Nothing
CA in biplotEZ"
In biplotEZ: EZ-to-Use Biplots
knitr::opts_chunk$set(
fig.height = 6, fig.width = 7,
collapse = TRUE,
comment = "#>"
)
library(biplotEZ)
What is a Correspondence Analysis?
In its simplest form Correspondence Analysis (CA) aims to expose the
association between two categorical variables by utilising a two-way
frequency table. Numerous variants of CA are available for the
application to diverse problems, the interested reader is referred to:
@UB2011, @BehLombardo2014.
In this vignette, focus will be placed on three EZ-to-use versions based on the Pearson residuals (@UB2011 :300).
Now, the two-way frequency table is also referred to as the data matrix:
$\mathbf{X}:r\times c$. This data matrix is different from the
continuous case used for the PCA() and CVA() examples, as it
represents the cross-tabulations of two categorical variables (i.e.
factors), each with a finite number of levels (i.e response values). The
elements of the data matrix represent the frequency of the co-occurrence
of two particular levels of the two variables. Consider the
HairEyeColor data set in R, which summarises the hair and eye color
of male and female statistics students. For the purpose of this example
only the male students will be considered:
X <- HairEyeColor[,,2]
X
The grand total of the table $N$ is obtained from the total of all
frequencies:
$$
\sum_{r=1}^{R}\sum_{c=1}^{C}x_{rc}=N
$$
N <- sum(X)
N
It is common to work with the proportions rather than the frequencies
in terms of the correspondence matrix, $\mathbf{P}$:
$$
\mathbf{P}=\frac{\mathbf{X}}{N}
$$
P <- X/N
P
Other useful summaries of $\mathbf{P}$ include the row and column masses (for arbitrary row and column $r$ and $c$, respectively), also expressed as diagonal matrices:
$$
\mathbf{r}r = \sum{c=1}^{C}p_{rc}; \hspace{0.5 cm}
\mathbf{c}c = \sum{r=1}^{R}p_{rc}\
\mathbf{r}=\mathbf{P1}; \hspace{0.5 cm} \mathbf{c}=\mathbf{P}^\prime\mathbf{1}
$$
rMass <- rowSums(P)
rMass
cMass <- colSums(P)
cMass
Diagonal matrices:
$$
\mathbf{D_r}=\text{diag}(\mathbf{r}); \hspace{0.5 cm} \mathbf{D_c}=\text{diag}(\mathbf{c})
$$
Dr <- diag(apply(P, 1, sum))
Dr
Dc <- diag(apply(P, 2, sum))
Dc
In order to obtain the first form of the row and column coordinates, the singular value decomposition (SVD) of the matrix of standardised Pearson residuals ($\mathbf{S}$) is computed:
$$
\begin{aligned}
\text{SVD}(\mathbf{S}) &= \text{SVD}\left(\mathbf{D_r^{-\frac{1}{2}}}(\mathbf{P}-\mathbf{rc^\prime})\mathbf{D_c^{-\frac{1}{2}}}\right)\&= \mathbf{U\Lambda V^\prime}
\end{aligned}
$$
The return value for the Standardised pearson residuals is Smat and the singular value decomposition, SVD.
Smat <- sqrt(solve(Dr))%*%(P-(Dr %*%matrix(1, nrow = nrow(X),
ncol = ncol(X)) %*% Dc))%*%sqrt(solve(Dc))
svd.out <- svd(Smat)
svd.out
This is linked to the $\chi^2$-statistic to determine whether the two categorical variables (i.e. the rows and columns of the contingency table) are independent. The expected frequencies represented by the product of the row and column masses ($\mathbf{rc^\prime}$).
Furthermore, since the weights of certain objects might be substantially different from others which could result in a distorted approximation in lower dimension, the $\chi^2$-distance, also referred to as the weighted Euclidean distance, is rather used to measure distances in CA.
This is an intuitive decision as it follows from the $\chi^2$-statistic to test the independence between two categorical variables, in this case the independence between the rows and columns of the contingency table. (@BehLombardo2014, @Greenacre2017).
