Lecture .mono[004]

class: center, middle, inverse, title-slide

.title[
# Lecture .mono[004]
]
.subtitle[
## Regression strikes back
]
.author[
### Edward Rubin
]

---

exclude: true

---
layout: true
# Admin

---
class: inverse, middle

---
name: admin-today
## Today

.b[In-class]

- A roadmap (where are we going?)
- Linear regression and model selection

---
name: admin-soon
# Admin

## Upcoming

.b[Readings]

- .note[Today]
  - .it[ISL] Ch. 3 and 6.1
- .note[Next]
  - .it[ISL] Ch. 6 and 4

.b[Problem set] Next problem set very soon!

---
name: admin-roadmap
layout: false
# Roadmap
## Where are we?

We've essentially covered the central topics in statistical learning.super[.pink[†]]

- Prediction and inference
- Supervised .it[vs.] unsupervised methods
- Regression and classification problems
- The dangers of overfitting
- The bias-variance tradeoff
- Model assessment
- Holdouts, validation sets, and cross validation.super[.pink[††]]
- Model training and tuning
- Simulation

.footnote[
.pink[†] Plus a few of the "basic" methods: OLS regression and KNN.
<br>.pink[††] And the bootstrap!
]

---
# Roadmap
## Where are we going?

Next, we will cover many common machine-learning algorithms, _e.g._,

- Decision trees
- Random forests and ensemble techniques
- SVM
- Neural nets
- Clustering

But first, we return to good old .hi-orange[linear regression]—in a new light...

- Linear regression
- Variable/model selection and LASSO/Ridge regression
- .it[Plus:] Logistic regression and discriminant analysis
---
# Roadmap
## Why return to regression?

.hi[Motivation 1]
<br>We have new tools. It might help to first apply them in a .b[familiar] setting.

.hi[Motivation 2]
<br>We have new tools. Maybe linear regression will be (even) .b[better now?]
<br>.it[E.g.], did (cross) validation help you beat your old model?

.hi[Motivation 3]
> many fancy statistical learning approaches can be seen as .b[generalizations or extensions of linear regression].

.pad-left[.grey-light.it[Source: ISL, p. 59; emphasis added]]

---
layout: true
# Linear regression

---
class: inverse, middle

---
name: regression-intro
## Regression regression

.note[Recall] Linear regression "fits" coefficients `$\color{#e64173}{\beta}_0,\, \ldots,\, \color{#e64173}{\beta}_p$` for a model
$$
`\begin{align}
  \color{#FFA500}{y}_i = \color{#e64173}{\beta}_0 + \color{#e64173}{\beta}_1 x_{1,i} + \color{#e64173}{\beta}_2 x_{2,i} + \cdots + \color{#e64173}{\beta}_p x_{p,i} + \varepsilon_i
\end{align}`
$$
and is often applied in two distinct settings with fairly distinct goals:

1. .hi[Causal inference] estimates and interprets .pink[the coefficients].

1. .hi-orange[Prediction] focuses on accurately estimating .orange[outcomes].

Regardless of the goal, the way we "fit" (estimate) the model is the same.

---
## Fitting the regression line

As is the case with many statistical learning methods, regression focuses on minimizing some measure of loss/error.

$$
`\begin{align}
  e_i = \color{#FFA500}{y_i} - \color{#6A5ACD}{\hat{y}_i}
\end{align}`
$$

Linear regression uses the L.sub[2] loss function—also called .it[residual sum of squares] (RSS) or .it[sum of squared errors] (SSE)

$$
`\begin{align}
  \text{RSS} = e_1^2 + e_2^2 + \cdots + e_n^2 = \sum_{i=1}^n e_i^2
\end{align}`
$$

Specifically: OLS chooses the `$\color{#e64173}{\hat{\beta}_j}$` that .hi[minimize RSS].

