From 5f5ea4afc50144d906bc0a48a49b208fba8bb659 Mon Sep 17 00:00:00 2001 From: Michael Friendly Date: Thu, 30 Nov 2023 13:43:11 -0500 Subject: [PATCH] start on biplots --- 04-pca-biplot.qmd | 52 +++++++-- R/crime-ggbiplot.R | 119 +------------------ R/crime-pca.R | 156 +++++++++++++++++++++++++ bib/pkgs.bib | 38 ++++-- docs/04-pca-biplot.html | 81 +++++++++---- docs/90-references.html | 5 + docs/collinearity-ridge.html | 20 ++-- docs/eqcov.html | 2 +- docs/figs/fig-NC-candisc-1.png | Bin 275880 -> 273463 bytes docs/figs/fig-SC1-hecan-1.png | Bin 201259 -> 202055 bytes docs/figs/fig-cars-collin-biplot-1.png | Bin 176108 -> 176556 bytes docs/figs/fig-crime-vectors-1.png | Bin 93090 -> 92892 bytes docs/figs/fig-duncan-plot-model-1.png | Bin 232072 -> 231202 bytes docs/figs/fig-plot-prestige-mod-1.png | Bin 306472 -> 305757 bytes docs/hotelling.html | 2 +- docs/index.html | 4 +- docs/linear_models-plots.html | 5 +- docs/mlm-viz.html | 2 +- docs/search.json | 24 ++-- figs/fig-NC-candisc-1.png | Bin 275880 -> 273463 bytes figs/fig-SC1-hecan-1.png | Bin 201259 -> 202055 bytes figs/fig-cars-collin-biplot-1.png | Bin 176108 -> 176556 bytes figs/fig-crime-vectors-1.png | Bin 93090 -> 92892 bytes figs/fig-duncan-plot-model-1.png | Bin 232072 -> 231202 bytes figs/fig-plot-prestige-mod-1.png | Bin 306472 -> 305757 bytes 25 files changed, 323 insertions(+), 187 deletions(-) create mode 100644 R/crime-pca.R diff --git a/04-pca-biplot.qmd b/04-pca-biplot.qmd index 51324cf3..e03c411b 100644 --- a/04-pca-biplot.qmd +++ b/04-pca-biplot.qmd @@ -69,11 +69,14 @@ knitr::include_graphics("images/MV-juicer.png") Here, I concentrate on **principal components analysis** (PCA), whose goal reflects A Square's desire to see that sparkly cloud of points in $nD$ space in the plane showing the greatest variation (squeezing the most juice) among all other possible views. - This appealed to his sense of geometry, but left him wondering how the variables in that high-D cloud were related to the dimensions he could see in a best-fitting plane. + The idea of a **biplot**, showing the data points in the plane, together with thick pointed -arrows---variable vectors--- in one view is the other topic explained in this chapter (@sec-biplot). +arrows---variable vectors--- in one view is the other topic explained in this chapter (@sec-biplot). The biplot is the simplest example of a multivariate juicer. The essential idea is to project the cloud of data points in $n$ dimensions into the +2D space of principal components and simultaneously show how the original variables +relate to this space. For exploratory analysis to get an initial, incisive view of +a multivariate dataset, a biplot is often my first choice. **Packages** @@ -264,11 +267,13 @@ The components returned by various PCA methods have (confusingly) different name Then, a simple visualization is a plot of the proportion of variance for each component (or cumulative proportion) against the component number, usually called a **screeplot**. The idea, introduced by @Cattell1966, is that after the largest, dominant components, the remainder should resemble the rubble, or scree formed by rocks falling -from a cliff. From this plot, imagine +from a cliff. + +From this plot, imagine drawing a straight line through the plotted eigenvalues, starting with the largest one. The typical rough guidance is that the last point to fall on this line represents the last component to extract, the idea being that beyond this, the amount of additional variance explained is non-meaningful. Another rule of thumb is to choose the number of components to extract a desired proportion of total variance, usually in the range of 80 - 90%. -`stats::plot(crime.pca)` would give a bar plot of the variances of the components, however ggbiplot::ggscreeplot()` gives nicer and more flexible displays as shown in @fig-crime-ggscreeplot. +`stats::plot(crime.pca)` would give a bar plot of the variances of the components, however `ggbiplot::ggscreeplot()` gives nicer and more flexible displays as shown in @fig-crime-ggscreeplot. ```{r} #| label: fig-crime-ggscreeplot @@ -331,7 +336,7 @@ ellipse. ```{r} #| label: fig-crime-scores-plot12 -#| out-width: "80%" +#| out-width: "100%" #| fig-cap: "Plot of component scores on the first two principal components for the `crime` data. States are colored by `region`." crime.pca |> broom::augment(crime) |> # add