CA biplot
Coordinates
In order to construct a biplot in which the distances between the row and column coordinates are meaningful an asymmetric display should be constructed. This means that the contribution of the singular values should be different for the row and column coordinates. (@Gabriel1971)
The standard coordinates are expressed by:
$$
\begin{aligned}
\text{Row standard coordinates:} \hspace{0.5 cm}&\mathbf{U}\
\text{Column standard coordinates:} \hspace{0.5 cm}&\mathbf{V}
\end{aligned}
$$
The principal coordinates are expressed by:
$$
\begin{aligned}
\text{Row principal coordinates:} \hspace{0.5 cm}&\mathbf{U\Lambda}\
\text{Column principal coordinates:} \hspace{0.5 cm}&\mathbf{V\Lambda}
\end{aligned}
$$
By including the singular values the magnitude of the association between the variables are incorporated in the scaling of the coordinates.
In the ca() function the argument variant allows the user to choose between three types of CA biplots: Princ, Stand and Symmetric.
$$
\begin{aligned}
\text{Row coordinates:} \hspace{0.5 cm}&\mathbf{U\Lambda^\gamma}\
\text{Column coordinates:} \hspace{0.5 cm}&\mathbf{V\Lambda^{1-\gamma}}
\end{aligned}
$$
The row standard (i.e. column principal) coordinate biplot: Stand, results from $\gamma=0$.
The row principal (i.e. column standard) coordinate biplot: Princ, results from $\gamma=1$.
The symmetric plot in which row and column coordinates are scaled equally: Symmetric, results from $\gamma=0.5$.
The return value is rowcoor and colcoor, respectively.
gamma <- 0
rowcoor <- svd.out[[2]]%*%(diag(svd.out[[1]])^gamma)
colcoor <- svd.out[[3]]%*%(diag(svd.out[[1]])^(1-gamma))
Lambda scaling
As presented in @UB2011 :24, when constructing a biplot representing the rows of a coordinate matrix $\mathbf{A}$ and $\mathbf{B}^\prime$. Take note that the inner product is invariant when $\mathbf{A}$ and $\mathbf{B}$ are scaled inversely by $\lambda$.
$$
\mathbf{AB} = (\lambda\mathbf{A})(\mathbf{B}/\lambda)
$$
An arbitrary value of $\lambda$ can be selected or an optimal value could be to ensure that the average squared distance of the points in $\lambda\mathbf{A}$ and $\mathbf{B}/\lambda$ is equal.
$$
\lambda^4 =\frac{r}{c}\frac{||\mathbf{B}||^2}{||\mathbf{A}||^2}
$$
The default setting is to not apply lambda-scaling (i.e. $\lambda=1$).
The return value is lambda.val.
Measures of fit
quality, adequacy, row.predictivities and column.predictivities are available for CA biplots.
As explained in the biplotEZ vignette, the quality of the biplot is measured by the ratio of the variance explained (sum of the squared singular values of the utilised ($M$) components) and the total variance (sum of all squared singular values ($p$)).
$$
\frac{\sum_{m=1}^{M}\lambda_m^2}{\sum_{m=1}^{p}\lambda_m^2}
$$
The adequacy refers to the representation of the variables. In CA() the factor variable represented in the columns is treated as the variables and is calculated as explained in the biplotEZ vignette:
$$
\frac{diag(\mathbf{V}_r\mathbf{V}_r')}{diag(\mathbf{VV}')}= diag(\mathbf{V}_r\mathbf{V}_r')
$$
The predictivities provide a measure of how well the original values are recovered from the biplot. An element that is well represented will have a predictivity close to one, indicating that the row or column variable values from prediction is close to the observed values. If an element is poorly represented, the predicted values will be very different from the original values and the predictivity value will be close to zero.
The row.predictivities are calculated as follows (@UB2011 :299):
$$
diag(\mathbf{U}\mathbf{\Sigma}\mathbf{J}\mathbf{\Sigma}\mathbf{U}')[diag(\mathbf{U}\mathbf{\Sigma}\mathbf{\Sigma}\mathbf{U}')]^{-1}\
=diag(\mathbf{U}\mathbf{\Sigma}^2\mathbf{J}\mathbf{U}')[diag(\mathbf{U}\mathbf{\Sigma^2}\mathbf{U}')]^{-1}
$$
The col.predictivities are calculated as follows (@UB2011 :299):
$$
diag(\mathbf{V}\mathbf{\Sigma}^2\mathbf{J}\mathbf{V}')[diag(\mathbf{V}\mathbf{\Sigma^2}\mathbf{V}')]^{-1}
$$
The function CA()
The function CA() requires a two-way contingency table as input and will return an object of class CA and biplot. As this is not a standard data matrix as for PCA and CVA, scaling and centering is not allowed on the two-way contingency table and a warning will be given if either scale or center is specified as TRUE in biplot()`.
