[1] "playerID" "yearID" "stint" "teamID" "lgID" "G"
[7] "AB" "R" "H" "X2B" "X3B" "HR"
[13] "RBI" "SB" "CS" "BB" "SO" "IBB"
[19] "HBP" "SH" "SF" "GIDP" Visualizing density - I
Lecture 5
Histograms
Histogram of 200 random numbers generated from a \(\textsf{Normal}(\mu=-1, \sigma=0.5)\) and 400 generated from a \(\textsf{Normal}(\mu=2, \sigma=0.75)\):
Density Plots
What’s the difference? Histograms are counts of bins of observed data. Density plots are estimates of the unknown distribution.
Density Graph Example
Stacked Density Graph By Decade
home_runs |>
mutate(
decade = cut(
yearID,
breaks = seq(1920, 2030, 10),
labels = paste0(seq(1920, 2020, 10), "'s")
)
) |>
ggplot(aes(HR, fill = decade)) +
geom_density(position = "stack") +
labs(x = "Home runs per player per year")
If we stratify by decade, we can see the mode of the density graphs slowly creep forward, but it is difficult to see the tail of the distribution.
Overlapping Density Graphs By Decade
home_runs |>
mutate(
decade = cut(
yearID,
breaks = seq(1920, 2030, 10),
labels = paste0(seq(1920, 2020, 10), "'s")
)
) |>
ggplot(aes(HR, color = decade)) +
geom_density() +
labs(x = "Home runs per player per year")
The modes moving forward is a little more apparent now, but the graphs are too coupled to digest easily.
Density Graph with Conditional Probabilities
home_runs |>
mutate(
decade = cut(
yearID,
breaks = seq(1920, 2030, 10),
labels = paste0(seq(1920, 2020, 10), "'s")
)
) |>
ggplot(aes(x = HR, y = after_stat(count), fill = decade)) +
geom_density(position = "fill") +
geom_vline(xintercept = 60, linetype = "dashed") +
labs(x = "Home runs per player per year")
By using position = "fill" and y = after_stat(count) we graph the conditional probability of a decade given that a player has hit a certain number of home runs. We see that players would hit about 60 homeruns in the 20’s and 30’s, but that disappears until the 90’s and 2000’s. 1
Violin Plot
home_runs |>
filter(yearID %in% 1985:2005) |>
ggplot(aes(HR, x = factor(yearID))) +
geom_violin(draw_quantiles = c(0.25, 0.5, 0.75)) +
geom_jitter(
data = ~ filter(.x, HR >= 30),
height = 0, width = 0.1, alpha = 0.5
) +
scale_x_discrete(guide = guide_axis(n.dodge = 2)) +
labs(x = "Home runs per player per year")
Let’s examine the years near the change point, 1985 to 2005. All points shown are players that hit 30 or more home runs in a given year. It looks like around 1995 players started hitting a lot more home runs.
Ridge Plot
library(ggridges)
home_runs |>
filter(yearID %in% 1985:2010) |>
ggplot(aes(x = HR, y = factor(yearID))) +
stat_density_ridges(
mapping = aes(fill = factor(after_stat(quantile))),
geom = "density_ridges_gradient",
calc_ecdf = TRUE,
quantiles = c(.25, .50, .75, .95),
quantile_lines = TRUE,
scale = 2,
rel_min_height = 0.01
) +
scale_fill_viridis_d(
name = "Quantiles",
labels = c("0", "25th", "50th", "75th", "95th")
) +
geom_jitter(
data = ~ filter(.x, HR >= 30),
height = 0.2, width = 0, alpha = 0.3,
) +
scale_x_continuous(
name = "Home runs per player per year",
limits = c(0, 73)
) +
labs(y = "Year")
The quantiles also have a consistent increase, along with many more players hitting 30 or more home runs! In 1998 there was a home run record race between two players; this brought a lot of interest back into baseball. 1995 to about 2005 is known as the Steroid Era in baseball. During this time, players would take performance enhancing drugs freely as the league did not enforce the ban on them. League-wide testing began in 2003.
Sheather-Jones Example
Cautions
long_tailed_data <- tibble(random_values = rlnorm(1000, -3, 1))
ggplot2 generally handles this for you by putting bounds at the range of your data, but it can occasionally skip this depending on how complicated your graph becomes.
2D Density/Histograms
Players that hit lots of home runs tend to strikeout and walk more.
Density Graphs Summary
Pros:
- Visualize entire distribution
- Mean, median, variance, outliers, support, skewness, normality etc.
plot(density(Batting$HR))is usually the first thing I do when analyzing data
Cons:
- Sensitive to bandwidth choices
- Harder to communicate to non-statisticians
- Difficult to build yourself (use libraries!)