original dataset back in @@ -413,19 +418,50 @@ vectors |> What is shown in @fig-crime-vectors has the following interpretations: -* the lengths of the variable vectors, $||\mathbf{v}_i|| = \sqrt{\Sigma_{j = 1:2} \; v_{ij}^2}$ give the proportion of variance of each variable accounted for in a two-dimensional display. +(1) the lengths of the variable vectors, $||\mathbf{v}_i|| = \sqrt{\Sigma_{j = 1:2} \; v_{ij}^2}$ give the proportion of variance of each variable accounted for in a two-dimensional display. -* the value, $v_{ij}$, of the vector for variable $\mathbf{x}_i$ on component $j$ gives +(2) the value, $v_{ij}$, of the vector for variable $\mathbf{x}_i$ on component $j$ gives the correlation of that variable with the $j$th principal component. -* the cosine of the angle between two variable vectors, $\mathbf{v}_i$ and $\mathbf{v}_j$ +(3) the cosine of the angle between two variable vectors, $\mathbf{v}_i$ and $\mathbf{v}_j$ gives the approximation of the correlation between $\mathbf{x}_i$ and $\mathbf{x}_j$ that is shown in this space. This means that two variable vectors that point in the same direction are highly correlated, while variable vectors at right angles are approximately uncorrelated. +To illustrate point (1), the following indicates that almost 70% of the variance of +`murder` is represented in the the 2D plot shown in @fig-crime-scores-plot12, but only +40% of the variance of `robbery` is captured. + +```{r} +vectors |> select(label, PC1, PC2) |> + mutate(length = sqrt(PC1^2 + PC2^2)) +``` + + ## Biplots {#sec-biplot} +The biplot is a simple and powerful idea that came from the recognition that +you can overlay a plot of observation scores in a principal components analysis +with the information of the variable loadings (weights) to give a simultaneous display +that is easy to interpret. In this sense, a biplot is generalization of a scatterplot, +going from data space to PCA space, where the observations are shown by points and variables are shown by vectors (or scaled linear axes aligned with those vectors). + +Biplot methodolgy is far more general than I cover here. ... +**TODO**: mention categorical variables (category level points), correspondence analysis, CCA, MDS, ... + +The idea of the biplot was introduced by Ruben Gabriel [-@Gabriel:71] and later expanded in scope by @GowerHand:96. +**TODO**: Cite Greenacre, Biplots in Practice, Gower et al. 2010. + +### Constructing a biplot + +SVD: + +### Example + + + + ## Elliptical insights: Outlier detection The data ellipse (@sec-data-ellipse), or ellipsoid in more than 2D is fundamental in regression. But also, diff --git a/R/crime-ggbiplot.R b/R/crime-ggbiplot.R index 2e82ae1a..b425ca02 100644 --- a/R/crime-ggbiplot.R +++ b/R/crime-ggbiplot.R @@ -9,7 +9,7 @@ library(corrplot) library(patchwork) library(broom) -data(crime) +data(crime, package = "ggbiplot") crime |> dplyr::select(where(is.numeric)) |> @@ -23,123 +23,6 @@ crime.pca <- summary(crime.pca) -# show the eigenvalue decomposition -(crime.eig <- crime.pca |> - broom::tidy(matrix = "eigenvalues")) - -# add fitted line for smallest eigenvalues -p1 <- ggscreeplot(crime.pca) + - stat_smooth(data = crime.eig |> filter(PC>=4), - aes(x=PC, y=percent), method = "lm", - se = FALSE, - fullrange = TRUE) + - theme_bw(base_line_size = 14) - -p2 <- ggscreeplot(crime.pca, type = "cev") + - geom_hline(yintercept = c(0.8, 0.9), color = "blue") + - theme_bw(base_size = 14) - -p1 + p2 - -# tidy scores plot - -scores <- crime.pca |> purrr::pluck("x") -cov(scores) |> zapsmall() - -crime.pca |> - broom::augment(crime) |> # add original dataset back in - ggplot(aes(.fittedPC1, .fittedPC2, color = region)) + - geom_hline(yintercept = 0) + - geom_vline(xintercept = 0) + - geom_point(size = 1.5) + - geom_text(aes(label = st), nudge_x = 0.2) + - stat_ellipse(color = "grey") + - coord_fixed() - labs(x = "PC Dimension 1", y = "PC Dimnension 2") + - theme_minimal(base_size = 14) + - theme(legend.position = "top") + - -crime.pca |> - broom::augment(crime) |> - ggplot(aes(.fittedPC1, .fittedPC3, color = region, label = st)) + - geom_point(size = 1.5) + - geom_text(nudge_x = 0.2) + - geom_hline(yintercept = 0) + - geom_vline(xintercept = 0) + - labs(x = "PC Dimension 1", y = "PC Dimnension 3") + - theme_minimal(base_size = 14) + - theme(legend.position = "top") + - coord_fixed() - - - -# tidy