2D Quantitative Data

Prof Ron Yurko

2024-09-11

Reminders, previously, and today…

  • HW2 is due TONIGHT!

  • HW3 is posted and due next Wednesday Sept 18th

  • Finished up discussion of 1D quantitative visualizations

  • Discussed impact of bins on histograms

  • Covered ECDFs and connection to KS-tests

  • Walked through density estimation and ways of visualizing conditional distributions

TODAY:

  • Visualize 2D quantitative data

  • Discuss approaches for visualizing conditional and joint distributions

2D quantitative data

  • We’re working with two variables: \((X, Y) \in \mathbb{R}^2\), i.e., dataset with \(n\) rows and 2 columns

  • Goals:

    • describing the relationships between two variables

    • describing the conditional distribution \(Y | X\) via regression analysis

    • describing the joint distribution \(X,Y\) via contours, heatmaps, etc.

  • Few big picture ideas to keep in mind:

    • scatterplots are by far the most common visual

    • regression analysis is by far the most popular analysis (you have a whole class on this…)

    • relationships may vary across other variables, e.g., categorical variables

Making scatterplots with geom_point()

penguins |>
  ggplot(aes(x = flipper_length_mm, y = body_mass_g)) +
  geom_point()

Making scatterplots: ALWAYS adjust the alpha

penguins |>
  ggplot(aes(x = flipper_length_mm, y = body_mass_g)) +
  geom_point(alpha = 0.5)

Displaying trend lines: linear regression

penguins |>
  ggplot(aes(x = flipper_length_mm, y = body_mass_g)) +
  geom_point(alpha = 0.5) +
  geom_smooth(method = "lm")

Assessing assumptions of linear regression

Linear regression assumes \(Y_i \overset{iid}{\sim} N(\beta_0 + \beta_1 X_i, \sigma^2)\)

  • If this is true, then \(Y_i - \hat{Y}_i \overset{iid}{\sim} N(0, \sigma^2)\)

Plot residuals against \(\hat{Y}_i\), residuals vs fit plot

  • Used to assess linearity, any divergence from mean 0

  • Used to assess equal variance, i.e., if \(\sigma^2\) is homogenous across predictions/fits \(\hat{Y}_i\)

More difficult to assess the independence and fixed \(X\) assumptions

  • Make these assumptions based on subject-matter knowledge

Residual vs fit plots

lin_reg <- lm(body_mass_g ~ flipper_length_mm, data = penguins)

tibble(fits = fitted(lin_reg), residuals = residuals(lin_reg)) |>
  ggplot(aes(x = fits, y = residuals)) +
  geom_point(alpha = 0.5) +
  geom_hline(yintercept = 0, linetype = "dashed", color = "red")

Residual vs fit plots

tibble(fits = fitted(lin_reg), residuals = residuals(lin_reg)) |>
  ggplot(aes(x = fits, y = residuals)) +
  geom_point(alpha = 0.5) +
  geom_hline(yintercept = 0, linetype = "dashed", color = "red") +
  geom_smooth()

Local linear regression via LOESS

\(Y_i \overset{iid}{\sim} N(f(x), \sigma^2)\), where \(f(x)\) is some unknown function

In local linear regression, we estimate \(f(X_i)\):

\[\text{arg }\underset{\beta_0, \beta_1}{\text{min}} \sum_i^n w_i(x) \cdot \big(Y_i - \beta_0 - \beta_1 X_i \big)^2\]

geom_smooth() uses tri-cubic weighting:

\[w_i(d_i) = \begin{cases} (1 - |d_i|^3)^3, \text{ if } i \in \text{neighborhood of } x, \\ 0 \text{ if } i \notin \text{neighborhood of } x \end{cases}\]

  • \(d_i\) is the distance between \(x\) and \(X_i\) scaled to be between 0 and 1

  • span: decides proportion of observations in neighborhood (default is 0.75)

Displaying trend lines: LOESS

penguins |>
  ggplot(aes(x = flipper_length_mm, y = body_mass_g)) +
  geom_point(alpha = 0.5) + 
  geom_smooth()

