Distributions
and
Models

Data 304: Visualizing Data and Models

Setup

library(vegabrite)
library(tidyverse)
library(palmerpenguins)
vega_data <- altair::import_vega_data()

Visualizing distributions

Histograms

Ingredients:

  • bin aggregation, bar mark
Code 
vl_chart(width = 150, height = 60) |>
  vl_add_data(penguins) |>
  vl_encode_x("body_mass_g:Q", bin = list(maxbins = 30)) |>
  vl_encode_y(aggregate = "count") |>
  vl_encode_column("species:N") |>
  vl_encode_row("sex:N") |>
  vl_mark_bar()

Improvements?

Improved Histogram

Code 
vl_chart(width = 150, height = 60) |>
  vl_add_data(penguins |> filter(!is.na(sex))) |>
  vl_encode_x("body_mass_g:Q", bin = list(maxbins = 30)) |>
  vl_encode_y(aggregate = "count") |>
  vl_axis_x(title = NA, values = (2:6) * 1000) |>
  vl_scale_x(nice = TRUE) |>
  vl_axis_y(title = NA) |>
  vl_encode_column("species:N", title = NA) |>
  vl_encode_row("sex:N", title = NA) |>
  vl_mark_bar() |>
  vl_vconcat(
    vl_chart(height = 15, width = 500) |>
      vl_add_data(tibble(x = 1, y = 1)) |>
      vl_mark_text(size = 14) |>
      vl_encode_text(value = "body mass (g)") |>
      vl_encode_x("x:N", axis = FALSE) |>
      vl_encode_y("y:N", axis = FALSE) |>
      vl_config_view(stroke = "transparent")
      
  )

Freqency polygons

Ingredients:

  • bin aggregation (just like histograms), line mark
Code 
vl_chart(width = 400, height = 60) |>
  vl_add_data(penguins) |>
  vl_mark_line(opacity = 0.5) |>
  vl_encode_x("body_mass_g:Q", bin = list(maxbins = 30)) |>
  vl_encode_y(aggregate = "count") |>
  vl_axis_y(title = NA) |>
  vl_encode_color("species:N") |>
  vl_encode_row("sex:N")

Density plots

Usual method: kernel density estimation (kde)

Code 
set.seed(123)
S <- tibble(x = round(rgamma(10, shape = 2, rate = 0.1), 1))

kde_demo <- function(bw = 6) {
  base <-
    vl_chart(width = 300, height = 100) |>
    vl_add_properties(title = paste0("bandwidth: ", bw)) |>
    vl_add_data(S) |>
    vl_encode_y("density:Q") |>
    vl_scale_y(domain = c(0, .5))

  kde <- base |>
    vl_density(
      "x",
      as = list("value", "density"),
      bandwidth = bw,
      counts = TRUE,
      extent = c(0, 60)
    ) |>
    vl_mark_line(opacity = 2) |>
    vl_encode_color(value = "red") |>
    vl_encode_x("value:Q", title = NA)

  kernels <- base |>
    vl_density(
      "x",
      as = list("value", "density"),
      bandwidth = bw,
      groupby = list("x"),
      extent = c(-3, 60),
      counts = TRUE
    ) |>
    vl_mark_line(opacity = 0.5) |>
    vl_encode_detail("x:N") |>
    vl_encode_x("value:Q", title = NA)

  ticks <-
    base |>
    vl_mark_tick() |>
    vl_encode(x = "x") |>
    vl_encode_y(datum = 0)

  kernels + kde + ticks
}

((kde_demo(2) | kde_demo(4)) &
  (kde_demo(6) | kde_demo(9)))

Density plots

Ingredients:

  • Use density transform to create x/y values for density curve
  • Use line, area, etc. to display
  • bandwidth determines amount of smoothing
  • groupby to compute separate densities for each group
Code 
vl_chart(width = 400, height = 60) |>
  vl_add_data(penguins) |>
  vl_density("body_mass_g", groupby = list("species", "sex")) |>
  vl_mark_line(opacity = 0.5) |>
  vl_encode_x("value:Q") |>
  vl_encode_y("density:Q") |>
  vl_encode_color("species:N") |>
  vl_encode_row("sex:N")

Boxplots

Code 
vl_chart(width = 500) |>
  vl_add_data(penguins) |>
  vl_mark_boxplot() |>
  vl_encode_y("species:N") |>
  vl_encode_x("body_mass_g:Q", scale = list(zero = FALSE))

More Boxplots

Code 
vl_chart(width = 500) |>
  vl_add_data(penguins) |>
  vl_mark_boxplot(outliers = FALSE, median = FALSE, extent = "min-max") |>
  vl_encode_x("body_mass_g:Q", scale = list(zero = FALSE)) |>
  vl_encode_y("species:N") |>
  vl_encode_color("sex:N") |>
  vl_encode_yOffset("sex:N")

Visualizing Models

Data 304: Visualizing Data and Models

We have focused mainly on visualizing data so far.

  • easier
  • foundational to visualizing models

(Group) means are “models”

Examples: Two-sample t procedures, ANOVA

Method: aggregate or precompute

Warning

Beware the tendency to summarize everything with means (and to use bars to display them). This is a common approach to visualization, but it hides variation, which is the other side of the coin.

