**NOTE:** You should be able to render this document right away, but there will be ERROR messages in many of the code chunks until you complete them.


```{r}
#| include: false
# OPTIONAL: Set a more color blind friendly palette 
palette("Okabe-Ito")
scale_colour_discrete <- function(...) scale_colour_manual(values = palette())
scale_fill_discrete   <- function(...) scale_fill_manual(values = palette())

knitr::opts_chunk$set(
  collapse = TRUE, 
  warning = FALSE,
  message = FALSE, 
  error = TRUE,
  fig.height = 2.75, 
  fig.width = 4.25,
  fig.env='figure',
  fig.pos = 'h',
  fig.align = 'center')
```



# Small Group Discussion


> **READING**    
> 
> For more on this topic, you're encouraged to read [ISLR](https://www.statlearning.com/) Chapter 8.2.



Within the broader machine learning landscape, we left off by discussing **supervised classification techniques**:

- **build a model** of *categorical variable* y by predictors x       
    - parametric model: logistic regression
    - nonparametric models: KNN & trees

- **evaluate the model**        
    We can use CV & in-sample techniques to estimate the accuracy of our classification models.
    - for binary y: sensitivity, specificity, ROC curves
    - for y with any number of categories: overall accuracy rates, category specific accuracy rates






**GOAL**        

Add more nonparametric algorithms to our toolkit: **random forests** & **bagging**



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## EXAMPLE 1: Anticipation

What does the word "forest" mean to you?









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## EXAMPLE 2: Candy!!!

[fivethirtyeight article](https://fivethirtyeight.com/videos/the-ultimate-halloween-candy-power-ranking/)

[the experiment](http://walthickey.com/2017/10/18/whats-the-best-halloween-candy/)

```{r}
library(tidyverse)
library(tidymodels)
library(rpart)        # for building trees
library(rpart.plot)   # for plotting trees
library(randomForest) # for bagging & forests
library(infer)        # for resampling

library(fivethirtyeight)
data("candy_rankings")

head(candy_rankings)
```

```{r}
# What are the 6 most popular candies?


# The least popular?

```



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## EXAMPLE 3: Build an unpruned tree


*For demonstration purposes only* let's:

- define a `popularity` variable that categorizes the candies as "low", "medium", or "high" popularity
- delete the original `winpercent` variable
- rename variables to make them easier to read in a tree
- make the candy name a row label, not a predictor

```{r}
candy <- candy_rankings %>% 
  mutate(popularity = cut(winpercent, breaks = c(0, 40, 60, 100), labels = c("low", "med", "high"))) %>% 
  select(-winpercent) %>% 
  rename("price" = pricepercent, "sugar" = sugarpercent, "nutty" = peanutyalmondy, "wafer" = crispedricewafer) %>% 
  column_to_rownames("competitorname")
```




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Our goal is to model candy `popularity` by all possible predictors in our data.


```{r fig.height = 8, fig.width = 8}
# STEP 1: tree specification
tree_spec <- decision_tree() %>%
  set_mode("classification") %>% 
  set_engine(engine = "rpart") %>% 
  set_args(cost_complexity = 0, min_n = 2, tree_depth = 30)

# STEP 2: Build the tree! No tuning (hence no workflows) necessary.
original_tree <- tree_spec %>% 
  fit(popularity ~ ., data = candy)

# Plot the tree
original_tree %>% 
  extract_fit_engine() %>% 
  plot(margin = 0) 
original_tree %>% 
  extract_fit_engine() %>% 
  text(cex = 0.7)
```

Ideally, our classification algorithm would have both low bias and low variance:

- low variance = the results wouldn't change much if we changed up the data set
- low bias = within any data set, the predictions of y tend to have low error / high accuracy

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\

Unfortunately, like other **overfit** algorithms, unpruned trees don't enjoy both of these.
They have...

- low bias, low variance
- low bias, high variance
- high bias, low variance
- high bias, high variance






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## New Concept

**GOAL**

Maintain the low bias of an unpruned tree while decreasing variance.


**APPROACH**

Build a bunch of unpruned trees from different data.

This way, our final result isn't overfit to our sample data.


**THE RUB**

We only have 1 set of data...








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## EXAMPLE 4: Take a REsample of candy

We only have 1 sample of data.
But we can *re*sample it (basically pretending we have a different sample).
Let's each take our *own* unique candy resample:

- Take a sample of 85 candies from the original 85 candies, *with* replacement.
- Some data points will be sampled multiple times while others aren't sampled at all.
- On average, 2/3 of the original data points will show up in the resample and 1/3 will be left out.



