genridge

Generalized Ridge Trace Plots for Ridge Regression

https://github.com/friendly/genridge

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Generalized Ridge Trace Plots for Ridge Regression

README.Rmd

---
output: github_document
---



```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  warning = FALSE,
  comment = "#>",
  fig.path = "man/figures/README-",
  fig.height = 5,
  fig.width = 5
#  out.width = "100%"
)

options(digits = 4)
library(genridge)

```


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# genridge 

## Generalized Ridge Trace Plots for Ridge Regression


Version `r getNamespaceVersion("genridge")`

### What is ridge regression?


Consider the standard linear model,
$\mathbf{y} = \mathbf{X} \; \mathbf{\beta} + \mathbf{\epsilon}$
for $p$ predictors in a multiple regression.
In this context,
high multiple correlations among the predictors lead to well-known problems of collinearity
under ordinary least squares (OLS) estimation, which result in unstable estimates of the
parameters in β: standard errors are inflated and estimated coefficients tend to be too large
in absolute value on average.

Ridge regression is an instance of a class of techniques designed to obtain more favorable
predictions at the expense of some increase in bias, compared to ordinary least squares (OLS)
estimation.
An essential idea behind these methods is that the OLS estimates are constrained in
some way, shrinking them, on average, toward zero, to satisfy increased predictive accuracy.

The OLS estimates, which minimize the sum of squared residuals $RSS = \Sigma \mathbf{\epsilon}^2$  are given by:
$$
\widehat{\mathbf{\beta}}^{\mathrm{OLS}} = (\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{y} \; ,
$$
with $\widehat{\text{Var}} (\widehat{\mathbf{\beta}}^{\mathrm{OLS}}) = \widehat{\sigma}^2 (\mathbf{X}^\top \mathbf{X})^{-1}$.

Ridge regression replaces the standard residual sum of squares criterion with a penalized
form,

$$
\mathrm{RSS}(\lambda) = (\mathbf{y}-\mathbf{X} \mathbf{\beta})^\top  (\mathbf{y}-\mathbf{X} \mathbf{\beta}) + \lambda \mathbf{\beta}^\top \mathbf{\beta} \quad\quad (\lambda \ge 0) \: ,
$$

whose solution is easily seen to be:

$$
\widehat{\mathbf{\beta}}^{\mathrm{RR}}_k  = (\mathbf{X}^\top \mathbf{X} + \lambda \mathbf{I})^{-1} \mathbf{X}^\top \mathbf{y}  
$$

where $\lambda$ is the _shrinkage factor_ or _tuning constant_, penalizing larger coefficients. 
Shrinkage can also be expressed as the equivalent degrees of freedom, the trace of 
the analog of the "hat" matrix, $\text{tr}[(\mathbf{X}^\top \mathbf{X} + \lambda \mathbf{I})^{-1} \mathbf{X}^\top]$.
In general,

* The bias increases as λ increases,
* The sampling variance decreases as λ increases.

One goal of the `genridge` package is to provide visualization methods for these models to
help understand the tradeoff between bias and variance and choice of a shrinkage value $\lambda$.


### Package overview

The `genridge` package introduces generalizations of the standard univariate
ridge trace plot used in ridge regression and related methods (Friendly, 2011, 2013).  These graphical methods
show both bias (actually, shrinkage) and precision, by plotting the covariance ellipsoids of the estimated
coefficients, rather than just the estimates themselves.  2D and 3D plotting methods are provided,
both in the space of the predictor variables and in the transformed space of the PCA/SVD of the
predictors.  

### Details

This package provides computational support for the graphical methods described in Friendly (2013). Ridge regression models may be fit using the function `ridge`, which incorporates features of `MASS::lm.ridge()` and `ElemStatLearn::simple.ridge()`. In particular, the shrinkage factors in ridge regression may be specified either in terms of the constant ($\lambda$) added to the diagonal of $X^\top X$ matrix, or the equivalent number of degrees of freedom.

The following computational functions are provided:

* `ridge()` Calculates ridge regression estimates; returns an object of class `"ridge"`
* `pca.ridge()` Transform coefficients and covariance matrices to PCA/SVD space; returns an object of class `c("pcaridge", "ridge")`
* `vif.ridge()` Calculates VIFs for "ridge" objects
* `precision()` Calculates measures of precision and shrinkage

More importantly, the `ridge` functions also calculate and returns the associated covariance matrices of each of the ridge estimates, allowing precision to be studied and displayed graphically.

