From de8ef384d9e35581c4b7807eec447efe612155b6 Mon Sep 17 00:00:00 2001 From: Michael Friendly Date: Fri, 8 Dec 2023 14:21:29 -0500 Subject: [PATCH] more work on PCA math --- 04-pca-biplot.qmd | 73 ++++++++++++++++++++----- bib/pkgs.bib | 19 +++---- bib/references.bib | 10 ++++ docs/03-multivariate_plots.html | 4 +- docs/04-pca-biplot.html | 73 ++++++++++++++++++------- docs/90-references.html | 4 ++ 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-biplot3-1.png | Bin 301362 -> 307282 bytes docs/figs/fig-duncan-plot-model-1.png | Bin 232072 -> 231202 bytes docs/figs/fig-mtcars-biplot-1.png | Bin 226892 -> 224709 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 | 33 ++++++----- 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-biplot3-1.png | Bin 301362 -> 307282 bytes figs/fig-duncan-plot-model-1.png | Bin 232072 -> 231202 bytes figs/fig-mtcars-biplot-1.png | Bin 226892 -> 224709 bytes figs/fig-plot-prestige-mod-1.png | Bin 306472 -> 305757 bytes 27 files changed, 176 insertions(+), 75 deletions(-) diff --git a/04-pca-biplot.qmd b/04-pca-biplot.qmd index b28c3192..4f327c17 100644 --- a/04-pca-biplot.qmd +++ b/04-pca-biplot.qmd @@ -234,25 +234,63 @@ higher-dimensional ellipsoids. The brides maids were eigenvectors, pointing in a different directions as space would allow, each sized according to their associated eigenvalues. Attending the wedding were the ghosts of uncles, Leonhard Euler, Jean-Louis Lagrange, Augustin-Louis Cauchy and others who had earlier discovered the mathematical properties -of quadratic forms in relation to problems in physics. +of ellipses and quadratic forms in relation to problems in physics. The key idea in the statistical application was that, for a set of variables $\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_p$, -the $p \times p$ covariance matrix $\mathbf{S}$ could be expressed exactly as a matrix -product involving a matrix $\mathbf{V}$, whose columns are eigenvectors ($\mathbf{v}_i$) and a -diagonal matrix $\mathbf{\Lambda}$, whose diagonal elements ($\lambda_i$) are the corresponding eigenvalues, +the $p \times p$ covariance matrix $\mathbf{S}$ could be expressed **exactly** as a matrix +product involving a matrix $\mathbf{V}$, whose columns are _eigenvectors_ ($\mathbf{v}_i$) and a +diagonal matrix $\mathbf{\Lambda}$, whose diagonal elements ($\lambda_i$) are the corresponding _eigenvalues_. + +To explain this, it is helpful to use a bit of matrix math: -$$ \begin{align*} -\mathbf{S} & = \mathbf{V} \mathbf{\Lambda} \mathbf{V}^T \\ - & = \lambda_1 \mathbf{v}_1 \mathbf{v}_1^T + \lambda_2 \mathbf{v}_2 \mathbf{v}_2^T + ... + \lambda_p \mathbf{v}_p \mathbf{v}_p^T +\mathbf{S}_{p \times p} & = \mathbf{V}_{p \times p} \quad\quad \mathbf{\Lambda}_{p \times p} \quad\quad \mathbf{V}_{p \times p}^T \\ + & = \left( \mathbf{v}_1, \, \mathbf{v}_2, \,\dots, \, \mathbf{v}_p \right) + \begin{pmatrix} + \lambda_1 & & & \\ + & \lambda_2 & & \\ + & & \ddots & \\ + & & & \lambda_p + \end{pmatrix} + + \begin{pmatrix} + \mathbf{v}_1^T\\ + \mathbf{v}_2^T\\ + \vdots\\ + \mathbf{v}_p^T\\ + \end{pmatrix} + \\ + & = \lambda_1 \mathbf{v}_1 \mathbf{v}_1^T + \lambda_2 \mathbf{v}_2 \mathbf{v}_2^T + \cdots + \lambda_p \mathbf{v}_p \mathbf{v}_p^T \end{align*} -$$ + In this equation, -* The columns of $\mathbf{V}$ are the eigenvectors and they represent orthogonal (uncorrelated) directions in data space. The values $\mathbf{v}_i$ are the weights applied to the variables. -* The eigenvalues, $\lambda_i$ ... + +* The last line follows because $\mathbf{\Lambda}$ is a diagonal matrix, so $\mathbf{S}$ is expressed as a sum of outer products of each $\mathbf{v}_i$ with itself. +* The columns of $\mathbf{V}$ are the eigenvectors of $\mathbf{S}$. They are orthogonal +and of unit length, so $\mathbf{V}^T \mathbf{V} = \mathbf{I}$ and thus +they represent orthogonal (uncorrelated) directions in data space. + +* The