Take your resample:
    
```{r}
# Set the seed to YOUR phone number (just the numbers)
set.seed(5678325)

# Take a REsample of candies from our sample
my_candy <- sample_n(candy, size = nrow(candy), replace = TRUE)

# Check it out
head(my_candy, 3)
```

In the next exercise, we'll each build a tree of `popularity` using our own resample data.
First, check your intuition:

a. TRUE / FALSE: All of our trees will be the same.



b. TRUE / FALSE: Our trees will use the same *predictor* (but possibly a different cut-off) in the first split.



c. TRUE / FALSE: Our trees will use the same predictors in all splits.









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## EXAMPLE 5: Build & share YOUR tree

Build and plot a tree using your unique sample (`my_candy`):

```{r fig.width = 8, fig.height = 8}
# Build your tree
my_tree <- tree_spec %>% 
  fit(popularity ~ ., data = my_candy)

# Plot your tree
my_tree %>% 
  extract_fit_engine() %>% 
  plot(margin = 0) 
my_tree %>% 
  extract_fit_engine() %>% 
  text(cex = 0.7)
```

Use your tree to classify Baby Ruth, the 7th candy in the original data.

```{r}
my_tree %>% 
  predict(new_data = candy[7,])
```

Finally, share your results!

Record your prediction and paste a picture of your tree into [this document](https://docs.google.com/document/d/1S3l0dN1vPhy6N_OOom8Z91HPe4X9f-V-_EC0e9_L0do/edit?usp=sharing).







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## EXAMPLE 6: Using our FOREST

We now have a group of multiple trees -- a **forest**!
These trees...

- differ from resample to resample
- don't use the same predictor in each split (not even in the first split)!
- produce different `popularity` predictions for Baby Ruth

a. Based on our *forest* of trees (not just your 1 tree), what's your prediction for Baby Ruth's popularity?




b. What do you think are the advantages of predicting candy popularity using a forest instead of a single tree?




c. Can you anticipate any *drawbacks* of using forests instead of trees?

    








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# Notes

## BAGGING (Bootstrap AGGregatING) & RANDOM FORESTS  

To classify a categorical response variable y using a set of p predictors x:

- Take B *re*samples from the original sample.

- Use each resample to build an *unpruned* tree.    
    - For bagging: consider all p predictors in each split of each tree    
    - For forests: at each split in each tree, randomly select and consider only a *subset* of the predictors (often roughly p/2 or $\sqrt{p}$)

- Use each of the B trees to classify y at a set of predictor values x.    

- Average the classifications using a *majority vote*: classify y as the most common classification among the B trees.    




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## ENSEMBLE METHODS    

Bagging and random forest algorithms are **ensemble methods**.
They combine the outputs of *multiple* machine learning algorithms.
As a result, they decrease variability from sample to sample, hence provide more stable predictions / classifications than might be obtained by any algorithm alone.



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## EXAMPLE 7: pros & cons

a. Order trees, forests, & bagging algorithms from least to most computationally expensive.




b. What results will be easier to interpret: trees or forests?




c. Which of bagging or forests will produce a collection of trees that tend to look very similar to each other, and similar to the original tree? Hence which of these algorithms is more dependent on the sample data, thus will *vary* more if we change up the data?





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# Exercises



## Exercise 1: Tuning parameters (challenge)
    
Our random forest of `popularity` by all 11 possible predictors will depend upon 3 tuning parameters:

- `trees` = the number of trees in the forest
- `mtry` = number of predictors to randomly choose & consider at each split
- `min_n` = minimum number of data points in any leaf node of any tree

Check your intuition.

a. Does increasing the number of `trees` make the forest algorithm *more* or *less* variable from dataset to dataset? 
    
b. We have 11 possible predictors, and sqrt(11) is roughly 3. Recall: Would considering just 3 randomly chosen predictors in each split (instead of all 11) make the forest algorithm *more* or *less* variable from dataset to dataset?

c. Recall that using *unpruned* trees in our forest is important to maintaining low bias. Thus should `min_n` be small or big?



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## Exercise 2: Build the forest

Given that forests are relatively computationally expensive, we'll only build *one* forest using the following tuning parameters:

- `mtry = NULL`: this sets `mtry` to the default, which is sqrt(number of predictors)
- `trees = 500`
- `min_n = 2`

Fill in the below code to run this forest algorithm.

```{r}
# There's randomness behind the splits!
set.seed(253)

# STEP 1: Model Specification
rf_spec <- rand_forest()  %>%
  set_mode("___") %>%
  ___(engine = "ranger") %>% 
  ___(
    mtry = NULL,
    trees = 500,
    min_n = 2,
    probability = FALSE,    # Report classifications, not probability calculations
    importance = "impurity" # Use Gini index to measure variable importance
  )

# STEP 2: Build the forest
# There are no preprocessing steps or tuning, hence no need for a workflow!
candy_forest <- ___ %>% 
  fit(___, ___)
```
    