---
name: performance
## Performance

There's a large variety of ways to assess the fit.super[.pink[†]] of linear-regression models.
.footnote[
.pink[†] or predictive performance
]

.hi[Residual standard error] (.hi[RSE])
$$
`\begin{align}
  \text{RSE}=\sqrt{\dfrac{1}{n-p-1}\text{RSS}}=\sqrt{\dfrac{1}{n-p-1}\sum_{i=1}^{n} \left( y_i - \hat{y}_i \right)^2}
\end{align}`
$$

.hi[R-squared] (.hi[R.super[2]])
$$
`\begin{align}
  R^2 = \dfrac{\text{TSS} - \text{RSS}}{\text{TSS}} = 1 - \dfrac{\text{RSS}}{\text{TSS}} \quad \text{where} \quad \text{TSS} = \sum_{i=1}^{n} \left( y_i - \overline{y} \right)^2
\end{align}`
$$

---
## Performance and overfit

As we've seen throughout the course, we need to be careful .b[not to overfit].

.hi[R.super[2]] provides no protection against overfitting—and actually encourages it.
$$
`\begin{align}
  R^2 = 1 - \dfrac{\text{RSS}}{\text{TSS}}
\end{align}`
$$
.attn[Add a new variable:] RSS `$\downarrow$` and TSS is unchanged. Thus, R.super[2] increases.

.hi[RSE] .it[slightly] penalizes additional variables:
$$
`\begin{align}
  \text{RSE}=\sqrt{\dfrac{1}{n-p-1}\text{RSS}}
\end{align}`
$$
.attn[Add a new variable:] RSS `$\downarrow$` but `$p$` increases. Thus, RSE's change is uncertain.

---
exclude: true

```r
library(pacman)
p_load(parallel, stringr, data.table, magrittr, here)
# Set parameters
set.seed(123)
N = 2e3
n = 500
p = n - 1
# Generate data
X = matrix(data = rnorm(n = N*p), ncol = p)
β = matrix(data = rnorm(p, sd = 0.005), ncol = 1)
y = X %*% β + matrix(rnorm(N, sd = 0.01), ncol = 1)
# Create a data table
pop_dt = X %>% data.matrix() %>% as.data.table()
setnames(pop_dt, paste0("x", str_pad(1:p, 4, "left", 0)))
pop_dt[, y := y %>% unlist()]
# Subset
sub_dt = pop_dt[1:n,]
out_dt = pop_dt[(n+1):N,]
Nn = N - n
# For j in 1 to p: fit a model, record R2 and RSE
fit_dt = mclapply(X = seq(1, p, by = 5), mc.cores = 12, FUN = function(j) {
  # Fit a model with the the first j variables
  lm_j = lm(y ~ ., data = sub_dt[, c(1:j,p+1), with = F])
  # Unseen data performance
  y_hat = predict(lm_j, newdata = out_dt[, c(1:j,p+1), with = F])
  out_rss = sum((out_dt[,y] - y_hat)^2)
  out_tss = sum((out_dt[,y] - mean(out_dt[,y]))^2)
  # Return data table
  data.table(
    p = j,
    in_rse = summary(lm_j)$sigma,
    in_r2 = summary(lm_j)$r.squared,
    in_r2_adj = summary(lm_j)$adj.r.squared,
    in_aic = AIC(lm_j),
    in_bic = BIC(lm_j),
    out_rse = sqrt(1 / (Nn - j - 1) * out_rss),
    out_r2 = 1 - out_rss/out_tss,
    out_r2_adj = 1 - ((out_rss) / (Nn - j - 1)) / ((out_tss) / (Nn-1))
  )
}) %>% rbindlist()
# Save results
saveRDS(
  object = fit_dt,
  file = here("other-files", "overfit-data.rds")
)
```

```r
# Load the data
fit_dt = here("other-files", "overfit-data.rds") %>% read_rds()
```

---
layout: true
class: clear

---
name: overfit
class: middle

## Example
Let's see how .b[R.super[2]] and .b[RSE] perform with 500 very weak predictors.

To address overfitting, we can compare .b[in-] .it[vs.] .hi[out-of-sample] performance.

