Load necessary libraries

options(warnPartialMatchArgs = FALSE)  # don't want these warnings
library(magrittr)     # pipes
library(tibble)       # tibbles
library(dplyr)        # data wrangling
library(boot)         # for `cv.glm`
library(purrr)        # iteration
library(ggplot2)      # tidy plotting
library(ISLR)         # Auto data set

1 The Auto Data Set

Auto %<>% as.tibble()                     # Convert -> tibble
class(Auto)                               # Confirm Auto is now a tibble
[1] "tbl_df"     "tbl"        "data.frame"
names(Auto)                               # List variables
[1] "mpg"          "cylinders"    "displacement" "horsepower"   "weight"       "acceleration"
[7] "year"         "origin"       "name"        
Auto                                      # Print tibble; top 10 by default
Auto %>%                                  
  ggplot(aes(x = horsepower, y = mpg)) +  # Plot hp vs mpg
  geom_point(alpha = 0.5) +               # Create a scatter plot
  geom_smooth(method = 'loess')           # Fit a polynomial smoothing function

2 Cross-validation (by hand)

# Create a random partition into training and test sets
set.seed(1)
n     <- nrow(Auto)
train <- sample(1:n, n/2)         # Randomly select half of the numbers 1 to n

# Fit linear model -> mpg as a function of hp
stats::lm(mpg ~ horsepower, data = Auto, subset = train)  # linear in horsepower

Call:
stats::lm(formula = mpg ~ horsepower, data = Auto, subset = train)

Coefficients:
(Intercept)   horsepower  
    40.3404      -0.1617  
# Loop over polynomial power and calculate MSE on test set (-train)
purrr::map_dbl(1:5,
  ~stats::lm(mpg ~ stats::poly(horsepower, .x),  # fit lm
             data = Auto, subset = train) %>%  
    predict(., Auto) %>%                 # Predict for all samples
    magrittr::subtract(Auto$mpg) %>%     # Obtain differences from actual values
    magrittr::raise_to_power(2) %>%      # Square differences
    magrittr::extract(-train) %>%        # Ignore training predictions
    mean()                               # MSE
)
[1] 26.14142 19.82259 19.78252 19.99969 20.18225
# Set a different random seed to create a different random train/test partition
set.seed(2)
train <- sample(1:n, n/2)                # Randomly select half of the numbers 1 to n
purrr::map_dbl(1:5,
  ~stats::lm(mpg ~ stats::poly(horsepower, .x),
             data = Auto, subset = train) %>%  # fit lm
    predict(., Auto) %>%                 # Predict for all samples
    magrittr::subtract(Auto$mpg) %>%     # Obtain differences from actual values
    magrittr::raise_to_power(2) %>%      # Square differences
    magrittr::extract(-train) %>%        # Ignore training predictions
    mean()                               # MSE
)
[1] 23.29559 18.90124 19.25740 20.38538 20.29775

3 Leave-One-Out CV (boot::cv.glm)

# Linear regression using glm
glm_fit <- stats::glm(mpg ~ horsepower, data = Auto)
cv_err  <- boot::cv.glm(Auto, glm_fit)
cv_err$K            # Default K = n; i.e. LOOCV
[1] 392
cv_err$delta
[1] 24.23151 24.23114
# Loop over polynomial power
# raw = TRUE issue resolved
purrr::map_dbl(1:5, function(.x) {
  fit <- stats::glm(mpg ~ stats::poly(horsepower, degree = .x, raw = TRUE),
                    data = Auto)
  boot::cv.glm(Auto, fit) %>%
  purrr::pluck("delta") %>%     # Retrieve the `delta` element
  purrr::pluck(1)               # Extract the first component of `delta`
})
[1] 24.23151 19.24821 19.33498 19.42443 19.03321

4 K-fold CV (boot::cv.glm)

set.seed(17)
# K-folds chosen at random; here K = 10
# Loop over polynomial power
# Default: `raw = FALSE`
purrr::map_dbl(1:10, function(.x) {
  fit <- stats::glm(mpg ~ stats::poly(horsepower, degree = .x), data = Auto)
  boot::cv.glm(Auto, fit, K = 10) %>%
    purrr::pluck("delta") %>%
    purrr::pluck(1)
})
 [1] 169.3269 190.3656 194.8383 195.6951 192.9770 196.7003 197.5728 197.6055 195.7763
[10] 203.9586

5 Bootstrap (boot::boot)

Here we estimate the accuracy of a linear regression model via the bootstrap. We will first define a function to fit a linear model of mpg ~ horsepower given a dataset and a subset index (rows).

