222
#> [28,] -0.009136936 -0.051262516
#> [29,] -0.009136936 -0.051262516
#> [30,] 0.011012809 -0.047224225
#> [31,] 0.011012809 -0.047224225
#> [32,] 0.096928281 -0.030005560
#> [33,] 0.096928281 -0.030005560
#> [34,] 0.108263690 -0.027733785
#> [35,] 0.108263690 -0.027733785
#> [36,] 0.111997991 -0.026985379
#> [37,] 0.111997991 -0.026985379
#> [38,] 0.114926271 -0.026398510
#> [39,] 0.114926271 -0.026398510
#> [40,] 0.117224127 -0.025937988
#> [41,] 0.117224127 -0.025937988
#> [42,] 0.141851263 -0.021002364
#> [43,] 0.141851263 -0.021002364
#> [44,] 0.258354041 0.002346425
#> [45,] 0.258354041 0.002346425
#> [46,] 0.268912298 0.004462448
#> [47,] 0.268912298 0.004462448
#> [48,] 0.393498449 0.029431259
#> [49,] 0.393498449 0.029431259
#> [50,] 0.568136327 0.064431138
#> [51,] 0.568136327 0.064431138
#> [52,] 0.776452979 0.106180715
#>
#> [[12]]
#> [,1] [,2]
#> [1,] -0.67657397 -0.2539204
#> [2,] -0.54621440 -0.2751939
#> [3,] -0.54621440 -0.2751939
#> [4,] -0.46941208 -0.2877273
#> [5,] -0.46941208 -0.2877273
#> [6,] -0.44569328 -0.2915980
#> [7,] -0.44569328 -0.2915980
#> [8,] -0.36981973 -0.3039798
#> [9,] -0.36981973 -0.3039798
#> [10,] -0.34741309 -0.3076364
#> [11,] -0.34741309 -0.3076364
#> [12,] -0.33910412 -0.3089923
#> [13,] -0.33910412 -0.3089923
#> [14,] -0.19408038 -0.3326588
#> [15,] -0.19408038 -0.3326588
#> [16,] -0.14993621 -0.3398627
#> [17,] -0.14993621 -0.3398627
#> [18,] -0.14905921 -0.3400059
#> [19,] -0.14905921 -0.3400059
#> [20,] -0.13109177 -0.3429380
#> [21,] -0.13109177 -0.3429380
#> [22,] -0.12444805 -0.3440222
#> [23,] -0.12444805 -0.3440222
#> [24,] 0.03579740 -0.3701727
#> [25,] 0.03579740 -0.3701727
#> [26,] 0.06966143 -0.3756990
#> [27,] 0.06966143 -0.3756990
#> [28,] 0.07142246 -0.3759864
#> [29,] 0.07142246 -0.3759864
#> [30,] 0.16774700 -0.3917056
#> [31,] 0.16774700 -0.3917056
#> [32,] 0.22652528 -0.4012977
#> [33,] 0.22652528 -0.4012977
#> [34,] 0.36529448 -0.4239435
#> [35,] 0.36529448 -0.4239435
#> [36,] 0.50514096 -0.4467651
#> [37,] 0.50514096 -0.4467651
#> [38,] 0.77645298 -0.4910407
#>
#> [[13]]
#> [,1] [,2]
#> [1,] -0.67657397 0.16993965
#> [2,] -0.53821283 0.08242208
#> [3,] -0.53821283 0.08242208
#> [4,] -0.50340106 0.06040259
#> [5,] -0.50340106 0.06040259
#> [6,] -0.34439271 -0.04017496
#> [7,] -0.34439271 -0.04017496
#> [8,] -0.31402859 -0.05938117
#> [9,] -0.31402859 -0.05938117
#> [10,] -0.25041578 -0.09961818
#> [11,] -0.25041578 -0.09961818
#> [12,] -0.24511601 -0.10297044
#> [13,] -0.24511601 -0.10297044
#> [14,] -0.22691522 -0.11448299
#> [15,] -0.22691522 -0.11448299
#> [16,] -0.22191142 -0.11764804
#> [17,] -0.22191142 -0.11764804
#> [18,] -0.22070983 -0.11840809
#> [19,] -0.22070983 -0.11840809
#> [20,] -0.20428566 -0.12879686
#> [21,] -0.20428566 -0.12879686
#> [22,] -0.12350397 -0.17989370
#> [23,] -0.12350397 -0.17989370
#> [24,] -0.11521156 -0.18513890
#> [25,] -0.11521156 -0.18513890
#> [26,] -0.11174331 -0.18733267
#> [27,] -0.11174331 -0.18733267
#> [28,] -0.08096737 -0.20679938
#> [29,] -0.08096737 -0.20679938
#> [30,] -0.08085068 -0.20687319
#> [31,] -0.08085068 -0.20687319
#> [32,] 0.04853453 -0.28871321
#> [33,] 0.04853453 -0.28871321
#> [34,] 0.05322565 -0.29168049
#> [35,] 0.05322565 -0.29168049
#> [36,] 0.05628719 -0.29361700
#> [37,] 0.05628719 -0.29361700
#> [38,] 0.16158214 -0.36021922
#> [39,] 0.16158214 -0.36021922
#> [40,] 0.22652528 -0.40129770
#> [41,] 0.22652528 -0.40129770
#> [42,] 0.43694785 -0.53439628
#> [43,] 0.43694785 -0.53439628
#> [44,] 0.56776779 -0.61714382
#>
#> [[14]]
#> [,1] [,2]
#> [1,] -0.67657397 0.4114319677
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#> [3,] -0.62358867 0.3729524361
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#> [5,] -0.35689824 0.1792737857
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#> [44,] 0.01964094 -0.0941803502
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#> [46,] 0.08632527 -0.1426085267
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#> [48,] 0.12438738 -0.1702503806
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#> [55,] 0.50514096 -0.4467651418
#> [56,] 0.73974793 -0.6171438178
#>
#> [[15]]
#> [,1] [,2]
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#> [46,] 0.1318503060 -0.04463027
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#> [48,] 0.4374154667 -0.17259540
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#> [50,] 0.4702210998 -0.18633380
#> [51,] 0.4702210998 -0.18633380
#> [52,] 0.7764529788 -0.31457814
#>
#> [[16]]
#> [,1] [,2]
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#> [24,] -0.14149396 0.24814471
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#> [32,] -0.05180969 0.29466913
#> [33,] -0.05180969 0.29466913
#> [34,] -0.04446036 0.29848165
#> [35,] -0.04446036 0.29848165
#> [36,] 0.02550599 0.33477724
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#> [38,] 0.10432924 0.37566741
#> [39,] 0.10432924 0.37566741
#> [40,] 0.21403896 0.43258018
#> [41,] 0.21403896 0.43258018
#> [42,] 0.21542498 0.43329919
#> [43,] 0.21542498 0.43329919