---

.b[In-sample R.super[2]] mechanically increases as we add predictors.
<br>
.white[.b[Out-of-sample R.super[2]] does not.]

---

.b[In-sample R.super[2]] mechanically increases as we add predictors.
<br>
.hi[Out-of-sample R.super[2]] does not.

---
class: middle

What about RSE? Does its penalty .it[help]?

---

Despite its penalty for adding variables, .b[in-sample RSE] still can overfit,
<br>
.white[as evidenced by .b[out-of-sample RSE].]

---

Despite its penalty for adding variables, .b[in-sample RSE] still can overfit,
<br>
as evidenced by .hi[out-of-sample RSE].

---
layout: false
# Linear regression
## Penalization

RSE is not the only way to penalization the addition of variables..super[.pink[†]]

.footnote[
.pink[†] We'll talk about other penalization methods (LASSO and Ridge) shortly.
]

.b[Adjusted R.super[2]] is another .it[classic] solution.

$$
`\begin{align}
  \text{Adjusted }R^2 = 1 - \dfrac{\text{RSS}\color{#6A5ACD}{/(n - p - 1)}}{\text{TSS}\color{#6A5ACD}{/(n-1)}}
\end{align}`
$$

Adj. R.super[2] attempts to "fix" R.super[2] by .hi-purple[adding a penalty for the number of variables].

- `$\text{RSS}$` always decreases when a new variable is added.

- `$\color{#6A5ACD}{\text{RSS}/(n-p-1)}$` may increase or decrease with a new variable.

---
layout: true
class: clear

---

However, .b[in-sample adjusted R.super[2]] still can overfit.
<br>
.white[Illustrated by .b[out-of-sample R.super[2]].]

---

However, .b[in-sample adjusted R.super[2]] still can overfit.
<br>
Illustrated by .hi[out-of-sample adjusted R.super[2]].

---

Here are .b[in-sample] .b.orange[AIC] and .b.purple[BIC].
<br>
Neither in-sample metric seems to entirely guard against overfitting.

---
layout: true
# Model selection

---
name: better
## A better way?

R.super[2], adjusted R.super[2], and RSE each offer some flavor of model fit, but they appear .b[limited in their abilities to prevent overfitting].

We want a method to optimally select a (linear) model—balancing variance and bias and avoiding overfit.

We'll discuss two (related) methods today:

1. .hi[Subset selection] chooses a (sub)set of our `$p$` potential predictors

2. .hi[Shrinkage] fits a model using all `$p$` variables but "shrinks" its coefficients

---
name: subset-selection
## Subset selection

In .attn[subset selection], we

1. whittle down the `$p$` potential predictors (using some magic/algorithm)
1. estimate the chosen linear model using OLS

How do we do the *whittling* (selection)?
--
 We've got .b[options].

- .attn[Best subset selection] fits a model for every possible subset.
- .attn[Forward stepwise selection] starts with only an intercept and tries to build up to the best model (using some fit criterion).
- .attn[Backward stepwise selection] starts with all `$p$` variables and tries to drop variables until it hits the best model (using some fit criterion).
- .attn[Hybrid approaches] are what their name implies (_i.e._, hybrids).

---
name: best-subset
## Best subset selection

.attn[Best subset selection] is based upon a simple idea: Estimate a model for every possible subset of variables; then compare their performances.

.qa[Q] So what's the problem? (Why do we need other selection methods?)
--
<br>
.qa[A] "a model for .hi[every possible subset]" can mean .hi[a lot] `$\left( 2^p \right)$` of models.

.note[E.g.,]
- 10 predictors `$\rightarrow$` 1,024 models to fit
- 25 predictors `$\rightarrow$` >33.5 million models to fit
- 100 predictors `$\rightarrow$` ~1.5 trillion models to fit

Even with plentiful, cheap computational power, we can run into barriers.

