Author: Taiyun Wei, Viliam Simko
Date: 2024-10-14

# An Introduction to corrplot Package

#### Taiyun Wei, Viliam Simko

#### 2024-10-14

## Introduction

R package corrplot provides a visual exploratory tool on correlation matrix that supports automatic variable reordering to help detect hidden patterns among variables.

corrplot is very easy to use and provides a rich array of plotting options in visualization method, graphic layout, color, legend, text labels, etc. It also provides p-values and confidence intervals to help users determine the statistical significance of the correlations.

`corrplot()` has about 50 parameters, however the mostly common ones are only a few. We can get a correlation matrix plot with only one line of code in most scenes.

The mostly using parameters include `method`, `type`, `order`, `diag`, and etc.

There are seven visualization methods (parameter `method`) in corrplot package, named `'circle'`, `'square'`, `'ellipse'`, `'number'`, `'shade'`, `'color'`, `'pie'`. Color intensity of the glyph is proportional to the correlation coefficients by default color setting.

- `'circle'` and `'square'`, the areas of circles or squares show the absolute value of corresponding correlation coefficients.

- `'ellipse'`, the ellipses have their eccentricity parametrically scaled to the correlation value. It comes from D.J. Murdoch and E.D. Chow’s job, see in section References.

- `'number'`, coefficients numbers with different color.

- `'color'`, square of equal size with different color.

- `'shade'`, similar to `'color'`, but the negative coefficients glyphs are shaded. Method `'pie'` and `'shade'` come from Michael Friendly’s job.

- `'pie'`, the circles are filled clockwise for positive values, anti-clockwise for negative values.

`corrplot.mixed()` is a wrapped function for mixed visualization style, which can set the visual methods of lower and upper triangular separately.

There are three layout types (parameter `type`): `'full'`, `'upper'` and `'lower'`.

The correlation matrix can be reordered according to the correlation matrix coefficients. This is important to identify the hidden structure and pattern in the matrix.

    library(corrplot)

    ## corrplot 0.95 loaded

    M = cor(mtcars)
    corrplot(M, method = 'number') # colorful number

    corrplot(M, method = 'color', order = 'alphabet')

    corrplot(M) # by default, method = 'circle'

    corrplot(M, order = 'AOE') # after 'AOE' reorder

    corrplot(M, method = 'shade', order = 'AOE', diag = FALSE)

    corrplot(M, method = 'square', order = 'FPC', type = 'lower', diag = FALSE)

    corrplot(M, method = 'ellipse', order = 'AOE', type = 'upper')

    corrplot.mixed(M, order = 'AOE')

    corrplot.mixed(M, lower = 'shade', upper = 'pie', order = 'hclust')

## Reorder a correlation matrix

The details of four `order` algorithms, named `'AOE'`, `'FPC'`, `'hclust'`, `'alphabet'` are as following.

- `'AOE'` is for the angular order of the eigenvectors. It is calculated from the order of the angles \(a_i\),

  \[ a_i = \begin{cases} \arctan (e_{i2}/e_{i1}), & \text{if $e_{i1}>0$;} \newline \arctan (e_{i2}/e_{i1}) + \pi, & \text{otherwise.} \end{cases} \]

  where \(e_1\) and \(e_2\) are the largest two eigenvalues of the correlation matrix. See Michael Friendly (2002) for details.

- `'FPC'` for the first principal component order.

- `'hclust'` for hierarchical clustering order, and `'hclust.method'` for the agglomeration method to be used. `'hclust.method'` should be one of `'ward'`, `'ward.D'`, `'ward.D2'`, `'single'`, `'complete'`, `'average'`, `'mcquitty'`, `'median'` or `'centroid'`.

- `'alphabet'` for alphabetical order.

You can also reorder the matrix ‘manually’ via function `corrMatOrder()`.

If using `'hclust'`, `corrplot()` can draw rectangles around the plot of correlation matrix based on the results of hierarchical clustering.

    corrplot(M, order = 'hclust', addrect = 2)

    corrplot(M, method = 'square', diag = FALSE, order = 'hclust',
             addrect = 3, rect.col = 'blue', rect.lwd = 3, tl.pos = 'd')

R package seriation provides the infrastructure for ordering objects with an implementation of several seriation/sequencing/ordination techniques to reorder matrices, dissimilarity matrices, and dendrograms. For more information, see in section References.

