Builds the single mdsDisplay used for a linear regression biplot and documents
the associated regression-biplot fit and predictivity measures.
Regression biplots do not use the multi-mdsDisplay fit machinery available for
PCA/CVA displays: they have one fixed mdsDisplay (mdsDisplay_12),
append_mdsDisplay() and remove_mdsDisplay() are not supported,
and the only active toggle button is “Translated Axes”.
Usage
# S3 method for class 'regress'
wrap_bipl5(x)Arguments
- x
An object of class
biplotfrom the biplotEZ package withregress()method applied.
Details
For the linear regression biplot handled by this method, let
\(\mathbf{X}\in\mathbb{R}^{n\times p}\) denote the processed data
matrix stored in the biplot object after centring and any optional
scaling performed by biplot(), and let
\(\mathbf{Z}\in\mathbb{R}^{n\times 2}\) denote the externally supplied
display coordinates of the \(n\) samples. Write
\(\mathbf{Z} = [\mathbf{z}_1\ \mathbf{z}_2]\), where
\(\mathbf{z}_1\) and \(\mathbf{z}_2\) are the first and second displayed
coordinates respectively. In contrast to a PCA biplot, the sample map is taken
as given and the variable axes are then fitted to that map by multivariate
least squares. This is the regression-biplot point of view used in the biplot
literature for general low-dimensional sample maps (Gower and Hand, 1996;
Gower, Lubbe and le Roux, 2011).
The fitted linear model is $$\mathbf{X} = \mathbf{Z}\mathbf{H}^{\top} + \mathbf{E},$$ where, when \(\mathbf{Z}\) has full column rank, $$\mathbf{H}^{\top} = (\mathbf{Z}^{\top}\mathbf{Z})^{-1}\mathbf{Z}^{\top}\mathbf{X}.$$ Hence the fitted values are $$\widehat{\mathbf{X}} = \mathbf{Z}\mathbf{H}^{\top} = \mathbf{Z}(\mathbf{Z}^{\top}\mathbf{Z})^{-1}\mathbf{Z}^{\top}\mathbf{X} = \mathbf{P}_Z\mathbf{X},$$ where \(\mathbf{P}_Z\) is the orthogonal projector onto the column space of \(\mathbf{Z}\). More generally, if the supplied display coordinates are rank-deficient, the same fitted matrix \(\widehat{\mathbf{X}}\) is obtained by interpreting \(\mathbf{P}_Z\) as the orthogonal projector onto \(\mathrm{col}(\mathbf{Z})\). The regression biplot therefore displays the variables through the least-squares predictions obtained from the supplied 2D sample map (Gower and Hand, 1996; Gower, Lubbe and le Roux, 2011).
If \(\mathbf{h}_{(j)}\) denotes the \(j\)th column of
\(\mathbf{H}\), then the predicted value of variable \(j\) for sample
\(i\) is
$$\widehat{x}_{ij} = \mathbf{z}_i^{\top}\mathbf{h}_{(j)}.$$
The calibrated axis for variable \(j\) has direction
\(\mathbf{h}_{(j)}\), and the point on that axis corresponding to marker
value \(\mu\) is
$$\mathbf{p}_{\mu j} =
\frac{\mu}{\mathbf{h}_{(j)}^{\top}\mathbf{h}_{(j)}}\mathbf{h}_{(j)}.$$
This is the calibration formula used to place tick marks and to recover
predicted values from projections onto the displayed axis, in direct analogy
with calibrated-axis biplot constructions (Gabriel, 1971; Gower, Lubbe and
le Roux, 2011). All such predicted values are on the same centred/scaled scale
as the stored matrix \(\mathbf{X}\); if needed, they can be back-transformed
to the original variable scale using the means and standard deviations stored
in the input biplot object.
