Review

• A data visualisation consists of data symbols, guides, and labels.

• A data visualisation can help to answer questions.

  • An effective data visualisation will pose questions that the visual system is good at answering.

• We need to choose a mapping from data values to data symbols.

  • An effective data visualisation will have good mappings from data to data symbols.

Quantitative Data

• Visual features

• Data types

• Quantitative features

• Quantitative scales

• Feature mismatch

Visual Features

• Data symbols are the geometric shapes that represent data values.

• In {ggplot2}, our choice of geom selects a geometric shape.

• geom_col() produces a rectangle.

ggplot(crimeLevelTotal) + 
    geom_col(aes(x=total, y=level))

• In {ggplot2}, our choice of geom and aesthetic selects a visual feature.

• geom_col() plus x=total produces rectangles with different lengths.

ggplot(crimeLevelTotal) + 
    geom_col(aes(x=total, y=level))

• In {ggplot2}, our choice of geom and aesthetic selects a visual feature.

• geom_col() plus y=level produces rectangles at different positions.

ggplot(crimeLevelTotal) + 
    geom_col(aes(x=total, y=level))

Visual Features

• The combination of geom and aesthetic that we choose maps data values to a visual feature.

• Visual features are the properties of data symbols that our visual system identifies very rapidly, in parallel, and without conscious effort.

• Our task is to identify a good mapping from data values to visual features.

Visual Features

• Some common visual features are: position (or location), length (height or width), area, angle, colour, and shape.

  

Effective Visual Features

• How do we choose an effective visual feature to represent data values?

• It is important that we can map backwards from a visual feature to the data.

Effective Visual Features

• One idea is to select geoms and aesthetics so that data values are mapped to visual features that have the same type as the data.

• For visualising quantitative data, we need to identify which visual features are quantitative visual features.

Data Types

• Nominal data only allows us to calculate whether X is equal to a specific value.

• Ordinal data also allows us to calculate whether X is greater than a specific value.

• Ordinal and nominal data are collectively referred to as qualitative data and typically involve character values (text labels).

Nominal Data

• crimeGroup contains the total count of offenders aged 14-16, per year, broken down by ethnic group.

head(crimeGroup)

           group year count       pop      rate   yearDate
1          Māori 2011  5957  42960.00 1407.6156 2011-06-30
2       Pasifika 2011  1092  16947.80  654.0788 2011-06-30
3 European/Other 2011  5775 125542.24  466.9634 2011-06-30
6          Māori 2012  5458  43190.03 1287.8118 2012-06-30
7       Pasifika 2012  1040  16934.88  625.8260 2012-06-30
8 European/Other 2012  4831 125175.13  393.2976 2012-06-30

Nominal Data

• crimeGroup$group is an example of nominal data.

• We can calculate whether group is equal to a specific value.

head(crimeGroup$group == "Pasifika")

[1] FALSE  TRUE FALSE FALSE  TRUE FALSE

Ordinal Data

• crimeLevel contains the total count of offenders aged 14-16, per year, broken down by the level of crime.

head(crimeLevel)

        level year count      prop levelNumeric   yearDate
1         Low 2011  3986 30.619143            1 2011-06-30
2  Low-Medium 2011  4010 30.803503            2 2011-06-30
3      Medium 2011  1770 13.596559            3 2011-06-30
4 Medium-High 2011  2346 18.021201            4 2011-06-30
5        High 2011   906  6.959594            5 2011-06-30
6         Low 2012  3578 30.991771            1 2012-06-30

Ordinal Data

• crimeLevel$level is an example of ordinal data.