---
## Best subset selection

Computational constraints aside, we can implement .attn[best subset selection] as

.mono-small[
1. Define `$\mathcal{M}_0$` as the model with no predictors.

1. For `$k$` in 1 to `$p$`:

- Fit every possible model with `$k$` predictors.

- Define `$\mathcal{M}_k$` as the "best" model with `$k$` predictors.

1. Select the "best" model from `$\mathcal{M}_0,\,\ldots,\,\mathcal{M}_p$`.
]

As we've seen, RSS declines (and R.super[2] increases) with `$p$`, so we should use a cross-validated measure of model performance in step .mono[3]..super[.pink[†]]

.footnote[
.pink[†] Back to our distinction between test .it[vs.] training performance.
]

---
## Example dataset: `Credit`

We're going to use the `Credit` dataset from .it[ISL]'s R package `ISLR`.

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The `Credit` dataset has 400 observations on 12 variables.

---
## Example dataset: `Credit`

We need to pre-process the dataset before we can select a model...

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Now the dataset on has 400 observations on 12 variables (2,048 subsets).

---
exclude: true

---
layout: false
class: clear, middle

---
layout: false
class: clear, middle

---
layout: true
# Model selection

---
## Best subset selection

From here, you would

1. Estimate cross-validated error for each `$\mathcal{M}_k$`.

1. Choose the `$\mathcal{M}_k$` that minimizes the CV error.

1. Train the chosen model on the full dataset.

---
## Best subset selection

.b[Warnings]
- Computationally intensive
- Selected models may not be "right" (squared terms with linear terms)
- You need to protect against overfitting when choosing across `$\mathcal{M}_k$`
- Also should worry about overfitting when `$p$` is "big"
- Dependent upon the variables (transformations) you provide

.b[Benefits]
- Comprehensive search across provided variables
- Resulting model—when estimated with OLS—has OLS properties
- Can be applied to other (non-OLS) estimators

---
name: stepwise
## Stepwise selection

.attn[Stepwise selection] provides a less computational intensive alternative to best subset selection.

The basic idea behind .attn[stepwise selection]

.mono-small[
1. Start with an arbitrary model.
1. Try to find a "better" model by adding/removing variables.
1. Repeat.
1. Stop when you have the best model. (Or choose the best model.)
]

The two most-common varieties of stepwise selection:
- .attn[Forward] starts with only intercept `$\left( \mathcal{M}_0 \right)$` and adds variables
- .attn[Backward] starts with all variables `$\left( \mathcal{M}_p \right)$` and removes variables

---
name: forward
## Forward stepwise selection

The process...

.mono-small[
1. Start with a model with only an intercept (no predictors), `$\mathcal{M}_0$`.

1. For `$k=0,\,\ldots,\,p$`:

- Estimate a model for each of the remaining `$p-k$` predictors, separately adding the predictors to model `$\mathcal{M}_k$`.

- Define `$\mathcal{M}_{k+1}$` as the "best" model of the `$p-k$` models.

1. Select the "best" model from `$\mathcal{M}_0,\,\ldots,\, \mathcal{M}_p$`.
]

What do we mean by "best"?
<br> .mono-small[2:] .it[best] is often RSS or R.super[2].
<br> .mono-small[3:] .it[best] should be a cross-validated fit criterion.

---
layout: false
class: clear

.hi-purple[Forward stepwise selection] with `caret` in R

```r
train_forward = train(
  y = credit_dt[["balance"]],
  x = credit_dt %>% dplyr::select(-balance),
  trControl = trainControl(method = "cv", number = 5),
  method = "leapForward",
  tuneGrid = expand.grid(nvmax = 1:11)
)
```

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]

.col-right[

]

---
name: backward
# Model selection
## Backward stepwise selection

The process for .attn[backward stepwise selection] is quite similar...