5.1 Sampling with replacement

# What does sample with replacement do?
# Sampling with replacement is what the bootstrap is all about!
set.seed(3)
sample(1:10, 10, replace = FALSE)
 [1]  2  8  4  3  9  6  1  5 10  7
sample(1:10, 10, replace = TRUE)
 [1] 6 6 6 6 9 9 2 8 9 3

5.2 Create a boot function

boot_fn <- function(data, index) {
  lm(mpg ~ horsepower, data = data, subset = index) %>%
    purrr::pluck("coefficients")
}

5.3 Perform bootstrap fits (by hand)

# Using all the data
boot_fn(Auto, 1:392)
(Intercept)  horsepower 
 39.9358610  -0.1578447 
# Make the simulation reproducible
set.seed(1)

# First sample
sample(1:392, 392, replace = TRUE) %>% boot_fn(Auto, .)
(Intercept)  horsepower 
 38.7387134  -0.1481952 
# Second sample
sample(1:392, 392, replace = TRUE) %>% boot_fn(Auto, .)
(Intercept)  horsepower 
 40.0383086  -0.1596104

5.4 Use boot::boot()

# 1000 bootstrap samples; gets tedious, so use `boot::boot`
boot::boot(Auto, boot_fn, R = 1000)

ORDINARY NONPARAMETRIC BOOTSTRAP


Call:
boot::boot(data = Auto, statistic = boot_fn, R = 1000)


Bootstrap Statistics :
      original      bias    std. error
t1* 39.9358610  0.02972191 0.860007896
t2* -0.1578447 -0.00030823 0.007404467
# Compare with linear regression
# (differences indicate some assumptions may be broken)
lm(mpg ~ horsepower, data = Auto) %>%
  summary() %>%
  purrr::pluck("coefficients")
              Estimate  Std. Error   t value      Pr(>|t|)
(Intercept) 39.9358610 0.717498656  55.65984 1.220362e-187
horsepower  -0.1578447 0.006445501 -24.48914  7.031989e-81

5.5 Repeat with a quadratic model

# Redefine `boot_fn()`
boot_fn2 <- function(data, index) {
  lm(mpg ~ horsepower + I(horsepower^2), data = data, subset = index) %>%
    purrr::pluck("coefficients")
}

# Make the simulation reproducible
set.seed(1)
boot(Auto, boot_fn2, 1000)

ORDINARY NONPARAMETRIC BOOTSTRAP


Call:
boot(data = Auto, statistic = boot_fn2, R = 1000)


Bootstrap Statistics :
        original        bias     std. error
t1* 56.900099702  6.098115e-03 2.0944855842
t2* -0.466189630 -1.777108e-04 0.0334123802
t3*  0.001230536  1.324315e-06 0.0001208339
# Compare with standard error from linear regression
lm(mpg ~ horsepower + I(horsepower^2), data = Auto) %>%
  summary() %>%
  purrr::pluck("coefficients")
                    Estimate   Std. Error   t value      Pr(>|t|)
(Intercept)     56.900099702 1.8004268063  31.60367 1.740911e-109
horsepower      -0.466189630 0.0311246171 -14.97816  2.289429e-40
I(horsepower^2)  0.001230536 0.0001220759  10.08009  2.196340e-21

Created on 2020-01-30 by Rmarkdown (v1.10) and R version 3.5.0 (2018-04-23).

---
title: 'STAA 577: Laboratory Five </br> Resampling methods (`tidyverse`)'
author: 'Adapted by Tavener & Field </br> From: James, Witten, Hastie and Tibshirani'
date: "`r format(Sys.Date(), '%e %B %Y')`"
output:
  html_notebook:
    code_folding: show
    number_sections: yes
    toc: yes
    toc_float:
      collapsed: no
ratio: '9:16'
tables: yes
fontsize: 12pt
---


### Load necessary libraries {-}
```{r setup, message = FALSE, warning = FALSE}
options(warnPartialMatchArgs = FALSE)  # don't want these warnings
library(magrittr)     # pipes
library(tibble)       # tibbles
library(dplyr)        # data wrangling
library(boot)         # for `cv.glm`
library(purrr)        # iteration
library(ggplot2)      # tidy plotting
library(ISLR)         # Auto data set
```

---------------------------

# The `Auto` Data Set
```{r data, include = TRUE}
Auto %<>% as.tibble()                     # Convert -> tibble
class(Auto)                               # Confirm Auto is now a tibble
names(Auto)                               # List variables
Auto                                      # Print tibble; top 10 by default
Auto %>%                                  
  ggplot(aes(x = horsepower, y = mpg)) +  # Plot hp vs mpg
  geom_point(alpha = 0.5) +               # Create a scatter plot
  geom_smooth(method = 'loess')           # Fit a polynomial smoothing function
```