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## Exercise 3: Use the forest for prediction

Use the forest to predict the `popularity` level for Baby Ruth. (Remember that its real `popularity` is "med".)

```{r}
candy_forest %>% 
  predict(new_data = candy[7,])
```
    


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## Exercise 4: Evaluating forests (concepts)

But how *good* is our forest at classifying candy popularity?
To this end, we could evaluate 3 types of forest predictions.

a. Why don't **in-sample** predictions, i.e. asking how well our forest classifies our sample candies, give us an "honest" assessment of our forest's performance?




b. Instead, suppose we used **10-fold cross-validation (CV)** to estimate how well our forest classifies *new* candies. In this process, how many total *trees* would we need to construct?  




c. Alternatively, we can estimate how well our forest classifies *new* candies using the **out-of-bag (OOB) error rate**. Since we only use a *resample* of data points to build any given tree in the forest, the "out-of-bag" data points that do *not* appear in a tree's resample are natural test cases for that tree. The OOB error rate is the proportion or percent of these out-of-bag test cases that are misclassified by their tree. How many total trees would we need to construct to calculate the OOB error rate?


d. Moving forward, we'll use OOB and *not* CV to evaluate forest performance. Why?
    


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## Exercise 5: Evaluating forests (implementation)

a. Report and interpret the estimated `OOB prediction error`.

```{r}
candy_forest
``` 




b. The **test** or **OOB confusion matrix** provides more detail. Use this to confirm the OOB prediction error from part a. HINT: Remember to calculate *error* (1 - accuracy), not *accuracy*.       

```{r}
# NOTE: t() transposes the confusion matrix so that 
# the columns and rows are in the usual order
candy_forest %>% 
  extract_fit_engine() %>% 
  pluck("confusion.matrix") %>% 
  t()
```




c. Which level of candy popularity was least accurately classified by our forest?





d. Check out the **in-sample** confusion matrix. In general, are the in-sample predictions better or worse than the OOB predictions?

```{r}
# The cbind() includes the original candy data
# alongside their predicted popularity levels
candy_forest %>% 
  predict(new_data = candy) %>% 
  cbind(candy) %>% 
  conf_mat(
    truth = popularity,
    estimate = .pred_class
  )
```

  
    






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## Exercise 6: Variable importance

Variable importance metrics, averaged over all trees, measure the strength of the 11 predictors in classifying candy `popularity`:

```{r}
# Print the metrics
candy_forest %>%
  extract_fit_engine() %>%
  pluck("variable.importance") %>% 
  sort(decreasing = TRUE)

# Plot the metrics
library(vip)
candy_forest %>% 
  vip(geom = "point", num_features = 11)
```


a. If you're a candy connoisseur, does this ranking make some contextual sense to you?   




b. The only 2 *quantitative* predictors, `sugar` and `price`, have the highest importance metrics. This *could* simply be due to their quantitative structure: trees tend to favor predictors with lots of unique values. *Explain*. HINT: A tree's binary splits are identified by considering *every* possible cut / split point in *every* possible predictor.








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## Exercise 7: Classification regions

Just like any classification model, forests divide our data points into classification regions.
Let's explore this idea using some *simulated* data that illustrate some important contrasts.^[citation: https://daviddalpiaz.github.io/r4sl/ensemble-methods.html#tree-versus-ensemble-boundaries]
Import and plot the data:

```{r}
# Import data
simulated_data <- read.csv("https://kegrinde.github.io/stat253_coursenotes/data/circle_sim.csv") %>% 
  mutate(class = as.factor(class))

# Plot data
ggplot(simulated_data, aes(y = X2, x = X1, color = class)) + 
  geom_point() + 
  theme_minimal()
```

a. Below is a classification **tree** of `class` by `X1` and `X2`. What do you think its classification regions will look like?        

```{r}
# Build the (default) tree
circle_tree <- decision_tree() %>%
  set_mode("classification") %>% 
  set_engine(engine = "rpart") %>% 
  fit(class ~ ., data = simulated_data)

circle_tree %>% 
  extract_fit_engine() %>% 
  rpart.plot()
```




b. Check your intuition. Were you right?        

```{r}
# THIS IS ONLY DEMO CODE.
# Plot the tree classification regions
examples <- data.frame(X1 = seq(-1, 1, len = 100), X2 = seq(-1, 1, len = 100)) %>% 
  expand.grid()
circle_tree %>% 
  predict(new_data = examples) %>% 
  cbind(examples) %>% 
  ggplot(aes(y = X2, x = X1, color = .pred_class)) + 
    geom_point() + 
    labs(title = "tree classification regions") + 
    theme_minimal()
```



c. If we built a **forest** model of `class` by `X1` and `X2`, what do you think the classification regions will look like?





d. Check your intuition. Were you right?        

```{r}
# THIS IS ONLY DEMO CODE.
# Build the forest
circle_forest <- rf_spec %>% 
  fit(class ~ ., data = simulated_data)

# Plot the tree classification regions
circle_forest %>% 
  predict(new_data = examples) %>% 
  cbind(examples) %>% 
  ggplot(aes(y = X2, x = X1, color = .pred_class)) + 
    geom_point() + 
    labs(title = "forest classification regions") + 
    theme_minimal()
```


e. Reflect on what you've observed here!






