cv.Rmd

---
title: "Cross-validating regression models"
subtitle: "Introduction to the cv package"
author: "John Fox and Georges Monette"
date: "`r Sys.Date()`"
package: cv
output: 
  rmarkdown::html_vignette:
  fig_caption: yes
bibliography: ["cv.bib"]
csl: apa.csl
vignette: >
  %\VignetteIndexEntry{Cross-validating regression models}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  message = TRUE,
  warning = TRUE,
  fig.align = "center",
  fig.height = 6,
  fig.width = 7,
  fig.path = "fig/",
  dev = "png",
  comment = "#>" #,
)

# save some typing
knitr::set_alias(w = "fig.width",
                 h = "fig.height",
                 cap = "fig.cap")

# colorize text: use inline as `r colorize(text, color)`
colorize <- function(x, color) {
  if (knitr::is_latex_output()) {
    sprintf("\\textcolor{%s}{%s}", color, x)
  } else if (knitr::is_html_output()) {
    sprintf("<span style='color: %s;'>%s</span>", color,
      x)
  } else x
}


.opts <- options(digits = 5)
```

This vignette covers the basics of using the **cv** package for cross-validation. The first, and major, section of the vignette consists of examples that fit linear and generalized linear models to data sets with independently sampled cases. Brief sections follow on replicating cross-validation, manipulating the objects produced by `cv()` and related functions, and employing parallel computations.

There are several other vignettes associated with the **cv** package: on cross-validating mixed-effects models; on cross-validating model-selection procedures; and on technical details, such as computational procedures.

## Cross-validation

Cross-validation (CV) is an essentially simple and intuitively reasonable approach to estimating the predictive accuracy of regression models. CV is developed in many standard sources on regression modeling and "machine learning"---we particularly recommend @JamesEtAl:2021 [Secs. 5.1, 5.3]---and so we will describe the method only briefly here before taking up computational issues and some examples. See @ArlotCelisse:2010 for a wide-ranging, if technical, survey of cross-validation and related methods that emphasizes the statistical properties of CV.

Validating research by replication on independently collected data is a common scientific norm. Emulating this process in a single study by data-division is less common: The data are randomly divided into two, possibly equal-size, parts; the first part is used to develop and fit a statistical model; and then the second part is used to assess the adequacy of the model fit to the first part of the data. Data-division, however, suffers from two problems: (1) Dividing the data decreases the sample size and thus increases sampling error; and (2), even more disconcertingly, particularly in smaller samples, the results can vary substantially based on the random division of the data: See @Harrell:2015 [Sec. 5.3] for this and other remarks about data-division and cross-validation.

Cross-validation speaks to both of these issues. In CV, the data are randomly divided as equally as possible into several, say $k$, parts, called "folds." The statistical model is fit $k$ times, leaving each fold out in turn. Each fitted model is then used to predict the response variable for the cases in the omitted fold. A CV criterion or "cost" measure, such as the mean-squared error ("MSE") of prediction, is then computed using these predicted values. In the extreme $k = n$, the number of cases in the data, thus omitting individual cases and refitting the model $n$ times---a procedure termed "leave-one-out (LOO) cross-validation."

Because the $n$ models are each fit to $n - 1$ cases, LOO CV produces a nearly unbiased estimate of prediction error. The $n$ regression models are highly statistical dependent, however, based as they are on nearly the same data, and so the resulting estimate of prediction error has relatively large variance. In contrast, estimated prediction error for $k$-fold CV with $k = 5$ or $10$ (commonly employed choices) are somewhat biased but have smaller variance. It is also possible to correct $k$-fold CV for bias (see below).

## Examples

### Polynomial regression for the `Auto` data

The data for this example are drawn from the **ISLR2** package for R, associated with @JamesEtAl:2021. The presentation here is close (though not identical) to that in the original source [@JamesEtAl:2021 Secs. 5.1, 5.3], and it demonstrates the use of the `cv()` function in the **cv** package.[^boot] 

[^boot]: @JamesEtAl:2021 use the `cv.glm()` function in the **boot** package [@CantyRipley2022; @DavisonHinkley:1997]. Despite its name, `cv.glm()` is an independent function and not a method of a `cv()` generic function.

The `Auto` dataset contains information about 392 cars:

```{r Auto}
data("Auto", package="ISLR2")
head(Auto)
dim(Auto)
```
With the exception of `origin` (which we don't use here), these variables are largely self-explanatory, except possibly for units of measurement: for details see `help("Auto", package="ISLR2")`.

We'll focus here on the relationship of `mpg` (miles per gallon) to `horsepower`, as displayed in the following scatterplot:

```{r mpg-horsepower-scatterplot}
#| out.width = "100%",
#| fig.height = 6,
#| fig.cap = "`mpg` vs `horsepower` for the `Auto` data"
plot(mpg ~ horsepower, data=Auto)
```
The relationship between the two variables is monotone, decreasing, and nonlinear. Following @JamesEtAl:2021, we'll consider approximating the relationship by a polynomial regression, with the degree of the polynomial $p$ ranging from 1 (a linear regression) to 10.[^log-trans] Polynomial fits  for $p = 1$ to $5$ are shown in the following figure:

[^log-trans]: Although it serves to illustrate the use of CV, a polynomial is probably not the best choice here. Consider, for example the scatterplot for log-transformed `mpg` and `horsepower`, produced by `plot(mpg ~ horsepower, data=Auto, log="xy")` (execution of which is left to the reader).

```{r mpg-horsepower-scatterplot-polynomials}
#| out.width = "100%",
#| fig.height = 5,
#| fig.cap = "`mpg` vs `horsepower` for the `Auto` data"
plot(mpg ~ horsepower, data = Auto)
horsepower <- with(Auto,
                   seq(min(horsepower), max(horsepower),
                       length = 1000))
for (p in 1:5) {
  m <- lm(mpg ~ poly(horsepower, p), data = Auto)
  mpg <- predict(m, newdata = data.frame(horsepower = horsepower))
  lines(horsepower,
        mpg,
        col = p + 1,
        lty = p,
        lwd = 2)
}
legend(
  "topright",
  legend = 1:5,
  col = 2:6,
  lty = 1:5,
  lwd = 2,
  title = "Degree",
  inset = 0.02
)
```
The linear fit is clearly inappropriate; the fits for $p = 2$ (quadratic) through $4$ are very similar; and the fit for $p = 5$ may over-fit the data by chasing one or two relatively high `mpg` values at the right (but see the CV results reported below).

The following graph shows two measures of estimated (squared) error as a function of polynomial-regression degree: The mean-squared error ("MSE"), defined as $\mathsf{MSE} = \frac{1}{n}\sum_{i=1}^n (y_i - \widehat{y}_i)^2$, and the usual residual variance, defined as $\widehat{\sigma}^2 = \frac{1}{n - p - 1} \sum_{i=1}^n (y_i - \widehat{y}_i)^2$. The former necessarily declines with $p$ (or, more strictly, can't increase with $p$), while the latter gets slightly larger for the largest values of $p$, with the "best" value, by a small margin, for $p = 7$.

```{r mpg-horsepower-MSE-se}
#| out.width = "100%",
#| fig.height = 5,
#| fig.cap = "Estimated squared error as a function of polynomial degree, $p$"
library("cv") # for mse() and other functions

var <- mse <- numeric(10)
for (p in 1:10) {
  m <- lm(mpg ~ poly(horsepower, p), data = Auto)
  mse[p] <- mse(Auto$mpg, fitted(m))
  var[p] <- summary(m)$sigma ^ 2
}

plot(
  c(1, 10),
  range(mse, var),
  type = "n",
  xlab = "Degree of polynomial, p",
  ylab = "Estimated Squared Error"
)
lines(
  1:10,
  mse,
  lwd = 2,
  lty = 1,
  col = 2,
  pch = 16,
  type = "b"
)
lines(
  1:10,
  var,
  lwd = 2,
  lty = 2,
  col = 3,
  pch = 17,
  type = "b"
)
legend(
  "topright",
  inset = 0.02,
  legend = c(expression(hat(sigma) ^ 2), "MSE"),
  lwd = 2,
  lty = 2:1,
  col = 3:2,
  pch = 17:16
)
```
The code for this graph uses the `mse()` function from the **cv** package to compute the MSE for each fit.

#### Using `cv()`

The generic `cv()` function has an `"lm"` method, which by default performs $k = 10$-fold CV:
```{r cv-lm-1}
m.auto <- lm(mpg ~ poly(horsepower, 2), data = Auto)
summary(m.auto)