Variant="Princ"
The default CA biplot is a row principal coordinate biplot:
biplot(HairEyeColor[,,2], center = FALSE) |> CA() |> plot()
Variant="Stand"
To construct the CA biplot for row standard coordinates:
ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Stand") |>
plot()
Variant="Symmetric"
To construct the symmetric CA map:
ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |>
CA(variant = "Symmetric") |> plot()
ca.out$lambda.val
Measures of fit
The fit.mesaures() function should be utilised to obtain the specific fit measures explained above.
ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |>
CA(variant = "Symmetric") |> fit.measures()
print("Quality")
ca.out$quality
print("Adequacy")
ca.out$adequacy
print("Row predictivities")
ca.out$row.predictivities
print("Column predictivities")
ca.out$col.predictivities
Interpolating new samples
Adding cross-tabulations of the two categorical variables to the plot is facilitated by the function interpolate(). Note that the additional variables to be interpolated did not contribute to the construction of the biplot. This is the reason why @Greenacre2017 term these supplementary points.
The function interpolate() accepts a matrix or data frame containing the samples and variables to be interpolated. The argument newdata containing the samples to be interpolated needs to have a similar structure to the data set sent to biplot(). If biplot() received a data frame, newdata can be either another data frame or a matrix containing the subset of numerical variables.
The function newsamples() operates similar to samples() and enables aesthetic changes to the new samples.
biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Symmetric") |>
samples(pch = c(0,2)) |> interpolate(newdata = HairEyeColor[,,1]) |>
newsamples(col = c("orange","purple"), pch = c(15,17)) |> plot()
Aesthetics and legend
The sample() function should be utilised to specify the colours, plotting characters and expansion of the samples.
biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Princ") |>
samples(col = c("cyan","purple"), pch = c(15,17), label.side = c("bottom","top"),
label.cex = 1) |> legend.type(samples = TRUE, new = TRUE) |> plot()
biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Symmetric") |>
samples(col = c("forestgreen","magenta"), pch = c(12,17),
label.side = c("top","bottom")) |>
legend.type(samples = TRUE) |> plot()
Additional example
Consider the South African Crime data set 2008, extracted from the South African police website (http://www.saps.gov.za/). @UB2011 :312.
SACrime <- matrix(c(1235,432,1824,1322,573,588,624,169,629,34479,16833,46993,30606,13670,
16849,15861,9898,24915,2160,939,5257,4946,722,1271,881,775,1844,5946,
4418,15117,10258,5401,4273,4987,1956,10639,29508,15705,62703,37203,
11857,18855,14722,4924,42376,604,156,7466,3889,203,664,291,5,923,19875,
19885,57153,29410,11024,12202,10406,5431,32663,7086,4193,22152,9264,3760,
4752,3863,1337,8578,7929,4525,12348,24174,3198,1770,7004,2201,45985,764,
427,1501,1197,215,251,345,213,1850,3515,879,3674,4713,696,835,917,422,2836,
88,59,174,76,31,61,117,32,257,5499,2628,8073,6502,2816,2635,3017,1020,4000,
8939,4501,50970,24290,2447,5907,5528,1175,14555),nrow=9, ncol=14)
dimnames(SACrime) <- list(paste(c("ECpe", "FrSt", "Gaut", "KZN", "Limp", "Mpml", "NWst", "NCpe",
"WCpe")), paste(c("Arsn", "AGBH", "AtMr", "BNRs", "BRs", "CrJk",
"CmAs", "CmRb", "DrgR", "InAs", "Mrd", "PubV",
"Rape", "RAC" )))
names(dimnames(SACrime))[[1]] <- "Provinces"
names(dimnames(SACrime))[[2]] <- "Crimes"
SACrime
biplot(SACrime, center = FALSE) |>
CA(variant = "Symmetric", lambda.scal = TRUE) |>
samples(col = c("cyan","purple"), pch = c(15,17), label.side = c("bottom","top")) |>
legend.type(samples = TRUE, new = TRUE) |> plot()
References
Try the biplotEZ package in your browser
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biplotEZ documentation built on April 4, 2025, 2:20 a.m.