#> [44,] 0.21679420 0.43400948
#> [45,] 0.21679420 0.43400948
#> [46,] 0.37681731 0.51702272
#> [47,] 0.37681731 0.51702272
#> [48,] 0.77645298 0.72433683
#>
#> [[17]]
#> [,1] [,2]
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#> [22,] -0.134635876 0.084624966
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#> [24,] -0.083052807 0.064918509
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#> [33,] -0.032680826 0.045674728
#> [34,] -0.004237262 0.034808336
#> [35,] -0.004237262 0.034808336
#> [36,] -0.004047237 0.034735740
#> [37,] -0.004047237 0.034735740
#> [38,] 0.001256994 0.032709347
#> [39,] 0.001256994 0.032709347
#> [40,] 0.091391096 -0.001724894
#> [41,] 0.091391096 -0.001724894
#> [42,] 0.092369359 -0.002098623
#> [43,] 0.092369359 -0.002098623
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#> [45,] 0.115556890 -0.010957035
#> [46,] 0.141851263 -0.021002364
#> [47,] 0.141851263 -0.021002364
#> [48,] 0.391792125 -0.116488129
#> [49,] 0.391792125 -0.116488129
#> [50,] 0.523746463 -0.166899098
#> [51,] 0.523746463 -0.166899098
#> [52,] 0.776452979 -0.263441435
#>
#> [[18]]
#> [,1] [,2]
#> [1,] -0.569656581 -0.6171438178
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#> [37,] -0.019404224 0.1183117197
#> [38,] -0.001769820 0.1418814866
#> [39,] -0.001769820 0.1418814866
#> [40,] 0.006568661 0.1530265220
#> [41,] 0.006568661 0.1530265220
#> [42,] 0.039593002 0.1971661504
#> [43,] 0.039593002 0.1971661504
#> [44,] 0.042161297 0.2005988779
#> [45,] 0.042161297 0.2005988779
#> [46,] 0.045940968 0.2056507065
#> [47,] 0.045940968 0.2056507065
#> [48,] 0.049782806 0.2107856242
#> [49,] 0.049782806 0.2107856242
#> [50,] 0.053492045 0.2157433144
#> [51,] 0.053492045 0.2157433144
#> [52,] 0.213327502 0.4293759683
#> [53,] 0.213327502 0.4293759683
#> [54,] 0.216794200 0.4340094830
#> [55,] 0.216794200 0.4340094830
#> [56,] 0.230005490 0.4516674105
#> [57,] 0.230005490 0.4516674105
#> [58,] 0.517467609 0.8358831340
#>
#> [[19]]
#> [,1] [,2]
#> [1,] -0.195752319 -0.61714382
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#> [47,] -0.063617969 0.18253899
#> [48,] -0.056417018 0.22611946
#> [49,] -0.056417018 0.22611946
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#> [51,] -0.048271754 0.27541494
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#> [53,] -0.044460363 0.29848165
#> [54,] -0.009186021 0.51196349
#> [55,] -0.009186021 0.51196349
#> [56,] 0.044336339 0.83588313
#>
#> [[20]]
#> [,1] [,2]
#> [1,] -0.1512571155 0.83588313
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#> [18,] -0.0208330272 0.12608757
#> [19,] -0.0208330272 0.12608757
#> [20,] -0.0194042237 0.11831172
#> [21,] -0.0194042237 0.11831172
#> [22,] -0.0156656075 0.09796538
#> [23,] -0.0156656075 0.09796538
#> [24,] -0.0117069084 0.07642130
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#> [26,] -0.0040773803 0.03489979
#> [27,] -0.0040773803 0.03489979
#> [28,] -0.0040472371 0.03473574
#> [29,] -0.0040472371 0.03473574
#> [30,] -0.0031265201 0.02972500
#> [31,] -0.0031265201 0.02972500
#> [32,] -0.0024134026 0.02584407
#> [33,] -0.0024134026 0.02584407
#> [34,] 0.0004227342 0.01040921
#> [35,] 0.0004227342 0.01040921
#> [36,] 0.0110128086 -0.04722422
#> [37,] 0.0110128086 -0.04722422
#> [38,] 0.0196409407 -0.09418035
#> [39,] 0.0196409407 -0.09418035
#> [40,] 0.0266625120 -0.13239323
#> [41,] 0.0266625120 -0.13239323
#> [42,] 0.0388814896 -0.19889150
#> [43,] 0.0388814896 -0.19889150
#> [44,] 0.0547061534 -0.28501268
#> [45,] 0.0547061534 -0.28501268
#> [46,] 0.0562871878 -0.29361700
#> [47,] 0.0562871878 -0.29361700
#> [48,] 0.0714224636 -0.37598638
#> [49,] 0.0714224636 -0.37598638
#> [50,] 0.0766781337 -0.40458886
#> [51,] 0.0766781337 -0.40458886
#> [52,] 0.0823131921 -0.43525604
#> [53,] 0.0823131921 -0.43525604
#> [54,] 0.1157348543 -0.61714382
#>
#> [[21]]
#> [,1] [,2]
#> [1,] -0.451985395 0.8358831340
#> [2,] -0.143291971 0.3600361035
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#> [[22]]
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#> [[23]]
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#> [[24]]
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#>
#> [[25]]
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#>
#> [[26]]
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#> [56,] 0.128302982 -0.617143818
#>
#> [[27]]
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#> [27,] -0.03791741 0.0672353127
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#> [29,] -0.01170691 0.0764212962
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#> [32,] 0.02860645 0.0905499009
#> [33,] 0.02860645 0.0905499009
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#> [35,] 0.03098861 0.0913847770
#> [36,] 0.03300444 0.0920912649
#> [37,] 0.03300444 0.0920912649
#> [38,] 0.07045275 0.1052157591
#> [39,] 0.07045275 0.1052157591
#> [40,] 0.12239992 0.1234216608
#> [41,] 0.12239992 0.1234216608
#> [42,] 0.14733439 0.1321604342
#> [43,] 0.14733439 0.1321604342
#> [44,] 0.19543757 0.1490191346
#> [45,] 0.19543757 0.1490191346