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## Exercise 8: If you finish early

Do one of the following:        

- Check out the optional "Deeper learning" section below on another ensemble method: **boosting**. (it is one of the options for group assignment 2)

- Check out group assignment 2 on Moodle.

- Work on homework.






    




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# Deeper learning (optional)

*Extreme gradient boosting*, or **XGBoost**, is yet another ensemble algorithm for regression and classification.

We'll consider the big picture here.

If you want to dig deeper:

- Section 8.2.3 of the book provides a more detailed background
- Julia Silge's [blogpost on predicting home runs](https://juliasilge.com/blog/baseball-racing/) provides an example of implementing XGBoost using `tidymodels`.


The **big picture**:

- Like bagging and forests, boosting combines predictions from B different trees.

- BUT these trees aren't built from B different resamples. Boosting trees are grown *sequentially*, each tree *slowly learning* from the previous trees in the sequence to *improve* in areas where the previous trees didn't do well. Loosely speaking, data points with larger misclassification rates among previous trees are given more weight in building future trees.

- Unlike in bagging and forests, trees with better performance are given more weight in making future classifications. 


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\


**Bagging vs boosting**

- Bagging typically helps decrease variance, but not bias. Thus it is useful in scenarios where other algorithms are unstable and overfit to the sample data.

- Boosting typically helps decrease bias, but not variance. Thus it is useful in scenarios where other algorithms are stable, but overly simple. 



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# R code

Suppose we want to build a forest or bagging algorithm of some categorical response variable `y` using predictors `x1` and `x2` in our `sample_data`.

```{r eval = FALSE}
# Load packages
library(tidymodels)
library(rpart)
library(rpart.plot)

# Resolves package conflicts by preferring tidymodels functions
tidymodels_prefer()
```



\
\


**Make sure that y is a factor variable**

```{r eval = FALSE}
sample_data <- sample_data %>% 
  mutate(y = as.factor(y))
```


\
\


**Build the forest / bagging model**

We'll typically use the following tuning parameters:

- `trees` = 500 (the more trees we use, the less variable the forest)
- `min_n` = 2 (the smaller we allow the leaf nodes to be, the less pruned, hence less biased our forest will be)
- `mtry`
    - for forests: `mtry = NULL` (the default) will use the "floor", or biggest integer below, sqrt(number of predictors)
    - for bagging: set `mtry` to the number of predictors

```{r eval = FALSE}
# STEP 1: Model Specification
rf_spec <- rand_forest()  %>%
  set_mode("classification") %>%
  set_engine(engine = "ranger") %>% 
  set_args(
    mtry = ___,
    trees = 500,
    min_n = 2,
    probability = FALSE, # give classifications, not probability calculations
    importance = "impurity" # use Gini index to measure variable importance
  )

# STEP 2: Build the forest or bagging model
# There are no preprocessing steps or tuning, hence no need for a workflow!
ensemble_model <- rf_spec %>% 
  fit(y ~ x1 + x2, data = sample_data)
```

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\


**Use the model to make predictions / classifications**


```{r eval = FALSE}
# Put in a data.frame object with x1 and x2 values (at minimum)
ensemble_model %>% 
  predict(new_data = ___)  
```




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\


**Examine variable importance**

```{r eval = FALSE}
# Print the metrics
ensemble_model %>%
  extract_fit_engine() %>%
  pluck("variable.importance") %>% 
  sort(decreasing = TRUE)

# Plot the metrics
# Plug in the number of top predictors you wish to plot
# (The upper limit varies by application!)
library(vip)
ensemble_model %>% 
  vip(geom = "point", num_features = ___)
```



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\



**Evaluate the classifications**

```{r eval = FALSE}
# Out-of-bag (OOB) prediction error
ensemble_model

# OOB confusion matrix
ensemble_model %>% 
  extract_fit_engine() %>% 
  pluck("confusion.matrix") %>% 
  t()

# In-sample confusion matrix
ensemble_model %>% 
  predict(new_data = sample_data) %>% 
  cbind(sample_data) %>% 
  conf_mat(
    truth = y,
    estimate = .pred_class
  )
```

  