This provides the support for the main plotting functions in the package:

* `traceplot()`: Traditional univariate ridge trace plots
* `plot.ridge()`: Bivariate ridge trace plots, showing the covariance ellipse of the estimated coefficients.
* `pairs.ridge()`: All pairwise bivariate ridge trace plots
* `plot3d.ridge()`: 3D ridge trace plots with ellipsoids
* `plot.precision()`: Plots a measure of precsion vs. one of shrinkage
* `plot.vif.ridge()`: Plots variance inflation factors

In addition, the `pca()` method for `"ridge"` objects transforms the coefficients and covariance matrices of a ridge object from predictor space to the equivalent, but more interesting space of the PCA of $X^\top X$ or the SVD of $X$. The main plotting functions also work for these objects, of class `c("ridge", "pcaridge")`.

* `biplot.pcaridge()`: Adds variable vectors to the bivariate plots of coefficients in PCA space


Finally, the functions `precision()` and `vif.ridge()` provide other useful measures and plots.

## Installation

+-------------------+-----------------------------------------------------------------------------+
| CRAN version      | `install.packages("genridge")`                                              |
+-------------------+-----------------------------------------------------------------------------+
| R-universe        | `install.packages("genridge", repos = c('https://friendly.r-universe.dev')` |
+-------------------+-----------------------------------------------------------------------------+
| Development       | `remotes::install_github("friendly/genridge")`                              |
| version           |                                                                             |
+-------------------+-----------------------------------------------------------------------------+

## Examples



The classic example for ridge regression is Longley's (1967) data, consisting of 7 economic variables, observed yearly from 1947 to 1962 (n=16), in the data frame `datasets::longley`.
The goal is to predict `Employed` from `GNP`, `Unemployed`, `Armed.Forces`, `Population`, `Year`,
`GNP.deflator`.

These data, constructed to illustrate numerical problems in least squares software at the time, are (purposely) perverse, in that: 

* each variable is a time series so that there is clearly a lack of independence among predictors.
* worse, there is also some _structural collinearity_ among the variables `GNP`, `Year`, `GNP.deflator`, `Population`, e.g., `GNP.deflator` is a multiplicative factor to account for inflation.

```{r longley1}
data(longley)
str(longley)
```

Shrinkage values, can be specified using either $\lambda$ (where $\lambda = 0$ corresponds to OLS),
or equivalent effective degrees of freedom.
This quantifies the tradeoff between bias and variance for predictive modeling, 
where OLS has low bias, but can have large predictive variance.

`ridge()` returns a matrix containing the coefficients for each predictor for each shrinkage value
and other quantities.
```{r longley2}
lambda <- c(0, 0.005, 0.01, 0.02, 0.04, 0.08)
lridge <- ridge(Employed ~ GNP + Unemployed + Armed.Forces + Population + Year + GNP.deflator, 
		data=longley, lambda=lambda)
lridge
```



### Variance Inflation Factors

The effects of collinearity can be measured by a variance inflation factor (VIF), the ratio
of the sampling variances of the coefficients, relative to what they would be if all
predictors were uncorrelated, given by 
$$
\text{VIF}(\beta_i) = \frac{1}{1 - R^2_{i | \text{others}}} \; ,
$$
where "others" represents all other predictors except $X_i$.

`vif()` for a `"ridge"` object calculates variance inflation factors for all
values of the ridge constant. You can see that for OLS, nearly all VIF values
are dangerously high. With a ridge factor of 0.04 or greater, variance inflation has been
considerably reduced for a few of the predictors.

```{r vif}
vridge <- vif(lridge)
vridge
```

`vif()` returns a `"vif.ridge"` object for which there is a plot method:

```{r plot-vif}
clr <-  c("black", "red", "darkgreen","blue", "cyan4", "magenta")
pch <- c(15:18, 7, 9)
plot(vridge, X = "df", Y="sqrt",
     col=clr, pch=pch, cex = 1.2,
     xlim = c(4, 6.5))
```