#> [46,] 0.21496411 0.1558625935
#> [47,] 0.21496411 0.1558625935
#> [48,] 0.75091430 0.3436968343
#> [49,] 0.75091430 0.3436968343
#> [50,] 0.77645298 0.3526473635
#>
#> [[28]]
#> [,1] [,2]
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#> [20,] -0.096999122 0.05120771
#> [21,] -0.096999122 0.05120771
#> [22,] -0.085999077 0.06202199
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#> [26,] -0.080005035 0.06791481
#> [27,] -0.080005035 0.06791481
#> [28,] -0.055454998 0.09205027
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#> [31,] -0.046163093 0.10118526
#> [32,] -0.044521893 0.10279874
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#> [39,] -0.003746872 0.14288519
#> [40,] -0.002937279 0.14368111
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#> [42,] 0.006568661 0.15302652
#> [43,] 0.006568661 0.15302652
#> [44,] 0.045823000 0.19161797
#> [45,] 0.045823000 0.19161797
#> [46,] 0.052290756 0.19797650
#> [47,] 0.052290756 0.19797650
#> [48,] 0.063964119 0.20945273
#> [49,] 0.063964119 0.20945273
#> [50,] 0.130965985 0.27532313
#> [51,] 0.130965985 0.27532313
#> [52,] 0.192569193 0.33588603
#> [53,] 0.192569193 0.33588603
#> [54,] 0.376817314 0.51702272
#> [55,] 0.376817314 0.51702272
#> [56,] 0.701154878 0.83588313
#>
#> [[29]]
#> [,1] [,2]
#> [1,] -0.211268853 -0.617143818
#> [2,] -0.178956687 -0.515495926
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#> [6,] -0.124448046 -0.344022170
#> [7,] -0.124448046 -0.344022170
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#> [10,] -0.080850678 -0.206873191
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#> [12,] -0.080824211 -0.206789931
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#> [18,] -0.032059607 -0.053385817
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#> [23,] -0.013415304 0.005265593
#> [24,] -0.010346583 0.014919202
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#> [32,] 0.011825627 0.084668734
#> [33,] 0.011825627 0.084668734
#> [34,] 0.014448964 0.092921249
#> [35,] 0.014448964 0.092921249
#> [36,] 0.019560479 0.109001099
#> [37,] 0.019560479 0.109001099
#> [38,] 0.025596219 0.127988383
#> [39,] 0.025596219 0.127988383
#> [40,] 0.045823000 0.191617966
#> [41,] 0.045823000 0.191617966
#> [42,] 0.048182374 0.199040103
#> [43,] 0.048182374 0.199040103
#> [44,] 0.050595927 0.206632681
#> [45,] 0.050595927 0.206632681
#> [46,] 0.053492045 0.215743314
#> [47,] 0.053492045 0.215743314
#> [48,] 0.060679714 0.238354346
#> [49,] 0.060679714 0.238354346
#> [50,] 0.083487465 0.310103165
#> [51,] 0.083487465 0.310103165
#> [52,] 0.104329243 0.375667412
#> [53,] 0.104329243 0.375667412
#> [54,] 0.250624128 0.835883134
#>
#> [[30]]
#> [,1] [,2]
#> [1,] -0.072607553 0.8358831340
#> [2,] -0.018729702 0.5607069637
#> [3,] -0.018729702 0.5607069637
#> [4,] -0.009186021 0.5119634907
#> [5,] -0.009186021 0.5119634907
#> [6,] 0.010918959 0.4092791496
#> [7,] 0.010918959 0.4092791496
#> [8,] 0.025505989 0.3347772361
#> [9,] 0.025505989 0.3347772361
#> [10,] 0.038520446 0.2683070941
#> [11,] 0.038520446 0.2683070941
#> [12,] 0.047609001 0.2218881345
#> [13,] 0.047609001 0.2218881345
#> [14,] 0.049782806 0.2107856242
#> [15,] 0.049782806 0.2107856242
#> [16,] 0.050595927 0.2066326810
#> [17,] 0.050595927 0.2066326810
#> [18,] 0.052290756 0.1979764995
#> [19,] 0.052290756 0.1979764995
#> [20,] 0.055164574 0.1832987392
#> [21,] 0.055164574 0.1832987392
#> [22,] 0.062295437 0.1468785123
#> [23,] 0.062295437 0.1468785123
#> [24,] 0.070452755 0.1052157591
#> [25,] 0.070452755 0.1052157591
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#> [27,] 0.074840492 0.0828057957
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#> [40,] 0.103675637 -0.0644670602
#> [41,] 0.103675637 -0.0644670602
#> [42,] 0.124387379 -0.1702503806
#> [43,] 0.124387379 -0.1702503806
#> [44,] 0.128833064 -0.1929563072
#> [45,] 0.128833064 -0.1929563072
#> [46,] 0.149275011 -0.2973616709
#> [47,] 0.149275011 -0.2973616709
#> [48,] 0.161582143 -0.3602192181
#> [49,] 0.161582143 -0.3602192181
#> [50,] 0.167746996 -0.3917056376
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#> [52,] 0.211726463 -0.6163267259
#> [53,] 0.211726463 -0.6163267259
#> [54,] 0.211886445 -0.6171438178
#>
#> [[31]]
#> [,1] [,2]
#> [1,] -0.676573973 0.813773162
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#> [3,] -0.435309022 0.489164309
#> [4,] -0.283495026 0.284906845
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#> [11,] -0.162877946 0.122623133
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#> [13,] -0.144630171 0.098071746
#> [14,] -0.134635876 0.084624966
#> [15,] -0.134635876 0.084624966
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#> [17,] -0.132935526 0.082337238
#> [18,] -0.115582934 0.058990270
#> [19,] -0.115582934 0.058990270
#> [20,] -0.104394661 0.043937058
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#> [22,] -0.099197431 0.036944468
#> [23,] -0.099197431 0.036944468
#> [24,] -0.089770729 0.024261353
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#> [27,] -0.086054721 0.019261667
