**Econometrics and Free Software**, and kindly contributed to R-bloggers)

## Introduction

This blog posts will use several packages from the

`{tidymodels}`

collection of packages, namely

`{recipes}`

,

`{rsample}`

and

`{parsnip}`

to train a random forest the tidy way. I will

also use `{mlrMBO}`

to tune the hyper-parameters of the random forest.

## Set up

Let’s load the needed packages:

```
library("tidyverse")
library("tidymodels")
library("parsnip")
library("brotools")
library("mlbench")
```

Load the data, included in the `{mlrbench}`

package:

`data("BostonHousing2")`

I will train a random forest to predict the housing price, which is the `cmedv`

column:

`head(BostonHousing2)`

```
## town tract lon lat medv cmedv crim zn indus chas nox
## 1 Nahant 2011 -70.9550 42.2550 24.0 24.0 0.00632 18 2.31 0 0.538
## 2 Swampscott 2021 -70.9500 42.2875 21.6 21.6 0.02731 0 7.07 0 0.469
## 3 Swampscott 2022 -70.9360 42.2830 34.7 34.7 0.02729 0 7.07 0 0.469
## 4 Marblehead 2031 -70.9280 42.2930 33.4 33.4 0.03237 0 2.18 0 0.458
## 5 Marblehead 2032 -70.9220 42.2980 36.2 36.2 0.06905 0 2.18 0 0.458
## 6 Marblehead 2033 -70.9165 42.3040 28.7 28.7 0.02985 0 2.18 0 0.458
## rm age dis rad tax ptratio b lstat
## 1 6.575 65.2 4.0900 1 296 15.3 396.90 4.98
## 2 6.421 78.9 4.9671 2 242 17.8 396.90 9.14
## 3 7.185 61.1 4.9671 2 242 17.8 392.83 4.03
## 4 6.998 45.8 6.0622 3 222 18.7 394.63 2.94
## 5 7.147 54.2 6.0622 3 222 18.7 396.90 5.33
## 6 6.430 58.7 6.0622 3 222 18.7 394.12 5.21
```

Only keep relevant columns:

```
boston <- BostonHousing2 %>%
select(-medv, -town, -lon, -lat) %>%
rename(price = cmedv)
```

I remove `town`

, `lat`

and `lon`

because the information contained in the column `tract`

is enough.

To train and evaluate the model’s performance, I split the data in two.

One data set, which I call the training set, will be further split into two down below. I won’t

touch the second data set, the test set, until the very end.

```
train_test_split <- initial_split(boston, prop = 0.9)
housing_train <- training(train_test_split)
housing_test <- testing(train_test_split)
```

I want to train a random forest to predict price of houses, but random forests have so-called

hyperparameters, which are parameters that cannot be estimated, or learned, from the data. Instead,

these parameters have to be chosen by the analyst. In order to choose them, you can

use values from the literature that seemed to have worked well (like is done in Macro-econometrics)

or you can further split the train set into two, create a grid of hyperparameter, train the model

on one part of the data for all values of the grid, and compare the predictions of the models on the

second part of the data. You then stick with the model that performed the best, for example, the

model with lowest RMSE. The thing is, you can’t estimate the true value of the RMSE with only

one value. It’s like if you wanted to estimate the height of the population by drawing one single

observation from the population. You need a bit more observations. To approach the true value of the

RMSE for a give set of hyperparameters, instead of doing one split, I’ll do 30. I then

compute the average RMSE, which implies training 30 models for each combination of the values of the

hyperparameters I am interested in.

First, let’s split the training data again, using the `mc_cv()`

function from `{rsample}`

package.

This function implements Monte Carlo cross-validation:

`validation_data <- mc_cv(housing_train, prop = 0.9, times = 30)`

What does `validation_data`

look like?

`validation_data`

```
## # # Monte Carlo cross-validation (0.9/0.1) with 30 resamples
## # A tibble: 30 x 2
## splits id
##
```
## 1 Resample01
## 2 Resample02
## 3 Resample03
## 4 Resample04
## 5 Resample05
## 6 Resample06
## 7 Resample07
## 8 Resample08
## 9 Resample09
## 10 Resample10
## # ... with 20 more rows

Let’s look further down:

`validation_data$splits[[1]]`

`## <411/45/456>`

The first value is the number of rows of the first set, the second value of the second, and the third

was the original amount of values in the training data, before splitting again.

