This workflow demonstrates how tidyfit
can be used to easily compare a large number of regularized regression methods in R
. Using the Boston house prices data set, the analysis shows how Bayesian methods strongly outperform most alternatives:
library(dplyr); library(tidyr); library(purrr) # Data wrangling
library(ggplot2); library(stringr) # Plotting
library(tidyfit) # AutoML modeling
The Boston house prices data set (MASS::Boston
) presents a popular test case for regression algorithms. In this article, I assess the relative performance of 15 different linear regression techniques using tidyfit::regress
.
Let’s begin by loading the data, and splitting it into equally sized training and test sets:
data < MASS::Boston
# For reproducibility
set.seed(123)
ix_tst < sample(1:nrow(data), round(nrow(data)*0.5))
data_trn < data[ix_tst,]
data_tst < data[ix_tst,]
as_tibble(data)
#> # A tibble: 506 × 14
#> crim zn indus chas nox rm age dis rad tax ptratio black
#> <dbl> <dbl> <dbl> <int> <dbl> <dbl> <dbl> <dbl> <int> <dbl> <dbl> <dbl>
#> 1 0.00632 18 2.31 0 0.538 6.58 65.2 4.09 1 296 15.3 397.
#> 2 0.0273 0 7.07 0 0.469 6.42 78.9 4.97 2 242 17.8 397.
#> 3 0.0273 0 7.07 0 0.469 7.18 61.1 4.97 2 242 17.8 393.
#> 4 0.0324 0 2.18 0 0.458 7.00 45.8 6.06 3 222 18.7 395.
#> 5 0.0690 0 2.18 0 0.458 7.15 54.2 6.06 3 222 18.7 397.
#> 6 0.0298 0 2.18 0 0.458 6.43 58.7 6.06 3 222 18.7 394.
#> 7 0.0883 12.5 7.87 0 0.524 6.01 66.6 5.56 5 311 15.2 396.
#> 8 0.145 12.5 7.87 0 0.524 6.17 96.1 5.95 5 311 15.2 397.
#> 9 0.211 12.5 7.87 0 0.524 5.63 100 6.08 5 311 15.2 387.
#> 10 0.170 12.5 7.87 0 0.524 6.00 85.9 6.59 5 311 15.2 387.
#> # ℹ 496 more rows
#> # ℹ 2 more variables: lstat <dbl>, medv <dbl>
The aim is to predict the median value of owneroccupied homes ('medv'
in the data set) using the remaining columns.^{1} We will assess performance using \(R^2\), which allows intuitive comparisons of insample and outofsample fit.
A simple regression
To establish a baseline, we fit an OLS regression model and examine the test and training set Rsquared. The predict
function can be used to obtain predictions, and yardstick
is used to assess performance. The regression is estimated using tidyfit::regress
. tidyfit::m
is a generic model wrapper that is capable of fitting a large number of different algorithms:
model_frame < data_trn %>%
regress(medv ~ ., OLS = m("lm"))
Let’s create a helper function to assess performance. The helper function wraps predict
and prepares in and outofsample performance stats:
assess < function(model_frame, data_trn, data_tst) {
oos < model_frame %>%
predict(data_tst) %>%
group_by(model, .add = TRUE) %>%
yardstick::rsq_trad(truth, prediction) %>%
mutate(type = "Outofsample") %>%
arrange(desc(.estimate))
is < model_frame %>%
predict(data_trn) %>%
group_by(model, .add = TRUE) %>%
yardstick::rsq_trad(truth, prediction) %>%
mutate(type = "Insample")
return(bind_rows(oos, is))
}
plotter < function(df) {
df %>%
mutate(lab = round(.estimate, 2)) %>%
mutate(model = str_wrap(model, 12)) %>%
mutate(model = factor(model, levels = unique(.$model))) %>%
ggplot(aes(model, .estimate)) +
geom_point(aes(color = type), size = 2.5, shape = 4) +
geom_label(aes(label = lab, color = type), size = 2, nudge_x = 0.35) +
theme_bw() +
scale_color_manual(values = c("firebrick", "darkblue")) +
theme(legend.title = element_blank(),
axis.title.x = element_blank()) +
coord_cartesian(ylim = c(0.65, 0.95))
}
Now we plot the resulting metrics:
model_frame %>%
assess(data_trn, data_tst) %>%
plotter
The insample fit of the regression is naturally better than outofsample, but the difference is not large. The results can be improved substantially by adding interaction terms between features. This leads to a vast improvement in insample fit (again, this is a logical consequence of including a large number of regressors), but leads to no improvement in outofsample fit, suggesting substantial overfit:
model_frame < data_trn %>%
regress(medv ~ .*., OLS = m("lm"))
model_frame %>%
assess(data_trn, data_tst) %>%
plotter
Regularized regression estimators
Regularization controls overfit by introducing parameter bias in exchange for a reduction in estimator variance. A large number of approaches to regularization exist that can be summarized in at least 5 categories:
 Best subset selection
 L1/L2 penalized regression
 Latent factor regression
 Other machine learning
 Bayesian regression
tidyfit
includes linear estimators from each of these categories, as summarized below:
Best subset  L1/L2 penalized  Latent factors  Machine Learning  Bayesian 