#> [28,] -0.069756023 -0.002667344
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#> [31,] -0.063355181 -0.011279327
#> [32,] -0.039453583 -0.043437625
#> [33,] -0.039453583 -0.043437625
#> [34,] -0.032059607 -0.053385817
#> [35,] -0.032059607 -0.053385817
#> [36,] -0.030461323 -0.055536222
#> [37,] -0.030461323 -0.055536222
#> [38,] -0.026814208 -0.060443217
#> [39,] -0.026814208 -0.060443217
#> [40,] 0.002755649 -0.100227848
#> [41,] 0.002755649 -0.100227848
#> [42,] 0.004845522 -0.103039659
#> [43,] 0.004845522 -0.103039659
#> [44,] 0.026662512 -0.132393231
#> [45,] 0.026662512 -0.132393231
#> [46,] 0.074347920 -0.196551351
#> [47,] 0.074347920 -0.196551351
#> [48,] 0.149275011 -0.297361671
#> [49,] 0.149275011 -0.297361671
#> [50,] 0.226525277 -0.401297697
#> [51,] 0.226525277 -0.401297697
#> [52,] 0.386952521 -0.617143818
#>
#> [[32]]
#> [,1] [,2]
#> [1,] -0.676573973 0.8262610645
#> [2,] -0.477734763 0.6120575218
#> [3,] -0.477734763 0.6120575218
#> [4,] -0.140434685 0.2486942212
#> [5,] -0.140434685 0.2486942212
#> [6,] -0.139428139 0.2476098986
#> [7,] -0.139428139 0.2476098986
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#> [11,] -0.076501708 0.1798211344
#> [12,] -0.067320132 0.1699300971
#> [13,] -0.067320132 0.1699300971
#> [14,] -0.065945980 0.1684497638
#> [15,] -0.065945980 0.1684497638
#> [16,] -0.061279062 0.1634222325
#> [17,] -0.061279062 0.1634222325
#> [18,] -0.027600268 0.1271410734
#> [19,] -0.027600268 0.1271410734
#> [20,] -0.023859962 0.1231117531
#> [21,] -0.023859962 0.1231117531
#> [22,] -0.019404224 0.1183117197
#> [23,] -0.019404224 0.1183117197
#> [24,] 0.002271677 0.0949609194
#> [25,] 0.002271677 0.0949609194
#> [26,] 0.011825627 0.0846687340
#> [27,] 0.011825627 0.0846687340
#> [28,] 0.022746390 0.0729041219
#> [29,] 0.022746390 0.0729041219
#> [30,] 0.036486887 0.0581018955
#> [31,] 0.036486887 0.0581018955
#> [32,] 0.037287563 0.0572393514
#> [33,] 0.037287563 0.0572393514
#> [34,] 0.089925996 0.0005335387
#> [35,] 0.089925996 0.0005335387
#> [36,] 0.091222336 -0.0008629695
#> [37,] 0.091222336 -0.0008629695
#> [38,] 0.092369359 -0.0020986230
#> [39,] 0.092369359 -0.0020986230
#> [40,] 0.112668375 -0.0239661463
#> [41,] 0.112668375 -0.0239661463
#> [42,] 0.114926271 -0.0263985102
#> [43,] 0.114926271 -0.0263985102
#> [44,] 0.121325930 -0.0332926723
#> [45,] 0.121325930 -0.0332926723
#> [46,] 0.131850306 -0.0446302680
#> [47,] 0.131850306 -0.0446302680
#> [48,] 0.261416602 -0.1842081689
#> [49,] 0.261416602 -0.1842081689
#> [50,] 0.505140960 -0.4467651418
#> [51,] 0.505140960 -0.4467651418
#> [52,] 0.663298762 -0.6171438178
#>
#> [[33]]
#> [,1] [,2]
#> [1,] -0.676573973 0.484463917
#> [2,] -0.517189725 0.403548228
#> [3,] -0.517189725 0.403548228
#> [4,] -0.382071023 0.334951594
#> [5,] -0.382071023 0.334951594
#> [6,] -0.283495026 0.284906845
#> [7,] -0.283495026 0.284906845
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#> [9,] -0.175912636 0.230289759
#> [10,] -0.098250954 0.190862723
#> [11,] -0.098250954 0.190862723
#> [12,] -0.076501708 0.179821134
#> [13,] -0.076501708 0.179821134
#> [14,] -0.066708460 0.174849329
#> [15,] -0.066708460 0.174849329
#> [16,] -0.065029406 0.173996912
#> [17,] -0.065029406 0.173996912
#> [18,] -0.063681174 0.173312446
#> [19,] -0.063681174 0.173312446
#> [20,] -0.025994985 0.154180042
#> [21,] -0.025994985 0.154180042
#> [22,] -0.011840931 0.146994356
#> [23,] -0.011840931 0.146994356
#> [24,] -0.003746872 0.142885190
#> [25,] -0.003746872 0.142885190
#> [26,] -0.001769820 0.141881487
#> [27,] -0.001769820 0.141881487
#> [28,] 0.025596219 0.127988383
#> [29,] 0.025596219 0.127988383
#> [30,] 0.046517674 0.117367045
#> [31,] 0.046517674 0.117367045
#> [32,] 0.070452755 0.105215759
#> [33,] 0.070452755 0.105215759
#> [34,] 0.091822081 0.094367047
#> [35,] 0.091822081 0.094367047
#> [36,] 0.096095931 0.092197313
#> [37,] 0.096095931 0.092197313
#> [38,] 0.134169301 0.072868345
#> [39,] 0.134169301 0.072868345
#> [40,] 0.223165022 0.027687280
#> [41,] 0.223165022 0.027687280
#> [42,] 0.238827944 0.019735577
#> [43,] 0.238827944 0.019735577
#> [44,] 0.268912298 0.004462448
#> [45,] 0.268912298 0.004462448
#> [46,] 0.596940036 -0.162069633
#> [47,] 0.596940036 -0.162069633
#> [48,] 0.776452979 -0.253204194
#>
#> [[34]]
#> [,1] [,2]
#> [1,] -6.765740e-01 0.834955464
#> [2,] -5.005749e-01 0.678217832
#> [3,] -5.005749e-01 0.678217832
#> [4,] -1.432920e-01 0.360036103
#> [5,] -1.432920e-01 0.360036103
#> [6,] -9.957669e-02 0.321105046
#> [7,] -9.957669e-02 0.321105046
#> [8,] -6.323610e-02 0.288741587
#> [9,] -6.323610e-02 0.288741587
#> [10,] -5.361242e-02 0.280171121
#> [11,] -5.361242e-02 0.280171121
#> [12,] -4.827175e-02 0.275414945
#> [13,] -4.827175e-02 0.275414945
#> [14,] -4.226678e-05 0.232463716
#> [15,] -4.226678e-05 0.232463716
#> [16,] 3.279240e-02 0.203222491
#> [17,] 3.279240e-02 0.203222491
#> [18,] 3.310607e-02 0.202943153
#> [19,] 3.310607e-02 0.202943153
#> [20,] 3.959300e-02 0.197166150
#> [21,] 3.959300e-02 0.197166150