variable vectors - -crime.pca |> purrr::pluck("rotation") - -crime.pca |> - tidy(matrix = "rotation") - - - -# define arrow style for plotting -arrow_style <- arrow( - angle = 20, ends = "first", type = "closed", length = grid::unit(8, "pt") -) - -# try to plot the unit circle -r <- 1 -theta <- c(seq(-pi, pi, length = 100)) -cir <- data.frame(PC1 = r * cos(theta), PC2 = r * sin(theta)) -circle <- geom_path(data = circle, color = "grey") - -# plot rotation matrix -crime.pca |> - tidy(matrix = "rotation") |> - tidyr::pivot_wider(names_from = "PC", - names_prefix = "PC", - values_from = "value") |> - ggplot(aes(PC1, PC2)) + - geom_segment(xend = 0, yend = 0, arrow = arrow_style) + - geom_text( - aes(label = column), - hjust = 1, nudge_x = -0.02, - color = "brown") + - xlim(-0.8, .2) + ylim(-.7, 0.6) + - coord_fixed() + # fix aspect ratio to 1:1 - theme_minimal(base_size = 14) - - -# do this without pivot_wider - -vectors <- crime.pca |> - purrr::pluck("rotation") |> - as.data.frame() |> - mutate(PC1 = -1 * PC1, PC2 = -1 * PC2) |> # reflect axes - tibble::rownames_to_column(var = "label") -vectors - -vectors |> - ggplot(aes(PC1, PC2)) + - geom_hline(yintercept = 0) + - geom_vline(xintercept = 0) + - geom_segment(xend = 0, yend = 0, linewidth=1, arrow = arrow_style) + - geom_text(aes(label = label), - size = 5, - hjust = "outward", - nudge_x = 0.05, - color = "brown") + - xlim(-0.4, 0.9) + ylim(-0.8, 0.8) + - coord_fixed() + # fix aspect ratio to 1:1 - theme_minimal(base_size = 14) - -# interpreting variable vectors -vectors[, 2:3]^2 |> rowSums() |> sqrt() -vectors |> select(label, PC1, PC2) |> mutate(length = sqrt(PC1^2 + PC2^2)) - -# correlations -crime |> select(murder, auto) |> cor() - #biplot(crime.pca) diff --git a/R/crime-pca.R b/R/crime-pca.R new file mode 100644 index 00000000..a63ae02e --- /dev/null +++ b/R/crime-pca.R @@ -0,0 +1,156 @@ +#' --- +#' title: crime data - pca +#' --- + +library(ggplot2) +library(ggbiplot) +library(dplyr) +library(corrplot) +library(patchwork) +library(broom) + +data(crime, package = "ggbiplot") + +crime |> + dplyr::select(where(is.numeric)) |> + cor() |> + corrplot(method = "ellipse", tl.srt = 0, tl.col = "black", mar = rep(.5, 4)) + +crime.pca <- + crime |> + dplyr::select(where(is.numeric)) |> + prcomp(scale. = TRUE) + +summary(crime.pca) + +# show the eigenvalue decomposition +(crime.eig <- crime.pca |> + broom::tidy(matrix = "eigenvalues")) + +# add fitted line for smallest eigenvalues +p1 <- ggscreeplot(crime.pca) + + stat_smooth(data = crime.eig |> filter(PC>=4), + aes(x=PC, y=percent), method = "lm", + se = FALSE, + fullrange = TRUE) + + theme_bw(base_line_size = 14) + +p2 <- ggscreeplot(crime.pca, type = "cev") + + geom_hline(yintercept = c(0.8, 0.9), color = "blue") + + theme_bw(base_size = 14) + +p1 + p2 + +# tidy scores plot + +scores <- crime.pca |> purrr::pluck("x") +cov(scores) |> zapsmall() + +crime.pca |> + broom::augment(crime) |> # add original dataset back in + ggplot(aes(.fittedPC1, .fittedPC2, color = region)) + + geom_hline(yintercept = 0) + + geom_vline(xintercept = 0) + + geom_point(size = 1.5) + + geom_text(aes(label = st), nudge_x = 0.2) + + stat_ellipse(color = "grey") + + coord_fixed() +labs(x = "PC Dimension 1", y = "PC Dimnension 2") + + theme_minimal(base_size = 14) + + theme(legend.position = "top") + + + crime.pca |> + broom::augment(crime) |> + ggplot(aes(.fittedPC1, .fittedPC3, color = region, label = st)) + + geom_point(size = 1.5) + + geom_text(nudge_x = 0.2) + + geom_hline(yintercept = 0) + + geom_vline(xintercept = 0) + + labs(x = "PC Dimension 1", y = "PC Dimnension 3") + + theme_minimal(base_size = 14) + + theme(legend.position = "top") + + coord_fixed() + + + +# tidy variable vectors + +crime.pca |> purrr::pluck("rotation") + +crime.pca |> + tidy(matrix = "rotation") + + + +# define arrow style for plotting +arrow_style <- arrow( + angle = 20, ends = "first", type = "closed", length = grid::unit(8, "pt") +) + +# try to plot the unit circle +r <- 1 +theta <- c(seq(-pi, pi, length = 100)) +cir <- data.frame(PC1 = r * cos(theta), PC2 = r * sin(theta)) +circle <- geom_path(data = circle, color = "grey") + +# plot rotation matrix +crime.pca |> + tidy(matrix = "rotation") |> + tidyr::pivot_wider(names_from = "PC", + names_prefix = "PC", + values_from = "value") |> + ggplot(aes(PC1, PC2)) + + geom_segment(xend = 0, yend = 0, arrow = arrow_style) + + geom_text( + aes(label = column), + hjust = 1, nudge_x = -0.02, + color = "brown") + + xlim(-0.8, .2) + ylim(-.7, 0.6) + + coord_fixed() + # fix aspect ratio to 1:1 + theme_minimal(base_size = 14) + + +# do this without pivot_wider + +vectors <- crime.pca |> + purrr::pluck("rotation") |> + as.data.frame() |> + mutate(PC1 = -1 * PC1, PC2 = -1 * PC2) |> # reflect axes + tibble::rownames_to_column(var = "label") +vectors + +vectors |> + ggplot(aes(PC1, PC2)) + + geom_hline(yintercept = 0) + + geom_vline(xintercept = 0) + + geom_segment(xend = 0, yend = 0, linewidth=1, arrow = arrow_style) + + geom_text(aes(label = label), + size = 5, + hjust = "outward", + nudge_x = 0.05, + color = "brown") + + xlim(-0.4, 0.9) + ylim(-0.8, 0.8) + + coord_fixed() + # fix aspect ratio to 1:1 + theme_minimal(base_size = 14) + +# interpreting variable vectors +vectors[, 2:3]^2 |> rowSums() |> sqrt() +vectors |> select(label, PC1, PC2) |> mutate(length = sqrt(PC1^2 + PC2^2)) + +matlib::len(vectors) +matlib::len(t(vectors)) + +# correlations of +crime.pca |> + broom::augment(crime) + +# correlations +crime |> select(murder, burglary, larceny, auto) |> cor() +crime |> select(larceny, auto) |> cor() + +# angles between vectors +vec <- vectors <- crime.pca |> + purrr::pluck("rotation") +vec <- vec[, 1:2] +vec %*% t(vec) +matlib::angle(vec[1,], vec[7,]) |> cos() diff --git a/bib/pkgs.bib b/bib/pkgs.bib index 06992e17..7ae83e2a 100644 --- a/bib/pkgs.bib +++ b/bib/pkgs.bib @@ -38,7 +38,7 @@ @Manual{R-car title = {car: Companion to Applied Regression}, author = {John Fox and Sanford Weisberg and Brad Price}, year = {2023}, - note = {R package version 3.1-4}, + note = {R package version 3.1-3}, url = {https://r-forge.r-project.org/projects/car/}, } @@ -82,14 +82,6 @@ @Manual{R-corrplot url = {https://github.com/taiyun/corrplot}, } -@Manual{R-datasauRus, - title = {datasauRus: Datasets from the Datasaurus Dozen}, - author = {Rhian Davies and Steph Locke and Lucy {D'Agostino McGowan}}, - year = {2022}, - note = {R package version 0.1.6}, - url = {https://github.com/jumpingrivers/datasauRus}, -} - @Manual{R-datawizard, title = {datawizard: Easy Data Wrangling and Statistical Transformations}, author = {Indrajeet Patil and Etienne Bacher and Dominique Makowski and Daniel Lüdecke and Mattan S. Ben-Shachar and Brenton M. Wiernik}, @@ -139,6 +131,14 @@ @Manual{R-factoextra url = {http://www.sthda.com/english/rpkgs/factoextra}, } +@Manual{R-FactoMineR, + title = {FactoMineR: Multivariate Exploratory Data Analysis and Data Mining}, + author = {Francois Husson and Julie Josse and Sebastien Le and Jeremy Mazet}, + year = {2023}, + note = {R package version 2.9}, + url = {http://factominer.free.fr}, +} + @Manual{R-forcats, title = {forcats: Tools for Working with Categorical Variables (Factors)}, author = {Hadley Wickham}, @@ -147,6 +147,13 @@ @Manual{R-forcats url = {https://forcats.tidyverse.org/}, } +@Manual{R-genridge, + title = {genridge: Generalized Ridge Trace Plots for Ridge Regression}, + author = {Michael Friendly}, + year = {2023}, + note = {R package version 0.7.0}, + url = {https://friendly.github.io/genridge/}, +} @Manual{R-geomtextpath, title = {geomtextpath: Curved Text in ggplot2}, @@ -201,7 +208,7 @@ @Manual{R-heplots title = {heplots: Visualizing Hypothesis Tests in Multivariate Linear Models}, author = {Michael Friendly and John Fox and Georges Monette}, year = {2023}, - note = {R package version 1.6.1}, + note = {R package version 1.6.0}, url = {http://friendly.github.io/heplots/}, } @@ -478,6 +485,17 @@ @Article{effectsize2020 url = {https://doi.org/10.21105/joss.02815}, } +@Article{FactoMineR2008, + title = {{FactoMineR}: A Package for Multivariate Analysis}, + author = {S\'ebastien L\^e and Julie Josse and Fran\c{c}ois Husson}, + journal = {Journal of Statistical Software}, + year = {2008}, + volume = {25}, + number = {1}, + pages = {1--18}, + doi = {10.18637/jss.v025.i01}, +} + @Book{ggplot22016, author = {Hadley Wickham}, title = {ggplot2: Elegant Graphics for Data Analysis}, diff --git a/docs/04-pca-biplot.html b/docs/04-pca-biplot.html index c8d14a84..4bacca20 100644 --- a/docs/04-pca-biplot.html +++ b/docs/04-pca-biplot.html @@ -257,7 +257,13 @@
  • 4.2.5 Visualizing PCA scores and variable vectors
  • -
  • 4.3 Biplots
  • +
  • +4.3 Biplots + +
  • 4.4 Elliptical insights: Outlier detection
  • @@ -304,8 +310,8 @@