columns $\mathbf{v}_i$ are the weights applied to the variables to produce the scores on +the principal components. For example, the first principal component is the weighted sum: + +$$ +PC1 = v_{11} \mathbf{x}_1 + v_{12} \mathbf{x}_2 + \cdots + v_{1p} \mathbf{x}_p +$$ + +* The eigenvalues, $\lambda_1, \lambda_2, \dots, \lambda_p$ are the variances of the the components, because +$\mathbf{v}_i^T \;\mathbf{S} \; \mathbf{v}_i = \lambda_i$. + +* It is usually the case that the variables $\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_p$ are linearly +independent, which means that none of these is an exact linear combination of the others. In this case, +all eigenvalues $\lambda_i$ are positive and the covariance matrix $\mathbf{S}$ is said to have **rank** $p$. + +* Here is the key fact: If the eigenvalues are arranged in order, so that $\lambda_1 > \lambda_2 > \dots > \lambda_p$, then the first $d$ components give a $d$-dimensional approximation to $\mathbf{S}$, which accounts for $\sigma_i^d \lambda_i / \sigma_i^p \lambda_i$ of the total variance. For the case of two variables, $\mathbf{x}_1$ and $\mathbf{x}_2$ @fig-pca-rotation shows the -transformation ... +transformation from data space to component space. The eigenvectors, $\mathbf{v}_1, \mathbf{v}_2$ +are the major and minor +axes of the data ellipse, whose lengths are the square roots $\sqrt{\lambda_1}, \sqrt{\lambda_2}$ +of the eigenvalues. ```{r} #| label: fig-pca-rotation #| out-width: "100%" @@ -266,8 +304,15 @@ transformation ... 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, ...). +The **FactoMineR** package [@R-FactoMineR; @Husson-etal-2017] +has extensive capabilities for exploratory analysis of multivariate data (PCA, correspondence analysis, cluster analysis). + +A particular strength of FactoMineR +for PCA is that it allows the inclusion of _supplementary variables_ (which can be categorical +or quantitative) and _supplementary points_ for individuals. These are not used in the analysis, +but are projected into the plots to facilitate interpretation. For example, in the analysis of +the `crime` data described below, it would be useful to have measures of other characteristics +of the U.S. states, such as poverty and average level of education. Unfortunately, although all of these performing similar calculations, the options for analysis and the details of the result they return differ. @@ -811,6 +856,8 @@ Yes, the correlation of number of forward gears (`gear`) and number of carburato in the upper left and lower right corners is moderately positive (0.27) while all the others in their off-diagonal blocks are negative. +## Application: Eigenfaces + ## 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/bib/pkgs.bib b/bib/pkgs.bib index 0b726567..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}, @@ -155,6 +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}, @@ -209,7 +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/}, } diff --git a/bib/references.bib b/bib/references.bib index 0d60e67b..e0602b95 100644 --- a/bib/references.bib +++ b/bib/references.bib @@ -808,6 +808,16 @@ @Article{Hotelling:1931 publisher = {Institute of Mathematical Statistics}, } +@Book{Husson-etal-2017, + author = {Francois Husson and Sebastien Le and Jérôme Pagès}, + publisher = {Chapman & Hall}, + title = {Exploratory Multivariate Analysis by Example Using R}, + year = {2017}, + address = {New York}, + doi = {10.1201/b21874}, + url = {https://dokumen.pub/exploratory-multivariate-analysis-by-example-using-r-1nbsped-1439835802-9781439835807.html}, +} + @ARTICLE{Inselberg:1985, author = {A. Inselberg}, title = {The Plane with Parallel Coordinates}, diff --git a/docs/03-multivariate_plots.html b/docs/03-multivariate_plots.html index 4e8eb664..33976e96 100644 --- a/docs/03-multivariate_plots.html +++ b/docs/03-multivariate_plots.html @@ -1154,7 +1154,7 @@

