# pre: an R package for deriving prediction rule ensembles

**pre** is an **R** package for deriving prediction rule ensembles for
binary, multinomial, (multivariate) continuous, count and survival
responses. Input variables may be numeric, ordinal and categorical. An
extensive description of the implementation and functionality is
provided in Fokkema (2017). The package largely implements the algorithm
for deriving prediction rule ensembles as described in Friedman &
Popescu (2008), with several adjustments:

- The package is completely
**R**based, allowing users better access to the results and more control over the parameters used for generating the prediction rule ensemble. - The unbiased tree induction algorithms of Hothorn, Hornik, & Zeileis (2006) is used for deriving prediction rules, by default. Alternatively, the (g)lmtree algorithm of Zeileis, Hothorn, & Hornik (2008) can be employed, or the classification and regression tree (CART) algorithm of Breiman, Friedman, Olshen, & Stone (1984).
- The package supports a wider range of response variable types.
- The initial ensembles may be generated as in bagging, boosting and/or random forests.
- Hinge functions of predictor variables may be included as
baselearners, as in the multivariate adaptive regression splines
method of Friedman (1991), using function
`gpe()`

.

Note that **pre** is under development, and much work still needs to be
done. Below, a short introductory example is provided. Fokkema (2017)
provides an extensive description of the fitting procedures implemented
in function `pre()`

and example analyses with more extensive
explanations.

## Example: Predicting ozone levels

To get a first impression of how function `pre()`

works, we will fit a
prediction rule ensemble to predict Ozone levels using the `airquality`

dataset. We fit a prediction rule ensemble using function `pre()`

:

```
library("pre")
airq <- airquality[complete.cases(airquality), ]
set.seed(42)
airq.ens <- pre(Ozone ~ ., data = airq)
```

Note that it is necessary to set the random seed, to allow for later replication of the results, because the fitting procedure depends on random sampling of training observations.

We can print the resulting ensemble (alternatively, we could use the
`print`

method):

```
airq.ens
#>
#> Final ensemble with cv error within 1se of minimum:
#> lambda = 3.543968
#> number of terms = 12
#> mean cv error (se) = 352.3834 (99.13981)
#>
#> cv error type : Mean-Squared Error
#>
#> rule coefficient description
#> (Intercept) 68.48270406 1
#> rule191 -10.97368179 Wind > 5.7 & Temp <= 87
#> rule173 -10.90385520 Wind > 5.7 & Temp <= 82
#> rule42 -8.79715538 Wind > 6.3 & Temp <= 84
#> rule204 7.16114780 Wind <= 10.3 & Solar.R > 148
#> rule10 -4.68646144 Temp <= 84 & Temp <= 77
#> rule192 -3.34460037 Wind > 5.7 & Temp <= 87 & Day <= 23
#> rule51 -2.27864287 Wind > 5.7 & Temp <= 84
#> rule93 2.18465676 Temp > 77 & Wind <= 8.6
#> rule74 -1.36479546 Wind > 6.9 & Temp <= 84
#> rule28 -1.15326093 Temp <= 84 & Wind > 7.4
#> rule25 -0.70818399 Wind > 6.3 & Temp <= 82
#> rule166 -0.04751152 Wind > 6.9 & Temp <= 82
```

The firest few lines of the printed results provide the penalty
parameter value (*λ*) employed for selecting the final ensemble. By
default, the ‘1-SE’ rule is used for selecting *λ*; this default can be
overridden by employing the `penalty.par.val`

argument of the `print`

method and other functions in the package. Note that the printed
cross-validated error is calculated using the same data as was used for
generating the rules and likely provides an overly optimistic estimate
of future prediction error. To obtain a more realistic prediction error
estimate, we will use function `cvpre()`

later on.

Next, the printed results provide the rules and linear terms selected in
the final ensemble, with their estimated coefficients. For rules, the
`description`

column provides the conditions. For linear terms (which
were not selected in the current ensemble), the winsorizing points used
to reduce the influence of outliers on the estimated coefficient would
be printed in the `description`

column. The `coefficient`

column
presents the estimated coefficient. These are regression coefficients,
reflecting the expected increase in the response for a unit increase in
the predictor, keeping all other predictors constant. For rules, the
coefficient thus reflects the difference in the expected value of the
response when the conditions of the rule are met, compared to when they
are not.

