# CA in biplotEZ

# 1 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: Gower, Lubbe, and Roux (2011), Beh and Lombardo (2014).

In this vignette, focus will be placed on three EZ-to-use versions based on the Pearson residuals (Gower, Lubbe, and Roux (2011) :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
    #>        Eye
    #> Hair    Brown Blue Hazel Green
    #>   Black    36    9     5     2
    #>   Brown    66   34    29    14
    #>   Red      16    7     7     7
    #>   Blond     4   64     5     8

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
    #> [1] 313

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
    #>        Eye
    #> Hair          Brown        Blue       Hazel       Green
    #>   Black 0.115015974 0.028753994 0.015974441 0.006389776
    #>   Brown 0.210862620 0.108626198 0.092651757 0.044728435
    #>   Red   0.051118211 0.022364217 0.022364217 0.022364217
    #>   Blond 0.012779553 0.204472843 0.015974441 0.025559105

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
    #>     Black     Brown       Red     Blond
    #> 0.1661342 0.4568690 0.1182109 0.2587859

    cMass <- colSums(P)
    cMass
    #>      Brown       Blue      Hazel      Green
    #> 0.38977636 0.36421725 0.14696486 0.09904153

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
    #>           [,1]     [,2]      [,3]      [,4]
    #> [1,] 0.1661342 0.000000 0.0000000 0.0000000
    #> [2,] 0.0000000 0.456869 0.0000000 0.0000000
    #> [3,] 0.0000000 0.000000 0.1182109 0.0000000
    #> [4,] 0.0000000 0.000000 0.0000000 0.2587859

    Dc <- diag(apply(P, 2, sum))
    Dc
    #>           [,1]      [,2]      [,3]       [,4]
    #> [1,] 0.3897764 0.0000000 0.0000000 0.00000000
    #> [2,] 0.0000000 0.3642173 0.0000000 0.00000000
    #> [3,] 0.0000000 0.0000000 0.1469649 0.00000000
    #> [4,] 0.0000000 0.0000000 0.0000000 0.09904153

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
    #> $d
    #> [1] 5.499629e-01 1.806424e-01 7.541748e-02 7.966568e-17
    #>
    #> $u
    #>            [,1]       [,2]        [,3]      [,4]
    #> [1,] -0.3832205  0.7807477 -0.27828196 0.4075956
    #> [2,] -0.3195599 -0.2982325  0.59335474 0.6759209
    #> [3,] -0.1736837 -0.5332073 -0.75320188 0.3438181
    #> [4,]  0.8490333  0.1310737 -0.05635808 0.5087101
    #>
    #> $v
    #>             [,1]       [,2]        [,3]      [,4]
    #> [1,] -0.61837991  0.4547976 -0.14487614 0.6243207
    #> [2,]  0.75797511  0.2307003  0.08963172 0.6035041
    #> [3,] -0.20612962 -0.5898626  0.68015284 0.3833600
    #> [4,]  0.02430218 -0.6260980 -0.71300012 0.3147086

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. (Beh and Lombardo (2014), Greenacre (2017)).

# 2 CA biplot

## 2.1 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. (Gabriel (1971)) 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))

## 2.2 Lambda scaling

As presented in Gower, Lubbe, and Roux (2011) :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`.

## 2.3 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 (Gower, Lubbe, and Roux (2011) :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 (Gower, Lubbe, and Roux (2011) :299):

\[ diag(\mathbf{V}\mathbf{\Sigma}^2\mathbf{J}\mathbf{V}')[diag(\mathbf{V}\mathbf{\Sigma^2}\mathbf{V}')]^{-1} \]

# 3 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()`.

## 3.1 `Variant="Princ"`

The default CA biplot is a row principal coordinate biplot:

    biplot(HairEyeColor[,,2], center = FALSE) |> CA() |> plot()

## 3.2 `Variant="Stand"`

To construct the CA biplot for row standard coordinates:

    ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Stand") |>
      plot()

## 3.3 `Variant="Symmetric"`

To construct the symmetric CA map:

    ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |>
      CA(variant = "Symmetric") |> plot()

    ca.out$lambda.val
    #> [1] 1

## 3.4 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")
    #> [1] "Quality"

    ca.out$quality
    #> [1] 0.9833094

    print("Adequacy")
    #> [1] "Adequacy"

    ca.out$adequacy
    #>        Brown   Blue  Hazel  Green
    #> Dim 1 0.3824 0.5745 0.0425 0.0006
    #> Dim 2 0.5892 0.6277 0.3904 0.3926
    #> Dim 3 0.6102 0.6358 0.8530 0.9010
    #> Dim 4 1.0000 1.0000 1.0000 1.0000

    print("Row predictivities")
    #> [1] "Row predictivities"

    ca.out$row.predictivities
    #>        Black  Brown    Red  Blond
    #> Dim 1 0.6860 0.8630 0.4219 0.9974
    #> Dim 2 0.9932 0.9441 0.8508 0.9999
    #> Dim 3 1.0000 1.0000 1.0000 1.0000
    #> Dim 4 1.0000 1.0000 1.0000 1.0000

    print("Column predictivities")
    #> [1] "Column predictivities"

    ca.out$col.predictivities
    #>        Brown   Blue  Hazel  Green
    #> Dim 1 0.9439 0.9898 0.4789 0.0113
    #> Dim 2 0.9990 0.9997 0.9020 0.8177
    #> Dim 3 1.0000 1.0000 1.0000 1.0000
    #> Dim 4 1.0000 1.0000 1.0000 1.0000

## 3.5 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 Greenacre (2017) 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()

## 3.6 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()

## 3.7 Additional example

Consider the South African Crime data set 2008, extracted from the South African police website (http://www.saps.gov.za/). Gower, Lubbe, and Roux (2011) :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
    #>          Crimes
    #> Provinces Arsn  AGBH AtMr  BNRs   BRs CrJk  CmAs  CmRb  DrgR InAs  Mrd PubV
    #>      ECpe 1235 34479 2160  5946 29508  604 19875  7086  7929  764 3515   88
    #>      FrSt  432 16833  939  4418 15705  156 19885  4193  4525  427  879   59
    #>      Gaut 1824 46993 5257 15117 62703 7466 57153 22152 12348 1501 3674  174
    #>      KZN  1322 30606 4946 10258 37203 3889 29410  9264 24174 1197 4713   76
    #>      Limp  573 13670  722  5401 11857  203 11024  3760  3198  215  696   31
    #>      Mpml  588 16849 1271  4273 18855  664 12202  4752  1770  251  835   61
    #>      NWst  624 15861  881  4987 14722  291 10406  3863  7004  345  917  117
    #>      NCpe  169  9898  775  1956  4924    5  5431  1337  2201  213  422   32
    #>      WCpe  629 24915 1844 10639 42376  923 32663  8578 45985 1850 2836  257
    #>          Crimes
    #> Provinces Rape   RAC
    #>      ECpe 5499  8939
    #>      FrSt 2628  4501
    #>      Gaut 8073 50970
    #>      KZN  6502 24290
    #>      Limp 2816  2447
    #>      Mpml 2635  5907
    #>      NWst 3017  5528
    #>      NCpe 1020  1175
    #>      WCpe 4000 14555

    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

Beh, E, and Rosaria Lombardo. 2014. “Correspondence Analysis.” Theory, Paractice and New Strategies.
Gabriel, K. R. 1971. “The Biplot Graphic Display of Matrices with Application to Principal Component Analysis.” Biometrika, 453–67.
Gower, J. C., S. Lubbe, and N. J. le Roux. 2011. Understanding Biplots. Wiley.
Greenacre, M. J. 2017. Correspondence Analysis in Practice. CRC press.