#> [22,] 4.582300e-02 0.191617966
#> [23,] 4.582300e-02 0.191617966
#> [24,] 5.516457e-02 0.183298739
#> [25,] 5.516457e-02 0.183298739
#> [26,] 7.318631e-02 0.167249311
#> [27,] 7.318631e-02 0.167249311
#> [28,] 1.223999e-01 0.123421661
#> [29,] 1.223999e-01 0.123421661
#> [30,] 1.276418e-01 0.118753482
#> [31,] 1.276418e-01 0.118753482
#> [32,] 1.334621e-01 0.113570150
#> [33,] 1.334621e-01 0.113570150
#> [34,] 1.415979e-01 0.106324730
#> [35,] 1.415979e-01 0.106324730
#> [36,] 1.895892e-01 0.063585633
#> [37,] 1.895892e-01 0.063585633
#> [38,] 2.271917e-01 0.030098380
#> [39,] 2.271917e-01 0.030098380
#> [40,] 2.388279e-01 0.019735577
#> [41,] 2.388279e-01 0.019735577
#> [42,] 2.583540e-01 0.002346425
#> [43,] 2.583540e-01 0.002346425
#> [44,] 3.917921e-01 -0.116488129
#> [45,] 3.917921e-01 -0.116488129
#> [46,] 4.535956e-01 -0.171527798
#> [47,] 4.535956e-01 -0.171527798
#> [48,] 4.702211e-01 -0.186333799
#> [49,] 4.702211e-01 -0.186333799
#> [50,] 7.764530e-01 -0.459051512
#>
#> [[35]]
#> [,1] [,2]
#> [1,] -0.67657397 0.38666961
#> [2,] -0.62358867 0.37295244
#> [3,] -0.62358867 0.37295244
#> [4,] -0.46620135 0.33220701
#> [5,] -0.46620135 0.33220701
#> [6,] -0.37277352 0.30801983
#> [7,] -0.37277352 0.30801983
#> [8,] -0.28349503 0.28490684
#> [9,] -0.28349503 0.28490684
#> [10,] -0.14149396 0.24814471
#> [11,] -0.14149396 0.24814471
#> [12,] -0.13942814 0.24760990
#> [13,] -0.13942814 0.24760990
#> [14,] -0.07785477 0.23166940
#> [15,] -0.07785477 0.23166940
#> [16,] -0.06021123 0.22710173
#> [17,] -0.06021123 0.22710173
#> [18,] -0.05641702 0.22611946
#> [19,] -0.05641702 0.22611946
#> [20,] -0.03835453 0.22144332
#> [21,] -0.03835453 0.22144332
#> [22,] -0.01766214 0.21608635
#> [23,] -0.01766214 0.21608635
#> [24,] 0.03266193 0.20305813
#> [25,] 0.03266193 0.20305813
#> [26,] 0.03310607 0.20294315
#> [27,] 0.03310607 0.20294315
#> [28,] 0.04216130 0.20059888
#> [29,] 0.04216130 0.20059888
#> [30,] 0.04818237 0.19904010
#> [31,] 0.04818237 0.19904010
#> [32,] 0.05229076 0.19797650
#> [33,] 0.05229076 0.19797650
#> [34,] 0.08507624 0.18948879
#> [35,] 0.08507624 0.18948879
#> [36,] 0.15598731 0.17113088
#> [37,] 0.15598731 0.17113088
#> [38,] 0.19118955 0.16201750
#> [39,] 0.19118955 0.16201750
#> [40,] 0.20975823 0.15721032
#> [41,] 0.20975823 0.15721032
#> [42,] 0.21496411 0.15586259
#> [43,] 0.21496411 0.15586259
#> [44,] 0.37013493 0.11569099
#> [45,] 0.37013493 0.11569099
#> [46,] 0.56813633 0.06443114
#> [47,] 0.56813633 0.06443114
#> [48,] 0.77645298 0.01050081
#>
#> [[36]]
#> [,1] [,2]
#> [1,] -0.67657397 0.05323447
#> [2,] -0.53821283 0.08242208
#> [3,] -0.53821283 0.08242208
#> [4,] -0.40800600 0.10988952
#> [5,] -0.40800600 0.10988952
#> [6,] -0.32050157 0.12834879
#> [7,] -0.32050157 0.12834879
#> [8,] -0.31140887 0.13026691
#> [9,] -0.31140887 0.13026691
#> [10,] -0.29436724 0.13386189
#> [11,] -0.29436724 0.13386189
#> [12,] -0.27449189 0.13805464
#> [13,] -0.27449189 0.13805464
#> [14,] -0.26586831 0.13987380
#> [15,] -0.26586831 0.13987380
#> [16,] -0.18792085 0.15631700
#> [17,] -0.18792085 0.15631700
#> [18,] -0.10698382 0.17339086
#> [19,] -0.10698382 0.17339086
#> [20,] -0.07650171 0.17982113
#> [21,] -0.07650171 0.17982113
#> [22,] -0.06580980 0.18207662
#> [23,] -0.06580980 0.18207662
#> [24,] -0.06361797 0.18253899
#> [25,] -0.06361797 0.18253899
#> [26,] -0.05085888 0.18523055
#> [27,] -0.05085888 0.18523055
#> [28,] -0.03241539 0.18912125
#> [29,] -0.03241539 0.18912125
#> [30,] 0.03246298 0.20280749
#> [31,] 0.03246298 0.20280749
#> [32,] 0.03310607 0.20294315
#> [33,] 0.03310607 0.20294315
#> [34,] 0.04594097 0.20565071
#> [35,] 0.04594097 0.20565071
#> [36,] 0.05059593 0.20663268
#> [37,] 0.05059593 0.20663268
#> [38,] 0.06396412 0.20945273
#> [39,] 0.06396412 0.20945273
#> [40,] 0.09979011 0.21701031
#> [41,] 0.09979011 0.21701031
#> [42,] 0.16943647 0.23170238
#> [43,] 0.16943647 0.23170238
#> [44,] 0.34926634 0.26963791
#> [45,] 0.34926634 0.26963791
#> [46,] 0.43964953 0.28870446
#> [47,] 0.43964953 0.28870446
#> [48,] 0.77645298 0.35975394
#>
#> [1] "Awe"
#> [1] 54
#> [1] "Awe"
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#> [1] 58
#> [1] "Awe"
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#> [1] "Awe"
#> [1] 50
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#> [1] 56
#> [1] "Awe"
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#> [1] "Awe"
#> [1] 38
#> [1] "Awe"
#> [1] 44
#> [1] "Awe"
#> [1] 56
#> [1] "Awe"
#> [1] 52
#> [1] "Awe"
#> [1] 48
#> [1] "Awe"
#> [1] 52
#> [1] "Awe"
#> [1] 58
#> [1] "Awe"
#> [1] 56
#> [1] "Awe"
#> [1] 54
#> [1] "Awe"
#> [1] 52
#> [1] "Awe"
#> [1] 50
#> [1] "Awe"
#> [1] 52
#> [1] "Awe"
#> [1] 52
#> [1] "Awe"
#> [1] 56
#> [1] "Awe"
#> [1] 56
#> [1] "Awe"
#> [1] 50
#> [1] "Awe"
#> [1] 56
#> [1] "Awe"
#> [1] 54
#> [1] "Awe"
#> [1] 54
#> [1] "Awe"
#> [1] 52
#> [1] "Awe"
#> [1] 52
#> [1] "Awe"
#> [1] 48
#> [1] "Awe"
#> [1] 50
#> [1] "Awe"
#> [1] 48
#> [1] "Awe"
#> [1] 48
fit.measures()
and summary()
There is a number of fit measures that are specific to CVA biplots.