    -

    Here, I concentrate on principal components analysis (PCA), whose goal reflects A Square’s desire to see that sparkly cloud of points in \(nD\) space in the plane showing the greatest variation (squeezing the most juice) among all other possible views.

    -

    This appealed to his sense of geometry, but left him wondering how the variables in that high-D cloud were related to the dimensions he could see in a best-fitting plane. The idea of a biplot, showing the data points in the plane, together with thick pointed arrows—variable vectors— in one view is the other topic explained in this chapter (Section 4.3).

    +

    Here, I concentrate on principal components analysis (PCA), whose goal reflects A Square’s desire to see that sparkly cloud of points in \(nD\) space in the plane showing the greatest variation (squeezing the most juice) among all other possible views. This appealed to his sense of geometry, but left him wondering how the variables in that high-D cloud were related to the dimensions he could see in a best-fitting plane.

    +

    The idea of a biplot, showing the data points in the plane, together with thick pointed arrows—variable vectors— in one view is the other topic explained in this chapter (Section 4.3). The biplot is the simplest example of a multivariate juicer. The essential idea is to project the cloud of data points in \(n\) dimensions into the 2D space of principal components and simultaneously show how the original variables relate to this space. For exploratory analysis to get an initial, incisive view of a multivariate dataset, a biplot is often my first choice.

    Packages

    In this chapter I use the following packages. Load them now:

    @@ -368,7 +374,7 @@

    4.2.2 Mathematics and geometry of PCA

    4.2.3 Finding principal components

    -

    In R, principal components analysis is most easily carried out using stats::prcomp() or stats::princomp() or similar functions in other packages such as FactomineR::PCA(). The FactoMineR package (R-FactoMineR?) has extensive capabilities for exploratory analysis of multivariate data (PCA, correspondence analysis, cluster analysis, …).

    +

    In R, principal components analysis is most easily carried out using stats::prcomp() or stats::princomp() or similar functions in other packages such as FactomineR::PCA(). The FactoMineR package (Husson et al. 2023) has extensive capabilities for exploratory analysis of multivariate data (PCA, correspondence analysis, cluster analysis, …).

    Unfortunately, although all of these performing similar calculations, the options for analysis and the details of the result they return differ …

    The important options for analysis include:

      @@ -428,8 +434,9 @@

      #> 6 6 0.471 0.0317 0.982 #> 7 7 0.352 0.0177 1

    -

    Then, a simple visualization is a plot of the proportion of variance for each component (or cumulative proportion) against the component number, usually called a screeplot. The idea, introduced by Cattell (1966), is that after the largest, dominant components, the remainder should resemble the rubble, or scree formed by rocks falling from a cliff. From this plot, imagine drawing a straight line through the plotted eigenvalues, starting with the largest one. The typical rough guidance is that the last point to fall on this line represents the last component to extract, the idea being that beyond this, the amount of additional variance explained is non-meaningful. Another rule of thumb is to choose the number of components to extract a desired proportion of total variance, usually in the range of 80 - 90%.