We can reorder the matrix via seriation package and then corrplot it. Here are some examples.

    library(seriation)
    list_seriation_methods('matrix')

    ##  [1] "AOE"       "BEA"       "BEA_TSP"   "CA"        "Heatmap"   "Identity"
    ##  [7] "LLE"       "Mean"      "PCA"       "PCA_angle" "Random"    "Reverse"

    list_seriation_methods('dist')

    ##  [1] "ARSA"           "BBURCG"         "BBWRCG"         "Enumerate"
    ##  [5] "GSA"            "GW"             "GW_average"     "GW_complete"
    ##  [9] "GW_single"      "GW_ward"        "HC"             "HC_average"
    ## [13] "HC_complete"    "HC_single"      "HC_ward"        "Identity"
    ## [17] "MDS"            "MDS_angle"      "OLO"            "OLO_average"
    ## [21] "OLO_complete"   "OLO_single"     "OLO_ward"       "QAP_2SUM"
    ## [25] "QAP_BAR"        "QAP_Inertia"    "QAP_LS"         "R2E"
    ## [29] "Random"         "Reverse"        "SGD"            "SPIN_NH"
    ## [33] "SPIN_STS"       "Sammon_mapping" "Spectral"       "Spectral_norm"
    ## [37] "TSP"            "VAT"            "isoMDS"         "isomap"
    ## [41] "metaMDS"        "monoMDS"

    data(Zoo)
    Z = cor(Zoo[, -c(15, 17)])

    dist2order = function(corr, method, ...) {
      d_corr = as.dist(1 - corr)
      s = seriate(d_corr, method = method, ...)
      i = get_order(s)
      return(i)
    }

Methods `'PCA_angle'` and `'HC'` in seriation, are same as `'AOE'` and `'hclust'` separately in `corrplot()` and `corrMatOrder()`.

Here are some plots after seriation.

    # Fast Optimal Leaf Ordering for Hierarchical Clustering
    i = dist2order(Z, 'OLO')
    corrplot(Z[i, i], cl.pos = 'n')

    # Quadratic Assignment Problem
    i = dist2order(Z, 'QAP_2SUM')
    corrplot(Z[i, i], cl.pos = 'n')

    # Multidimensional Scaling
    i = dist2order(Z, 'MDS_nonmetric')

    ## Warning in get_seriation_method("dist", method): seriation method
    ## 'MDS_nonmetric' is now deprecated and will be removed in future releases. Using
    ## `isoMDS`

    corrplot(Z[i, i], cl.pos = 'n')

    # Simulated annealing
    i = dist2order(Z, 'ARSA')
    corrplot(Z[i, i], cl.pos = 'n')

    # TSP solver
    i = dist2order(Z, 'TSP')
    corrplot(Z[i, i], cl.pos = 'n')

    # Spectral seriation
    i = dist2order(Z, 'Spectral')
    corrplot(Z[i, i], cl.pos = 'n')

`corrRect()` can add rectangles on the plot with three ways(parameter `index`, `name` and `namesMat`) after `corrplot()`. We can use pipe operator `*>%` in package `magrittr` with more convenience. Since R 4.1.0, `|>` is supported without extra package.

    library(magrittr)

    # Rank-two ellipse seriation, use index parameter
    i = dist2order(Z, 'R2E')
    corrplot(Z[i, i], cl.pos = 'n') %>% corrRect(c(1, 9, 15))

    # use name parameter
    # Since R 4.1.0, the following one line code works:
    # corrplot(M, order = 'AOE') |> corrRect(name = c('gear', 'wt', 'carb'))
    corrplot(Z, order = 'AOE') %>%
      corrRect(name = c('tail', 'airborne', 'venomous', 'predator'))

    # use namesMat parameter
    r = rbind(c('eggs', 'catsize', 'airborne', 'milk'),
              c('catsize', 'eggs', 'milk', 'airborne'))
    corrplot(Z, order = 'hclust') %>% corrRect(namesMat = r)

## Change color spectra, color-legend and text-legend

We can get sequential and diverging colors from `COL1()` and `COL2()`. The color palettes are borrowed from `RColorBrewer` package.