The measures are computed with the function fit.measures()
and the results are displayed by the function
summary()
.
Canonical variate analysis can be considered as a transformation of the original variables to the canonical space followed by constructing a PCA biplot of canonical variables. The matrix of class means \(\bar{\mathbf{X}} = (\mathbf{G'G})^{-1} \mathbf{G'X}\) is transformed to \(\mathbf{\bar{X}L}\) where \(\mathbf{L}\) is a non-singular matrix such that \(\mathbf{LL'=W}^{-1}\). Pricipal component analysis finds the orthogonal matrix \(\mathbf{V}\) such that
\[ \mathbf{(L'\bar{X}'C\bar{X}L)V=V \Lambda} \]
where \(\mathbf{M = LV}\) as defined in section 1. The predicted values for the class means is given by
\[ \mathbf{\hat{\bar{X}}} = \mathbf{\bar{X}MJ}\mathbf{M}^{-1}. \]
Based on the two-step process described above, there are two measures of quality of fit. The quality of the approximation of the canonical variables \(\mathbf{\bar{X}L}\) in the \(2\)-dimensional display is given by
\[ Quality (canonical \: variables) = \frac{tr(\mathbf{\Lambda J})}{tr(\mathbf{\Lambda)}} \] and the quality of the approximation of the original variables \(\mathbf{\bar{X}}\) in the 2D CVA biplot is given by
\[ Quality (original \: variables) = \frac{tr(\mathbf{\Lambda J})}{tr(\mathbf{\Lambda)}} \]
The adequacy with which each of the variables is represented in the biplot is given by the elementwise ratios
\[ Adequacy = \frac{diag(\mathbf{MJM'})}{diag(\mathbf{MM'})}. \]
The axis and class mean predictivities are defined in terms of the weighted class means.
The elementwise ratios for the predictivity of each of the axes are given by
\[ axis \: predictivity = \frac{diag(\mathbf{\hat{\bar{X}}}'\mathbf{C\hat{\bar{X}}})}{diag(\mathbf{\bar{X}}'\mathbf{C\bar{X}})}. \]
Similarly for each of the class means the elementwise ratio is computed from
\[ class \: predictivity = \frac{diag(\mathbf{C}^{\frac{1}{2}}\mathbf{\hat{\bar{X}}}'\mathbf{W^{-1}}\mathbf{\hat{\bar{X}}}\mathbf{C}^{\frac{1}{2}})}{diag(\mathbf{C}^{\frac{1}{2}}\mathbf{\bar{X}}'\mathbf{W^{-1}}\mathbf{\bar{X}}\mathbf{C}^{\frac{1}{2}})}. \]
We define the matrix of samples as deviations from their class means as
\[ (\mathbf{I-H})\mathbf{X}=(\mathbf{I}_n-\mathbf{G}(\mathbf{G'G})^{-1}\mathbf{G}')\mathbf{X} \]
where \(\mathbf{H} = \mathbf{G}(\mathbf{G'G})^{-1}\mathbf{G}'\).
The within class axis predictivity is computed as the elementwise ratios
\[ within \: class \: axis \: predictivity = \frac{diag(\mathbf{\hat{X}}'(\mathbf{I-H)\hat{X}})}{diag(\mathbf{X}'(\mathbf{I-H)X})}. \]
Unlike PCA biplots, sample predictivity for CVA biplots are computed for the observations expressed as deviations from their class means. The elementwise ratios is obtained from
\[
within \: class \: axis \: predictivity =
\frac{diag(\mathbf{(I-H)\hat{X}}\mathbf{W}^{-1}\mathbf{\hat{X}'(I-H)})}{diag(\mathbf{(I-H)X}\mathbf{W}^{-1}\mathbf{X'(I-H)})}.
\] To display the fit measures, we create a biplot
object with the measures added by the function
fit.measures()
and call summary()
.
obj <- biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.division) |>
fit.measures() |>
plot()
summary (obj)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
#> 9 classes: New England Middle Atlantic South Atlantic East South Central West South Central East North Central West North Central Mountain Pacific
#>
#> Quality of fit of canonical variables = 70.7%
#> Quality of fit of original variables = 70.5%
#> Adequacy of variables:
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.41716176 0.15621549 0.16136381 0.09759664 0.19426796 0.55332679 0.50497634
#> Area
#> 0.40661470
#> Axis predictivity:
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.1859124 0.4019427 0.8195756 0.6925389 0.7685373 0.9506355 0.7819324
#> Area
#> 0.8458143
#> Class predictivity:
#> New England Middle Atlantic South Atlantic East South Central
#> 0.7922047 0.6570417 0.8191791 0.8777759
#> West South Central East North Central West North Central Mountain
#> 0.7416085 0.6370315 0.3265978 0.6825966
#> Pacific
#> 0.6700194
#> Within class axis predictivity:
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.04212318 0.09357501 0.25675620 0.19900223 0.29474972 0.75215233 0.31027358
#> Area
#> 0.12741853
#> Within class sample predictivity:
#> Alabama Alaska Arizona Arkansas California
#> 0.722548912 0.163442379 0.333341120 0.268976273 0.229139828
#> Colorado Connecticut Delaware Florida Georgia
#> 0.264963758 0.082284385 0.593415987 0.461070888 0.636531435
#> Hawaii Idaho Illinois Indiana Iowa
#> 0.015640188 0.113711473 0.338612599 0.389208196 0.507060148
#> Kansas Kentucky Louisiana Maine Maryland
#> 0.784831952 0.314119027 0.078465054 0.008388471 0.306141816
#> Massachusetts Michigan Minnesota Mississippi Missouri
#> 0.076563044 0.218470793 0.645446212 0.046129058 0.710971640
#> Montana Nebraska Nevada New Hampshire New Jersey
#> 0.086279776 0.810374638 0.090490164 0.298187909 0.003496353
#> New Mexico New York North Carolina North Dakota Ohio
#> 0.007134343 0.024268121 0.422776032 0.446240464 0.277262145
#> Oklahoma Oregon Pennsylvania Rhode Island South Carolina
#> 0.450104680 0.108636860 0.033945796 0.415029328 0.261568299
#> South Dakota Tennessee Texas Utah Vermont
#> 0.134881180 0.247921823 0.110537439 0.500454605 0.159941068
#> Virginia Washington West Virginia Wisconsin Wyoming
#> 0.310439564 0.030877305 0.066303623 0.295499472 0.474458397
The call to biplot()
, CVA()
and
fit.measures()
is required to (a) create an object of class
biplot
, (b) extend the object to class CVA
and
(c) compute the fit measures. The call to the function
plot()
is optional. It is further possible to select which
fit measures to display in the summary()
function where all
measures default to TRUE
.