    -

    stats::plot(crime.pca) would give a bar plot of the variances of the components, however ggbiplot::ggscreeplot()` gives nicer and more flexible displays as shown in Figure 4.5.

    +

    Then, a simple visualization is a plot of the proportion of variance for each component (or cumulative proportion) against the component number, usually called a screeplot. The idea, introduced by Cattell (1966), is that after the largest, dominant components, the remainder should resemble the rubble, or scree formed by rocks falling from a cliff.

    +

    From this plot, imagine drawing a straight line through the plotted eigenvalues, starting with the largest one. The typical rough guidance is that the last point to fall on this line represents the last component to extract, the idea being that beyond this, the amount of additional variance explained is non-meaningful. Another rule of thumb is to choose the number of components to extract a desired proportion of total variance, usually in the range of 80 - 90%.

    +

    stats::plot(crime.pca) would give a bar plot of the variances of the components, however ggbiplot::ggscreeplot() gives nicer and more flexible displays as shown in Figure 4.5.

    p1 <- ggscreeplot(crime.pca) +
       stat_smooth(data = crime.eig |> filter(PC>=4), 
    @@ -503,7 +510,7 @@ 

    theme(legend.position = "top")

    -

    +

    Figure 4.6: Plot of component scores on the first two principal components for the crime data. States are colored by region.
    @@ -514,13 +521,13 @@

    crime.pca |> purrr::pluck("rotation")
     #>             PC1     PC2     PC3     PC4     PC5     PC6     PC7
    -#> murder   -0.300 -0.6292  0.1782 -0.2321  0.5381  0.2591  0.2676
    -#> rape     -0.432 -0.1694 -0.2442  0.0622  0.1885 -0.7733 -0.2965
    -#> robbery  -0.397  0.0422  0.4959 -0.5580 -0.5200 -0.1144 -0.0039
    -#> assault  -0.397 -0.3435 -0.0695  0.6298 -0.5067  0.1724  0.1917
    -#> burglary -0.440  0.2033 -0.2099 -0.0576  0.1010  0.5360 -0.6481
    -#> larceny  -0.357  0.4023 -0.5392 -0.2349  0.0301  0.0394  0.6017
    -#> auto     -0.295  0.5024  0.5684  0.4192  0.3698 -0.0573  0.1470
    +#> murder -0.300 -0.6292 -0.1782 0.2321 0.5381 0.2591 0.2676 +#> rape -0.432 -0.1694 0.2442 -0.0622 0.1885 -0.7733 -0.2965 +#> robbery -0.397 0.0422 -0.4959 0.5580 -0.5200 -0.1144 -0.0039 +#> assault -0.397 -0.3435 0.0695 -0.6298 -0.5067 0.1724 0.1917 +#> burglary -0.440 0.2033 0.2099 0.0576 0.1010 0.5360 -0.6481 +#> larceny -0.357 0.4023 0.5392 0.2349 0.0301 0.0394 0.6017 +#> auto -0.295 0.5024 -0.5684 -0.4192 0.3698 -0.0573 0.1470

    But note something important in this output: All of the weights for the first component are negative. In PCA, the directions of the eigenvectors are completely arbitrary, in the sense that the vector \(-\mathbf{v}_i\) gives the same linear combination as \(\mathbf{v}_i\), but with its’ sign reversed. For interpretation, it is useful (and usually recommended) to reflect the loadings to a positive orientation by multiplying them by -1.

    To reflect the PCA loadings and get them into a convenient format for plotting with ggplot(), it is necessary to do a bit of processing, including making the row.names() into an explicit variable for the purpose of labeling.

    @@ -554,7 +561,8 @@

    geom_vline(xintercept = 0) + geom_segment(xend = 0, yend = 0, linewidth = 1, - arrow = arrow_style) + + arrow = arrow_style, + color = "brown") + geom_text(aes(label = label), size = 5, hjust = "outward", @@ -567,23 +575,45 @@

    -
    Figure 4.7: Plot of component loadings the first two principal components for the crime data.
    +
    Figure 4.7: Plot of component loadings the first two principal components for the crime data. These are interpreted as the contributions of the variables to the components.

    What is shown in Figure 4.7 has the following interpretations:

    -
      +
      1. the lengths of the variable vectors, \(||\mathbf{v}_i|| = \sqrt{\Sigma_{j = 1:2} \; v_{ij}^2}\) give the proportion of variance of each variable accounted for in a two-dimensional display.

      2. the value, \(v_{ij}\), of the vector for variable \(\mathbf{x}_i\) on component \(j\) gives the correlation of that variable with the \(j\)th principal component.

