The Language and Fitting of Models



Data Science in a Box

datasciencebox.org

Cornell College
DSC 223 - Fall 2022

October 11th, 2022

The Language of Models

What is a model?

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!)

Data: Paris Paintings

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

Two paintings very rich in composition, of a beautiful execution, and whose merit is very remarkable, each 17 inches 3 lines high, 23 inches wide; the first, painted on wood, comes from the Cabinet of Madame la Comtesse de Verrue; it represents a departure for the hunt: it shows in the front a child on a white horse, a man who gives the horn to gather the dogs, a falconer and other figures nicely distributed across the width of the painting; two horses drinking from a fountain; on the right in the corner a lovely country house topped by a terrace, on which people are at the table, others who play instruments; trees and fabriques pleasantly enrich the background.

Auction Data Code

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, Antoine"
$ 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
$ lrgfont           <dbl> 0
$ relig             <dbl> 0
$ landsALL          <dbl> 1
$ lands_sc          <dbl> 0
$ lands_elem        <dbl> 1
$ lands_figs        <dbl> 1
$ lands_ment        <dbl> 0
$ arch              <dbl> 1
$ mytho             <dbl> 0
$ peasant           <dbl> 0
$ othgenre          <dbl> 0
$ singlefig         <dbl> 0
$ portrait          <dbl> 0
$ still_life        <dbl> 0
$ discauth          <dbl> 0
$ history           <dbl> 0
$ allegory          <dbl> 0
$ pastorale         <dbl> 0
$ other             <dbl> 0
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, Antoine"
$ 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
$ lrgfont           <dbl> 0
$ relig             <dbl> 0
$ landsALL          <dbl> 1
$ lands_sc          <dbl> 0
$ lands_elem        <dbl> 1
$ lands_figs        <dbl> 1
$ lands_ment        <dbl> 0
$ arch              <dbl> 1
$ mytho             <dbl> 0
$ peasant           <dbl> 0
$ othgenre          <dbl> 0
$ singlefig         <dbl> 0
$ portrait          <dbl> 0
$ still_life        <dbl> 0
$ discauth          <dbl> 0
$ history           <dbl> 0
$ allegory          <dbl> 0
$ pastorale         <dbl> 0
$ other             <dbl> 0

Modeling the relationship between variables

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)"
  )

… 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", "#523178")) +
  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)

Question

The plot below displays the relationship between height and width of paintings. The only difference from the previous plots is that it uses a smaller alpha value, making the points somewhat transparent. What feature is apparent in this plot that was not (as) apparent in the previous plots? What might be the reason for this feature?

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

How, if at all, does the relationship between width and height of paintings vary by whether or not they have any landscape elements?

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 \(y\)

Fitting and Interpreting Models

Models with numerical explanatory variables

Data: Paris Paintings

pp <- read_csv("data/paris-paintings.csv", na = c("n/a", "", "NA"))
  • Number of observations: 3393
  • Number of variables: 61

Goal: Predict height from width

\[\widehat{height}_{i} = \beta_0 + \beta_1 \times width_{i}\]

Step 1: Specify model

linear_reg()
Linear Regression Model Specification (regression)

Computational engine: lm

Step 2: Set model fitting engine

linear_reg() %>%
  set_engine("lm") # lm: linear model
Linear Regression Model Specification (regression)

Computational engine: lm

Step 3: Fit model & estimate parameters

… using formula syntax

linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ Width_in, data = pp)
parsnip model object


Call:
stats::lm(formula = Height_in ~ Width_in, data = data)

Coefficients:
(Intercept)     Width_in  
     3.6214       0.7808

A closer look at model output

parsnip model object


Call:
stats::lm(formula = Height_in ~ Width_in, data = data)

Coefficients:
(Intercept)     Width_in  
     3.6214       0.7808

\[\widehat{height}_{i} = 3.6214 + 0.7808 \times width_{i}\]

A tidy look at model output

linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ Width_in, data = pp) %>%
  tidy()
# A tibble: 2 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)    3.62    0.254        14.3 8.82e-45
2 Width_in       0.781   0.00950      82.1 0

\[\widehat{height}_{i} = 3.62 + 0.781 \times width_{i}\]

Slope and intercept

\[\widehat{height}_{i} = 3.62 + 0.781 \times width_{i}\] - Slope: For each additional inch the painting is wider, the height is expected to be higher, on average, by 0.781 inches.

