1 Introduction

The enpls package offers an algorithmic framework for measuring feature importance, detecting outliers, and ensemble modeling based on (sparse) partial least squares regression. The key functions included in the package are listed in the table below.

Next, we will use the data from (Wang et al. 2015) to demonstrate the general workflow of enpls. The dataset contains 1,000 compounds, each characterized by 80 molecular descriptors. The response is the octanol/water partition coefficient at pH 7.4 (logD7.4).

Let’s load the data and take a look at it:

library("enpls")

## Registered S3 methods overwritten by 'ggplot2':
##   method         from 
##   [.quosures     rlang
##   c.quosures     rlang
##   print.quosures rlang

library("ggplot2")

data("logd1k")
x <- logd1k$x
y <- logd1k$y
head(x)[, 1:5]

##   BalabanJ  BertzCT   Chi0  Chi0n  Chi0v
## 1    1.949  882.760 16.845 13.088 13.088
## 2    1.970  781.936 15.905 13.204 14.021
## 3    2.968  343.203  9.845  7.526  7.526
## 4    2.050 1133.679 19.836 15.406 15.406
## 5    2.719  437.346 12.129  9.487  9.487
## 6    2.031  983.304 19.292 15.289 15.289

head(y)

## [1] -0.96 -0.92 -0.90 -0.83 -0.82 -0.79

2 Model Fitting

Here we fit the ensemble sparse partial least squares to the data, so that the model complexity could usually be further reduced than vanilla partial least squares when we build each model.

set.seed(42)
fit <- enspls.fit(x, y, ratio = 0.7, reptimes = 20, maxcomp = 3)
y.pred <- predict(fit, newx = x)

df <- data.frame(y, y.pred)
ggplot(df, aes_string(x = "y", y = "y.pred")) +
  geom_abline(slope = 1, intercept = 0, colour = "darkgrey") +
  geom_point(size = 3, shape = 1, alpha = 0.8) +
  coord_fixed(ratio = 1) +
  xlab("Observed Response") +
  ylab("Predicted Response")

We used the fitted model to predict on the training data and plotted the predicted values against the true values.

The parameter ratio decides the sampling ratio for each Monte-Carlo run; maxcomp controls the maximum number of components included within each model; reptimes sets the times of Monte-Carlo resampling, we recommend setting it to a large number (500 by default).

One common parameter for all functions in enpls is parallel, it controls the number of CPU cores to use if you want to train the models in parallel.

3 Cross Validation

K-fold cross validation is a traditional way to measure the empirical predictive performance of the model. We can use function cv.enpls() or cv.enspls() to perform \(k\)-fold cross validation for the ensemble (sparse) partial least squares model.

Since the parameters (number of components and level of sparsity) are automatically tuned for each model in enpls, the cross validation here is used to see if certain combinations of parameters (specified by ratio, maxcomp, alpha, etc.) can produce ensemble models with better performance.

cv.fit <- cv.enspls(x, y,
  nfolds = 5, ratio = 0.7,
  reptimes = 10, maxcomp = 3, verbose = FALSE
)
cv.fit

## Cross Validation Result for Ensemble Sparse Partial Least Squares
## ---
## RMSE = 1.1410
## MAE = 0.913153
## Rsquare = 0.587959

plot(cv.fit)

The returned object gives three model performance evaluation metrics for the ensemble model: RMSE, MAE, and \(R^2\). Here we also plotted the predicted values for each test fold against the true response.

4 Feature Importance

To measure feature importance, simply use enpls.fs() or enspls.fs():

fs <- enspls.fs(x, y, ratio = 0.7, reptimes = 20, maxcomp = 3)
print(fs, nvar = 10)

## Variable Importance by Ensemble Sparse Partial Least Squares
## ---
##                     Importance
## Chi1n                 29.48614
## NumValenceElectrons   26.78683
## HeavyAtomCount        23.28745
## Chi1                  21.39361
## LabuteASA             21.27471
## SlogP_VSA2            20.93239
## Chi0n                 20.54714
## SlogP_VSA5            19.89059
## MolMR                 19.82387
## SMR_VSA5              19.19199

plot(fs, nvar = 10)

plot(fs, type = "boxplot", nvar = 10)

The top 10 most important features are ranked as above. The boxplot gives additional information about the coefficient stability of each feature. We can see the feature TPSA (Topological Polar Surface Area) has different pattern compared to others: it has large effect size, but the effect sizes also have a large variance. This indicates that TPSA is important for predicting logD7.4. However, such importance may vary on different subsets of the samples.

5 Outlier Detection

By using information from the prediction error distribution for each sample produced by many models, we can measure if the responses of particular samples are harder to predict than the others. Such measurements can help on identifying outliers in the dataset. Thus, they can be removed to get us a “clean” dataset before the actual modeling.

This could be done with enpls.od(), enspls.od() easily:

od <- enspls.od(x, y, ratio = 0.8, reptimes = 20, maxcomp = 3)
plot(od, prob = 0.05)

plot(od, criterion = "sd", sdtimes = 2)

The two plots showed that several samples in our dataset might be outlier candidates, based on two different criterions. The samples in each area of the plots represent different types of outliers.

6 Applicability Domain Evaluation

Model applicability domain measures how well the predictive model (PLS/Sparse PLS model) we built on the training set performs on external test sets. A certain type of perturbation (such as bootstrapping, jackknifing) is applied to the samples or the variables of the training set, and we could get many different predictions for all samples in all test sets (and of course, including the training set) using each sub-model built with each perturbated training set. The general evaluation strategy design and comparisons were analyzed in Kaneko and Funatsu (2014).

The functions enpls.ad() and enspls.ad() could help us evaluate the model applicability domain. Here we constructed two “pseudo” test sets from the original logd1k dataset for demonstration:

# remove low variance variables
x <- x[, -c(17, 52, 59)]

# make training set
x.tr <- x[1:500, ]
y.tr <- y[1:500]

# make two test sets
x.te <- list(
  "test.1" = x[501:700, ],
  "test.2" = x[701:800, ]
)
y.te <- list(
  "test.1" = y[501:700],
  "test.2" = y[701:800]
)

ad <- enspls.ad(x.tr, y.tr, x.te, y.te,
  maxcomp = 3, space = "variable", method = "mc",
  ratio = 0.8, reptimes = 50
)
plot(ad)

Additionally, by using

plot(ad, type = "interactive")

we will get an interactive plot, which could help us better explore the model applicability domain by supporting zooming-in / inspecting which sample each point represents interactively. The interactive plot is based on plotly, and it only requires an HTML viewer to be correctly rendered.

7 Conclusion

Ensemble learning approaches are not only powerful for improving base learner’s predictive performance but also capable of accomplishing model diagnostic tasks, such as measuring the importance of features. It would be interesting to see if such ideas could be applied to more relevant topics and further facilitate the predictive modeling tasks.