Data Visualization

class: center, middle, inverse, title-slide

# Data Visualization
## Visualizing 1D categorical and continuous variables
### June 9th, 2022

---

## New dataset - 2021 MVP Shohei Ohtani's batted balls

Created dataset of batted balls by the American League MVP Shohei Ohtani in 2021 season using [`baseballr`](http://billpetti.github.io/baseballr/):

```r
library(tidyverse)
ohtani_batted_balls <- 
  read_csv("http://www.stat.cmu.edu/cmsac/sure/2022/materials/data/sports/xy_examples/ohtani_2021_batted_balls.csv")
head(ohtani_batted_balls)
```

```
## # A tibble: 6 x 7
##   pitch_type batted_ball_type  hit_x hit_y exit_velocity launch_angle outcome  
##   <chr>      <chr>             <dbl> <dbl>         <dbl>        <dbl> <chr>    
## 1 FC         line_drive        89.7  144.          113.            20 home_run 
## 2 CH         fly_ball           3.35  83.9          83.9           55 field_out
## 3 CH         fly_ball         -65.6  126.          102.            38 field_out
## 4 CU         ground_ball       39.2   50.4          82.5            8 field_out
## 5 FC         fly_ball         -37.6  138.          101.            23 field_out
## 6 KC         popup            -51.9   41.6          84             65 field_out
```

--
- each row / observation is a batted ball from Ohtani's 2021 season

--
- __Categorical__ / qualitative variables: `pitch_type`, `batted_ball_type`, `outcome`

--
- __Continuous__ / quantitative variables: `hit_x`, `hit_y`, `exit_velocity`, `launch_angle`

---

## Visualizing 1D categorical data

How can we summarize `batted_ball_type` and other categorical variables?

.pull-left[

- We make a __bar chart__ with [`geom_bar()`](https://ggplot2.tidyverse.org/reference/geom_bar.html)

```r
ohtani_batted_balls %>%
  ggplot(aes(x = batted_ball_type)) +
* geom_bar() +
  theme_bw()
```

- Only map `batted_ball_type` to the x-axis

- Counts of each type are displayed on y-axis...

]
.pull-right[
<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-1-1.png" width="504" />
]

---

## Remember statistical summaries!

.center[![](https://d33wubrfki0l68.cloudfront.net/70a3b18a1128c785d8676a48c005ee9b6a23cc00/7283c/images/visualization-stat-bar.png)]

From [Chapter 3 of `R` for Data Science](https://r4ds.had.co.nz/data-visualisation.html)

---

## What does a bar chart show?

__Marginal distribution__: probability that categorical variable X (e.g., `batted_ball_type`) takes each particular value x (e.g. `fly_ball`).
--
  _So how do we display the individual probabilities?_
  
--

.pull-left[

```r
ohtani_batted_balls %>%
  ggplot(aes(x = batted_ball_type)) +
* geom_bar(aes(y = after_stat(count) / sum(after_stat(count)))) +
  theme_bw()
```

- [`after_stat()`](https://ggplot2.tidyverse.org/reference/aes_eval.html) indicates the aesthetic mapping is performed after the statistical transformation

- Use `after_stat(count)` to access the `stat_count()` called by `geom_bar()`

- __We can code this in a more clear way__

]
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<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-2-1.png" width="504" />
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---

## Compute and display the proportions directly

.pull-left[

```r
ohtani_batted_balls %>%
* group_by(batted_ball_type) %>%
* summarize(count = n()) %>%
* ungroup() %>%
* mutate(total = sum(count),
*        prop = count / total) %>%
  ggplot(aes(x = batted_ball_type)) +
* geom_bar(aes(y = prop),
*          stat = "identity") +
  theme_bw()
```

- Category counts give info about sample size, but this could be labeled in the chart

- Proportions `$=$` the __probability mass function__ (PMF) for __discrete__ variables 
  - e.g. `$P$` (`batted_ball_type` `$=$` `fly_ball`)

]
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<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-3-1.png" width="504" />
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---

## Population versus sample...

We have the __population__ of Ohtani's batted balls in the 2021 season
--
`$\Rightarrow$` __we know the true probabilities__:

- `$P$` (`batted_ball_type` `$=$` `fly_ball`) 
- `$P$` (`batted_ball_type` `$=$` `ground_ball`)
- `$P$` (`batted_ball_type` `$=$` `line_drive`)
- `$P$` (`batted_ball_type` `$=$` `popup`)

--
_What if we pretend this is a sample from all hypothetical Ohtani 2021 seasons_?