How should we call these two new data sets? The author of `{rsample}`

, Max Kuhn, talks about

the *analysis* and the *assessment* sets:

rsample convention for now but I intend on using it everywhere. Reusing training and testing is insane.

— Max Kuhn (@topepos) November 24, 2018

Now, in order to continue I need pre-process the data. I will do this in three steps.

The first and the second step are used to center and scale the numeric variables and the third step

converts character and factor variables to dummy variables. This is needed because I will train a

random forest, which cannot handle factor variables directly. Let’s define a recipe to do that,

and start by pre-processing the testing set. I write a wrapper function around the recipe,

because I will need to apply this recipe to various data sets:

```
simple_recipe <- function(dataset){
recipe(price ~ ., data = dataset) %>%
step_center(all_numeric()) %>%
step_scale(all_numeric()) %>%
step_dummy(all_nominal())
}
```

Once the recipe is defined, I can use the `prep()`

function, which estimates the parameters from

the data which are needed to process the data. For example, for centering, `prep()`

estimates

the mean which will then be subtracted from the variables. With `bake()`

the estimates are then

applied on the data:

```
testing_rec <- prep(simple_recipe(housing_test), testing = housing_test)
test_data <- bake(testing_rec, newdata = housing_test)
```

It is important to split the data before using `prep()`

and `bake()`

, because if not, you will

use observations from the test set in the `prep()`

step, and thus introduce knowledge from the test

set into the training data. This is called data leakage, and must be avoided. This is why it is

necessary to first split the training data into an analysis and an assessment set, and then also

pre-process these sets separately. However, the `validation_data`

object cannot now be used with

`recipe()`

, because it is not a dataframe. No worries, I simply need to write a function that extracts

the analysis and assessment sets from the `validation_data`

object, applies the pre-processing, trains

the model, and returns the RMSE. This will be a big function, at the center of the analysis.

But before that, let’s run a simple linear regression, as a benchmark. For the linear regression, I will

not use any CV, so let’s pre-process the training set:

```
trainlm_rec <- prep(simple_recipe(housing_train), testing = housing_train)
trainlm_data <- bake(trainlm_rec, newdata = housing_train)
linreg_model <- lm(price ~ ., data = trainlm_data)
broom::augment(linreg_model, newdata = test_data) %>%
rmse(price, .fitted)
```

```
## # A tibble: 1 x 3
## .metric .estimator .estimate
##
```
## 1 rmse standard 0.469

`broom::augment()`

adds the predictions to the `test_data`

in a new column, `.fitted`

. I won’t

use this trick with the random forest, because there is no `augment()`

method for random forests

from the `{ranger}`

which I’ll use. I’ll add the predictions to the data myself.

Ok, now let’s go back to the random forest and write the big function:

```
my_rf <- function(mtry, trees, split, id){
analysis_set <- analysis(split)
analysis_prep <- prep(simple_recipe(analysis_set), training = analysis_set)
analysis_processed <- bake(analysis_prep, newdata = analysis_set)
model <- rand_forest(mtry = mtry, trees = trees) %>%
set_engine("ranger", importance = 'impurity') %>%
fit(price ~ ., data = analysis_processed)
assessment_set <- assessment(split)
assessment_prep <- prep(simple_recipe(assessment_set), testing = assessment_set)
assessment_processed <- bake(assessment_prep, newdata = assessment_set)
tibble::tibble("id" = id,
"truth" = assessment_processed$price,
"prediction" = unlist(predict(model, new_data = assessment_processed)))
}
```

The `rand_forest()`

function is available from the `{parsnip}`

package. This package provides an

unified interface to a lot of other machine learning packages. This means that instead of having to

learn the syntax of `range()`

and `randomForest()`

and, and… you can simply use the `rand_forest()`

function and change the `engine`

argument to the one you want (`ranger`

, `randomForest`

, etc).

Let’s try this function:

```
results_example <- map2_df(.x = validation_data$splits,
.y = validation_data$id,
~my_rf(mtry = 3, trees = 200, split = .x, id = .y))
```

`head(results_example)`

```
## # A tibble: 6 x 3
## id truth prediction
##
```
## 1 Resample01 0.0235 -0.104
## 2 Resample01 -0.135 -0.0906
## 3 Resample01 -0.378 -0.158
## 4 Resample01 -0.232 0.0623
## 5 Resample01 -0.0859 0.0173
## 6 Resample01 0.169 0.303

I can now compute the RMSE when `mtry`

= 3 and `trees`

= 200:

```
results_example %>%
group_by(id) %>%
rmse(truth, prediction) %>%
summarise(mean_rmse = mean(.estimate)) %>%
pull
```

`## [1] 0.4319164`

The random forest has already lower RMSE than the linear regression. The goal now is to lower this

RMSE by tuning the `mtry`

and `trees`

hyperparameters. For this, I will use Bayesian Optimization

methods implemented in the `{mlrMBO}`

package.