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Note that we can't provide technical support on individual packages. You should contact the package authors for that.
knitr::opts_chunk$set( fig.height = 6, fig.width = 7, collapse = TRUE, comment = "#>" )
library(biplotEZ)
What is a Correspondence Analysis?
In its simplest form Correspondence Analysis (CA) aims to expose the association between two categorical variables by utilising a two-way frequency table. Numerous variants of CA are available for the application to diverse problems, the interested reader is referred to: @UB2011, @BehLombardo2014.
In this vignette, focus will be placed on three EZ-to-use versions based on the Pearson residuals (@UB2011 :300).
Now, the two-way frequency table is also referred to as the data matrix:
$\mathbf{X}:r\times c$. This data matrix is different from the
continuous case used for the PCA() and CVA() examples, as it
represents the cross-tabulations of two categorical variables (i.e.
factors), each with a finite number of levels (i.e response values). The
elements of the data matrix represent the frequency of the co-occurrence
of two particular levels of the two variables. Consider the
HairEyeColor data set in R, which summarises the hair and eye color
of male and female statistics students. For the purpose of this example
only the male students will be considered:
X <- HairEyeColor[,,2] X
The grand total of the table $N$ is obtained from the total of all frequencies:
$$ \sum_{r=1}^{R}\sum_{c=1}^{C}x_{rc}=N $$
N <- sum(X) N
It is common to work with the proportions rather than the frequencies in terms of the correspondence matrix, $\mathbf{P}$:
$$ \mathbf{P}=\frac{\mathbf{X}}{N} $$
P <- X/N P
Other useful summaries of $\mathbf{P}$ include the row and column masses (for arbitrary row and column $r$ and $c$, respectively), also expressed as diagonal matrices:
$$ \mathbf{r}r = \sum{c=1}^{C}p_{rc}; \hspace{0.5 cm} \mathbf{c}c = \sum{r=1}^{R}p_{rc}\ \mathbf{r}=\mathbf{P1}; \hspace{0.5 cm} \mathbf{c}=\mathbf{P}^\prime\mathbf{1} $$
rMass <- rowSums(P) rMass cMass <- colSums(P) cMass
Diagonal matrices:
$$ \mathbf{D_r}=\text{diag}(\mathbf{r}); \hspace{0.5 cm} \mathbf{D_c}=\text{diag}(\mathbf{c}) $$
Dr <- diag(apply(P, 1, sum)) Dr Dc <- diag(apply(P, 2, sum)) Dc
In order to obtain the first form of the row and column coordinates, the singular value decomposition (SVD) of the matrix of standardised Pearson residuals ($\mathbf{S}$) is computed:
$$
\begin{aligned}
\text{SVD}(\mathbf{S}) &= \text{SVD}\left(\mathbf{D_r^{-\frac{1}{2}}}(\mathbf{P}-\mathbf{rc^\prime})\mathbf{D_c^{-\frac{1}{2}}}\right)\&= \mathbf{U\Lambda V^\prime}
\end{aligned}
$$
The return value for the Standardised pearson residuals is Smat and the singular value decomposition, SVD.
Smat <- sqrt(solve(Dr))%*%(P-(Dr %*%matrix(1, nrow = nrow(X), ncol = ncol(X)) %*% Dc))%*%sqrt(solve(Dc)) svd.out <- svd(Smat) svd.out
This is linked to the $\chi^2$-statistic to determine whether the two categorical variables (i.e. the rows and columns of the contingency table) are independent. The expected frequencies represented by the product of the row and column masses ($\mathbf{rc^\prime}$).
Furthermore, since the weights of certain objects might be substantially different from others which could result in a distorted approximation in lower dimension, the $\chi^2$-distance, also referred to as the weighted Euclidean distance, is rather used to measure distances in CA. This is an intuitive decision as it follows from the $\chi^2$-statistic to test the independence between two categorical variables, in this case the independence between the rows and columns of the contingency table. (@BehLombardo2014, @Greenacre2017).