For \(n > 1000\), mgcv::gam() is used with formula = y ~ s(x, bs = "cs") and method = "REML"

Displaying trend lines: LOESS

penguins |>
  ggplot(aes(x = flipper_length_mm, y = body_mass_g)) +
  geom_point(alpha = 0.5) + 
  geom_smooth(span = .1)

Can also update formula within plot

penguins |>
  ggplot(aes(x = flipper_length_mm, y = body_mass_g)) +
  geom_point(alpha = 0.5) + 
  geom_smooth(method = "lm", formula = y ~ x + I(x^2))

Exercise: check the updated residual plot with this model

What about focusing on the joint distribution?

Example dataset of pitches thrown by baseball superstar Shohei Ohtani

ohtani_pitches |>
  ggplot(aes(x = plate_x, y = plate_z)) +
  geom_point(alpha = 0.2) +
  coord_fixed() +
  theme_bw()

Going from 1D to 2D density estimation

In 1D: estimate density \(f(x)\), assuming that \(f(x)\) is smooth:

\[ \hat{f}(x) = \frac{1}{n} \sum_{i=1}^n \frac{1}{h} K_h(x - x_i) \]

In 2D: estimate joint density \(f(x_1, x_2)\)

\[\hat{f}(x_1, x_2) = \frac{1}{n} \sum_{i=1}^n \frac{1}{h_1h_2} K(\frac{x_1 - x_{i1}}{h_1}) K(\frac{x_2 - x_{i2}}{h_2})\]

In 1D there was one bandwidth, now we have two bandwidths

  • \(h_1\): controls smoothness as \(X_1\) changes, holding \(X_2\) fixed
  • \(h_2\): controls smoothness as \(X_2\) changes, holding \(X_1\) fixed

Again Gaussian kernels are the most popular…

So how do we display densities for 2D data?

How to read contour plots?

Best known in topology: outlines (contours) denote levels of elevation

Display 2D contour plot

ohtani_pitches |>
  ggplot(aes(x = plate_x, y = plate_z)) +
  geom_point(alpha = 0.2) +
  geom_density2d() +
  coord_fixed() +
  theme_bw()

Display 2D contour plot

ohtani_pitches |>
  ggplot(aes(x = plate_x, y = plate_z)) +
  geom_density2d() +
  coord_fixed() +
  theme_bw()

Display 2D contour plot

ohtani_pitches |>
  ggplot(aes(x = plate_x, y = plate_z)) +
  stat_density2d(aes(fill = after_stat(level)), geom = "polygon") +
  coord_fixed() +
  scale_fill_gradient(low = "darkblue", high = "darkorange") +
  theme_bw()

Visualizing grid heat maps

ohtani_pitches |>
  ggplot(aes(x = plate_x, y = plate_z)) +
  stat_density2d(aes(fill = after_stat(density)), 
                 geom = "tile", contour = FALSE) + 
  coord_fixed() +
  scale_fill_gradient(low = "white", high = "red") +
  theme_bw()

Alternative idea: hexagonal binning

ohtani_pitches |>
  ggplot(aes(x = plate_x, y = plate_z)) +
  geom_hex() +
  coord_fixed() +
  scale_fill_gradient(low = "darkblue", high = "darkorange") + 
  theme_bw()

LeBron James’ shots from hoopR

lebron_shots <- read_csv("https://raw.githubusercontent.com/ryurko/DataViz-Class-Data/main/lebron_shots.csv")
lebron_shots |>
  ggplot(aes(x = coordinate_x, y = coordinate_y)) +
  geom_point(alpha = 0.4) +
  geom_density2d(binwidth = 0.0001) + 
  coord_fixed() +
  theme_bw()

Recap and next steps

  • Use scatterplots to visualize 2D quantitative

  • Be careful of over-plotting! May motivate contours or hexagonal bins…

  • Discussed approaches for visualizing conditional relationships

  • HW2 is due TONIGHT!

  • HW3 is posted due next Wednesday Sept 18th