Using the `plot`

method, we can plot the rules in the ensemble as simple
decision trees. Here, we will request the nine most important
baselearners through specification of the `nterms`

argument. Through the
`cex`

argument, we specify the size of the node and path labels:

`plot(airq.ens, nterms = 9, cex = .5)`

Using the `coef`

method, we can obtain the estimated coefficients for
each of the baselearners (we only print the first six terms here for
space considerations):

```
coefs <- coef(airq.ens)
coefs[1:6, ]
#> rule coefficient description
#> 201 (Intercept) 68.482704 1
#> 167 rule191 -10.973682 Wind > 5.7 & Temp <= 87
#> 150 rule173 -10.903855 Wind > 5.7 & Temp <= 82
#> 39 rule42 -8.797155 Wind > 6.3 & Temp <= 84
#> 179 rule204 7.161148 Wind <= 10.3 & Solar.R > 148
#> 10 rule10 -4.686461 Temp <= 84 & Temp <= 77
```

We can generate predictions for new observations using the `predict`

method (only the first six predicted values are printed here for space
considerations):

```
predict(airq.ens, newdata = airq[1:6, ])
#> 1 2 3 4 7 8
#> 32.53896 24.22456 24.22456 24.22456 31.38570 24.22456
```

Using function `cvpre()`

, we can assess the expected prediction error of
the fitted PRE through *k*-fold cross validation (*k* = 10, by default,
which can be overridden through specification of the `k`

argument):

```
set.seed(43)
airq.cv <- cvpre(airq.ens)
#> $MSE
#> MSE se
#> 369.2010 88.7574
#>
#> $MAE
#> MAE se
#> 13.64524 1.28985
```

The results provide the mean squared error (MSE) and mean absolute error
(MAE) with their respective standard errors. These results are saved for
later use in `aiq.cv$accuracy`

. The cross-validated predictions, which
can be used to compute alternative estimates of predictive accuracy, are
saved in `airq.cv$cvpreds`

. The folds to which observations were
assigned are saved in `airq.cv$fold_indicators`

.

## Additional tools for interpretation

Package **pre** provides several additional tools for interpretation of
the final ensemble. These may be especially helpful for complex
ensembles containing many rules and linear terms.

### Importance measures

We can assess the relative importance of input variables as well as
baselearners using the `importance()`

function:

`imps <- importance(airq.ens, round = 4)`

As we already observed in the printed ensemble, the plotted variable
importances indicate that Temperature and Wind are most strongly
associated with Ozone levels. Solar.R and Day are also associated with
Ozone levels, but much less strongly. Variable Month is not plotted,
which means it obtained an importance of zero, indicating that it is not
associated with Ozone levels. We already observed this in the printed
ensemble: Month did not appear in any of the selected terms. The
variable and baselearner importances are saved for later use in
`imps$varimps`

and `imps$baseimps`

, respectively.

### Explaining individual predictions

We can obtain explanations of the predictions for individual
observations using function `explain()`

:

```
par(mfrow = c(1, 2))
expl <- explain(airq.ens, newdata = airq[1:2, ], cex = .8)
```

The values of the rules and linear terms for each observation are saved
in `expl$predictors`

, their contributions in `expl$contribution`

and the
predicted values in `expl$predicted.value`

.

### Partial dependence plots

We can obtain partial dependence plots to assess the effect of single
predictor variables on the outcome using the `singleplot()`

function:

`singleplot(airq.ens, varname = "Temp")`

We can obtain partial dependence plots to assess the effects of pairs of
predictor variables on the outcome using the `pairplot()`

function:

`pairplot(airq.ens, varnames = c("Temp", "Wind"))`

Note that creating partial dependence plots is computationally intensive
and computation time will increase fast with increasing numbers of
observations and numbers of variables. `**R**`

package `**plotmo**`

(Milborrow (2018)) provides more efficient functions for plotting
partial dependence, which also support `pre`

models.

If the final ensemble does not contain many terms, inspecting individual
rules and linear terms through the `print`

method may be more
informative than partial dependence plots. One of the main advantages of
prediction rule ensembles is their interpretability: the predictive
model contains only simple functions of the predictor variables (rules
and linear terms), which are easy to grasp. Partial dependence plots are
often much more useful for interpretation of complex models, like random
forests for example.