Exhaustive search (leaps ) 
Lasso (glmnet ) 
Principal components regression (pls ) 
Gradient Boosting (mboost ) 
Bayesian regression (arm ) 
Forward selection (leaps ) 
Ridge (glmnet ) 
Partial least squares (pls ) 
Hierarchical feature regression (hfr ) 
Bayesian model averaging (BMS ) 
Backward selection (leaps ) 
ElasticNet (glmnet ) 
Support vector regression (e1071 ) 
Bayesian Lasso (monomvn ) 

Sequential Replacement (leaps ) 
Adaptive Lasso (glmnet ) 
Bayesian Ridge (monomvn ) 

Genetic Algorithm (‘gaselect’)  Spike and Slab Regression (‘BoomSpikeSlab’) 
In the next code chunk, 17 of the available regression techniques are fitted to the data set. Hyperparameters are optimized using a 10fold cross validation powered by rsample
in the background. As a practical aside, it is possible to set a future
plan to ensure hyperparameter tuning is performed on a parallelized backend.^{2} This is done using plan(multisession(workers = n_workers))
, where n_workers
is the number of available cores. Furthermore, a progress bar could be activated using progressr::handlers(global = TRUE)
.
model_frame_all < data_trn %>%
regress(medv ~ .*.,
OLS = m("lm"),
BAYES = m("bayes"),
BMA = m("bma", iter = 10000),
SEQREP = m("subset", method = "seqrep", IC = "AIC"),
`FORWARD SELECTION` = m("subset", method = "forward", IC = "AIC"),
`BACKWARD SELECTION` = m("subset", method = "backward", IC = "AIC"),
LASSO = m("lasso"),
BLASSO = m("blasso"),
SPIKESLAB = m("spikeslab", niter = 10000),
RIDGE = m("ridge"),
BRIDGE = m("bridge"),
ELASTICNET = m("enet"),
ADALASSO = m("adalasso", lambda_ridge = c(0.001, 0.01, 0.1)),
PCR = m("pcr"),
PLSR = m("plsr"),
HFR = m("hfr"),
`GRADIENT BOOSTING` = m("boost"),
SVR = m("svm"),
GENETIC = m("genetic", populationSize = 1000, numGenerations = 50, statistic = "AIC", maxVariables = 20),
.cv = "vfold_cv", .cv_args = list(v = 10))
Let’s use yardstick
again to calculate insample and outofsample performance statistics. The models are arranged descending order of test set performance:
eval_frame < model_frame_all %>%
assess(data_trn, data_tst)
eval_frame %>%
plotter +
theme(legend.position = "top",
axis.text.x = element_text(angle = 90))
The Bayesian regression techniques overwhelmingly outperform the remaining methods, with the exception of the hierarchical feature regression, genetic algorithm and the adaptive lasso, both of which achieve relatively high outofsample performance. The subset selection algorithms achieve relatively good results, while the penalized and latent factor regressions perform worst. All methods improve on the OLS results.
A glimpse at the backend
tidyfit
makes it exceedingly simple to fit different regularized linear regressions. The package ensures that the input and output of the modeling engine, tidyfit::m
are standardized and model results are comparable. For instance, whenever necessary, features are standardized with the coefficients transformed back to the original scale. Hyperparameter grids are set to reasonable starting values and can be passed manually to tidyfit::m
.
The package further includes a tidyfit::classify
method that assumes a binomial or multinomial response. Many of the methods implemented can handle Gaussian and binomial responses. Furthermore, the tidyfit::regress
and tidyfit::classify
methods seamlessly integrate with grouped tibbles, making it extremely powerful to estimate a large number of regressions for different data groupings. Groups as well as response distributions are automatically respected by the predict
method.

Have a look at
?MASS::Boston
for definitions of the features.↩︎ 
Parallelizing happens at the level of cross validation slices, so always consider if the overhead of parallel computation is worth it. For instance, parallelizing will likely be very useful with
.cv = "loo_cv"
but can be less useful with.cv = "vfold_cv"
.↩︎