### Univariate trace plots

A standard, univariate, `traceplot()` simply plots the estimated coefficients for each predictor
against the shrinkage factor, $\lambda$.

```{r longley-tp1}
#| fig.width = 7,
#| echo = -1,
#| fig.cap = "Univariate ridge trace plot for the coefficients of predictors of Employment in Longley’s data via ridge regression, with ridge constants k = 0, 0.005, 0.01, 0.02, 0.04, 0.08."
par(mar=c(4, 4, 1, 1)+ 0.1)
traceplot(lridge, xlim = c(-0.02, 0.08))
```

The dotted lines show choices for the ridge
constant by two commonly used criteria to balance bias against precision
due to **HKB**: Hoerl, Kennard, and Baldwin (1975) and **LW**: Lawless and Wang (1976). 
These values (along with a generalized cross-validation value GCV) are also stored
in the `"ridge"` object,

```{r HKB}
c(HKB=lridge$kHKB, LW=lridge$kLW, GCV=lridge$kGCV)
# these are also stored in 'criteria' for use with plotting methods
criteria <- lridge$criteria
```

These values seem rather small, but note that the coefficients for `Year` and `GNP` are 
shrunk considerably.

### Alternative plot
It is sometimes easier to interpret the plot when coefficients are plotted against the equivalent
degrees of freedom, where $\lambda = 0$ corresponds to 6 degrees of freedom in the parameter
space of six predictors. Note that the values of $\lambda$ chosen here were approximately 
chosen on a log scale. Using the scaling `X="df"` here makes the points more nearly equally
spaced.

```{r longley-tp2}
#| fig.width = 7,
#| echo = -1,
#| fig.cap = "Univariate ridge trace plot of coefficients against effective degrees of freedom."
par(mar=c(4, 4, 1, 1)+ 0.1)
traceplot(lridge, X="df", xlim = c(4, 6.5))
```

**But wait: This is the wrong plot!** These plots show the trends in increased bias associated with larger $\lambda$, but they do **not**
show the accompanying decrease in variance (increase in precision).
For that, we need to consider the variances and covariances of the estimated coefficients.
The univariate trace plot is the wrong graphic form for what is essentially a _multivariate_ problem,
where we would like to visualize how _both_ coefficients and their variances change with
$\lambda$.



### Bivariate trace plots

The bivariate analog of the trace plot suggested by Friendly (2013) plots **bivariate
confidence ellipses** for pairs of coefficients. Their centers, $(\widehat{\beta}_i, \widehat{\beta}_j)$ show the estimated coefficients, and their size and shape
indicate sampling variance, $\widehat{\text{Var}} (\mathbf{\widehat{\beta}}_{ij})$.
Here, we plot those for `GNP` against
four of the other predictors.

```{r longley-plot-ridge}
#| out.width = "80%",
#| fig.show = "hold",
#| fig.cap = "Bivariate ridge trace plots for the coefficients of four predictors against the coefficient for GNP in Longley’s data, with λ = 0, 0.005, 0.01, 0.02, 0.04, 0.08. In most cases, the coefficients are driven toward zero, but the bivariate plot also makes clear the reduction in variance, as well as the bivariate path of shrinkage."
op <- par(mfrow=c(2,2), mar=c(4, 4, 1, 1)+ 0.1)
clr <-  c("black", "red", "darkgreen","blue", "cyan4", "magenta")
pch <- c(15:18, 7, 9)
lambdaf <- c(expression(~widehat(beta)^OLS), ".005", ".01", ".02", ".04", ".08")

for (i in 2:5) {
	plot(lridge, variables=c(1,i), 
	     radius=0.5, cex.lab=1.5, col=clr, 
	     labels=NULL, fill=TRUE, fill.alpha=0.2)
	text(lridge$coef[1,1], lridge$coef[1,i], 
	     expression(~widehat(beta)^OLS), cex=1.5, pos=4, offset=.1)
	text(lridge$coef[-1,c(1,i)], lambdaf[-1], pos=3, cex=1.3)
}
par(op)
```

As can be seen, the coefficients for each pair of predictors trace a path generally in toward
the origin $(0, 0)$, and the covariance ellipses get smaller, indicating increased precision.

The `pairs()` method for `"ridge"` objects shows all pairwise views in scatterplot matrix form.

```{r longley-pairs}
#| out.width = "90%",
#| fig.cap = "Scatterplot matrix of bivariate ridge trace plots"
pairs(lridge, radius=0.5, diag.cex = 2, 
      fill = TRUE, fill.alpha = 0.1)
```

See Friendly et-al. (2013) for other examples of how elliptical thinking can lead to insights
in statistical problems.