  • Intercept: Paintings that are 0 inches wide are expected to be 3.62 inches high, on average. (Does this make sense?)

Correlation does not imply causation

Remember this when interpreting model coefficients

Parameter estimation

Linear model with a single predictor

  • We’re interested in \(\beta_0\) (population parameter for the intercept) and \(\beta_1\) (population parameter for the slope) in the following model:

\[\hat{y}_{i} = \beta_0 + \beta_1~x_{i}\] - Tough luck, you can’t have them…

  • So we use sample statistics to estimate them:

\[\hat{y}_{i} = b_0 + b_1~x_{i}\]

Least squares regression

  • The regression line minimizes the sum of squared residuals.
  • If \(e_i = y_i - \hat{y}_i\), then, the regression line minimizes \(\sum_{i = 1}^n e_i^2\).

Visualizing residuals

Visualizing residuals (cont.)

Visualizing residuals (cont.)

Properties of least squares regression

  • The regression line goes through the center of mass point, the coordinates corresponding to average \(x\) and average \(y\), \((\bar{x}, \bar{y})\):

\[\bar{y} = b_0 + b_1 \bar{x} ~ \rightarrow ~ b_0 = \bar{y} - b_1 \bar{x}\]

  • The slope has the same sign as the correlation coefficient: \(b_1 = r \frac{s_y}{s_x}\)

  • The sum of the residuals is zero: \(\sum_{i = 1}^n e_i = 0\)

  • The residuals and \(x\) values are uncorrelated

Models with categorical explanatory variables

Categorical predictor with 2 levels

# A tibble: 3,393 × 3
   name      Height_in landsALL
   <chr>         <dbl>    <dbl>
 1 L1764-2          37        0
 2 L1764-3          18        0
 3 L1764-4          13        1
 4 L1764-5a         14        1
 5 L1764-5b         14        1
 6 L1764-6           7        0
 7 L1764-7a          6        0
 8 L1764-7b          6        0
 9 L1764-8          15        0
10 L1764-9a          9        0
11 L1764-9b          9        0
12 L1764-10a        16        1
13 L1764-10b        16        1
14 L1764-10c        16        1
15 L1764-11         20        0
16 L1764-12a        14        1
17 L1764-12b        14        1
18 L1764-13a        15        1
19 L1764-13b        15        1
20 L1764-14         37        0
# … with 3,373 more rows
  • landsALL = 0: No landscape features
  • landsALL = 1: Some landscape features

Height & landscape features

linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ factor(landsALL), data = pp) %>%
  tidy()
# A tibble: 2 × 5
  term              estimate std.error statistic  p.value
  <chr>                <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)          22.7      0.328      69.1 0       
2 factor(landsALL)1    -5.65     0.532     -10.6 7.97e-26

Height & landscape features

\[\widehat{Height_{in}} = 22.7 - 5.645~landsALL\]

  • Slope: Paintings with landscape features are expected, on average, to be 5.645 inches shorter than paintings that without landscape features
    • Compares baseline level (landsALL = 0) to the other level (landsALL = 1)
  • Intercept: Paintings that don’t have landscape features are expected, on average, to be 22.7 inches tall