      3. -
      4. the angle between two variable vectors, \(\mathbf{v}_i\) and \(\mathbf{v}_j\) gives the approximation of the correlation between \(\mathbf{x}_i\) and \(\mathbf{x}_j\) that is shown in this space.

      5. -

    +
  • the cosine of the angle between two variable vectors, \(\mathbf{v}_i\) and \(\mathbf{v}_j\) gives the approximation of the correlation between \(\mathbf{x}_i\) and \(\mathbf{x}_j\) that is shown in this space. This means that two variable vectors that point in the same direction are highly correlated, while variable vectors at right angles are approximately uncorrelated.

  • + +

    To illustrate point (1), the following indicates that almost 70% of the variance of murder is represented in the the 2D plot shown in Figure 4.6, but only 40% of the variance of robbery is captured.

    +
    +
    vectors |> select(label, PC1, PC2) |> 
    +  mutate(length = sqrt(PC1^2 + PC2^2))
    +#>      label   PC1     PC2 length
    +#> 1   murder 0.300  0.6292  0.697
    +#> 2     rape 0.432  0.1694  0.464
    +#> 3  robbery 0.397 -0.0422  0.399
    +#> 4  assault 0.397  0.3435  0.525
    +#> 5 burglary 0.440 -0.2033  0.485
    +#> 6  larceny 0.357 -0.4023  0.538
    +#> 7     auto 0.295 -0.5024  0.583
    +
    +

    4.3 Biplots

    -

    +

    The biplot is a simple and powerful idea that came from the recognition that you can overlay a plot of observation scores in a principal components analysis with the information of the variable loadings (weights) to give a simultaneous display that is easy to interpret. In this sense, a biplot is generalization of a scatterplot, going from data space to PCA space, where the observations are shown by points and variables are shown by vectors (or scaled linear axes aligned with those vectors).

    +

    Biplot methodolgy is far more general than I cover here. … TODO: mention categorical variables (category level points), correspondence analysis, CCA, MDS, …

    +

    The idea of the biplot was introduced by Ruben Gabriel (1971) and later expanded in scope by Gower and Hand (1996). TODO: Cite Greenacre, Biplots in Practice, Gower et al. 2010.

    +

    +4.3.1 Constructing a biplot

    +

    SVD:

    +

    +4.3.2 Example

    +

    4.4 Elliptical insights: Outlier detection

    The data ellipse (Section 3.1.4), or ellipsoid in more than 2D is fundamental in regression. But also, as Pearson showed, it is key to understanding principal components analysis, where the principal component directions are simply the axes of the ellipsoid of the data. As such, observations that are unusual in data space may not stand out in univariate views of the variables, but will stand out in principal component space, usually on the smallest dimension.

    As an illustration, I created a dataset of \(n = 100\) observations with a linear relation, \(y = x + \mathcal{N}(0, 1)\) and then added two discrepant points at (1.5, -1.5), (-1.5, 1.5).

    -
    set.seed(123345)
    +
    set.seed(123345)
     x <- c(rnorm(100),             1.5, -1.5)
     y <- c(x[1:100] + rnorm(100), -1.5, 1.5)
    @@ -618,9 +648,18 @@

    Friendly, Michael, and Howard Wainer. 2021. A History of Data Visualization and Graphic Communication. Cambridge, MA: Harvard University Press. https://doi.org/10.4159/9780674259034.

    +
    +Gabriel, K. R. 1971. “The Biplot Graphic Display of Matrices with Application to Principal Components Analysis.” Biometrics 58 (3): 453–67. +
    Galton, Francis. 1886. “Regression Towards Mediocrity in Hereditary Stature.” Journal of the Anthropological Institute 15: 246–63. http://www.jstor.org/cgi-bin/jstor/viewitem/09595295/dm995266/99p0374f/0.
    +
    +Gower, J. C., and D. J. Hand. 1996. Biplots. London: Chapman & Hall. +
    +
    +Husson, Francois, Julie Josse, Sebastien Le, and Jeremy Mazet. 2023. FactoMineR: Multivariate Exploratory Data Analysis and Data Mining. http://factominer.free.fr. +
    Pearson, Karl. 1896. “Contributions to the Mathematical Theory of Evolution—III, Regression, Heredity and Panmixia.” Philosophical Transactions of the Royal Society of London, A, 187: 253–318.
    diff --git a/docs/90-references.html b/docs/90-references.html index e85f0ad0..31413f66 100644 --- a/docs/90-references.html +++ b/docs/90-references.html @@ -575,6 +575,11 @@