Notice: the middle color getting from `COL2()` is fixed to `'#FFFFFF'`(white), thus we can visualizing element 0 with white color.

- `COL1()`: Get sequential colors, suitable for visualize a non-negative or non-positive matrix (e.g. matrix in [0, 20], or [-100, -10], or [100, 500]).
- `COL2()`: Get diverging colors, suitable for visualize a matrix which elements are partly positive and partly negative (e.g. correlation matrix in [-1, 1], or [-20, 100]).

The colors of the correlation plots can be customized by `col` in `corrplot()`. They are distributed uniformly in `col.lim` interval.

- `col`: vector, the colors of glyphs. They are distributed uniformly in `col.lim` interval. By default,
  - If `is.corr` is `TRUE`, `col` will be `COL2('RdBu', 200)`.
  - If `is.corr` is `FALSE`,
    - and `corr` is a non-negative or non-positive matrix, `col` will be `COL1('YlOrBr', 200)`;
    - otherwise (elements are partly positive and partly negative), `col` will be `COL2('RdBu', 200)`.
- `col.lim`: the limits (x1, x2) interval for assigning color by `col`. By default,
  - `col.lim` will be `c(-1, 1)` when `is.corr` is `TRUE`,
  - `col.lim` will be `c(min(corr), max(corr))` when `is.corr` is `FALSE`.
  - NOTICE: if you set `col.lim` when `is.corr` is `TRUE`, the assigning colors are still distributed uniformly in [-1, 1], it only affect the display on color-legend.
- `is.corr`: logical, whether the input matrix is a correlation matrix or not. The default value is `TRUE`. We can visualize a non-correlation matrix by setting `is.corr = FALSE`.

Here all diverging colors from `COL2()` and sequential colors from `COL1()` are shown below.

Diverging colors:

Sequential colors:

Usage of `COL1()` and `COL2()`:

    COL1(sequential = c("Oranges", "Purples", "Reds", "Blues", "Greens",
                        "Greys", "OrRd", "YlOrRd", "YlOrBr", "YlGn"), n = 200)

    COL2(diverging = c("RdBu", "BrBG", "PiYG", "PRGn", "PuOr", "RdYlBu"), n = 200)

In addition, function `colorRampPalette()` is very convenient for generating color spectrum.

Parameters group `cl.*` is for color-legend. The common-using are:

- `cl.pos` is for the position of color labels. It is character or logical. If character, it must be one of `'r'` (means right, default if `type='upper'` or `'full'`), `'b'` (means bottom, default if `type='lower'`) or `'n'`(means don’t draw color-label).
- `cl.ratio` is to justify the width of color-legend, 0.1~0.2 is suggested.

Parameters group `tl.*` is for text-legend. The common-using are:

- `tl.pos` is for the position of text labels. It is character or logical. If character, it must be one of `'lt'`, `'ld'`, `'td'`, `'d'`, `'l'` or `'n'`. `'lt'`(default if `type='full'`) means left and top, `'ld'`(default if `type='lower'`) means left and diagonal, `'td'`(default if `type='upper'`) means top and diagonal(near), `'d'` means diagonal, `'l'` means left, `'n'` means don’t add text-label.
- `tl.cex` is for the size of text label (variable names).
- `tl.srt` is for text label string rotation in degrees.

    corrplot(M, order = 'AOE', col = COL2('RdBu', 10))

    corrplot(M, order = 'AOE', addCoef.col = 'black', tl.pos = 'd',
             cl.pos = 'n', col = COL2('PiYG'))

    corrplot(M, method = 'square', order = 'AOE', addCoef.col = 'black', tl.pos = 'd',
             cl.pos = 'n', col = COL2('BrBG'))

    ## bottom color legend, diagonal text legend, rotate text label
    corrplot(M, order = 'AOE', cl.pos = 'b', tl.pos = 'd',
             col = COL2('PRGn'), diag = FALSE)