ggpairs() function from the GGally package (Schloerke et al., 2021). This allows different plot types to be shown in the lower and upper triangles and in the diagonal cells of the plot matrix. As well, aesthetics such as color and shape can be used within the plots to distinguish groups directly. As illustrated below, you can define custom functions to control exactly what is plotted in any panel.

+

In the ggplot2 framework, these displays are implemented using the ggpairs() function from the GGally package (Schloerke et al., 2023). This allows different plot types to be shown in the lower and upper triangles and in the diagonal cells of the plot matrix. As well, aesthetics such as color and shape can be used within the plots to distinguish groups directly. As illustrated below, you can define custom functions to control exactly what is plotted in any panel.

The basic, default plot shows scatterplots for pairs of continuous variables in the lower triangle and the values of correlations in the upper triangle. A combination of a discrete and continuous variables is plotted as histograms in the lower triangle and boxplots in the upper triangle. Figure 3.29 includes sex to illustrate the combinations.

Code
ggpairs(peng, columns=c(3:6, 7),
@@ -1445,7 +1445,7 @@ 

https://lattice.r-forge.r-project.org/

-Schloerke, B., Cook, D., Larmarange, J., Briatte, F., Marbach, M., Thoen, E., Elberg, A., & Crowley, J. (2021). GGally: Extension to ggplot2. https://ggobi.github.io/ggally/ +Schloerke, B., Cook, D., Larmarange, J., Briatte, F., Marbach, M., Thoen, E., Elberg, A., & Crowley, J. (2023). GGally: Extension to ggplot2. https://ggobi.github.io/ggally/
Scott, D. W. (1992). Multivariate density estimation: Theory, practice, and visualization. Wiley. diff --git a/docs/04-pca-biplot.html b/docs/04-pca-biplot.html index a73be80a..2aaf4ad5 100644 --- a/docs/04-pca-biplot.html +++ b/docs/04-pca-biplot.html @@ -266,7 +266,8 @@
  • 4.4 Application: Variable ordering for data displays
  • -
  • 4.5 Elliptical insights: Outlier detection
  • +
  • 4.5 Application: Eigenfaces
  • +
  • 4.6 Elliptical insights: Outlier detection
  • @@ -406,15 +407,43 @@

    TODO: Simple PCA example using workers data

    4.2.2 Mathematics and geometry of PCA

    -

    As the ideas of principal components developed, there was a happy marriage of Galton’s geometrical intuition and Pearson’s mathematical analysis. The best men at the wedding were ellipses and higher-dimensional ellipsoids. The brides maids were eigenvectors, pointing in as many different directions as space would allow, each sized according to their associated eigenvalues. Attending the wedding were the ghosts of uncles, Leonhard Euler, Jean-Louis Lagrange, Augustin-Louis Cauchy and others who had earlier discovered the mathematical properties of quadratic forms in relation to problems in physics.

    -

    The key idea in the statistical application was that, for a set of variables \(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_p\), the \(p \times p\) covariance matrix \(\mathbf{S}\) could be expressed exactly as a matrix product involving a matrix \(\mathbf{V}\), whose columns are eigenvectors (\(\mathbf{v}_i\)) and a diagonal matrix \(\mathbf{\Lambda}\), whose diagonal elements (\(\lambda_i\)) are the corresponding eigenvalues,

    +

    As the ideas of principal components developed, there was a happy marriage of Galton’s geometrical intuition and Pearson’s mathematical analysis. The best men at the wedding were ellipses and higher-dimensional ellipsoids. The brides maids were eigenvectors, pointing in as many different directions as space would allow, each sized according to their associated eigenvalues. Attending the wedding were the ghosts of uncles, Leonhard Euler, Jean-Louis Lagrange, Augustin-Louis Cauchy and others who had earlier discovered the mathematical properties of ellipses and quadratic forms in relation to problems in physics.

    +

    The key idea in the statistical application was that, for a set of variables \(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_p\), the \(p \times p\) covariance matrix \(\mathbf{S}\) could be expressed exactly as a matrix product involving a matrix \(\mathbf{V}\), whose columns are eigenvectors (\(\mathbf{v}_i\)) and a diagonal matrix \(\mathbf{\Lambda}\), whose diagonal elements (\(\lambda_i\)) are the corresponding eigenvalues.