• We can calculate whether level is greater than a specific value.

head(crimeLevel$level > "Medium")

[1] FALSE FALSE FALSE  TRUE  TRUE FALSE

• We can also calculate whether level is equal to a specific value.

head(crimeLevel$level == "Medium")

[1] FALSE FALSE  TRUE FALSE FALSE FALSE

• A character vector can take any value.

  char <- c("Low", "Medium", "High")
  char

  [1] "Low"    "Medium" "High"  

• A factor has a set of valid levels.

  fac <- factor(char)
  fac

  [1] Low    Medium High  
  Levels: High Low Medium

• An ordered factor has an order to the levels.

  ordered(fac, levels=char)

  [1] Low    Medium High  
  Levels: Low < Medium < High

Data Type

• Interval data also allows us to calculate the difference between values.

• Ratio data also allows us to calculate the ratio between values.

• Ratio and interval data are collectively referred to as quantitative data and involve numeric values.
  

Interval and Ratio Data

• crimeLevel$year is an example of interval data.

• We can calculate the difference between year values.

head(crimeLevel$year - 2000)

[1] 11 11 11 11 11 12

• We can also calculate whether year is greater than a specific value or whether year is equal to a specific value.

head(crimeLevel$year > 2010)

[1] TRUE TRUE TRUE TRUE TRUE TRUE

head(crimeLevel$year == 2010)

[1] FALSE FALSE FALSE FALSE FALSE FALSE

• crimeLevel$count is an example of ratio data.

• We can calculate the ratio of count values.

crimeLevel$count[1] / crimeLevel$count[3]

[1] 2.251977

• We can also calculate the difference between count values and whether values are greater or equal to a specific value.

crimeLevel$count[1] - crimeLevel$count[3]

[1] 2216

crimeLevel$count[1] > crimeLevel$count[3]

[1] TRUE

crimeLevel$count[1] == crimeLevel$count[3]

[1] FALSE

Effective Visual Features

• One idea is to select geoms and aesthetics so that data values are mapped to visual features that have the same type as the data.

• For visualising quantitative data, we need to identify which visual features are quantitative visual features.

Quantitative Features

• Position, length, area, and angle are all examples of features that are able to represent quantitative values.

• In all of these cases, this includes ratio data because zero can be meaningfully represented by the feature.

  

Quantitative Features

• Colour and shape are examples of dimenions that are not appropriate for representing quantitative values.

  

Quantitative Data

• The crimeLevelTotal data frame contains the total count of offenders (from 2010 to 2020) for different levels of severity of crime.

• total is an example of quantitative data.

  crimeLevelTotal

  # A tibble: 5 × 3
    level       total   prop
    <ord>       <int>  <dbl>
  1 Low         24242 0.283 
  2 Low-Medium  21513 0.251 
  3 Medium      13823 0.161 
  4 Medium-High 18315 0.214 
  5 High         7793 0.0909

Quantitative Data

• If we map total to a quantitative visual feature, we can answer questions like:

  • What is the total for Low offences?
  • What is the difference in total for Low versus Low-Medium offences?
  • What is the ratio of total offenders for Medium-High versus High offences?

We can visualise quantitative data using the position of points.

• What is the total for Low offences?
• What is the difference in total for Low versus Low-Medium offences?

Quantitative Features in {ggplot2}

• We need to select a geom and an aesthetic that map data values to a quantitative visual feature:

  • geom_point() and the x and y aesthetics map data values to position.

  • geom_col() and the y aesthetic maps data values to length.

  • geom_point() and the size aesthetic map data values to area.

Quantitative Features in {ggplot2}

ggplot(crimeLevelTotal) + 
    geom_point(aes(x=total, y=level), size=2)

We can visualise quantitative data using the length of bars.

• What is the ratio of total offenders for Medium-High versus High offences?

Quantitative Features in {ggplot2}

ggplot(crimeLevelTotal) + 
    geom_col(aes(x=total, y=level))

We can visualise quantitative data using the area of points.

• Which is larger: the difference in total between High and Medium-High offences, or the difference in total between Medium-High and Medium offences?

Quantitative Features in {ggplot2}

ggplot(crimeLevelTotal) + 
    geom_point(aes(x="", y=level, size=total), shape=1)

We can visualise quantitative data using the angle/area of wedges.

• What proportion of total offences were Low offences?