------------------------------


# Cross-validation (by hand)

```{r cv}
# Create a random partition into training and test sets
set.seed(1)
n     <- nrow(Auto)
train <- sample(1:n, n/2)         # Randomly select half of the numbers 1 to n

# Fit linear model -> mpg as a function of hp
stats::lm(mpg ~ horsepower, data = Auto, subset = train)  # linear in horsepower

# Loop over polynomial power and calculate MSE on test set (-train)
purrr::map_dbl(1:5,
  ~stats::lm(mpg ~ stats::poly(horsepower, .x),  # fit lm
             data = Auto, subset = train) %>%  
    predict(., Auto) %>%                 # Predict for all samples
    magrittr::subtract(Auto$mpg) %>%     # Obtain differences from actual values
    magrittr::raise_to_power(2) %>%      # Square differences
    magrittr::extract(-train) %>%        # Ignore training predictions
    mean()                               # MSE
)

# Set a different random seed to create a different random train/test partition
set.seed(2)
train <- sample(1:n, n/2)                # Randomly select half of the numbers 1 to n
purrr::map_dbl(1:5,
  ~stats::lm(mpg ~ stats::poly(horsepower, .x),
             data = Auto, subset = train) %>%  # fit lm
    predict(., Auto) %>%                 # Predict for all samples
    magrittr::subtract(Auto$mpg) %>%     # Obtain differences from actual values
    magrittr::raise_to_power(2) %>%      # Square differences
    magrittr::extract(-train) %>%        # Ignore training predictions
    mean()                               # MSE
)
```



# Leave-One-Out CV (`boot::cv.glm`)

```{r LOOCV}
# Linear regression using glm
glm_fit <- stats::glm(mpg ~ horsepower, data = Auto)
cv_err  <- boot::cv.glm(Auto, glm_fit)
cv_err$K            # Default K = n; i.e. LOOCV
cv_err$delta
```


```{r LOOCV_poly, eval = FALSE, echo = FALSE}
# Loop over polynomial power
# This chunk not currently run
purrr::map_dbl(1:5, function(.x) {
  fit <- stats::glm(mpg ~ stats::poly(horsepower, degree = .x),
                    data = Auto)
  boot::cv.glm(Auto, fit) %>%
  purrr::pluck("delta") %>%     # Retrieve the `delta` element
  purrr::pluck(1)               # Extract the first component of `delta`
})
```


```{r LOOCV_poly_2}
# Loop over polynomial power
# raw = TRUE issue resolved
purrr::map_dbl(1:5, function(.x) {
  fit <- stats::glm(mpg ~ stats::poly(horsepower, degree = .x, raw = TRUE),
                    data = Auto)
  boot::cv.glm(Auto, fit) %>%
  purrr::pluck("delta") %>%     # Retrieve the `delta` element
  purrr::pluck(1)               # Extract the first component of `delta`
})
```


-------------------------


# K-fold CV (`boot::cv.glm`)

```{r k-fold_cross_validation_orthogonal_polynomials}
set.seed(17)
# K-folds chosen at random; here K = 10
# Loop over polynomial power
# Default: `raw = FALSE`
purrr::map_dbl(1:10, function(.x) {
  fit <- stats::glm(mpg ~ stats::poly(horsepower, degree = .x), data = Auto)
  boot::cv.glm(Auto, fit, K = 10) %>%
    purrr::pluck("delta") %>%
    purrr::pluck(1)
})
```

--------------------

# Bootstrap (`boot::boot`)

Here we estimate the accuracy of a linear regression model via the bootstrap.
We will first define a function to fit a linear model of `mpg ~ horsepower`
given a dataset and a subset index (rows).

## Sampling with replacement
```{r Sampling_with_replacement}
# What does sample with replacement do?
# Sampling with replacement is what the bootstrap is all about!
set.seed(3)
sample(1:10, 10, replace = FALSE)
sample(1:10, 10, replace = TRUE)
```

## Create a boot function
```{r bootstrap_function}
boot_fn <- function(data, index) {
  lm(mpg ~ horsepower, data = data, subset = index) %>%
    purrr::pluck("coefficients")
}
```

## Perform bootstrap fits (by hand)
```{r bootstrap}
# Using all the data
boot_fn(Auto, 1:392)

# Make the simulation reproducible
set.seed(1)

# First sample
sample(1:392, 392, replace = TRUE) %>% boot_fn(Auto, .)

# Second sample
sample(1:392, 392, replace = TRUE) %>% boot_fn(Auto, .)
```


## Use `boot::boot()`
```{r bootstrap with boot::boot}
# 1000 bootstrap samples; gets tedious, so use `boot::boot`
boot::boot(Auto, boot_fn, R = 1000)

# Compare with linear regression
# (differences indicate some assumptions may be broken)
lm(mpg ~ horsepower, data = Auto) %>%
  summary() %>%
  purrr::pluck("coefficients")
```

## Repeat with a quadratic model

```{r boot2}
# Redefine `boot_fn()`
boot_fn2 <- function(data, index) {
  lm(mpg ~ horsepower + I(horsepower^2), data = data, subset = index) %>%
    purrr::pluck("coefficients")
}

# Make the simulation reproducible
set.seed(1)
boot(Auto, boot_fn2, 1000)

# Compare with standard error from linear regression
lm(mpg ~ horsepower + I(horsepower^2), data = Auto) %>%
  summary() %>%
  purrr::pluck("coefficients")
```


-------------------------

Created on `r Sys.Date()` by [Rmarkdown](https://github.com/rstudio/rmarkdown)
(v`r utils::packageVersion("rmarkdown")`) and `r R.version$version.string`.