### Visualizing the bias-variance tradeoff

The function `precision()` calculates a number of measures of the effect of shrinkage of the
coefficients on the estimated sampling variance. Larger shrinkage $\lambda$ should lead
to smaller $\widehat{\text{Var}} (\mathbf{\widehat{\beta}})$, indicating increased precision.
See: `help(precision)` for details.

```{r precision}
precision(lridge)
```

`norm.beta` $= \lVert\mathbf{\beta}\rVert / \max{\lVert\mathbf{\beta}\rVert}$ is a measure of shrinkage, and `det`
$= \log{| \text{Var}(\mathbf{\beta}) |}$,
is a measure of variance of the coefficients (inverse of precision). Plotting these against
each other gives a direct view of the tradeoff between bias and precision.

```{r precision-plot}
#| fig.show = "hold",
#| fig.cap = "Plot of log(Variance) vs. shrinkage to show the tradeoff between bias and variance."
pridge <- precision(lridge)
op <- par(mar=c(4, 4, 1, 1) + 0.2)
library(splines)
with(pridge, {
	plot(norm.beta, det, type="b", 
	cex.lab=1.25, pch=16, cex=1.5, col=clr, lwd=2,
  xlab='shrinkage: ||b|| / max(||b||)',
	ylab='variance: log |Var(b)|')
	text(norm.beta, det, lambdaf, cex=1.25, pos=c(rep(2,length(lambda)-1),4), xpd = TRUE)
	text(min(norm.beta), max(det), "log |Variance| vs. Shrinkage", cex=1.5, pos=4)
	})
mod <- lm(cbind(det, norm.beta) ~ bs(lambda, df=5), data=pridge)
x <- data.frame(lambda=c(lridge$kHKB, lridge$kLW))
fit <- predict(mod, x)
points(fit[,2:1], pch=15, col=gray(.50), cex=1.5)
text(fit[,2:1], c("HKB", "LW"), pos=3, cex=1.5, col=gray(.50))
par(op)
```

These plots are now provided in the `plot()` method for (class `"precision"`) objects returned by `precision()`.
This plots the measure `norm.beta` on the horizontal axis vs. any of 
the variance measures `det`, `trace`, or `max.eig`, and labels the points with either `k` or `df`.
See `help("precision")` for the definitions of these variance measures.

A plot similar to that above can be produced as shown below, but here labeling the points with
effective degrees of freedom. The shape of the curve is quite similar.

```{r precision-plot2,echo=-1}
#| fig.cap = "Plot of det(Variance) vs. shrinkage (`norm.beta`) to show the tradeoff between bias and variance using the `plot()` method for `'precision'` objects. Points are labeled with the effective degrees of freedom."
op <- par(mar=c(4, 4, 1, 1) + 0.2)
plot(pridge, labels = "df", label.prefix="df:", criteria = criteria)
```



## Low-rank views

Just as principal components analysis gives low-dimensional views of a data set, PCA can
be useful to understand ridge regression.

The  `pca` method transforms a `"ridge"` object
from parameter space, where the estimated coefficients are
$\beta_k$ with covariance matrices $\Sigma_k$, to the
principal component space defined by the right singular vectors, $V$,
of the singular value decomposition of the scaled predictor matrix, $X$.

```{r pca-traceplot}
#| echo = -1
par(mar=c(4, 4, 1, 1)+ 0.1)
plridge <- pca(lridge)
plridge
traceplot(plridge)
```

What is perhaps surprising is that the coefficients for the first 4 components are not shrunk at all.
Rather, the effect of shrinkage is seen only on the _last two dimensions_.
These are the
directions that contribute most to collinearity, for which other visualization methods have
been proposed (Friendly & Kwan 2009).