A regression biplot admits a natural family of predictivity measures on the variable side. Let \(\mathbf{x}_{(j)}\) denote column \(j\) of \(\mathbf{X}\), let \(\widehat{\mathbf{x}}_{(j)}\) denote column \(j\) of \(\widehat{\mathbf{X}}\), and let \(\mathbf{e}_{(j)} = \mathbf{x}_{(j)} - \widehat{\mathbf{x}}_{(j)}\). Since \(\widehat{\mathbf{X}} = \mathbf{P}_Z\mathbf{X}\) is an orthogonal projection, the residual matrix satisfies $$\widehat{\mathbf{X}}^{\top}\mathbf{E} = \mathbf{0},$$ and therefore $$\mathbf{X}^{\top}\mathbf{X} = \widehat{\mathbf{X}}^{\top}\widehat{\mathbf{X}} + (\mathbf{X} - \widehat{\mathbf{X}})^{\top} (\mathbf{X} - \widehat{\mathbf{X}}).$$ This is the variable-side, or Type B, orthogonality that justifies variance-accounted-for ratios for the columns of \(\mathbf{X}\); it is the same side of the orthogonality argument that underlies column-wise predictivities in the biplot literature (Gower, Lubbe and le Roux, 2011; Greenacre, 2010).
The predictivity of variable \(j\) is therefore defined by $$\phi_j = \frac{\|\widehat{\mathbf{x}}_{(j)}\|^2} {\|\mathbf{x}_{(j)}\|^2} = 1 - \frac{\|\mathbf{x}_{(j)} - \widehat{\mathbf{x}}_{(j)}\|^2} {\|\mathbf{x}_{(j)}\|^2}, \qquad j=1,\ldots,p.$$ Thus \(\phi_j\) is the proportion of the sum of squares of variable \(j\) reproduced by the regression biplot, equivalently the ordinary multiple-regression \(R^2\) obtained by regressing variable \(j\) on the displayed coordinates \(\mathbf{Z}\). Each \(\phi_j\) lies in \([0,1]\); values near one indicate that the variable is well predicted by the displayed map, while values near zero indicate that the variable is poorly reproduced by the chosen display (Greenacre, 2010).
A natural overall quality-of-display measure is the proportion of total sum
of squares reproduced by the display,
$$R^2_{\mathrm{disp}} =
\frac{\|\widehat{\mathbf{X}}\|_F^2}{\|\mathbf{X}\|_F^2}
=
1 - \frac{\|\mathbf{X} - \widehat{\mathbf{X}}\|_F^2}{\|\mathbf{X}\|_F^2}.$$
Because the column-wise decomposition above is orthogonal, this overall
quality can be written as a weighted average of the variable predictivities:
$$R^2_{\mathrm{disp}} =
\sum_{j=1}^{p} w_j \phi_j,$$
where
$$w_j =
\frac{\|\mathbf{x}_{(j)}\|^2}{\|\mathbf{X}\|_F^2},
\qquad
\sum_{j=1}^{p} w_j = 1.$$
Hence variables with larger sums of squares contribute more to the overall
quality. In particular, if the original call to biplot() used
scale = TRUE, so that all processed variables have equal sums of
squares, then the weights are equal and
$$R^2_{\mathrm{disp}} = \frac{1}{p}\sum_{j=1}^{p}\phi_j.$$
This weighted-average interpretation is often the most natural way to read the
overall regression-biplot quality, since it combines the separate variable
predictivities into a single display-wide summary (Greenacre, 2010).
The quantities \(\phi_j\) and \(R^2_{\mathrm{disp}}\) depend only on the fitted projection \(\mathbf{P}_Z\mathbf{X}\) and therefore only on the subspace \(\mathrm{col}(\mathbf{Z})\). They do not depend on any particular basis chosen for that subspace. In particular, the variable predictivities \(\phi_j\) do not require any QR decomposition.
