Modelling

• Use models to explain the relationship between variables and to make predictions
• For now we will focus on linear models (but remember there are many many other types of models too!)

Paris Paintings

pp <- read_csv("data/paris-paintings.csv", na = c("n/a", "", "NA"))

• Source: Printed catalogues of 28 auction sales in Paris, 1764 - 1780
• Data curators Sandra van Ginhoven and Hilary Coe Cronheim (who were PhD students in the Duke Art, Law, and Markets Initiative at the time of putting together this dataset) translated and tabulated the catalogues
• 3393 paintings, their prices, and descriptive details from sales catalogues over 60 variables

Auctions today

Auctions back in the day

Paris auction market

Départ pour la chasse

Auction catalog text

pp %>%
  filter(name == "R1777-89a") %>%
  glimpse()

## Rows: 1
## Columns: 61
## $ name              <chr> "R1777-89a"
## $ sale              <chr> "R1777"
## $ lot               <chr> "89"
## $ position          <dbl> 0.3755274
## $ dealer            <chr> "R"
## $ year              <dbl> 1777
## $ origin_author     <chr> "D/FL"
## $ origin_cat        <chr> "D/FL"
## $ school_pntg       <chr> "D/FL"
## $ diff_origin       <dbl> 0
## $ logprice          <dbl> 8.575462
## $ price             <dbl> 5300
## $ count             <dbl> 1
## $ subject           <chr> "D\x8epart pour la chasse"
## $ authorstandard    <chr> "Wouwerman, Philips"
## $ artistliving      <dbl> 0
## $ authorstyle       <chr> NA
## $ author            <chr> "Philippe Wouwermans"
## $ winningbidder     <chr> "Langlier, Jacques for Poullain, Anto…
## $ winningbiddertype <chr> "DC"
## $ endbuyer          <chr> "C"
...

...
## $ Interm            <dbl> 1
## $ type_intermed     <chr> "D"
## $ Height_in         <dbl> 17.25
## $ Width_in          <dbl> 23
## $ Surface_Rect      <dbl> 396.75
## $ Diam_in           <dbl> NA
## $ Surface_Rnd       <dbl> NA
## $ Shape             <chr> "squ_rect"
## $ Surface           <dbl> 396.75
## $ material          <chr> "bois"
## $ mat               <chr> "b"
## $ materialCat       <chr> "wood"
## $ quantity          <dbl> 1
## $ nfigures          <dbl> 0
## $ engraved          <dbl> 0
## $ original          <dbl> 0
## $ prevcoll          <dbl> 1
## $ othartist         <dbl> 0
## $ paired            <dbl> 1
## $ figures           <dbl> 0
## $ finished          <dbl> 0
...

Heights

ggplot(data = pp, aes(x = Height_in)) +
  geom_histogram(binwidth = 5) +
  labs(x = "Height, in inches", y = NULL)

Widths

ggplot(data = pp, aes(x = Width_in)) +
  geom_histogram(binwidth = 5) +
  labs(x = "Width, in inches", y = NULL)

Models as functions

• We can represent relationships between variables using functions
• A function is a mathematical concept: the relationship between an output and one or more inputs
  • Plug in the inputs and receive back the output
  • Example: The formula \(y = 3x + 7\) is a function with input \(x\) and output \(y\). If \(x\) is \(5\), \(y\) is \(22\), \(y = 3 \times 5 + 7 = 22\)

Height as a function of width

ggplot(data = pp, aes(x = Width_in, y = Height_in)) +
  geom_point() +
  geom_smooth(method = "lm") +
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  )

## Warning: Removed 258 rows containing non-finite values
## (stat_smooth).

## Warning: Removed 258 rows containing missing values (geom_point).

... without the measure of uncertainty

ggplot(data = pp, aes(x = Width_in, y = Height_in)) +
  geom_point() +
  geom_smooth(method = "lm", 
              se = FALSE) +
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  )

... with different cosmetic choices

ggplot(data = pp, aes(x = Width_in, y = Height_in)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE,
              color = "#8E2C90", linetype = "dashed", size = 3) +
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  )