    +

    To explain this, it is helpful to use a bit of matrix math:

    +

    \[\begin{align*} +\mathbf{S}_{p \times p} & = \mathbf{V}_{p \times p} \quad\quad \mathbf{\Lambda}_{p \times p} \quad\quad \mathbf{V}_{p \times p}^T \\ + & = \left( \mathbf{v}_1, \, \mathbf{v}_2, \,\dots, \, \mathbf{v}_p \right) + \begin{pmatrix} + \lambda_1 & & & \\ + & \lambda_2 & & \\ + & & \ddots & \\ + & & & \lambda_p + \end{pmatrix} + + \begin{pmatrix} + \mathbf{v}_1^T\\ + \mathbf{v}_2^T\\ + \vdots\\ + \mathbf{v}_p^T\\ + \end{pmatrix} + \\ + & = \lambda_1 \mathbf{v}_1 \mathbf{v}_1^T + \lambda_2 \mathbf{v}_2 \mathbf{v}_2^T + \cdots + \lambda_p \mathbf{v}_p \mathbf{v}_p^T +\end{align*}\]

    +

    In this equation,

    +
      +
    • The last line follows because \(\mathbf{\Lambda}\) is a diagonal matrix, so \(\mathbf{S}\) is expressed as a sum of outer products of each \(\mathbf{v}_i\) with itself.

    • +
    • The columns of \(\mathbf{V}\) are the eigenvectors of \(\mathbf{S}\). They are orthogonal and of unit length, so \(\mathbf{V}^T \mathbf{V} = \mathbf{I}\) and thus they represent orthogonal (uncorrelated) directions in data space.

    • +
    • The columns \(\mathbf{v}_i\) are the weights applied to the variables to produce the scores on the principal components. For example, the first principal component is the weighted sum:

    • +

    \[ -\begin{align*} -\mathbf{S} & = \mathbf{V} \mathbf{\Lambda} \mathbf{V}^T \\ - & = \lambda_1 \mathbf{v}_1 \mathbf{v}_1^T + \lambda_2 \mathbf{v}_2 \mathbf{v}_2^T + ... + \lambda_p \mathbf{v}_p \mathbf{v}_p^T -\end{align*} -\] In this equation, * The columns of \(\mathbf{V}\) are the eigenvectors and they represent orthogonal (uncorrelated) directions in data space. The values \(\mathbf{v}_i\) are the weights applied to the variables. * The eigenvalues, \(\lambda_i\)

    -

    For the case of two variables, \(\mathbf{x}_1\) and \(\mathbf{x}_2\) Figure 4.6 shows the transformation …

    +PC1 = v_{11} \mathbf{x}_1 + v_{12} \mathbf{x}_2 + \cdots + v_{1p} \mathbf{x}_p +\]

    +
      +
    • The eigenvalues, \(\lambda_1, \lambda_2, \dots, \lambda_p\) are the variances of the the components, because \(\mathbf{v}_i^T \;\mathbf{S} \; \mathbf{v}_i = \lambda_i\).

    • +
    • It is usually the case that the variables \(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_p\) are linearly independent, which means that none of these is an exact linear combination of the others. In this case, all eigenvalues \(\lambda_i\) are positive and the covariance matrix \(\mathbf{S}\) is said to have rank \(p\).

    • +
    • Here is the key fact: If the eigenvalues are arranged in order, so that \(\lambda_1 > \lambda_2 > \dots > \lambda_p\), then the first \(d\) components give a \(d\)-dimensional approximation to \(\mathbf{S}\), which accounts for \(\sigma_i^d \lambda_i / \sigma_i^p \lambda_i\) of the total variance.

    • +
    +

    For the case of two variables, \(\mathbf{x}_1\) and \(\mathbf{x}_2\) Figure 4.6 shows the transformation from data space to component space. The eigenvectors, \(\mathbf{v}_1, \mathbf{v}_2\) are the major and minor axes of the data ellipse, whose lengths are the square roots \(\sqrt{\lambda_1}, \sqrt{\lambda_2}\) of the eigenvalues.

    @@ -425,7 +454,8 @@

    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 (Husson et al., 2023) 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., 2017, 2023) has extensive capabilities for exploratory analysis of multivariate data (PCA, correspondence analysis, cluster analysis).