Data value = model mean + residual

Regression models

Vega-Lite can handle some basic regression models with the regression transform.

Code 
base <-
  vl_chart() |>
  vl_add_data(penguins) |>
  vl_encode_x("body_mass_g:Q", scale = list(zero = FALSE)) |>
  vl_encode_y("flipper_length_mm:Q", scale = list(zero = FALSE))

points <- base |> vl_mark_circle(opacity = 0.3, size = 15)
line <- base |>
  vl_regression("body_mass_g", on = "flipper_length_mm") |>
  vl_mark_line()

(points + line) |>
  vl_add_properties(width = 500)

LOESS

LOESS = LOcally Estimated Scatterplot Smoothing

Big ideas:

  • fit many (regression) models using “nearby” points
  • weight nearer points more than farther points
Code 
base <-
  vl_chart() |>
  vl_add_data(penguins) |>
  vl_encode_x("body_mass_g:Q", scale = list(zero = FALSE)) |>
  vl_encode_y("flipper_length_mm:Q", scale = list(zero = FALSE)) 

points <- base |> vl_mark_circle(opacity = 0.3, size = 15) 

line <- function(bw = 0.2, color = "steelblue", opacity = 0.7) {
  base |> 
    vl_loess("body_mass_g", on = "flipper_length_mm", bandwidth = bw) |>
    vl_mark_line(color = color, opacity = opacity) |>
    vl_add_properties(title = paste0("bandwidth = ", bw))
}

vl_concat(
  points +  line(0.1, "red"),
  points +  line(0.2, "forestgreen"),
  points +  line(0.5, "navy"),
  points +  line(0.8, "black")
)

Principles for visualizing models

  1. Consider showing both the data and the model.

    • Can help us assess fit.

  2. Include representation of model uncertainy.

  3. Perform model diagnostics by visualizing data related to the model.

    • Examples: residuals plots, auto-correlation plots
    • May need to first create the appropriate data set, then do the visualization
      • Model precitions/fits: data frame of x/etc/y tuples

  4. Surprise and Scale

    Visualization can surprise you, but it doesn’t scale well. Modeling scales well, but it can’t surprise you. (Hadley Wickham, paraphrased)

    • We need both, and to go back and forth between the two.

broom::tidy()

library(broom)

model <- 
  lm(flipper_length_mm ~ body_mass_g + sex + species, 
     data = penguins)

tidy(model)
# A tibble: 5 × 5
  term              estimate std.error statistic   p.value
  <chr>                <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)      165.       3.18         51.7  1.01e-159
2 body_mass_g        0.00655  0.000931      7.04 1.15e- 11
3 sexmale            2.48     0.854         2.90 3.97e-  3
4 speciesChinstrap   5.54     0.785         7.06 9.92e- 12
5 speciesGentoo     18.0      1.44         12.5  1.46e- 29

broom::augment()

augment(model) |> head(2) |> gt::gt()
.rownames flipper_length_mm body_mass_g sex species .fitted .resid .hat .sigma .cooksd .std.resid
1 181 3750 male Adelie 191.6287 -10.628721 0.01240207 5.320866 0.010054358 -2.0008066
2 186 3800 female Adelie 189.4785 -3.478497 0.01539750 5.350125 0.001345147 -0.6558059
Code 
base <-
  vl_chart() |>
  vl_encode_x("body_mass_g:Q", scale = list(zero = FALSE), title = NA) |>
  vl_encode_y("flipper_length_mm:Q", scale = list(zero = FALSE))

points <-
  base |>
  vl_mark_point()

line <-
  base |>
  vl_encode_y(".fitted:Q", title = NA) |>
  vl_mark_line()

area <- base |>
  vl_mark_area(opacity = 0.5) |>
  vl_encode_y2("upper:Q") |>
  vl_encode_y("lower:Q")

(points + area + line) |>
  vl_facet_row("sex:N", title = NA) |>
  vl_facet_column("species:N", title = NA) |>
  vl_add_data(
    augment(model) |>
      mutate(upper = .fitted + .sigma) |>
      mutate(lower = .fitted - .sigma)
  ) |>
  vl_add_properties(height = 40, width = 150)

Your turn

Exercise 1 The anscombe data set (in Vega Data Sets, also in R) contains 4 x-y pairs.

names(anscombe)
[1] "x1" "x2" "x3" "x4" "y1" "y2" "y3" "y4"

Make a graphic showing scatter plots with regression lines for all four pairs. What is the point of this (artifical) data set?

Exercise 2 Make a list of visualizations “of models” that you have seen (in other classes, in papers, etc.). Describe them using the grammar of graphics and the data required to make them. (You may find it handy to draw a sketch for each.)

Your turn

Exercise 3 Open one (or more) of the graphics in these slides that uses a transform and inspect the data in the data viewer to see what the data look like post transformation.

Exercise 4 Return to the penguins data and create a scatter plot with multiple regression or loess lines. What groups of penguins should you use.

Exercise 5 Describe a qq-plot in terms of the grammar of graphics and the data you need to make one. (You migth start by reminding yourself – or asking someone – what a qq-plot is.)

Create a qq-plot (you may choose the data; perhaps use one of the features of penguins) using the quantile transform.