## Bayesian hyperparameter optimization

I will re-use the code from above, and define a function that does everything from pre-processing

to returning the metric I want to minimize by tuning the hyperparameters, the RMSE:

```
tuning <- function(param, validation_data){
mtry <- param[1]
trees <- param[2]
results <- purrr::map2_df(.x = validation_data$splits,
.y = validation_data$id,
~my_rf(mtry = mtry, trees = trees, split = .x, id = .y))
results %>%
group_by(id) %>%
rmse(truth, prediction) %>%
summarise(mean_rmse = mean(.estimate)) %>%
pull
}
```

This is exactly the code from before, but it now returns the RMSE. Let’s try the function

with the values from before:

`tuning(c(3, 200), validation_data)`

`## [1] 0.4330951`

Let’s also plot the value of RMSE for `mtry = 3`

and `trees`

from 200 to 300. This takes some

time, because I need to evaluate this costly function 100 times. If evaluating the function was

cheap, I could have made a 3D plot by varying values of `mtry`

too, but then again if evaluating

the function was cheap, I would run an exhaustive grid search to find the hyperparameters instead of

using Bayesian optimization.

```
plot_points <- crossing("mtry" = 3, "trees" = seq(200, 300))
plot_data <- plot_points %>%
mutate(value = map_dbl(seq(200, 300), ~tuning(c(3, .), validation_data)))
```

```
plot_data %>%
ggplot(aes(y = value, x = trees)) +
geom_line(colour = "#82518c") +
theme_blog() +
ggtitle("RMSE for mtry = 3")
```

For `mtry = 3`

the minimum seems to lie around 255. The function to minimize is not smooth at all.

I now follow the code that can be found in the arxiv paper to

run the optimization. I think I got the gist of the paper, but I did not understand everything yet.

For now, I am still experimenting with the library at the moment, but from what I understand, a

simpler model, called the surrogate model, is used to look for promising points and to evaluate the

value of the function at these points. This seems somewhat similar (in spirit) to the

*Indirect Inference* method as described in Gourieroux, Monfort, Renault.

Let’s first load the package and create the function to optimize:

`library("mlrMBO")`

```
fn <- makeSingleObjectiveFunction(name = "tuning",
fn = tuning,
par.set = makeParamSet(makeIntegerParam("x1", lower = 3, upper = 8),
makeIntegerParam("x2", lower = 50, upper = 500)))
```

This function is based on the function I defined before. The parameters to optimize are also

defined as are their bounds. I will look for `mtry`

between the values of 3 and 8, and `trees`

between 50 and 500.

Now comes the part I didn’t quite get.

```
# Create initial random Latin Hypercube Design of 10 points
library(lhs)# for randomLHS
des <- generateDesign(n = 5L * 2L, getParamSet(fn), fun = randomLHS)
```

I think this means that these 10 points are the points used to start the whole process. I did not

understand why they have to be sampled from a hypercube, but ok. Then I choose the surrogate model,

a random forest too, and predict the standard error. Here also, I did not quite get why the

standard error can be an option.

```
# Specify kriging model with standard error estimation
surrogate <- makeLearner("regr.ranger", predict.type = "se", keep.inbag = TRUE)
```

Here I define some options:

```
# Set general controls
ctrl <- makeMBOControl()
ctrl <- setMBOControlTermination(ctrl, iters = 10L)
ctrl <- setMBOControlInfill(ctrl, crit = makeMBOInfillCritEI())
```

And this is the optimization part:

```
# Start optimization
result <- mbo(fn, des, surrogate, ctrl, more.args = list("validation_data" = validation_data))
```