    ## text labels rotated 45 degrees and  wider color legend with numbers right aligned
    corrplot(M, type = 'lower', order = 'hclust', tl.col = 'black',
             cl.ratio = 0.2, tl.srt = 45, col = COL2('PuOr', 10))

    ## remove color legend, text legend and principal diagonal glyph
    corrplot(M, order = 'AOE', cl.pos = 'n', tl.pos = 'n',
             col = c('white', 'black'), bg = 'gold2')

## Visualize non-correlation matrix, NA value and math label

We can visualize a non-correlation matrix by set `is.corr=FALSE`, and assign colors by `col.lim`. If the matrix have both positive and negative values, the matrix transformation keep every values positiveness and negativeness.

If your matrix is rectangular, you can adjust the aspect ratio with the `win.asp` parameter to make the matrix rendered as a square.

    ## matrix in [20, 26], grid.col
    N1 = matrix(runif(80, 20, 26), 8)
    corrplot(N1, is.corr = FALSE, col.lim = c(20, 30), method = 'color', tl.pos = 'n',
             col = COL1('YlGn'), cl.pos = 'b', addgrid.col = 'white', addCoef.col = 'grey50')

    ## matrix in [-15, 10]
    N2 = matrix(runif(80, -15, 10), 8)

    ## using sequential colors, transKeepSign = FALSE
    corrplot(N2, is.corr = FALSE, transKeepSign = FALSE, method = 'color', col.lim = c(-15, 10),
             tl.pos = 'n', col = COL1('YlGn'), cl.pos = 'b', addCoef.col = 'grey50')

    ## using diverging colors, transKeepSign = TRUE (default)
    corrplot(N2, is.corr = FALSE, col.lim = c(-15, 10),
             tl.pos = 'n', col = COL2('PiYG'), cl.pos = 'b', addCoef.col = 'grey50')

    ## using diverging colors
    corrplot(N2, is.corr = FALSE, method = 'color', col.lim = c(-15, 10), tl.pos = 'n',
             col = COL2('PiYG'), cl.pos = 'b', addCoef.col = 'grey50')

Notice: when `is.corr` is `TRUE`, `col.lim` only affect the color legend If you change it, the color on correlation matrix plot is still assigned on `c(-1, 1)`

    # when is.corr=TRUE, col.lim only affect the color legend display
    corrplot(M/2)

    corrplot(M/2, col.lim=c(-0.5, 0.5))

By default, corrplot renders NA values as `'?'` characters. Using `na.label` parameter, it is possible to use a different value (max. two characters are supported).

Since version `0.78`, it is possible to use plotmath expression in variable names. To activate plotmath rendering, prefix your label with `'$'`.

    M2 = M
    diag(M2) = NA
    colnames(M2) = rep(c('$alpha+beta', '$alpha[0]', '$alpha[beta]'),
                       c(4, 4, 3))
    rownames(M2) = rep(c('$Sigma[i]^n', '$sigma',  '$alpha[0]^100', '$alpha[beta]'),
                       c(2, 4, 2, 3))
    corrplot(10*abs(M2), is.corr = FALSE, col.lim = c(0, 10), tl.cex = 1.5)

## Visualize p-value and confidence interval

`corrplot()` can also visualize p-value and confidence interval on the correlation matrix plot. Here are some important parameters.

About p-value:

- `p.mat` is the p-value matrix, if `NULL`, parameter `sig.level`, `insig, pch`, `pch.col`, `pch.cex` are invalid.
- `sig.level` is significant level, with default value 0.05. If the p-value in `p-mat` is bigger than `sig.level`, then the corresponding correlation coefficient is regarded as insignificant. If `insig` is `'label_sig'`, `sig.level` can be an increasing vector of significance levels, in which case `pch` will be used once for the highest p-value interval and multiple times (e.g. `'*'`, `'**'`, `'***'`) for each lower p-value interval.
- `insig` Character, specialized insignificant correlation coefficients, `'pch'` (default), `'p-value'`, `'blank',` `'n'`, or `'label_sig'`. If `'blank'`, wipe away the corresponding glyphs; if `'p-value'`, add p-values the corresponding glyphs; if `'pch'`, add characters (see pch for details) on corresponding glyphs; if `'n'`, don’t take any measures; if `'label_sig'`, mark significant correlations with `pch` (see `sig.level`).
- `pch` is for adding character on the glyphs of insignificant correlation coefficients (only valid when insig is `'pch'`). See `?par` .