    +

    A particular strength of FactoMineR for PCA is that it allows the inclusion of supplementary variables (which can be categorical or quantitative) and supplementary points for individuals. These are not used in the analysis, but are projected into the plots to facilitate interpretation. For example, in the analysis of the crime data described below, it would be useful to have measures of other characteristics of the U.S. states, such as poverty and average level of education.

    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:

      @@ -575,13 +605,13 @@

      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.

    @@ -851,8 +881,10 @@

    But wait, there is something else to be seen in Figure 4.15. Can you see one cell that doesn’t fit with the rest?

    Yes, the correlation of number of forward gears (gear) and number of carburators (carb) in the upper left and lower right corners is moderately positive (0.27) while all the others in their off-diagonal blocks are negative.

    -

    -4.5 Elliptical insights: Outlier detection

    +

    +4.5 Application: Eigenfaces

    +

    +4.6 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).

    @@ -921,6 +953,9 @@

    Husson, F., Josse, J., Le, S., & Mazet, J. (2023). FactoMineR: Multivariate exploratory data analysis and data mining. http://factominer.free.fr

    +
    +Husson, F., Le, S., & Pagès, J. (2017). Exploratory multivariate analysis by example using r. Chapman & Hall. https://doi.org/10.1201/b21874 +
    Pearson, K. (1896). Contributions to the mathematical theory of evolution—III, regression, heredity and panmixia. Philosophical Transactions of the Royal Society of London, 187, 253–318.
    diff --git a/docs/90-references.html b/docs/90-references.html index 390566cc..498ee466 100644 --- a/docs/90-references.html +++ b/docs/90-references.html @@ -585,6 +585,10 @@

    References

    Husson, F., Josse, J., Le, S., & Mazet, J. (2023). FactoMineR: Multivariate exploratory data analysis and data mining. http://factominer.free.fr +
    +Husson, F., Le, S., & Pagès, J. (2017). Exploratory multivariate +analysis by example using r. Chapman & Hall. https://doi.org/10.1201/b21874 +
    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 3b1b8f9f..647b8076 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 & 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 & 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
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     #> 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 87493570..06fe10dc 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.