obj <- biplot(state.x77, scaled = TRUE) |>
CVA(classes = state.region) |>
fit.measures()
summary (obj, adequacy = FALSE, within.class.axis.predictivity = FALSE,
within.class.sample.predictivity = FALSE)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
#> 4 classes: Northeast South North Central West
#>
#> Quality of fit of canonical variables = 91.9%
#> Quality of fit of original variables = 95.3%
#> Axis predictivity:
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.9873763 0.9848608 0.8757913 0.9050208 0.9955088 0.9970346 0.9558192
#> Area
#> 0.9344651
#> Class predictivity:
#> Northeast South North Central West
#> 0.8031465 0.9985089 0.6449906 0.9988469
It was mentioned that the eigen equation (1) has \(min(p, G-1)\) non-zero eigenvalues. This implies that the CVA biplot for \(G=2\) groups, reduces to a single dimension. If we write
\[
\mathbf{M} = \begin{bmatrix}
\mathbf{m}_1 & \mathbf{M}^*
\end{bmatrix}
\] the columns of \(\mathbf{M}^*\) forms a basis for the
orthogonal complement of the canonical space defined by \(\mathbf{m}\)_1. The argument
low.dim
determines how to uniquely define the second and
third dimensions. By default low.dim = "sample.opt"
which
selects the dimensions by minimising total squared reconstruction error
for samples.
The representation of the canonical variates \(\bar{\mathbf{Z}} = \bar{\mathbf{X}}\mathbf{m}_1\) are exact in the first dimension, but not the representation of the individual samples \({\mathbf{Z}} = {\mathbf{X}}\mathbf{m}_1\). If we define \(\mathbf{\hat{X}} = \mathbf{XMJ}\mathbf{M}^{-1}\) with \(\mathbf{J}\) a square matrix of zeros except for a \(1\) in the first diagonal position, then the total square reconstruction error for samples is given by
\[ TSRES = tr{(\mathbf{X}-\mathbf{\hat{X}})'(\mathbf{X}-\mathbf{\hat{X}})}. \] Define \[ \mathbf{M}^{-1} = \begin{bmatrix} \mathbf{M}^{(1)}:(G-1) \times p \\ \mathbf{M}^{(2)}: (p-G+1) \times p \end{bmatrix} \]
then \(TSRES\) is minimised when
\[ \mathbf{M}^{opt} = \begin{bmatrix} \mathbf{M}_1 & \mathbf{M}^*\mathbf{V} \end{bmatrix} \]
where $\mathbf{V}$ is the matrix of right singular vectors of $\mathbf{M}^{(2)}\mathbf{M}^{(2)'}$.
state.2group <- ifelse(state.division == "New England" |
state.division == "Middle Atlantic" |
state.division == "South Atlantic" |
state.division == "Pacific",
"Coastal", "Central")
biplot (state.x77) |> CVA (state.2group) |> legend.type(means=TRUE) |> plot()
#> Warning in CVA.biplot(biplot(state.x77), state.2group): The dimension of the
#> canonical space < dim.biplot sample.opt method used for additional
#> dimension(s).
@leRouxLubbe2024 discuss an alternative method for obtaining additional dimensions. When assuming underlying normal distributions, the Bhattacharyya distance can be optimised. This method is specific to the two class case and cannot be utilised to find a third dimension in a 3D CVA biplot with three classes.
biplot (state.x77) |> CVA (state.2group, low.dim="Bha") |> legend.type(means=TRUE) |> plot()
#> Warning in CVA.biplot(biplot(state.x77), state.2group, low.dim = "Bha"): The
#> dimension of the canonical space < dim.biplot Bhattacharyya.dist method used
#> for additional dimension(s).
Similar to the variance decomposition in CVA, analysis of distance decomposes the total sum of squared distances into a sum of squared distances between class means component and a sum of squared distances within classes component.
Consider any Euclidean embeddable distance metric \(\psi_{ij}=\psi(\mathbf{x}_i,\mathbf{x}_j)\). For a Euclidean embeddable metric it is possible to find high dimensional coordinates \(\mathbf{y}_i\) and \(\mathbf{y}_j\) such that the Euclidean distance between \(\mathbf{y}_i\) and \(\mathbf{y}_j\) is equal to \(\psi_{ij}\). Let the matrix \(\mathbf{\tilde\Psi}\) contain the values \(-\frac{1}2{}\psi_{ij}^2\) and similarly \(\mathbf{\tilde\Delta}\) the values \(-\frac{1}2{}\delta_{hk}^2\) where \(\delta_{hk}\) represent the distance between class means \(h\) and \(k\).
\[ \mathbf{T} = \mathbf{B} + \mathbf{W} \]
\[ \mathbf{1'\tilde\Psi1} = \mathbf{n'\tilde\Delta n} + \sum_{k=1}^{G} \frac{n}{n_k} \mathbf{g}_k'\mathbf{\tilde\Psi}\mathbf{g}_k \] where \(\mathbf{n}=\mathbf{(G'G)1}\). Thus, AoD differs from CVA in allowing any Euclidean embeddable measure of inter-class distance. As with CVA, these distances may be represented in maps with point representing the class means, supplemented by additional points representing the within-group variation. Principal coordinate analysis is performed, only on the \(G \times G\) matrix \(\mathbf{\tilde\Delta}\).
By default linear regression biplot axes are fitted to the plot. Alternatively, spline axes can be constructed.