Relationship between height and school

linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ school_pntg, data = pp) %>%
  tidy()
# A tibble: 7 × 5
  term            estimate std.error statistic p.value
  <chr>              <dbl>     <dbl>     <dbl>   <dbl>
1 (Intercept)        14.0       10.0     1.40  0.162  
2 school_pntgD/FL     2.33      10.0     0.232 0.816  
3 school_pntgF       10.2       10.0     1.02  0.309  
4 school_pntgG        1.65      11.9     0.139 0.889  
5 school_pntgI       10.3       10.0     1.02  0.306  
6 school_pntgS       30.4       11.4     2.68  0.00744
7 school_pntgX        2.87      10.3     0.279 0.780

Dummy variables

# A tibble: 7 × 5
  term            estimate std.error statistic p.value
  <chr>              <dbl>     <dbl>     <dbl>   <dbl>
1 (Intercept)        14.0       10.0     1.40  0.162  
2 school_pntgD/FL     2.33      10.0     0.232 0.816  
3 school_pntgF       10.2       10.0     1.02  0.309  
4 school_pntgG        1.65      11.9     0.139 0.889  
5 school_pntgI       10.3       10.0     1.02  0.306  
6 school_pntgS       30.4       11.4     2.68  0.00744
7 school_pntgX        2.87      10.3     0.279 0.780
  • When the categorical explanatory variable has many levels, they’re encoded to dummy variables
  • Each coefficient describes the expected difference between heights in that particular school compared to the baseline level

Categorical predictor with 3+ levels

school_pntg D_FL F G I S X
A 0 0 0 0 0 0
D/FL 1 0 0 0 0 0
F 0 1 0 0 0 0
G 0 0 1 0 0 0
I 0 0 0 1 0 0
S 0 0 0 0 1 0
X 0 0 0 0 0 1 
# A tibble: 3,393 × 3
   name      Height_in school_pntg
   <chr>         <dbl> <chr>      
 1 L1764-2          37 F          
 2 L1764-3          18 I          
 3 L1764-4          13 D/FL       
 4 L1764-5a         14 F          
 5 L1764-5b         14 F          
 6 L1764-6           7 I          
 7 L1764-7a          6 F          
 8 L1764-7b          6 F          
 9 L1764-8          15 I          
10 L1764-9a          9 D/FL       
11 L1764-9b          9 D/FL       
12 L1764-10a        16 X          
13 L1764-10b        16 X          
14 L1764-10c        16 X          
15 L1764-11         20 D/FL       
16 L1764-12a        14 D/FL       
17 L1764-12b        14 D/FL       
18 L1764-13a        15 D/FL       
19 L1764-13b        15 D/FL       
20 L1764-14         37 F          
# … with 3,373 more rows

Relationship between height and school

# A tibble: 7 × 5
  term            estimate std.error statistic p.value
  <chr>              <dbl>     <dbl>     <dbl>   <dbl>
1 (Intercept)        14.0       10.0     1.40  0.162  
2 school_pntgD/FL     2.33      10.0     0.232 0.816  
3 school_pntgF       10.2       10.0     1.02  0.309  
4 school_pntgG        1.65      11.9     0.139 0.889  
5 school_pntgI       10.3       10.0     1.02  0.306  
6 school_pntgS       30.4       11.4     2.68  0.00744
7 school_pntgX        2.87      10.3     0.279 0.780
  • Austrian school (A) paintings are expected, on average, to be 14 inches tall.
  • Dutch/Flemish school (D/FL) paintings are expected, on average, to be 2.33 inches taller than Austrian school paintings.
  • French school (F) paintings are expected, on average, to be 10.2 inches taller than Austrian school paintings.
  • German school (G) paintings are expected, on average, to be 1.65 inches taller than Austrian school paintings.
  • Italian school (I) paintings are expected, on average, to be 10.3 inches taller than Austrian school paintings.
  • Spanish school (S) paintings are expected, on average, to be 30.4 inches taller than Austrian school paintings.
  • Paintings whose school is unknown (X) are expected, on average, to be 2.87 inches taller than Austrian school paintings.

🔗 dsc223-fb2-2022.github.io