--
__Empirical distribution__: We __estimate__ the __true marginal__ distribution with __observed (sample) data__

`$\Rightarrow$` Estimate `$P$` (`batted_ball_type` = `$C_j$`) with `$\hat{p}_j$` for each category `$C_j$` (e.g. `$\hat{p}_{\texttt{fly_ball}}$`)

--
Compute __standard error__ for each `$\hat{p}_j$`:

$$
SE(\hat{p}_j) = \sqrt{\frac{\hat{p}_j ( 1 - \hat{p}_j)}{n}}
$$

For large `$n$` `$\Rightarrow$` `$\approx$` 95% __confidence interval (CI)__: `$\hat{p}_j +/- 2 \cdot SE(\hat{p}_j)$`

---

## Add confidence intervals to bar chart

.pull-left[

```r
ohtani_batted_balls %>%
  group_by(batted_ball_type) %>% 
  summarize(count = n()) %>% 
  ungroup() %>% 
  mutate(total = sum(count),
         prop = count / total,
*        se = sqrt(prop * (1 - prop) / total),
*        lower = prop - 2 * se,
*        upper = prop + 2 * se) %>%
  ggplot(aes(x = batted_ball_type)) +
  geom_bar(aes(y = prop),
           stat = "identity") + 
* geom_errorbar(aes(ymin = lower,
*                   ymax = upper),
*               color = "red") +
  theme_bw()
```

__Be careful about your interpration of CIs...__

_You should remember to label your charts!_

]
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<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-4-1.png" width="504" />
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---

## Fun with factors using [`forcats`](https://forcats.tidyverse.org/)

.pull-left[

```r
ohtani_batted_balls %>%
  group_by(batted_ball_type) %>% 
  summarize(count = n()) %>% 
  ungroup() %>% 
  mutate(total = sum(count),
         prop = count / total,
         se = sqrt(prop * (1 - prop) / total),  
         lower = prop - 2 * se, 
         upper = prop + 2 * se,
*        batted_ball_type =
*          fct_reorder(batted_ball_type,
*                      prop)) %>%
  ggplot(aes(x = batted_ball_type)) +
  geom_bar(aes(y = prop),
           stat = "identity") + 
  geom_errorbar(aes(ymin = lower, 
                    ymax = upper),
                color = "red") + 
  theme_bw()
```

]
.pull-right[
<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-5-1.png" width="504" />
]

---

## Did you say pie chart?

.center[![](https://cdn.vox-cdn.com/uploads/chorus_asset/file/3500550/0003VL-9150.0.jpeg)]

--
__This is the only pie chart I will show you all summer__

---

## Describing 1D continuous data

How can we summarize `exit_velocity` and other continuous variables?

--
- __Center__: mean, median, number and location of modes

- __Spread__: range (max - min), quantiles, variance (standard deviation), etc.

- __Shape__: skew vs symmetry, outliers, heavy vs light tails, etc.

--
- Compute basic summary statistics

```r
summary(ohtani_batted_balls$exit_velocity)
```

```
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   27.50   83.75   96.00   93.26  105.55  119.00      27
```

```r
sd(ohtani_batted_balls$exit_velocity)
```

```
## [1] NA
```

---

## Box plots visualize summary statistics

.pull-left[

- We make a __box plot__ with [`geom_boxplot()`](https://ggplot2.tidyverse.org/reference/geom_boxplot.html)

```r
ohtani_batted_balls %>%
* ggplot(aes(y = exit_velocity)) +
* geom_boxplot(aes(x = "")) +
  theme_bw() +
* coord_flip()
```

- __Pros__:
  - Displays outliers, percentiles, spread, skew
  - Useful for side-by-side comparison (tomorrow)

- __Cons__:
  - Does not display the full distribution shape!
  - Does not display modes
  
_Why use `aes(x = "")` inside `geom_boxplot()`?_

]
.pull-right[
<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-6-1.png" width="504" />
]

---

## Histograms display 1D continuous distributions

.pull-left[

- We make __histograms__ with [`geom_histogram()`](https://ggplot2.tidyverse.org/reference/geom_histogram.html)

```r
ohtani_batted_balls %>%
* ggplot(aes(x = exit_velocity)) +
* geom_histogram() +
  theme_bw() 
```

$$
\text{# total obs.} = \sum_{j=1}^k \text{# obs. in bin }j
$$

- __Pros__:
  - Displays full shape of distribution
  - Easy to interpret

- __Cons__:
  - Have to choose number of bins and bin locations (will revisit later)
  
]
.pull-right[
<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-7-1.png" width="504" />
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---

## Display the data points directly with beeswarm plots

.pull-left[

- We make a __beeswarm plot__ using the  [`ggbeeswarm` package](https://github.com/eclarke/ggbeeswarm)

```r
library(ggbeeswarm)
ohtani_batted_balls %>%
  ggplot(aes(y = exit_velocity)) + 
* geom_beeswarm(aes(x = ""),
*               cex = 3) +
  theme_bw() +
  coord_flip()
```

- __Pros__:
  - Displays each data point
  - Easy to view full shape of distribution

- __Cons__:
  - Can be overbearing with large datasets
  - Which algorithm for arranging points?