CA biplot
Coordinates
In order to construct a biplot in which the distances between the row and column coordinates are meaningful an asymmetric display should be constructed. This means that the contribution of the singular values should be different for the row and column coordinates. (@Gabriel1971) The standard coordinates are expressed by:
$$ \begin{aligned} \text{Row standard coordinates:} \hspace{0.5 cm}&\mathbf{U}\ \text{Column standard coordinates:} \hspace{0.5 cm}&\mathbf{V} \end{aligned} $$
The principal coordinates are expressed by:
$$ \begin{aligned} \text{Row principal coordinates:} \hspace{0.5 cm}&\mathbf{U\Lambda}\ \text{Column principal coordinates:} \hspace{0.5 cm}&\mathbf{V\Lambda} \end{aligned} $$
By including the singular values the magnitude of the association between the variables are incorporated in the scaling of the coordinates.
In the ca() function the argument variant allows the user to choose between three types of CA biplots: Princ, Stand and Symmetric.
$$
\begin{aligned}
\text{Row coordinates:} \hspace{0.5 cm}&\mathbf{U\Lambda^\gamma}\
\text{Column coordinates:} \hspace{0.5 cm}&\mathbf{V\Lambda^{1-\gamma}}
\end{aligned}
$$
The row standard (i.e. column principal) coordinate biplot: Stand, results from $\gamma=0$.
The row principal (i.e. column standard) coordinate biplot: Princ, results from $\gamma=1$.
The symmetric plot in which row and column coordinates are scaled equally: Symmetric, results from $\gamma=0.5$.
The return value is rowcoor and colcoor, respectively.
gamma <- 0 rowcoor <- svd.out[[2]]%*%(diag(svd.out[[1]])^gamma) colcoor <- svd.out[[3]]%*%(diag(svd.out[[1]])^(1-gamma))
Lambda scaling
As presented in @UB2011 :24, when constructing a biplot representing the rows of a coordinate matrix $\mathbf{A}$ and $\mathbf{B}^\prime$. Take note that the inner product is invariant when $\mathbf{A}$ and $\mathbf{B}$ are scaled inversely by $\lambda$.
$$ \mathbf{AB} = (\lambda\mathbf{A})(\mathbf{B}/\lambda) $$ An arbitrary value of $\lambda$ can be selected or an optimal value could be to ensure that the average squared distance of the points in $\lambda\mathbf{A}$ and $\mathbf{B}/\lambda$ is equal.
$$ \lambda^4 =\frac{r}{c}\frac{||\mathbf{B}||^2}{||\mathbf{A}||^2} $$
The default setting is to not apply lambda-scaling (i.e. $\lambda=1$).
The return value is lambda.val.
Measures of fit
quality, adequacy, row.predictivities and column.predictivities are available for CA biplots.
As explained in the biplotEZ vignette, the quality of the biplot is measured by the ratio of the variance explained (sum of the squared singular values of the utilised ($M$) components) and the total variance (sum of all squared singular values ($p$)).
$$
\frac{\sum_{m=1}^{M}\lambda_m^2}{\sum_{m=1}^{p}\lambda_m^2}
$$
The adequacy refers to the representation of the variables. In CA() the factor variable represented in the columns is treated as the variables and is calculated as explained in the biplotEZ vignette:
$$ \frac{diag(\mathbf{V}_r\mathbf{V}_r')}{diag(\mathbf{VV}')}= diag(\mathbf{V}_r\mathbf{V}_r') $$
The predictivities provide a measure of how well the original values are recovered from the biplot. An element that is well represented will have a predictivity close to one, indicating that the row or column variable values from prediction is close to the observed values. If an element is poorly represented, the predicted values will be very different from the original values and the predictivity value will be close to zero.
The row.predictivities are calculated as follows (@UB2011 :299):
$$ diag(\mathbf{U}\mathbf{\Sigma}\mathbf{J}\mathbf{\Sigma}\mathbf{U}')[diag(\mathbf{U}\mathbf{\Sigma}\mathbf{\Sigma}\mathbf{U}')]^{-1}\ =diag(\mathbf{U}\mathbf{\Sigma}^2\mathbf{J}\mathbf{U}')[diag(\mathbf{U}\mathbf{\Sigma^2}\mathbf{U}')]^{-1} $$
The col.predictivities are calculated as follows (@UB2011 :299):
$$ diag(\mathbf{V}\mathbf{\Sigma}^2\mathbf{J}\mathbf{V}')[diag(\mathbf{V}\mathbf{\Sigma^2}\mathbf{V}')]^{-1} $$
The function CA()
The function CA() requires a two-way contingency table as input and will return an object of class CA and biplot. As this is not a standard data matrix as for PCA and CVA, scaling and centering is not allowed on the two-way contingency table and a warning will be given if either scale or center is specified as TRUE in biplot()`.