To decompose the total display quality into separate contributions for the two displayed dimensions, this package applies an ordered orthogonalization of the supplied display coordinates. Specifically, define $$\mathbf{u}_1 = \mathbf{z}_1, \qquad \mathbf{q}_1 = \frac{\mathbf{u}_1}{\|\mathbf{u}_1\|}$$ whenever \(\mathbf{u}_1 \neq \mathbf{0}\), and then define $$\mathbf{u}_2 = \mathbf{z}_2 - \mathbf{q}_1\mathbf{q}_1^{\top}\mathbf{z}_2, \qquad \mathbf{q}_2 = \frac{\mathbf{u}_2}{\|\mathbf{u}_2\|}$$ whenever \(\mathbf{u}_2 \neq \mathbf{0}\). Equivalently, \(\mathbf{Q} = [\mathbf{q}_1\ \mathbf{q}_2]\) is obtained from the QR decomposition of \(\mathbf{Z}\), preserving the supplied column order. The vectors \(\mathbf{q}_1\) and \(\mathbf{q}_2\) are orthonormal and span the same display subspace as the nonzero columns of \(\mathbf{Z}\).
Because \(\mathbf{Q}\) and \(\mathbf{Z}\) span the same subspace, the orthogonal projector may also be written as $$\mathbf{P}_Z = \mathbf{Q}\mathbf{Q}^{\top}.$$ Consequently, $$\widehat{\mathbf{X}} = \mathbf{Q}\mathbf{Q}^{\top}\mathbf{X} = \mathbf{q}_1\mathbf{q}_1^{\top}\mathbf{X} + \mathbf{q}_2\mathbf{q}_2^{\top}\mathbf{X}$$ whenever both orthogonalized directions are present. Since \(\mathbf{q}_1^{\top}\mathbf{q}_2 = 0\), the two fitted parts are orthogonal and their sums of squares add. This yields the dimension-specific contributions $$R^2_1 = \frac{\|\mathbf{q}_1\mathbf{q}_1^{\top}\mathbf{X}\|_F^2} {\|\mathbf{X}\|_F^2},$$ and $$R^2_{2\mid 1} = \frac{\|\mathbf{q}_2\mathbf{q}_2^{\top}\mathbf{X}\|_F^2} {\|\mathbf{X}\|_F^2},$$ so that $$R^2_{\mathrm{disp}} = R^2_1 + R^2_{2\mid 1}$$ whenever the display space is two-dimensional.
Care should be taken when interpreting this decomposition. If the columns of \(\mathbf{Z}\) are already orthogonal, then the two displayed contributions correspond directly to the first and second supplied display axes. If the columns of \(\mathbf{Z}\) are not orthogonal, however, the decomposition is ordered. The first contribution \(R^2_1\) is attributable to the first supplied display coordinate \(\mathbf{z}_1\). The second contribution \(R^2_{2\mid 1}\) is attributable to the component of the second supplied display coordinate \(\mathbf{z}_2\) that is orthogonal to the first. Thus \(R^2_{2\mid 1}\) should be interpreted as the additional contribution of “Dim 2 given Dim 1”, not as the contribution of the raw second column of \(\mathbf{Z}\) considered in isolation. The ordering of the columns of \(\mathbf{Z}\) is therefore important for this decomposition.
The same ordered orthogonalization yields a decomposition of each variable's predictivity: $$\phi_j = \phi_{j1} + \phi_{j,2\mid 1},$$ where $$\phi_{j1} = \frac{\|\mathbf{q}_1\mathbf{q}_1^{\top}\mathbf{x}_{(j)}\|^2} {\|\mathbf{x}_{(j)}\|^2} = \frac{(\mathbf{q}_1^{\top}\mathbf{x}_{(j)})^2} {\|\mathbf{x}_{(j)}\|^2},$$ and $$\phi_{j,2\mid 1} = \frac{\|\mathbf{q}_2\mathbf{q}_2^{\top}\mathbf{x}_{(j)}\|^2} {\|\mathbf{x}_{(j)}\|^2} = \frac{(\mathbf{q}_2^{\top}\mathbf{x}_{(j)})^2} {\|\mathbf{x}_{(j)}\|^2}.$$ Thus \(\phi_{j1}\) is the part of variable \(j\)'s predictivity explained by the first supplied display dimension, while \(\phi_{j,2\mid 1}\) is the additional part explained by the second display dimension after removing its overlap with the first.
