boennecd / pre

R package for deriving Prediction Rule Ensembles

pre: an R package for deriving prediction rule ensembles

pre is an R package for deriving prediction rule ensembles for binary, continuous and count outcome variables. Input variables may be numeric, ordinal and nominal. The package largely implements the algorithm for deriving prediction rule ensembles as described in Friedman & Popescu (2008), with several adjustments:

1. 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.
2. The unbiased tree induction algorithm of Hothorn, Hornik, & Zeileis (2006) is used for deriving prediction rules, instead of the classification and regression tree (CART) algorithm, which suffers from biased variable selection.
3. The package allows for plotting the final rule ensemble as a collection of simple decision trees.
4. The initial ensembles may be generated as in bagging, boosting and/or random forests.
5. Hinge functions of predictor variables may be included as baselearners, like in the multivariate adaptive regression splines method of Friedman (1991), using the gpe() function.
6. Ensembles for outcome variables representing counts can be derived. Note that pre is under development, and much work still needs to be done.

Example: Prediction rule ensemble for predicting ozone levels

To get a first impression of how pre works, we will fit a prediction rule ensemble to predict Ozone levels using the airquality dataset. We can fit a prediction rule ensemble using the pre() function:

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

We can print the resulting ensemble (alternatively, we could use the print method):

print(airq.ens)
#> 
#> Final ensemble with cv error within 1se of minimum: 
#>   lambda =  1.607146
#>   number of terms = 14
#>   mean cv error (se) = 299.798 (74.28342)
#> 
#>   cv error type : Mean-Squared Error
#> 
#>          rule    coefficient                          description
#>   (Intercept)   62.658099734                                 <NA>
#>        rule72  -13.401881493              Wind > 5.7 & Temp <= 84
#>       rule216    8.166292702         Wind <= 10.3 & Solar.R > 148
#>       rule122    8.027236520                            Temp > 77
#>       rule213   -7.901556274              Wind > 5.1 & Temp <= 87
#>       rule201   -6.587690267  Wind > 5.7 & Temp <= 87 & Day <= 23
#>        rule25   -5.524545249              Wind > 6.3 & Temp <= 82
#>       rule179   -4.981266386              Wind > 5.7 & Temp <= 82
#>         rule3    4.927105788              Temp > 78 & Wind <= 6.3
#>       rule149   -3.427067580                Temp <= 87 & Wind > 8
#>       rule212    3.315627932                        Solar.R > 201
#>        rule89    2.588377937              Temp > 77 & Wind <= 8.6
#>        rule76   -2.112110981              Wind > 6.3 & Temp <= 84
#>       rule174   -1.315368661                           Wind > 6.9
#>       rule119   -0.009507402                Wind > 8 & Temp <= 76

We can plot the baselarners in the ensemble using the plot method:

plot(airq.ens, penalty.par.val = "lambda.1se", max.terms.plot = 9, cex = .6)

We can obtain the estimated coefficients for each of the baselearners using the coef method:

coefs <- coef(airq.ens)
coefs[1:10,]
#>            rule coefficient                         description
#> 194 (Intercept)   62.658100                                <NA>
#> 56       rule72  -13.401881             Wind > 5.7 & Temp <= 84
#> 170     rule216    8.166293        Wind <= 10.3 & Solar.R > 148
#> 95      rule122    8.027237                           Temp > 77
#> 167     rule213   -7.901556             Wind > 5.1 & Temp <= 87
#> 159     rule201   -6.587690 Wind > 5.7 & Temp <= 87 & Day <= 23
#> 16       rule25   -5.524545             Wind > 6.3 & Temp <= 82
#> 140     rule179   -4.981266             Wind > 5.7 & Temp <= 82
#> 3         rule3    4.927106             Temp > 78 & Wind <= 6.3
#> 116     rule149   -3.427068               Temp <= 87 & Wind > 8

We can assess the importance of input variables as well as baselearners using the importance() function:

importance(airq.ens, round = 4)

#> $varimps
#>   varname     imp
#> 1    Temp 14.9921
#> 2    Wind 13.5507
#> 3 Solar.R  3.6914
#> 4     Day  1.0646
#> 
#> $baseimps
#>       rule                         description    imp coefficient     sd
#> 1   rule72             Wind > 5.7 & Temp <= 84 6.0981    -13.4019 0.4550
#> 2  rule216        Wind <= 10.3 & Solar.R > 148 4.0533      8.1663 0.4963
#> 3  rule122                           Temp > 77 4.0120      8.0272 0.4998
#> 4  rule201 Wind > 5.7 & Temp <= 87 & Day <= 23 3.1939     -6.5877 0.4848
#> 5  rule213             Wind > 5.1 & Temp <= 87 3.1087     -7.9016 0.3934
#> 6   rule25             Wind > 6.3 & Temp <= 82 2.6332     -5.5245 0.4766
#> 7  rule179             Wind > 5.7 & Temp <= 82 2.3424     -4.9813 0.4702
#> 8    rule3             Temp > 78 & Wind <= 6.3 1.7825      4.9271 0.3618
#> 9  rule149               Temp <= 87 & Wind > 8 1.6771     -3.4271 0.4894
#> 10 rule212                       Solar.R > 201 1.6647      3.3156 0.5021
#> 11  rule89             Temp > 77 & Wind <= 8.6 1.1985      2.5884 0.4630
#> 12  rule76             Wind > 6.3 & Temp <= 84 0.9858     -2.1121 0.4667
#> 13 rule174                          Wind > 6.9 0.5439     -1.3154 0.4135
#> 14 rule119               Wind > 8 & Temp <= 76 0.0046     -0.0095 0.4796

We can generate predictions for new observations using the predict method:

predict(airq.ens, newdata = airquality[1:4,])
#>        1        2        3        4 
#> 28.99997 20.83368 17.39711 20.71273

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 a partial dependence plot to assess the effects of pairs of predictor variables on the outcome using the pairplot() function:

pairplot(airq.ens, varnames = c("Temp", "Wind"))
#> NOTE: function pairplot uses package 'akima', which has an ACM license. See also https://www.acm.org/publications/policies/software-copyright-notice.

We can assess the expected prediction error of the prediction rule ensemble through cross validation (10-fold, by default) using the cvpre() function:

set.seed(43)
airq.cv <- cvpre(airq.ens)
airq.cv$accuracy
#> $MSE
#>       MSE        se 
#> 365.97383  84.68306 
#> 
#> $MAE
#>       MAE        se 
#> 13.775890  1.265625

We can assess the presence of input variable interactions using the interact() and bsnullinteract() funtions:

set.seed(44)
nullmods <- bsnullinteract(airq.ens)
int <- interact(airq.ens, nullmods = nullmods, c("Temp", "Wind", "Solar.R"))

Including hinge functions

More complex prediction ensembles can be obtained using the gpe() function. The abbreviation gpe stands for generalized prediction ensembles, which may include hinge functions of the predictor variables as described in Friedman (1991), in addition to rules and/or linear terms. Addition of such hinge functions may improve predictive accuracy (but may also reduce interpretability). More information about fitting generalized prediction ensembles can be obtained by typing:

?gpe

References

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

Friedman, J. H., & Popescu, B. E. (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.