Variant="Princ"
The default CA biplot is a row principal coordinate biplot:
biplot(HairEyeColor[,,2], center = FALSE) |> CA() |> plot()
Variant="Stand"
To construct the CA biplot for row standard coordinates:
ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Stand") |> plot()
Variant="Symmetric"
To construct the symmetric CA map:
ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Symmetric") |> plot() ca.out$lambda.val
Measures of fit
The fit.mesaures() function should be utilised to obtain the specific fit measures explained above.
ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Symmetric") |> fit.measures() print("Quality") ca.out$quality print("Adequacy") ca.out$adequacy print("Row predictivities") ca.out$row.predictivities print("Column predictivities") ca.out$col.predictivities
Interpolating new samples
Adding cross-tabulations of the two categorical variables to the plot is facilitated by the function interpolate(). Note that the additional variables to be interpolated did not contribute to the construction of the biplot. This is the reason why @Greenacre2017 term these supplementary points.
The function interpolate() accepts a matrix or data frame containing the samples and variables to be interpolated. The argument newdata containing the samples to be interpolated needs to have a similar structure to the data set sent to biplot(). If biplot() received a data frame, newdata can be either another data frame or a matrix containing the subset of numerical variables.
The function newsamples() operates similar to samples() and enables aesthetic changes to the new samples.
biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Symmetric") |> samples(pch = c(0,2)) |> interpolate(newdata = HairEyeColor[,,1]) |> newsamples(col = c("orange","purple"), pch = c(15,17)) |> plot()
Aesthetics and legend
The sample() function should be utilised to specify the colours, plotting characters and expansion of the samples.
biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Princ") |> samples(col = c("cyan","purple"), pch = c(15,17), label.side = c("bottom","top"), label.cex = 1) |> legend.type(samples = TRUE, new = TRUE) |> plot()
biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Symmetric") |> samples(col = c("forestgreen","magenta"), pch = c(12,17), label.side = c("top","bottom")) |> legend.type(samples = TRUE) |> plot()
Additional example
Consider the South African Crime data set 2008, extracted from the South African police website (http://www.saps.gov.za/). @UB2011 :312.
SACrime <- matrix(c(1235,432,1824,1322,573,588,624,169,629,34479,16833,46993,30606,13670, 16849,15861,9898,24915,2160,939,5257,4946,722,1271,881,775,1844,5946, 4418,15117,10258,5401,4273,4987,1956,10639,29508,15705,62703,37203, 11857,18855,14722,4924,42376,604,156,7466,3889,203,664,291,5,923,19875, 19885,57153,29410,11024,12202,10406,5431,32663,7086,4193,22152,9264,3760, 4752,3863,1337,8578,7929,4525,12348,24174,3198,1770,7004,2201,45985,764, 427,1501,1197,215,251,345,213,1850,3515,879,3674,4713,696,835,917,422,2836, 88,59,174,76,31,61,117,32,257,5499,2628,8073,6502,2816,2635,3017,1020,4000, 8939,4501,50970,24290,2447,5907,5528,1175,14555),nrow=9, ncol=14) dimnames(SACrime) <- list(paste(c("ECpe", "FrSt", "Gaut", "KZN", "Limp", "Mpml", "NWst", "NCpe", "WCpe")), paste(c("Arsn", "AGBH", "AtMr", "BNRs", "BRs", "CrJk", "CmAs", "CmRb", "DrgR", "InAs", "Mrd", "PubV", "Rape", "RAC" ))) names(dimnames(SACrime))[[1]] <- "Provinces" names(dimnames(SACrime))[[2]] <- "Crimes" SACrime
biplot(SACrime, center = FALSE) |> CA(variant = "Symmetric", lambda.scal = TRUE) |> samples(col = c("cyan","purple"), pch = c(15,17), label.side = c("bottom","top")) |> legend.type(samples = TRUE, new = TRUE) |> plot()
References
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