If the supplied display coordinates are collinear, then \(\mathbf{u}_2 = \mathbf{0}\) and the effective display space is one-dimensional. In that case \(R^2_{2\mid 1} = 0\) and \(\phi_{j,2\mid 1} = 0\) for all variables.
In addition to the sum-of-squares fit measures above, this method may also report direct-reading error diagnostics in the sense of Alves (2012). The purpose of these diagnostics is different from that of the predictivities \(\phi_j\). The quantities \(\phi_j\), \(\phi_{j1}\), \(\phi_{j,2\mid 1}\) and \(R^2_{\mathrm{disp}}\) measure how much of the variation in \(\mathbf{X}\) is reproduced by the fitted regression biplot. By contrast, the Alves diagnostics measure how accurately values can be read directly from a displayed calibrated axis in the current two-dimensional map. Alves (2012) proposed this idea for predictive PCA biplots; in the present two-dimensional regression-biplot setting the same principle applies in a particularly simple form because there is only one displayed map.
For each sample \(i=1,\ldots,n\) and variable axis \(j=1,\ldots,p\), the reading taken from the displayed axis of variable \(j\) is precisely the fitted value $$\widehat{x}_{ij} = \mathbf{z}_i^{\top}\mathbf{h}_{(j)}.$$ The corresponding point on the calibrated axis is $$\mathbf{p}_{ij} = \frac{\widehat{x}_{ij}} {\mathbf{h}_{(j)}^{\top}\mathbf{h}_{(j)}}\mathbf{h}_{(j)},$$ obtained by substituting \(\mu = \widehat{x}_{ij}\) into the calibration formula. Thus the direct reading from the graph and the fitted value from the regression model coincide.
Let \(s_j\) denote the standard deviation used to standardize variable
\(j\). When scale = TRUE, the processed matrix \(\mathbf{X}\)
already has unit-variance columns and hence \(s_j = 1\). The
pointwise direct-reading error for sample \(i\) on variable axis
\(j\) is defined by
$$\delta_{ij} =
\frac{|x_{ij} - \widehat{x}_{ij}|}{s_j}.$$
If the processed matrix is already standardized, then
\(\delta_{ij} = |x_{ij} - \widehat{x}_{ij}|\); in that case
\(\delta_{ij}\) is the direct analogue of Alves's standard predictive error.
More generally, dividing by \(s_j\) expresses the discrepancy on a
comparable variable-wise scale. The quantity \(\delta_{ij}\) is therefore a
sample-by-axis direct-reading error.
The corresponding axis-level mean direct-reading error is $$\bar{\delta}_j = \frac{1}{n}\sum_{i=1}^{n}\delta_{ij} = \frac{1}{n}\sum_{i=1}^{n} \frac{|x_{ij} - \widehat{x}_{ij}|}{s_j}.$$ This is the two-dimensional regression-biplot analogue of the mean standard predictive error of Alves (2012). Small values of \(\bar{\delta}_j\) indicate that the calibrated axis for variable \(j\) supports accurate direct readings on average across the displayed observations, whereas large values indicate that direct readings from that axis are unreliable in the current display.
Let \(\tau_{\mathrm{axis}} > 0\) be a user-specified tolerance parameter for axis-level direct-reading error. Then the Alves selection rule specialized to the present two-dimensional regression biplot is $$\text{retain axis }j \quad\Longleftrightarrow\quad \bar{\delta}_j \le \tau_{\mathrm{axis}}.$$ Thus an axis is shown only when its average direct-reading error is at most the allowed tolerance. Larger values of \(\tau_{\mathrm{axis}}\) retain more axes and therefore produce denser displays; smaller values enforce stricter axis selection and lead to sparser, more conservative displays. In Alves (2012), values around 0.5 are discussed as a practical starting point in conventional settings, but no universal default should be assumed.