Other smoothing methods: gam

ggplot(data = pp, aes(x = Width_in, y = Height_in)) +
  geom_point() +
  geom_smooth(method = "gam",
              se = FALSE, color = "#8E2C90") + 
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  )

Other smoothing methods: loess

ggplot(data = pp, aes(x = Width_in, y = Height_in)) +
  geom_point() +
  geom_smooth(method = "loess",
              se = FALSE, color = "#8E2C90") + 
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  )

Vocabulary

• Response variable: Variable whose behavior or variation you are trying to understand, on the y-axis
• Explanatory variables: Other variables that you want to use to explain the variation in the response, on the x-axis
• Predicted value: Output of the model function
  • The model function gives the typical (expected) value of the response variable conditioning on the explanatory variables
  • Residuals: A measure of how far each case is from its predicted value (based on a particular model)
  • Residual = Observed value - Predicted value
  • Tells how far above/below the expected value each case is

Residuals

ht_wt_fit <- linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ Width_in, data = pp)
ht_wt_fit_tidy <- tidy(ht_wt_fit$fit) 
ht_wt_fit_aug  <- augment(ht_wt_fit$fit) %>%
  mutate(res_cat = ifelse(.resid > 0, TRUE, FALSE))
ggplot(data = ht_wt_fit_aug) +
  geom_point(aes(x = Width_in, y = Height_in, color = res_cat)) +
  geom_line(aes(x = Width_in, y = .fitted), size = 0.75, color = "#8E2C90") + 
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  ) +
  guides(color = "none") +
  scale_color_manual(values = c("#260b27", "#e6b0e7")) +
  geom_text(aes(x = 0, y = 150), label = "Positive residual", color = "#e6b0e7", hjust = 0, size = 8) +
  geom_text(aes(x = 150, y = 25), label = "Negative residual", color = "#260b27", hjust = 0, size = 8)

Landscape paintings

• Landscape painting is the depiction in art of landscapes – natural scenery such as mountains, valleys, trees, rivers, and forests, especially where the main subject is a wide view – with its elements arranged into a coherent composition.¹
  • Landscape paintings tend to be wider than they are long.
• Portrait painting is a genre in painting, where the intent is to depict a human subject.²
  • Portrait paintings tend to be longer than they are wide.

Multiple explanatory variables

ggplot(data = pp, aes(x = Width_in, y = Height_in, color = factor(landsALL))) +
  geom_point(alpha = 0.4) +
  geom_smooth(method = "lm", se = FALSE) +
  labs(
    title = "Height vs. width of paintings, by landscape features",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)",
    color = "landscape"
  ) +
  scale_color_manual(values = c("#E48957", "#071381"))

Extending regression lines

ggplot(data = pp, aes(x = Width_in, y = Height_in, color = factor(landsALL))) +
  geom_point(alpha = 0.4) +
  geom_smooth(method = "lm", se = FALSE,
              fullrange = TRUE) +
  labs(
    title = "Height vs. width of paintings, by landscape features",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)",
    color = "landscape"
  ) +
  scale_color_manual(values = c("#E48957", "#071381"))

Models - upsides and downsides

• Models can sometimes reveal patterns that are not evident in a graph of the data. This is a great advantage of modeling over simple visual inspection of data.
• There is a real risk, however, that a model is imposing structure that is not really there on the scatter of data, just as people imagine animal shapes in the stars. A skeptical approach is always warranted.

Variation around the model...

is just as important as the model, if not more!

Statistics is the explanation of variation in the context of what remains unexplained.

• The scatter suggests that there might be other factors that account for large parts of painting-to-painting variability, or perhaps just that randomness plays a big role.
• Adding more explanatory variables to a model can sometimes usefully reduce the size of the scatter around the model. (We'll talk more about this later.)

How do we use models?

• Explanation: Characterize the relationship between \(y\) and \(x\) via slopes for numerical explanatory variables or differences for categorical explanatory variables
• Prediction: Plug in \(x\), get the predicted \(y\)