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     #>  base, broom, candisc, car, carData, datasets, dplyr, ggplot2, graphics, grDevices, heplots, methods, stats, tidyr, utils
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    -
    #> Writing packages to  C:/R/Projects/Vis-MLM-book/bib/pkgs.txt
    +
    #> Writing packages to  C:/R/projects/Vis-MLM-book/bib/pkgs.txt
     #> 16  packages used here:
     #>  base, broom, car, carData, corpcor, datasets, dplyr, ggplot2, graphics, grDevices, heplots, Hotelling, methods, stats, tidyr, utils
    diff --git a/docs/index.html b/docs/index.html index 6c5c2f30..b8733327 100644 --- a/docs/index.html +++ b/docs/index.html @@ -5,7 +5,7 @@ - + Visualizing Multivariate Data and Models in R
      ;RmU5hyskH!oXp&Mx?(GMfqU2@COpkmS4c zna0?My7tC53e`T&yuS?!TpD8fCWbFInVU+^0H9Z8HUVc1wXhi+ZgZyg=HlAwN@uA< zQxBO3b(iIFtLLOyE1!fan%C|d3h)5pl9lHg8vQc=8S8infX3?S$^4&diRVh1Ns>dqlW$Fp?duD;4X*!7sp@5_INef5M`S)yls1%Pb(PMMN`)wwn=; zpsHHGQlPotf4%_K$BPAYIA%Bxqyg2j_Gl}$zwC0QIb8v)a(t)85Cv|T8wZKSwW>xP z-E~J_7;Yy}T%bRxr~j;rSoK*IFaEB@KDECTM_+)m^59RjBqxm6ylxHM_b6=`fOfl=Og0ZsA_L zy|=dA+jchy$-v!5$N2BWOL$qp@yePxjL@b||K5XusXXZxec6u;PJq(MKrK>=PaN_b zZemLv4vu%sU47)F8lg#ID-b+0eOB}XniqxSq|Vu5@vC^8$tzp{XD=UwGD&uT&!F0W z^Mv`8lAWKqgVF}-sgDe$?28u0-P2|*Mfh0(6DU=zC_WtLhGvvh#W|#q1}m#I*(`5lu() zwv&jOk>^mjQ1Z}huA47UhlB>CFj;mBmqGGMJa*;{Z0bAYcAy;A`<4uSOd2{ue(X$%_tne@q%#3?`MBVs z+ohY9x>O-PM!I!0jp_XG^GoyqWZ2I#?vDRTH?8rb?k-^RS>>#=41@%1z~E5&)vMc` z)u=L2OHCFB0Gy{i-q8r6h0s3=05j$cKUfr0KPxgtcJtV`ornUn1QN}TJZ+ykeaIsX zLpa|Qvq@5LJA#Gv`G9cH9*?*R_~_Tn#V8W`6B2kOY-v#V<78}?1@t~b3vQgd1Wv! zmdz!Jn6|e-ydP?=%1#eaq!U?`K0I6%$9I1c)r9m3v z*vG&8;+6*^%fgan0R=5@M&518CE~uDTft%4^Ur=5%CNX|gqvsx%6Nu}>4AH`ai@2` z>PJTO9Pyb0zp*plYi2ej1!ZMO0?*4lboX*nm^pATaBr1TwYtUsEH(g59H>(F#$Gwn zO5AvBllWaRh|wL*fYCZzw@?Bq<3KfALp<|;K#^RU!{gis1%Cw^j6P97W!{sHjCLLf zl=Q}#fbha?`3Uy#$vy50w*}l-U>)}*A>zbb4%suBNB`o}pDPQ?gmNM6yyF`Mf(6kM zr>0)mB(^!?jdjk-&e&pzC-n+n3LCA(KNj&9Fx9M5n@atN2OI_98kbZtl&eQ!r2c@J zODL9JEo3rw13&U(I0-qC;{Ylf6}o(huA1!@=U$Q=^!e-_GH=Wm!fKXyYm& z&iWW1o8RX>?gC<+0?I=9?Ib8;n$7s6p zAyQPXbm(}q(YT^ky!tI2Q-?AgK`K*`oI|+2^qS}1EYe!1PNt*uXb|LVK?3Ty;2**FM^25O0nQ-1A{X4`QO2J)Ey8nh&} z)I)nf(|@N}00ehV=qbVFoL~Qi2Ek(y&D+Ari)p@yNlhjZ9aKlzRPl+&1Ds^3391qejaoy;bKZdA|y3J-k*J7r-VM^uuTb%uO za|~f9xvgi!sv)g_1fce0;dLiXdIU88U(rWdhF{$BuZD4lO!}=d9_Q_1jkd$`E#P&j zhv@}DvEGpvzqGVyU9|YRLn_)GZwUVWuO5WfGH*0L&$+raT(qW&YvQX(HlVWq|M}l# zFpfirb@@QrjcKP9$me=8LvP|~pJ;zAZbtT1mdf$&XamXtx9S(%yx(?n+4W;XPSPh9 zj>ati&pltZl%_C>f%n~S9zg$K;6+@3Gy41Yf&W^mjeWw!6O&K2$ppyV;DmjDKO(@G z11mavG>+Pj?V}7ic1D!4#9ybn-9r7}@nNm6l4N=2HtiRoeTBX=UG58&l9d1IvuL<2 zc$B-#U$VrCupdOMCr)$7pSip97E6P8Pv$?Z!8-@dc(U8;e+O->=yPAC!D@|a@BE*$ zI|M7Uh4N2fmz7XrerGqfxAEQlerW>Ktv?gkV09dg=cH9H(5|z(#mI_X_UG8y)?0XX zH+Qs9gVH){{*JrDEIRIcF;A{4{kM%V^~072N8^=J>-w3@ zo`Z-QTFw7f(?yG`qAdb~M_%6BYM}lriz-a~pG%u-%;~wi<+%ng?l`ZnQP0lmrvA=> z60N$<$PMi{YZ-EJcomhG^=+4H!N_HX#2E@xB~w{IT28*#(FhDLA<^R~zUQOOBduNO zW!&-!UuKwk6tSJbu)RXF)2QNaB3Div5TAC9`F#xrQ}JhG;NEd-P(9_Gdr{Qd*yL{+ zc%XJaKu~J?yYnh~{GQCnzxoODF>(1{`E^{pn76m^-Rx?a@9)BS*iFat=Uc`X8fCV( zTB3dztE2;*svU>%OThe value of Hotelling’s \(T^2\) found by hotelling.test() is 64.17. The value of the equivalent \(F\) statistic found by Anova() is 28.9. Verify that Equation 8.2 gives this result.

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