cv(m.auto)
```
The `"lm"` method by default uses `mse()` as the CV criterion and the Woodbury matrix identity [@Hager:1989] to update the regression with each fold deleted without having literally to refit the model. (Computational details are discussed in a separate vignette.) The function reports the CV estimate of MSE, a biased-adjusted estimate of the MSE (the bias adjustment is explained in the final section), and the MSE is also computed for the original, full-sample regression. Because the division of the data into 10 folds is random, `cv()` explicitly (randomly) generates and saves a seed for R's pseudo-random number generator, to make the results replicable. The user can also specify the seed directly via the `seed` argument to `cv()`.

To perform LOO CV, we can set the `k` argument to `cv()` to the number of cases in the data, here `k=392`, or, more conveniently, to `k="loo"` or `k="n"`:

```{r cv.lm-2`}
cv(m.auto, k = "loo")
```
For LOO CV of a linear model, `cv()` by default uses the hatvalues from the model fit to the full data for the LOO updates, and reports only the CV estimate of MSE. Alternative methods are to use the Woodbury matrix identity or the "naive" approach of literally refitting the model with each case omitted. All three methods produce exact results for a linear model (within the precision of floating-point computations):
```{r cv.lm-3}
cv(m.auto, k = "loo", method = "naive")

cv(m.auto, k = "loo", method = "Woodbury")
```
The `"naive"` and `"Woodbury"` methods also return the bias-adjusted estimate of MSE and the full-sample MSE, but bias isn't an issue for LOO CV.


#### Comparing competing models

The `cv()` function also has a method that can be applied to a list of regression models for the same data, composed using the `models()` function. For $k$-fold CV, the same folds are used for the competing models, which reduces random error in their comparison. This result can also be obtained by specifying a common seed for R's random-number generator while applying `cv()` separately to each model, but employing a list of models is more convenient for both $k$-fold and LOO CV (where there is no random component to the composition of the $n$ folds).

We illustrate with the polynomial regression models of varying degree for the `Auto` data (discussed previously), beginning by fitting and saving the 10 models:
```{r polyomial-models}
for (p in 1:10) {
  command <- paste0("m.", p, "<- lm(mpg ~ poly(horsepower, ", p,
                    "), data=Auto)")
  eval(parse(text = command))
}
objects(pattern = "m\\.[0-9]")
summary(m.2) # for example, the quadratic fit
```
The convoluted code within the loop to produce the 10 models insures that the model formulas are of the form, e.g., `mpg ~ poly(horsepower, 2)` rather than `mpg ~ poly(horsepower, p)`, which will be useful to us in a separate vignette, where we consider cross-validating the model-selection process.

We then apply `cv()` to the list of 10 models (the `data` argument is required):
```{r polynomial-regression-CV}
# 10-fold CV
cv.auto.10 <- cv(
  models(m.1, m.2, m.3, m.4, m.5,
         m.6, m.7, m.8, m.9, m.10),
  data = Auto,
  seed = 2120
)
cv.auto.10[1:2] # for the linear and quadratic models

# LOO CV
cv.auto.loo <- cv(models(m.1, m.2, m.3, m.4, m.5,
                         m.6, m.7, m.8, m.9, m.10),
                  data = Auto,
                  k = "loo")
cv.auto.loo[1:2] # linear and quadratic models
```
Because we didn't supply names for the models in the calls to the `models()` function, the names `model.1`, `model.2`, etc., are generated by the function.

Finally, we extract and graph the adjusted MSEs for $10$-fold CV and the MSEs for LOO CV (see the section below on manipulating `"cv"` objects:
```{r polynomial-regression-CV-graph}
#| out.width = "100%",
#| fig.height = 5,
#| fig.cap = "Cross-validated 10-fold and LOO MSE as a function of polynomial degree, $p$"
cv.mse.10 <- as.data.frame(cv.auto.10, 
                           rows="cv",             
                           columns="criteria"
                           )$adjusted.criterion
cv.mse.loo <- as.data.frame(cv.auto.loo, 
                           rows="cv",             
                           columns="criteria"
                           )$criterion
plot(
  c(1, 10),
  range(cv.mse.10, cv.mse.loo),
  type = "n",
  xlab = "Degree of polynomial, p",
  ylab = "Cross-Validated MSE"
)
lines(
  1:10,
  cv.mse.10,
  lwd = 2,
  lty = 1,
  col = 2,
  pch = 16,
  type = "b"
)
lines(
  1:10,
  cv.mse.loo,
  lwd = 2,
  lty = 2,
  col = 3,
  pch = 17,
  type = "b"
)
legend(
  "topright",
  inset = 0.02,
  legend = c("10-Fold CV", "LOO CV"),
  lwd = 2,
  lty = 2:1,
  col = 3:2,
  pch = 17:16
)
```

Alternatively, we can use the `plot()` method for `"cvModList"` objects to compare the models, though with separate graphs for 10-fold and LOO CV:

```{r, polynomial-regression-CV-graph-2, fig.show="hold"}
#| out.width = "45%",
#| fig.height = 6,
#| fig.cap = "Cross-validated 10-fold and LOO MSE as a function of polynomial degree, $p$"
plot(cv.auto.10, main="Polynomial Regressions, 10-Fold CV",
     axis.args=list(labels=1:10), xlab="Degree of Polynomial, p")
plot(cv.auto.loo, main="Polynomial Regressions, LOO CV",
     axis.args=list(labels=1:10), xlab="Degree of Polynomial, p")
```

In this example, 10-fold and LOO CV produce generally similar results, and also results that are similar to those produced by the estimated error variance $\widehat{\sigma}^2$ for each model, reported above (except for the highest-degree polynomials, where the CV results more clearly suggest over-fitting).

### Logistic regression for the `Mroz` data

The `Mroz` data set from the **carData** package [associated with @FoxWeisberg:2019] has been used by several authors to illustrate binary logistic regression; see, in particular @FoxWeisberg:2019. The data were originally drawn from the U.S. Panel Study of Income Dynamics and pertain to married women. Here are a few cases in the data set:
```{r Mroz-data}
data("Mroz", package = "carData")
head(Mroz, 3)
tail(Mroz, 3)
```
The response variable in the logistic regression is `lfp`, labor-force participation, a factor coded `"yes"` or `"no"`. The remaining variables are predictors:

* `k5`, number of children 5 years old of younger in the woman's household;
* `k618`, number of children between 6 and 18 years old;
* `age`, in years;
* `wc`, wife's college attendance, `"yes"` or `"no"`;
* `hc`, husband's college attendance;
* `lwg`, the woman's log wage rate if she is employed, or her *imputed* wage rate, if she is not [a variable that @FoxWeisberg:2019 show is problematically defined]; and
* `inc`, family income, in $1000s, exclusive of wife's income.

We use the `glm()` function to fit a binary logistic regression to the `Mroz` data:
```{r Mroz-logistic-regresion}
m.mroz <- glm(lfp ~ ., data = Mroz, family = binomial)
summary(m.mroz)

BayesRule(ifelse(Mroz$lfp == "yes", 1, 0),
          fitted(m.mroz, type = "response"))
```
In addition to the usually summary output for a GLM, we show the result of applying the `BayesRule()` function from the **cv** package to predictions derived from the fitted model. Bayes rule, which predicts a "success" in a binary regression model when the fitted probability of success [i.e., $\phi = \Pr(y = 1)$] is $\widehat{\phi} \ge .5$ and a "failure" if $\widehat{\phi} \lt .5$.[^BayesRule] The first argument to `BayesRule()` is the binary {0, 1} response, and the second argument is the predicted probability of success. `BayesRule()` returns the proportion of predictions that are *in error*, as appropriate for a "cost" function. 

[^BayesRule]: `BayesRule()` does some error checking; `BayesRule2()` is similar, but omits the error checking, and so can be faster for large problems.

The value returned by `BayesRule()` is associated with an "attribute" named `"casewise loss"` and set to `"y != round(yhat)"`, signifying that the Bayes rule CV criterion is computed as the mean of casewise values, here 0 if the prediction for a case matches the observed value and 1 if it does not (signifying a prediction error). The `mse()` function for numeric responses is also calculated as a casewise average. Some other criteria, such as the median absolute error, computed by the `medAbsErr()` function in the **cv** package, aren't averages of casewise components. The distinction is important because, to our knowledge, the statistical theory of cross-validation, for example, in @DavisonHinkley:1997, @BatesHastieTibshirani:2023, and @ArlotCelisse:2010, is developed for CV criteria like MSE that are means of casewise components. As a consequence, we limit computation of bias adjustment and confidence intervals (see below) to criteria that are casewise averages.

In this example, the fitted logistic regression incorrectly predicts 31% of the responses; we expect this estimate to be optimistic given that the model is used to "predict" the data to which it is fit.

The `"glm"` method for `cv()` is largely similar to the `"lm"` method, although the default algorithm, selected explicitly by `method="exact"`, refits the model with each fold removed (and is thus equivalent to `method="naive"` for `"lm"` models). For generalized linear models, `method="Woodbury"` or (for LOO CV) `method="hatvalues"` provide approximate results (see the computational and technical vignette for details):