### Assessing presence of interactions

We can assess the presence of interactions between the input variables
using the `interact()`

and `bsnullinteract()`

funtions. Function
`bsnullinteract()`

computes null-interaction models (10, by default)
based on bootstrap-sampled and permuted datasets. Function `interact()`

computes interaction test statistics for each predictor variables
appearing in the specified ensemble. If null-interaction models are
provided through the `nullmods`

argument, interaction test statistics
will also be computed for the null-interaction model, providing a
reference null distribution.

Note that computing null interaction models and interaction test statistics is computationally very intensive, so running the following code will take some time:

```
set.seed(44)
nullmods <- bsnullinteract(airq.ens)
int <- interact(airq.ens, nullmods = nullmods)
```

The plotted variable interaction strengths indicate that Temperature and
Wind may be involved in interactions, as their observed interaction
strengths (darker grey) exceed the upper limit of the 90% confidence
interval (CI) of interaction stengths in the null interaction models
(lighter grey bar represents the median, error bars represent the 90%
CIs). The plot indicates that Solar.R and Day are not involved in any
interactions. Note that computation of null interaction models is
computationally intensive. A more reliable result can be obtained by
computing a larger number of boostrapped null interaction datasets, by
setting the `nsamp`

argument of function `bsnullinteract()`

to a larger
value (e.g., 100).

### Correlations between selected terms

We can assess correlations between the baselearners appearing in the
ensemble using the `corplot()`

function:

`corplot(airq.ens)`

# Including hinge functions (multivariate adaptive regression splines)

More complex prediction ensembles can be obtained using the `gpe()`

function. Abbreviation gpe stands for generalized prediction ensembles,
which can also include hinge functions of the predictor variables as
described in Friedman (1991), in addition to rules and/or linear terms.
Addition of hinge functions may further improve predictive accuracy. See
the following example:

```
set.seed(42)
airq.gpe <- gpe(Ozone ~ ., data = airquality[complete.cases(airquality),],
base_learners = list(gpe_trees(), gpe_linear(), gpe_earth()))
airq.gpe
#>
#> Final ensemble with cv error within 1se of minimum:
#> lambda = 3.229132
#> number of terms = 11
#> mean cv error (se) = 361.2152 (110.9785)
#>
#> cv error type : Mean-squared Error
#>
#> description coefficient
#> (Intercept) 65.52169487
#> Temp <= 77 -6.20973854
#> Wind <= 10.3 & Solar.R > 148 5.46410965
#> Wind > 5.7 & Temp <= 82 -8.06127416
#> Wind > 5.7 & Temp <= 84 -7.16921733
#> Wind > 5.7 & Temp <= 87 -8.04255470
#> Wind > 5.7 & Temp <= 87 & Day <= 23 -3.40525575
#> Wind > 6.3 & Temp <= 82 -2.71925536
#> Wind > 6.3 & Temp <= 84 -5.99085126
#> Wind > 6.9 & Temp <= 82 -0.04406376
#> Wind > 6.9 & Temp <= 84 -0.55827336
#> eTerm(Solar.R * h(9.7 - Wind), scale = 410) 9.91783318
#>
#> 'h' in the 'eTerm' indicates the hinge function
```

# References

Breiman, L., Friedman, J., Olshen, R., & Stone, C. (1984). Classification and regression trees. Boca Raton, FL: Chapman&Hall/CRC.

Fokkema, M. (2017). pre: An R package for fitting prediction rule
ensembles. *arXiv:1707.07149*. Retrieved from
https://arxiv.org/abs/1707.07149

Friedman, J. (1991). Multivariate adaptive regression splines. *The
Annals of Statistics*, *19*(1), 1–67.

Friedman, J., & Popescu, B. (2008). Predictive learning via rule
ensembles. *The Annals of Applied Statistics*, *2*(3), 916–954.
Retrieved from http://www.jstor.org/stable/30245114

Hothorn, T., Hornik, K., & Zeileis, A. (2006). Unbiased recursive
partitioning: A conditional inference framework. *Journal of
Computational and Graphical Statistics*, *15*(3), 651–674.

Milborrow, S. (2018). *plotmo: Plot a model’s residuals, response, and
partial dependence plots*. Retrieved from
https://CRAN.R-project.org/package=plotmo

Zeileis, A., Hothorn, T., & Hornik, K. (2008). Model-based recursive
partitioning. *Journal of Computational and Graphical Statistics*,
*17*(2), 492–514.