`result`

```
## Recommended parameters:
## x1=6; x2=381
## Objective: y = 0.393
##
## Optimization path
## 10 + 10 entries in total, displaying last 10 (or less):
## x1 x2 y dob eol error.message exec.time ei
## 11 6 370 0.3943479 1 NA
``` 8.913 -3.134568e-05
## 12 6 362 0.3950402 2 NA 8.844 -2.987934e-05
## 13 6 373 0.3939587 3 NA 8.939 -2.259674e-05
## 14 6 394 0.3962875 4 NA 9.342 -7.427682e-06
## 15 6 368 0.3944954 5 NA 8.760 -4.121337e-06
## 16 6 378 0.3938796 6 NA 8.949 -4.503591e-07
## 17 6 381 0.3934176 7 NA 9.109 -1.141853e-06
## 18 6 380 0.3948077 8 NA 9.026 -4.718394e-08
## 19 6 381 0.3932636 9 NA 9.022 -9.801395e-08
## 20 6 383 0.3953004 10 NA 9.184 -1.579619e-09
## error.model train.time prop.type propose.time se mean
## 11 0.014 infill_ei 0.449 0.0010924600 0.3955131
## 12 0.012 infill_ei 0.458 0.0007415920 0.3948705
## 13 0.012 infill_ei 0.460 0.0006116756 0.3947185
## 14 0.012 infill_ei 0.729 0.0003104694 0.3943572
## 15 0.023 infill_ei 0.444 0.0003446061 0.3945085
## 16 0.013 infill_ei 0.458 0.0002381887 0.3944642
## 17 0.013 infill_ei 0.492 0.0002106454 0.3943200
## 18 0.013 infill_ei 0.516 0.0002093524 0.3940764
## 19 0.014 infill_ei 0.756 0.0002481260 0.3941597
## 20 0.013 infill_ei 0.483 0.0001687982 0.3939285

So the recommended parameters are 6 for `mtry`

and 381 for `trees`

. The value of the RMSE is lower

than before, and equals 0.393.

Let’s now train the random forest on the training data with this values. First, I pre-process the

training data:

```
training_rec <- prep(simple_recipe(housing_train), testing = housing_train)
train_data <- bake(training_rec, newdata = housing_train)
```

Let’s now train our final model and predict the prices:

```
final_model <- rand_forest(mtry = 6, trees = 381) %>%
set_engine("ranger", importance = 'impurity') %>%
fit(price ~ ., data = train_data)
price_predict <- predict(final_model, new_data = select(test_data, -price))
```

Let’s transform the data back and compare the predicted prices to the true ones visually:

```
cbind(price_predict * sd(housing_train$price) + mean(housing_train$price),
housing_test$price)
```

```
## .pred housing_test$price
## 1 26.66387 24.0
## 2 18.05651 18.9
## 3 17.73167 15.0
## 4 19.58751 18.2
## 5 15.83327 13.9
## 6 15.23911 13.2
## 7 20.71109 20.0
## 8 24.57467 25.0
## 9 23.30817 21.4
## 10 22.49698 22.9
## 11 22.80554 21.4
## 12 19.12121 18.6
## 13 19.63354 19.3
## 14 18.33941 20.4
## 15 17.03028 14.4
## 16 45.58864 50.0
## 17 28.26440 32.0
## 18 28.48881 30.5
## 19 28.18277 31.1
## 20 45.12203 48.5
## 21 45.13364 50.0
## 22 20.95897 24.4
## 23 32.70919 31.6
## 24 18.88725 18.5
## 25 23.80696 22.0
## 26 24.05506 23.2
## 27 33.69617 37.3
## 28 26.93756 27.9
## 29 22.71415 23.9
## 30 24.88508 27.1
## 31 24.63913 22.5
## 32 30.73453 33.1
## 33 23.35807 22.1
## 34 21.89201 23.1
## 35 23.22168 23.1
## 36 20.15691 19.3
## 37 21.12050 22.7
## 38 12.88927 13.1
## 39 19.25563 10.4
## 40 11.32007 10.5
## 41 16.30650 13.1
## 42 14.14549 8.3
## 43 11.08126 5.0
## 44 14.14806 13.4
## 45 20.56828 17.8
## 46 14.43175 14.4
## 47 20.00864 19.5
## 48 13.46410 12.0
## 49 20.81910 21.8
## 50 21.72144 21.2
```

Let’s now compute the RMSE:

```
tibble::tibble("truth" = test_data$price,
"prediction" = unlist(price_predict)) %>%
rmse(truth, prediction)
```

```
## # A tibble: 1 x 3
## .metric .estimator .estimate
##
```
## 1 rmse standard 0.327

Very nice.

Hope you enjoyed! If you found this blog post useful, you might want to follow

me on twitter for blog post updates and

buy me an espresso.

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**Econometrics and Free Software**.

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