```{r cv-Mroz-10-fold}
cv(m.mroz, criterion = BayesRule, seed = 248)

cv(m.mroz,
   criterion = BayesRule,
   seed = 248,
   method = "Woodbury")
```
To ensure that the two methods use the same 10 folds, we specify the seed for R's random-number generator explicitly; here, and as is common in our experience, the `"exact"` and `"Woodbury"` algorithms produce nearly identical results. The CV estimates of prediction error are slightly higher than the estimate based on all of the cases.

The printed output includes a 95% confidence interval for the bias-adjusted Bayes rule CV criterion. @BatesHastieTibshirani:2023 show that these confidence intervals are unreliable for models fit to small samples, and by default `cv()` computes them only when the sample size is 400 or larger and when the CV criterion employed is an average of casewise components, as is the case for Bayes rule. See the final section of the vignette for details of the computation of confidence intervals for bias-adjusted CV criteria.

Here are results of applying LOO CV to the Mroz model, using both the exact and the approximate methods:
```{r cv-Mroz-LOO}
cv(m.mroz, k = "loo", criterion = BayesRule)

cv(m.mroz,
   k = "loo",
   criterion = BayesRule,
   method = "Woodbury")

cv(m.mroz,
   k = "loo",
   criterion = BayesRule,
   method = "hatvalues")
```
To the number of decimal digits shown, the three methods produce identical results for this example.

## Replicating cross-validation

Assuming that the number of cases $n$ is a multiple of the number of folds $k$---a slightly simplifying assumption---the number of possible partitions of cases into folds is $\frac{n!}{[(n/k)!]^k}$, a number that grows very large very quickly. For example, for $n = 10$ and $k = 5$, so that the folds are each of size $n/k = 2$, there are $113,400$ possible partitions; for $n=100$ and $k=5$, where $n/k = 20$, still a small problem, the number of possible partitions is truly astronomical, $1.09\times 10^{66}$.

Because the partition into folds that's employed is selected randomly, the resulting CV criterion estimates are subject to sampling error. (An exception is LOO cross-validation, which is not at all random.) To get a sense of the magnitude of the sampling error, we can repeat the CV procedure with different randomly selected partitions into folds. All of the CV functions in the **cv** package are capable of repeated cross-validation, with the number of repetitions controlled by the `reps` argument, which defaults to `1`.

Here, for example, is 10-fold CV for the Mroz logistic regression, repeated 5 times:

```{r mroz-reps}
cv(
  m.mroz,
  criterion = BayesRule,
  seed = 248,
  reps = 5,
  method = "Woodbury"
)
```
When `reps` > `1`, the result returned by `cv()` is an object of class `"cvList"`---literally a list of `"cv"` objects. The results are reported for each repetition and then averaged across repetitions, with the standard deviations of the CV criterion and the biased-adjusted CV criterion given in parentheses. In this example, there is therefore little variation across repetitions, increasing our confidence in the reliability of the results. 

Notice that the seed that's set in the `cv()` command pertains to the first repetition and the seeds for the remaining repetitions are then selected pseudo-randomly.[^replicates] Setting the first seed, however, makes the entire process easily replicable, and the seed for each repetition is stored in the corresponding element of the `"cvList"` object (which isn't, however, saved in the example).

[^replicates]: Because of the manner in which the computation is performed, the order of the replicates in the `"cvList"` object returned by `cv()` isn't the same as the order in which the replicates are computed. Each element of the result, however, is a `"cv"` object with the correct random-number seed saved, and so this technical detail can be safely ignored. The individual `"cv"` objects are printed in the order in which they are stored rather than the order in which they are computed.

It's also possible to replicate CV when comparing competing models via the `cv()` method for `"modList"` objects. Recall our comparison of polynomial regressions of varying degree fit to the `Auto` data; we performed 10-fold CV for each of 10 models. Here, we replicate that process 5 times for each model and graph the results:
```{r model-comparison-with-reps}
#| out.width = "100%",
#| fig.height = 5,
#| fig.cap = " Replicated cross-validated 10-fold CV as a function of polynomial degree, $p$"
cv.auto.reps <- cv(
  models(m.1, m.2, m.3, m.4, m.5,
         m.6, m.7, m.8, m.9, m.10),
  data = Auto,
  seed = 8004,
  reps = 5
)
plot(cv.auto.reps)
```
The graph shows both the average CV criterion and its range for each of the competing models.

## Manipulating "cv" and related objects

The `cv()` functions returns an object of class `"cv"`---or a closely related object, for example, of class `"cvList"`---which contains a variety of information about the results of a CV procedure. The **cv** package provides `as.data.frame()` methods to put this information in the form of data frames for further examination and analysis.[^Friendly] There is also a `summary()` method for extracting and summarizing the information in the resulting data frames.

[^Friendly]: These `as.data.frame()` methods were created at the suggestion of Michael Friendly of York University as a mechanism for making the output of the various `cv()` methods more accessible to the user.

We'll illustrate with the replicated CV that we performed for the 10 polynomial-regression models fit to the `Auto` data:
```{r recall-cv.auto.reps}
cv.auto.reps
class(cv.auto.reps)
```
In this case, there are 5 replications of 10-fold CV.

Converting `cv.auto.reps` into a data frame produces, by default:
```{r as.data.frame}
D <- as.data.frame(cv.auto.reps)
dim(D)
class(D)
```
The resulting data frame has $\mathsf{replications} \times (\mathsf{folds} + 1) \times \mathsf{models} = 5 \times (10 + 1) \times 10 = 550$ rows, the first few of which are
```{r}
head(D)
```
All of these rows pertain to the first model and the first replication, and fold = 0 indicates the overall results for the first replication of the first model.

The regression coefficients appear as columns in the data frame. Because the first model includes only an intercept and a linear polynomial term, the other coefficients are all `NA`.

It's possible to suppress the regression coefficients by specifying the argument `columns="criteria"` to `as.data.frame()`:
```{r }
D <- as.data.frame(cv.auto.reps, columns="criteria")
head(D)
head(subset(D, fold == 0))
```

The `summary()` method for `"cvDataFrame"` and related objects has a formula interface, and may be used, for example, as follows:
```{r summary.cvDataFrame}
summary(D, adjusted.criterion ~ model + rep) # fold "0" only
summary(D, criterion ~ model + rep, 
        include="folds") # mean over folds
summary(D, criterion ~ model + rep, fun=sd, 
        include="folds")
```
See `?summary.cvDataFrame` for details.

## Parallel computations

The CV functions in the **cv** package are all capable of performing parallel computations by setting the `ncores` argument (specifying the number of computer cores to be used) to a number > `1` (which is the default). Parallel computation can be advantageous for large problems, reducing the execution time of the program.

To illustrate, let's time model selection in Mroz's logistic regression, repeating the computation as performed previously (but with LOO CV to lengthen the calculation) and then doing it in parallel using 2 cores:
```{r parallel-computation, cache=TRUE}
system.time(
  m.mroz.sel.cv <- cv(
    selectStepAIC,
    Mroz,
    k = "loo",
    criterion = BayesRule,
    working.model = m.mroz,
    AIC = FALSE
  )
)

system.time(
  m.mroz.sel.cv.p <- cv(
    selectStepAIC,
    Mroz,
    k = "loo",
    criterion = BayesRule,
    working.model = m.mroz,
    AIC = FALSE,
    ncores = 2
  )
)
all.equal(m.mroz.sel.cv, m.mroz.sel.cv.p)
```
On our computer, the parallel computation with 2 cores is nearly twice as fast, and produces the same result as the non-parallel computation.

```{r coda, include = FALSE}
options(.opts)
```

## References