In addition to axis selection, Alves (2012) proposed a second tolerance parameter for individual sample-axis discrepancies. Let \(\tau_{\mathrm{units}} > 0\). A sample \(i\) is then flagged as an outlier with respect to axis \(j\) whenever $$\delta_{ij} > \tau_{\mathrm{units}}.$$ Such a flag indicates that, even if axis \(j\) is acceptable on average, the direct reading for sample \(i\) from that axis is poor in the current display. Alves (2012) discusses values around 0.75 as a practical starting point for this tolerance parameter, again subject to the application and the scale of the analysis.
Because this wrapper is tied to a single two-dimensional regression-biplot display, the quantities \(\delta_{ij}\) and \(\bar{\delta}_j\) are display-specific diagnostics. They are not measures of the quality of the underlying fitted subspace in the sum-of-squares sense; rather, they quantify the numerical accuracy of direct readings from the currently displayed axes. This distinction is central in Alves (2012), who emphasizes that direct-reading error is conceptually different from earlier axis-predictivity measures.
The Alves diagnostics and the regression-biplot predictivities are therefore complementary. The quantity \(\phi_j\) is a variance-accounted-for ratio justified by Type B orthogonality and answers the question: “How much of variable \(j\)'s sum of squares is reproduced by the displayed regression biplot?” The quantity \(\bar{\delta}_j\) is a mean absolute direct-reading error and answers the different question: “How accurately can values of variable \(j\) be read from the displayed calibrated axis?” Consequently, a variable may have high \(\phi_j\) and still have a non-negligible direct-reading error, while a variable with moderate \(\phi_j\) may nevertheless support acceptable average direct readings. In this implementation, the \(\phi_j\)-family is the primary set of sum-of-squares fit measures, whereas the Alves quantities \(\bar{\delta}_j\) and \(\delta_{ij}\) provide supplementary, display-specific diagnostics for axis selection and observation-level checking.
In contrast, a regression biplot does not in general satisfy the sample-side decomposition $$\mathbf{X}\mathbf{X}^{\top} = \widehat{\mathbf{X}}\widehat{\mathbf{X}}^{\top} + (\mathbf{X} - \widehat{\mathbf{X}}) (\mathbf{X} - \widehat{\mathbf{X}})^{\top}.$$ Consequently, PCA-style sample predictivities are not generally justified for a regression biplot. The principled sum-of-squares fit measures are the variable predictivities \(\phi_j\), the overall quality \(R^2_{\mathrm{disp}}\), and the ordered dimension-specific contributions described above, with the Alves direct-reading errors providing a distinct supplementary perspective on the quality of the displayed axes.
In the wrapped bipl5_biplot object, these formulas drive the bottom
display-quality label, the hover-time predicted values
\(\widehat{\mathbf{X}}\), and the calibrated linear axes stored in
mdsDisplay_12. Since the regression display is tied to one externally
supplied map, wrap_bipl5.regress() produces a single mdsDisplay only.
There is no PC/CV toggle and no separate PCA-style sample-fit panel.
References
Gabriel, K. R. (1971). The biplot graphical display of matrices with application to principal component analysis. Biometrika, 58(3), 453–467. doi:10.1093/biomet/58.3.453
Gower, J. C. and Hand, D. J. (1996). Biplots. London: Chapman \& Hall.
Gower, J. C., Lubbe, S. and le Roux, N. J. (2011). Understanding Biplots. Chichester: Wiley.
Greenacre, M. (2010). Biplots in Practice. Bilbao: BBVA Foundation.
la Grange, A., le Roux, N. and Gardner-Lubbe, S. (2009). BiplotGUI: Interactive Biplots in R. Journal of Statistical Software, 30(12), 1–37. doi:10.18637/jss.v030.i12
Alves, M. R. (2012). Evaluation of the predictive power of biplot axes to automate the construction and layout of biplots based on the accuracy of direct readings from common outputs of multivariate analyses: application to principal component analysis. Journal of Chemometrics, 26(5), 180–190. doi:10.1002/cem.2433