About confidence interval:

- `plotCI` is character for the method of plotting confidence interval. If `'n'`, don’t plot confidence interval. If `'rect'`, plot rectangles whose upper side means upper bound and lower side means lower bound respectively.
- `lowCI.mat` is the matrix of the lower bound of confidence interval.
- `uppCI.mat` is the Matrix of the upper bound of confidence interval.

We can get p-value matrix and confidence intervals matrix by `cor.mtest()` which returns a list containing:

- `p` is the p-values matrix.
- `lowCI` is the lower bound of confidence interval matrix.
- `uppCI` is the lower bound of confidence interval matrix.

    testRes = cor.mtest(mtcars, conf.level = 0.95)

    ## specialized the insignificant value according to the significant level
    corrplot(M, p.mat = testRes$p, sig.level = 0.10, order = 'hclust', addrect = 2)

    ## leave blank on non-significant coefficient
    ## add significant correlation coefficients
    corrplot(M, p.mat = testRes$p, method = 'circle', type = 'lower', insig='blank',
             addCoef.col ='black', number.cex = 0.8, order = 'AOE', diag=FALSE)

    ## leave blank on non-significant coefficient
    ## add all correlation coefficients
    corrplot(M, p.mat = testRes$p, method = 'circle', type = 'lower', insig='blank',
             order = 'AOE', diag = FALSE)$corrPos -> p1
    text(p1$x, p1$y, round(p1$corr, 2))

    ## add p-values on no significant coefficients
    corrplot(M, p.mat = testRes$p, insig = 'p-value')

    ## add all p-values
    corrplot(M, p.mat = testRes$p, insig = 'p-value', sig.level = -1)

    ## add significant level stars
    corrplot(M, p.mat = testRes$p, method = 'color', diag = FALSE, type = 'upper',
             sig.level = c(0.001, 0.01, 0.05), pch.cex = 0.9,
             insig = 'label_sig', pch.col = 'grey20', order = 'AOE')

    ## add significant level stars and cluster rectangles
    corrplot(M, p.mat = testRes$p, tl.pos = 'd', order = 'hclust', addrect = 2,
             insig = 'label_sig', sig.level = c(0.001, 0.01, 0.05),
             pch.cex = 0.9, pch.col = 'grey20')

Visualize confidence interval.

    # Visualize confidence interval
    corrplot(M, lowCI = testRes$lowCI, uppCI = testRes$uppCI, order = 'hclust',
             tl.pos = 'd', rect.col = 'navy', plotC = 'rect', cl.pos = 'n')

    # Visualize confidence interval and cross the significant coefficients
    corrplot(M, p.mat = testRes$p, lowCI = testRes$lowCI, uppCI = testRes$uppCI,
             addrect = 3, rect.col = 'navy', plotC = 'rect', cl.pos = 'n')

## References

- Michael Friendly (2002). Corrgrams: Exploratory displays for correlation matrices. The American Statistician, 56, 316–324.

- D.J. Murdoch, E.D. Chow (1996). A graphical display of large correlation matrices. The American Statistician, 50, 178–180.

- Michael Hahsler, Christian Buchta and Kurt Hornik (2020). seriation: Infrastructure for Ordering Objects Using Seriation. R package version 1.2-9. https://CRAN.R-project.org/package=seriation

- Hahsler M, Hornik K, Buchta C (2008). “Getting things in order: An introduction to the R package seriation.” Journal of Statistical Software, 25(3), 1-34. ISSN 1548-7660, doi: 10.18637/jss.v025.i03 (URL: https://doi.org/10.18637/jss.v025.i03), <URL: https://www.jstatsoft.org/v25/i03/>.