The `pairs()` plot illustrates the _joint_ effects: the principal components of
$\mathbf{X}$ are uncorrelated, so the ellipses are all aligned with the coordinate axes
and the ellipses largely coincide for dimensions 1 to 4:
```{r pca-pairs}
#| out.width = "80%"
pairs(plridge)
```

If we focus on the plot of dimensions `5:6`, we can see where all the shrinkage action
is in this representation. Generally, the predictors that are related to the smallest
dimension (6) are shrunk quickly at first.
```{r pca-dim56}
#| echo = -1
par(mar=c(4, 4, 1, 1)+ 0.1)
plot(plridge, variables=5:6, fill = TRUE, fill.alpha=0.2)
text(plridge$coef[, 5:6], 
	   label = lambdaf, 
     cex=1.5, pos=4, offset=.1)
```

### Biplot view

Finally, we can project the _predictor variables_ into the PCA space of the _smallest dimensions_,
where the shrinkage action mostly occurs to see how the predictor variables relate to these dimensions.

`biplot.pcaridge()` supplements the standard display of the covariance ellipsoids for a ridge regression problem in PCA/SVD space with labeled arrows showing the contributions of the original variables to the dimensions plotted. The length of the arrows reflects proportion of variance
that each predictors shares with the components.

The biplot view showing the dimensions corresponding to the two smallest singular values is particularly useful for understanding how the predictors contribute to shrinkage in ridge regression. Here, `Year` and `Population` largely contribute to `dim 5`; a contrast
between (`Year`, `Population`) and `GNP` contributes to `dim 6`.

```{r biplot}
#| fig.show="hold",
#| fig.cap: "Biplot view of the ridge trace plot for the smallest two dimensions, where the effects of shrinkage are most apparent."
op <- par(mar=c(4, 4, 1, 1) + 0.2)
biplot(plridge, radius=0.5, 
       ref=FALSE, asp=1, 
       var.cex=1.15, cex.lab=1.3, col=clr,
       fill=TRUE, fill.alpha=0.2, prefix="Dimension ")
text(plridge$coef[,5:6], lambdaf, pos=2, cex=1.3)
par(op)
```

## Other examples

The genridge package contains four data sets, each with its own examples;
e.g., you can try `example(Acetylene)`.

```{r datasets}
vcdExtra::datasets(package="genridge")
```



## References

Friendly, M. (2011). Generalized Ridge Trace Plots: Visualizing Bias _and_ Precision with the `genridge` R package. SCS Seminar, Jan., 2011. Slides:
[gentalk.pdf](http://euclid.psych.yorku.ca/datavis/papers/gentalk.pdf);
[gentalk-2x2.pdf](http://euclid.psych.yorku.ca/datavis/papers/gentalk-2x2.pdf)

Friendly, M. (2013).
The Generalized Ridge Trace Plot: Visualizing Bias _and_ Precision.
_Journal of Computational and Graphical Statistics_, **22**(1), 50-68,
[DOI link](http://dx.doi.org/10.1080/10618600.2012.681237), Online:
[genridge-jcgs.pdf](https://www.datavis.ca/papers/genridge-jcgs.pdf),
Supp. materials: [genridge-supp.zip](http://datavis.ca/papers/genridge-supp.zip)

Friendly, M., and Kwan, E. (2009), Where’s Waldo: Visualizing Collinearity Diagnostics, 
_The American Statistician_, **63**(1), 56–65,
[DOI link](https://doi.org/10.1198/tast.2009.0012),
Online: [viscollin-tast.pdf](http://datavis.ca/papers/viscollin-tast.pdf),
Supp. materials: [http://datavis.ca/papers/viscollin/](http://datavis.ca/papers/viscollin/).

Friendly, M., Monette, G., & Fox, J. (2013). Elliptical Insights: Understanding Statistical Methods Through Elliptical Geometry. _Statistical Science_, **28**(1), 1–39. https://doi.org/10.1214/12-STS402

Golub G.H., Heath M., Wahba G. (1979) Generalized cross-validation as a method for choosing a good ridge parameter. 
_Technometrics_, **21**:215–223. https://doi.org/10.2307/1268518.

Hoerl, A. E., Kennard, R. W., and Baldwin, K. F. (1975), 
Ridge Regression: Some Simulations, _Communications in Statistics_, **4**, 105–123.

Lawless, J. F., and Wang, P. (1976), A Simulation Study of Ridge and Other Regression Estimators, _Communications in Statistics_, **5**, 307–323.

Longley, J. W.  (1967) An appraisal of least-squares programs from the point of view of the user. 
_Journal of the American Statistical Association_,
**62**, 819–841.

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