_What does `cex = 3` do?_
]
.pull-right[
<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-8-1.png" width="504" />
]

---

## Smooth summary with violin plots

.pull-left[

- We make __violin plots__ with [`geom_violin()`](https://ggplot2.tidyverse.org/reference/geom_violin.html)

```r
ohtani_batted_balls %>%
  ggplot(aes(y = exit_velocity)) + 
* geom_violin(aes(x = "")) +
  theme_bw() +
  coord_flip()
```

- __Pros__:
  - Displays full shape of distribution
  - Can easily layer...
  
]
.pull-right[
<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-9-1.png" width="504" />
]

---

## Smooth summary with violin plots + box plots

.pull-left[

- We make __violin plots__ with [`geom_violin()`](https://ggplot2.tidyverse.org/reference/geom_violin.html)

```r
ohtani_batted_balls %>%
  ggplot(aes(y = exit_velocity,
             x = "")) + 
  geom_violin() + 
* geom_boxplot(width = .2) +
  theme_bw() +
  coord_flip()
```

- __Pros__:
  - Displays full shape of distribution
  - Can easily layer... with box plots on top
  
- __Cons__:
  - Summary of data via __density estimate__ 
  - Mirror image is duplicate information
  
]
.pull-right[
<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-10-1.png" width="504" />
]

---

### What do visualizations of continuous distributions display?

__Probability that continuous variable X takes a particular value is 0__

e.g. `$P$` (`exit_velocity` `$= 100$`) `$= 0$`, _why_?

--
Instead we use the __probability density function (PDF)__ to provide a __relative likelihood__

- Density estimation is the focus of lecture next Monday

--
For continuous variables we can use the __cumulative distribution function (CDF)__,

$$
F(x) = P(X \leq x)
$$

--
For `$n$` observations we can easily compute the __Empirical CDF (ECDF)__:

`$$\hat{F}_n(x)  = \frac{\text{# obs. with variable} \leq x}{n} = \frac{1}{n} \sum_{i=1}^{n}1(x_i \leq x)$$`

- where `$1()$` is the indicator function, i.e. `ifelse(x_i <= x, 1, 0)`

---

## Display full distribution with ECDF plot

.pull-left[

- We make __ECDF plots__ with [`stat_ecdf()`](https://ggplot2.tidyverse.org/reference/stat_ecdf.html)

```r
ohtani_batted_balls %>%
  ggplot(aes(x = exit_velocity)) + 
* stat_ecdf() +
  theme_bw()
```

- __Pros__:
  - ECDF displays all information in data (except for order)
  - As `$n \rightarrow \infty$`, our ECDF `$\hat{F}_n(x)$` converges to the true CDF `$F(x)$`
  - Easy to interpret...
  
- __Cons__:
  - ... and yet it's not as popular!
  
]
.pull-right[
<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-11-1.png" width="504" />
]

---

## Rug plots display raw data

.pull-left[

- We make a __rug plot__ with [`geom_rug()`](https://ggplot2.tidyverse.org/reference/geom_rug.html)

```r
ohtani_batted_balls %>%
* ggplot(aes(x = exit_velocity)) +
* geom_rug(alpha = 0.7) +
  theme_bw()
```

- __Pros__:
  - Displays raw data points
  - Useful supplement for summaries and 2D plots...

- __Cons__:
  - Can be overbearing for larger datasets
  
]
.pull-right[
<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-12-1.png" width="504" />
]

---

## Rug plots supplement other displays

.pull-left[

```r
ohtani_batted_balls %>%
* ggplot(aes(x = exit_velocity)) +
* geom_rug(alpha = 0.7) +
* geom_histogram() +
  theme_bw()
```

<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-13-1.png" width="504" />
  
]
.pull-right[

```r
ohtani_batted_balls %>%
* ggplot(aes(x = exit_velocity)) +
* geom_rug(alpha = 0.7) +
* stat_ecdf() +
  theme_bw()
```

<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-14-1.png" width="504" />
]

---

## Scatterplots for 2D continuous data

.pull-left[

- We make a __scatterplot__ with [`geom_point()`](https://ggplot2.tidyverse.org/reference/geom_point.html)

```r
ohtani_batted_balls %>%
* ggplot(aes(x = exit_velocity,
*            y = launch_angle)) +
* geom_point() +
* geom_rug(alpha = 0.4) +
  theme_bw()
```

_Easy to supplement with rug plots_

__Look at the plot__: what question would you want to ask, assuming you know something about baseball?

_To be continued..._
  
]
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<img src="03-1d2d-dataviz_files/figure-html/unnamed-chunk-15-1.png" width="504" />
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