Quantitative Features in {ggplot2}

• coord_polar() changes the mapping of x and y so that these aesthetics map data values to angle and radius.

• geom_col() with constant y produces a stacked barplot.

• geom_col() with constant y and coord_polar() produces a pie chart.

Quantitative Features in {ggplot2}

ggplot(crimeLevelTotal) +
    geom_col(aes(x=total, y="", fill=level))

ggplot(crimeLevelTotal) +
    geom_col(aes(x=total, y="", fill=level)) +
    coord_polar() 

Quantitative Scales

• The mapping from data values to a visual feature also depends on the scale of the mapping.

• How are data values converted to values on the visual feature?

  

The Principle of Unambiguity

• The scale should be chosen so that differences in the data are visible in the data visualisation.

  

Quantitative Scales

• {ggplot2} provides a reasonable default.

• We can control the x- and y-scale limits using scale_x_continuous() and scale_y_continuous().

  

Quantitative Scales

• Position, length, area, and angle are only ratio features if the position or length is relative to zero.

• This is particularly important for length, area, and angle because zero length, area, and angle naturally correspond to a data value of zero.

If we visualise ratio data using position of points, a zero position must be shown if we expect the viewer to make ratio comparisons.

Quantitative Scales

• We can control the x- and y-scale limits using scale_x_continuous() and scale_y_continuous().

ggplot(crimeLevelTotal) + 
    geom_point(aes(x=total, y=level), size=2) +
    scale_x_continuous(limits=c(0, NA))

If we visualise data using length of bars, the bars should always start at zero.

• This is the default behaviour for geom_col().

Area

• We perceive the area of a circle rather than the radius.

  

Ratio Features in {ggplot2}

• scale_size() (the default) maps values to area.

• scale_size_area() maps zero to an area of zero.

ggplot(crimeLevelTotal) +
    geom_point(aes("", level, size=total)) +
    scale_size_area()

Feature Mismatch

• If the visual feature cannot represent quantitative values, we lose information.

  

Feature Mismatch

• We may deliberately lose information if we are only asking simple questions.

• Which level of offence has the highest total?

An effective data visualisation will select a visual feature that allows the visual system to answer the question of interest.

Summary

• In {ggplot2}, the choice of geom plus aesthetic mapping dictates the visual feature that we end up using.

• It makes sense to map quantitative data to quantitative visual features.

  • position, length, area, and angle are quantitative visual features.

  • A zero reference is necessary for visualising ratio data.

Summary

• It makes sense to map quantitative data to quantitative visual features.

  • Quantitative data can be visualised with:

    • bars of different lengths.
    • points at different positions or with different areas.
    • pie wedges with different angles/areas.

• Caveat: we only need fully quantitative visual features if we are asking fully quantitative questions.

Exercise

• We want to compare the total of offences for different ethnic groups:

  • What is the ratio of Māori versus European totals?
  • Which ethnic groups have a similar totals?

  crimeGroupTotal[1:2]

  # A tibble: 3 × 2
    group          total
    <chr>          <int>
  1 European/Other 31274
  2 Māori          41787
  3 Pasifika        6993

Exercise

• What type of data are total and group?

• Write {ggplot2} code to produce data visualisations that:

  • map total to length.
  • map total to position.
  • map total to angle.
  • map total to colour.

• Which data visualisations allow us to answer the questions of interest?

Exercise

• We want to compare the count of offences for different ethnic groups in different years.

  head(crimeGroup[1:3], 10)

              group year count
  1           Māori 2011  5957
  2        Pasifika 2011  1092
  3  European/Other 2011  5775
  6           Māori 2012  5458
  7        Pasifika 2012  1040
  8  European/Other 2012  4831
  11          Māori 2013  4866
  12       Pasifika 2013   880
  13 European/Other 2013  3775
  16          Māori 2014  4054

Exercise

• Which visual features are count, group, and year being mapped to?

• Are these mappings effective?

  

Exercise

• Can you see anything wrong with this data visualisation?