.mono-small[
1. Start with a model that includes all `$p$` predictors: `$\mathcal{M}_p$`.

1. For `$k=p,\, p-1,\, \ldots,\,1$`:

- Estimate `$k$` models, where each model removes exactly one of the `$k$` predictors from `$\mathcal{M}_k$`.

- Define `$\mathcal{M}_{k-1}$` as the "best" of the `$k$` models.

1. Select the "best" model from `$\mathcal{M}_0,\,\ldots,\, \mathcal{M}_p$`.
]

What do we mean by "best"?
<br> .mono-small[2:] .it[best] is often RSS or R.super[2].
<br> .mono-small[3:] .it[best] should be a cross-validated fit criterion.

---
layout: false
class: clear

.hi-pink[Backward stepwise selection] with `caret` in R

```r
train_backward = train(
  y = credit_dt[["balance"]],
  x = credit_dt %>% dplyr::select(-balance),
  trControl = trainControl(method = "cv", number = 5),
  method = "leapBackward",
  tuneGrid = expand.grid(nvmax = 1:11)
)
```

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]

.col-right[

]

---
class: clear, middle

.note[Note:] .hi-purple[forward] and .hi-pink[backward] step. selection can choose different models.

---
name: stepwise-notes
# Model selection
## Stepwise selection

Notes on stepwise selection

- .b[Less computationally intensive] (relative to best subset selection)
  - With `$p=20$`, BSS fits 1,048,576 models.
  - With `$p=20$`, foward/backward selection fits 211 models.

- There is .b[no guarantee] that stepwise selection finds the best model.

- .b.it[Best] is defined by your fit criterion (as always).

- Again, .b[cross validation is key] to avoiding overfitting.

---
name: criteria
# Model selection
## Criteria

Which model you choose is a function of .b[how you define "best"].

And we have many options...
--
 We've seen RSS, (R)MSE, RSE, MA, R.super[2], Adj. R.super[2].

Of course, there's more. Each .hi-purple[penalizes] the `$d$` predictors differently.
$$
`\begin{align}
  C_p &= \frac{1}{n} \left( \text{RSS} + \color{#6A5ACD}{2 d \hat{\sigma}^2 }\right)
  \\[1ex]
  \text{AIC} &= \frac{1}{n\hat{\sigma}^2} \left( \text{RSS} + \color{#6A5ACD}{2 d \hat{\sigma}^2 }\right)
  \\[1ex]
  \text{BIC} &= \frac{1}{n\hat{\sigma}^2} \left( \text{RSS} + \color{#6A5ACD}{\log(n) d \hat{\sigma}^2 }\right)
\end{align}`
$$

---
# Model selection
## Criteria

> `$C_p$`, `$AIC$`, and `$BIC$` all have rigorous theoretical justifications... the adjusted `$R^2$` is not as well motivated in statistical theory

.grey-light.it[ISL, p. 213]

In general, we will stick with cross-validated criteria, but you still need to choose a selection criterion.

---
name: sources
layout: false
# Sources

These notes draw upon

- [An Introduction to Statistical Learning](http://faculty.marshall.usc.edu/gareth-james/ISL/) (*ISL*)<br>James, Witten, Hastie, and Tibshirani
---
# Table of contents

.col-left[
.smallest[
#### Admin
- [Today](#admin-today)
- [Upcoming](#admin-soon)
- [Roadmap](#admin-roadmap)

#### Linear regression
- [Regression regression](#regression-intro)
- [Performance](#performance)
- [Overfit](#overfit)

#### Model selection
- [A better way?](#better)
- [Subset selection](#subset-selection)
  - [Best subset selection](#best-subset)
  - [Stepwise selection](#stepwise)
    - [Forward](#forward)
    - [Backward](#backward)
    - [Notes](#stepwise-notes)
- [Criteria](#criteria)
]
]

.col-right[
.smallest[

#### Other
- [Sources/references](#sources)
]
]

---
exclude: true