lab0.Rmd

---
title: 'MA 415/615 -- Lab 0: Workflow example'
author: Spring 2023
date: "Week of 01-23-2023"
output: html_document
subtitle: Based on Lecture 2
---

```{r setup, include=FALSE}
options(show.signif.stars = FALSE)
knitr::opts_chunk$set(tidy = FALSE) # turn off formatR's tidy
knitr::opts_chunk$set(fig.path="figures/", out.width="70%", fig.align = "center")
```

# Introduction

This is a simple notebook documenting the basic analyses in Lecture 2 of MA 415 / MA 615.


## Setting up R

After starting RStudio, we load packages in `tidyverse`:

```{r tidyverse}
library(tidyverse)
```

If the package is not installed, we can do it with:

```{r install, eval=FALSE}
install.packages("tidyverse") # it may take a while!
```

# Exploring the data

We will be using the `mpg` dataset on fuel economy data:

```{r mpg}
mpg
```

We get a description of the dataset by issuing `help(mpg)` or just `?mpg`.

Our main *question of interest* is to find a relationship between `displ` (engine displacement in litres) and `hwy` (highway mileage). Initial visualization with a scatterplot:

```{r displhwy1}
ggplot(data = mpg, mapping = aes(x = displ, y = hwy)) + geom_point()
```

Mileage seems to *decrease* with engine size; what is the effect of `cyl` (number of cylinders)?

```{r displhwy2}
ggplot(data = mpg, mapping = aes(x = displ, y = hwy)) +
  geom_point(aes(color = cyl))
```

We notice that `cyl` has only a few values so it's best seen as a *factor* (categorical variable):

```{r displhwy3}
ggplot(data = mpg %>% mutate(cyl = factor(cyl)),
       aes(x = displ, y = hwy)) + geom_point(aes(color = cyl))
```


## A bit of modeling

There seems to be a functional relationship between `displ` and `hwy`:

```{r displhwy4, message=FALSE}
ggplot(mpg %>% mutate(cyl = factor(cyl)), aes(displ, hwy)) +
  geom_point(aes(color = cyl)) +
  geom_smooth()
```

We can attempt to find such a relationship using a linear model. We assume that if $H$ is `hwy` and $D$ is `displ` then, at least approximately on average, we have 

$$H \propto D^{\beta}$$ 
After taking the log of both sides, this yields the following *linear model*:

$$\log(H) = \alpha + \beta \cdot \log(D)$$
Fitting the model yields

```{r lm}
fit <- lm(log(hwy) ~ log(displ), data = mpg) 
coef(fit)
```
and thus, $\beta \approx -0.5$, which yields

$$H \propto 1 / \sqrt{D}.$$

A visual check seems to confirm that `hwy` has an approximately linear relationship with the inverse square root of `displ`:

```{r displhwy5, message=FALSE}
ggplot(mpg %>% mutate(cyl = factor(cyl)),
       aes(1 / sqrt(displ), hwy)) +
  geom_point(aes(color = cyl)) +
  geom_smooth()
```