    References

    Hotelling, Harold. 1931. “The Generalization of Student’s Ratio.” The Annals of Mathematical Statistics 2 (3): 360–78. https://doi.org/10.1214/aoms/1177732979. +
    +Husson, Francois, Julie Josse, Sebastien Le, and Jeremy Mazet. 2023. +FactoMineR: Multivariate Exploratory Data Analysis and Data +Mining. http://factominer.free.fr. +
    Inselberg, A. 1985. “The Plane with Parallel Coordinates.” The Visual Computer 1: 69–91. diff --git a/docs/collinearity-ridge.html b/docs/collinearity-ridge.html index cf37852f..92efdf53 100644 --- a/docs/collinearity-ridge.html +++ b/docs/collinearity-ridge.html @@ -294,7 +294,7 @@

    library(car)
     library(VisCollin)
    -library(genridge)
    +library(genridge)
     library(MASS)
     library(dplyr)
     library(factoextra)
    @@ -616,13 +616,13 @@ 

    #> [1] 2.070 0.911 0.809 0.367 0.245 0.189 #> #> Rotation (n x k) = (6 x 6): -#> PC1 PC2 PC3 PC4 PC5 PC6 -#> cylinder -0.454 0.1869 -0.168 0.659 -0.2711 0.4725 -#> engine -0.467 0.1628 -0.134 0.193 -0.0109 -0.8364 -#> horse -0.462 0.0177 0.123 -0.620 -0.6123 0.1067 -#> weight -0.444 0.2598 -0.278 -0.350 0.6860 0.2539 -#> accel 0.330 0.2098 -0.865 -0.143 -0.2774 -0.0337 -#> year 0.237 0.9092 0.335 -0.025 -0.0624 -0.0142

    +#> PC1 PC2 PC3 PC4 PC5 PC6 +#> cylinder -0.454 -0.1869 0.168 -0.659 -0.2711 -0.4725 +#> engine -0.467 -0.1628 0.134 -0.193 -0.0109 0.8364 +#> horse -0.462 -0.0177 -0.123 0.620 -0.6123 -0.1067 +#> weight -0.444 -0.2598 0.278 0.350 0.6860 -0.2539 +#> accel 0.330 -0.2098 0.865 0.143 -0.2774 0.0337 +#> year 0.237 -0.9092 -0.335 0.025 -0.0624 0.0142

    The standard deviations above are the square roots \(\sqrt{\lambda_j}\) of the eigenvalues of the correlation matrix, and are returned in the sdev component of the "prcomp" object. The eigenvectors are returned in the rotation component, whose directions are arbitrary. Because we are interested in seeing the relative magnitude of variable vectors, we are free to multiply them by any constant to make them more visible in relation to the scores for the cars.

    @@ -737,7 +737,7 @@

    Example: Interactions

    -

    The dataset genridge::Acetylene gives data from Marquardt and Snee (1975) on the yield of a chemical manufacturing process to produce acetylene in relation to reactor temperature (temp), the ratio of two components and the contact time in the reactor. A naive response surface model might suggest that yield is quadratic in time and there are potential interactions among all pairs of predictors.

    +

    The dataset genridge::Acetylene gives data from Marquardt and Snee (1975) on the yield of a chemical manufacturing process to produce acetylene in relation to reactor temperature (temp), the ratio of two components and the contact time in the reactor. A naive response surface model might suggest that yield is quadratic in time and there are potential interactions among all pairs of predictors.

    data(Acetylene, package = "genridge")
     acetyl.mod0 <- lm(yield ~ temp + ratio + time + I(time^2) + 
    @@ -797,7 +797,7 @@ 

    7.4.3 Bivariate ridge trace plots

    -
    #> Writing packages to  C:/R/Projects/Vis-MLM-book/bib/pkgs.txt
    +
    #> Writing packages to  C:/R/projects/Vis-MLM-book/bib/pkgs.txt
     #> 17  packages used here:
     #>  base, car, carData, datasets, dplyr, factoextra, genridge, ggplot2, ggrepel, graphics, grDevices, MASS, methods, patchwork, stats, utils, VisCollin
    diff --git a/docs/eqcov.html b/docs/eqcov.html index 9ad1ba04..598d1c21 100644 --- a/docs/eqcov.html +++ b/docs/eqcov.html @@ -350,7 +350,7 @@

    the connection between Levene-type tests and an ANOVA (of centered absolute differences) suggests a parallel with a multivariate extension of Levene-type tests and a MANOVA. We explore this with a version of Hypothesis-Error (HE) plots we have found useful for visualizing mean differences in MANOVA designs.

    -
    #> Writing packages to  C:/R/Projects/Vis-MLM-book/bib/pkgs.txt
    +
    #> Writing packages to  C:/R/projects/Vis-MLM-book/bib/pkgs.txt
     #> 15  packages used here:
     #>  base, broom, candisc, car, carData, datasets, dplyr, ggplot2, graphics, grDevices, heplots, methods, stats, tidyr, utils
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