#> Calculating spline axis for variable 1
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 2
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 3
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 4
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 5
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 6
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 7
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 8
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
As an illustration of a Euclidean embeddable distance metric, other than Euclidean distance itself, we can construct an AoD biplot with the square root of the Manhattan distance.
biplot(state.x77, scaled = TRUE) |>
AoD(classes = state.region, axes = "splines", dist.func=sqrtManhattan) |> plot()
#> Calculating spline axis for variable 1
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 2
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 3
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 4
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 5
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 6
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 7
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
#> Calculating spline axis for variable 8
#> Population Income Illiteracy Life Exp Murder HS Grad
#> Alabama 3615 3624 2.1 69.05 15.1 41.3
#> Alaska 365 6315 1.5 69.31 11.3 66.7
#> Arizona 2212 4530 1.8 70.55 7.8 58.1
#> Arkansas 2110 3378 1.9 70.66 10.1 39.9
#> California 21198 5114 1.1 71.71 10.3 62.6
#> Colorado 2541 4884 0.7 72.06 6.8 63.9
#> Connecticut 3100 5348 1.1 72.48 3.1 56.0
#> Delaware 579 4809 0.9 70.06 6.2 54.6
#> Florida 8277 4815 1.3 70.66 10.7 52.6
#> Georgia 4931 4091 2.0 68.54 13.9 40.6
#> Hawaii 868 4963 1.9 73.60 6.2 61.9
#> Idaho 813 4119 0.6 71.87 5.3 59.5
#> Illinois 11197 5107 0.9 70.14 10.3 52.6
#> Indiana 5313 4458 0.7 70.88 7.1 52.9
#> Iowa 2861 4628 0.5 72.56 2.3 59.0
#> Kansas 2280 4669 0.6 72.58 4.5 59.9
#> Kentucky 3387 3712 1.6 70.10 10.6 38.5
#> Louisiana 3806 3545 2.8 68.76 13.2 42.2
#> Maine 1058 3694 0.7 70.39 2.7 54.7
#> Maryland 4122 5299 0.9 70.22 8.5 52.3
#> Massachusetts 5814 4755 1.1 71.83 3.3 58.5
#> Michigan 9111 4751 0.9 70.63 11.1 52.8
#> Minnesota 3921 4675 0.6 72.96 2.3 57.6
#> Mississippi 2341 3098 2.4 68.09 12.5 41.0
#> Missouri 4767 4254 0.8 70.69 9.3 48.8
#> Montana 746 4347 0.6 70.56 5.0 59.2
#> Nebraska 1544 4508 0.6 72.60 2.9 59.3
#> Nevada 590 5149 0.5 69.03 11.5 65.2
#> New Hampshire 812 4281 0.7 71.23 3.3 57.6
#> New Jersey 7333 5237 1.1 70.93 5.2 52.5
#> New Mexico 1144 3601 2.2 70.32 9.7 55.2
#> New York 18076 4903 1.4 70.55 10.9 52.7
#> North Carolina 5441 3875 1.8 69.21 11.1 38.5
#> North Dakota 637 5087 0.8 72.78 1.4 50.3
#> Ohio 10735 4561 0.8 70.82 7.4 53.2
#> Oklahoma 2715 3983 1.1 71.42 6.4 51.6
#> Oregon 2284 4660 0.6 72.13 4.2 60.0
#> Pennsylvania 11860 4449 1.0 70.43 6.1 50.2
#> Rhode Island 931 4558 1.3 71.90 2.4 46.4
#> South Carolina 2816 3635 2.3 67.96 11.6 37.8
#> South Dakota 681 4167 0.5 72.08 1.7 53.3
#> Tennessee 4173 3821 1.7 70.11 11.0 41.8
#> Texas 12237 4188 2.2 70.90 12.2 47.4
#> Utah 1203 4022 0.6 72.90 4.5 67.3
#> Vermont 472 3907 0.6 71.64 5.5 57.1
#> Virginia 4981 4701 1.4 70.08 9.5 47.8
#> Washington 3559 4864 0.6 71.72 4.3 63.5
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5
#> Wyoming 376 4566 0.6 70.29 6.9 62.9
#> Frost Area
#> Alabama 2.000000e+01 50708
#> Alaska 1.520000e+02 566432
#> Arizona 1.500000e+01 113417
#> Arkansas 6.500000e+01 51945
#> California 2.000000e+01 156361
#> Colorado 1.660000e+02 103766
#> Connecticut 1.390000e+02 4862
#> Delaware 1.030000e+02 1982
#> Florida 1.100000e+01 54090
#> Georgia 6.000000e+01 58073
#> Hawaii 1.421085e-14 6425
#> Idaho 1.260000e+02 82677
#> Illinois 1.270000e+02 55748
#> Indiana 1.220000e+02 36097
#> Iowa 1.400000e+02 55941
#> Kansas 1.140000e+02 81787
#> Kentucky 9.500000e+01 39650
#> Louisiana 1.200000e+01 44930
#> Maine 1.610000e+02 30920
#> Maryland 1.010000e+02 9891
#> Massachusetts 1.030000e+02 7826
#> Michigan 1.250000e+02 56817
#> Minnesota 1.600000e+02 79289
#> Mississippi 5.000000e+01 47296
#> Missouri 1.080000e+02 68995
#> Montana 1.550000e+02 145587
#> Nebraska 1.390000e+02 76483
#> Nevada 1.880000e+02 109889
#> New Hampshire 1.740000e+02 9027
#> New Jersey 1.150000e+02 7521
#> New Mexico 1.200000e+02 121412
#> New York 8.200000e+01 47831
#> North Carolina 8.000000e+01 48798
#> North Dakota 1.860000e+02 69273
#> Ohio 1.240000e+02 40975
#> Oklahoma 8.200000e+01 68782
#> Oregon 4.400000e+01 96184
#> Pennsylvania 1.260000e+02 44966
#> Rhode Island 1.270000e+02 1049
#> South Carolina 6.500000e+01 30225
#> South Dakota 1.720000e+02 75955
#> Tennessee 7.000000e+01 41328
#> Texas 3.500000e+01 262134
#> Utah 1.370000e+02 82096
#> Vermont 1.680000e+02 9267
#> Virginia 8.500000e+01 39780
#> Washington 3.200000e+01 66570
#> West Virginia 1.000000e+02 24070
#> Wisconsin 1.490000e+02 54464
#> Wyoming 1.730000e+02 97203
#> attr(,"scaled:center")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> -4246.4200 -4435.8000 -1.1700 -70.8786 -7.3780 -53.1080
#> Frost Area
#> -104.4600 -70735.8800
#> attr(,"scaled:scale")
#> Population Income Illiteracy Life Exp Murder HS Grad
#> 2.239897e-04 1.627419e-03 1.640600e+00 7.449380e-01 2.708897e-01 1.238084e-01
#> Frost Area
#> 1.923785e-02 1.171958e-05
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.