two_continuous_var.qmd

# Two continuous variables

In this chapter, we will look at techniques that explore the relationships between two continuous variables.

## Scatterplot

### Basics and implications

For the following example, we use data set `SpeedSki`. 

```{r,fig.width=4.8, fig.height=3.6}
library(GDAdata)
library(ggplot2)

ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_point() +
  labs(x = "Birth year", y = "Speed achieved (km/hr)") +
  ggtitle("Skiers by birth year and speed achieved")
```

In our example, we simply use `geom_point` on variables `Year` and `Speed` to create the scatterplot. we try to capture if there is a relationship between the age of a player and the speed he/she can achieve. From the graph, it seems such relationship does not exist. Overall, scatterplots are very useful in understanding the correlation (or lack thereof) between variables. The scatterplot gives a good idea of whether that relationship is positive or negative and if there’s a correlation. However, don’t mistake correlation in a scatterplot for causation!

### Overplotting

In some situations a scatter plot faces the problem of overplotting as there are so many points overlapping. Consider the following example from class. To save time, we randomly sample 20% of the data in advance.

```{r,fig.width=4.8, fig.height=3.6}
library(dplyr)
library(ggplot2movies)

sample <- slice_sample(movies, prop = 0.2)

ggplot(sample,aes(x=votes,y=rating)) +
  geom_point() +
  ggtitle("Votes vs. rating") +
  theme_classic()
```

To create better visuals, we can use:

* Alpha blending - `alpha=...`

* Open circles - `pch=21`

* smaller circles - `size=...` or `shape="."`

```{r,fig.height=8}
library(gridExtra)

f1 <- ggplot(sample,aes(x=votes,y=rating)) +
  geom_point(alpha=0.3) +
  theme_classic() +
  ggtitle("Alpha blending")

f2 <- ggplot(sample,aes(x=votes,y=rating)) +
  geom_point(pch = 21) +
  theme_classic() +
  ggtitle("Open circle")

f3 <- ggplot(sample,aes(x=votes,y=rating)) +
  geom_point(size=0.5) +
  theme_classic() +
  ggtitle("Smaller circle")

grid.arrange(f1, f2, f3,nrow = 3)
```

Other methods that directly deal with the data:

* Randomly sample data - as shown in the first code chunk using `sample_n`

* Subset - split data into bins using `ntile(votes, 10)`

* Remove outliers 

* Transform to log scale

### Interactive scatterplot

You can create an interactive scatterplot using `plotly`. In the following example, we take 1% of the movie data set to present a better visual. We plotted the votes vs. rating and grouped by the year they are released. In this graph:

* You can hover on to the points to see the title of the movie

* You can double click on the year legend to look at a certain year

* You can zoom into a certain part of the graph to better understand the data points.

```{r,fig.width=6, fig.height=4.5}
library(plotly)

sample2 <- slice_sample(movies,prop=0.01) |>
  filter(year > 2000)

plot_ly(sample2, x = ~votes, y = ~rating,
        color = ~as.factor(year), text= ~title,
        hoverinfo = 'text') 

```

### Modifications

#### Contour lines

Contour lines give a sense of the density of the data at a glance.

For these contour maps, we will use the `SpeedSki` dataset.

Contour lines can be added to the plot using geom_density_2d() and contour lines work best when combined with other layers

```{r,fig.width=4.8, fig.height=3.6}
ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_density_2d(bins=5) +
  geom_point() +
  ggtitle("Scatter plot with contour line")
```

You can use `bins` to control the number of contour bins.

#### Scatterplot matrices

If you want to compare multiple parameters to each other, consider using a scatterplot matrix. This will allow you to show many comparisons in a compact and efficient manner.

For these scatterplot matrices, we use the `movies` dataset from the `ggplot2movies` package.

As a default, the base R plot() function will create a scatterplot matrix when given multiple variables:

```{r}
sample3 <- slice_sample(movies,prop=0.01) #sample data

splomvar <- sample3 |> 
  dplyr::select(length, budget, votes, rating, year)

plot(splomvar)
```

While this is quite useful for personal exploration of a dataset, it is **not** recommended for presentation purposes. Something called the [Hermann grid illusion](https://en.wikipedia.org/wiki/Grid_illusion){target="_blank"} makes this plot very difficult to examine.


## Heatmaps

### Basics and implications 

In the following example, we still use the `SpeedSki` data set.

```{r,fig.width=4.8, fig.height=3.6}
ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_bin2d() 
```

To create a heatmap, simply substitute `geom_point()` with `geom_bin2d()`. Generally, heat maps are like a combination of scatterplots and histograms: they allow you to compare different parameters while also seeing their relative distributions.

### Modifications

For the following section, we introduce some variations on heatmaps.

#### Change number of bins / binwidth

By default, `geom_bin2d()` use 30 bins. Similar to a histogram, we can change the number of bins or binwidth.

```{r,fig.width=4, fig.height=3}
ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_bin2d(binwidth = c(5,5)) +
  ggtitle("Changing binwidth")
```

Notice we are specifying the binwidth for both x and y axis.

#### Combine with a scatterplot

```{r,fig.width=4, fig.height=3}
ggplot(SpeedSki, aes(Year, Speed)) +
  geom_bin2d(binwidth = c(10, 10), alpha = .4) + 
  geom_point(size = 2) +
  ggtitle("Combined with scatterplot")
```

#### Change color scale

You can change the continuous scale of color 

```{r,fig.width=4, fig.height=3}
ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_bin2d() +
  ggtitle("Changing color scale") + 
  scale_fill_viridis_c()
```

#### Hex heatmap

One alternative is a hex heatmap. You can create the graph using `geom_hex`

```{r,fig.width=4, fig.height=3}
ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_hex(binswidth = c(10,10)) +
  ggtitle("Hex heatmap")
```

#### Alternative approach to color

If you look at all the previous examples, you might notice that lighter points correspond to more clustered points, which is somewhat counter-intuitive. The following example suggests an alternative approach in color scale.

```{r,fig.width=6, fig.height=4}
ggplot(SpeedSki, aes(Year, Speed)) + 
  geom_hex(bins=12) +
  scale_fill_gradient(low = "grey", high = "purple") +  
  theme_classic(18) +
  ggtitle("Alternative approach to color")
```