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10.1. Feature Types

Before we make any exploratory plots, we examine the features (also known as variables) of the data and decide on their type, which we call the feature type or variable type. Although there are multiple ways of delineating feature types, in this book we consider three basic ones:

1. Nominal feature: A feature that represents “named” categories, where the categories do not have a natural ordering, is called nominal. Examples include: political party affiliation (Democrat, Republican, Green, Other); American Kennel Club breed group (herding, hound, non-sporting, sporting, terrier, toy, working); and computer operating system (Windows, MacOS, Linux).

2. Ordinal feature: Measurements that represent ordered categories are called ordinal. Examples of ordinal features are: T-shirt size (small, medium, large); Likert-scale response (disagree, neutral, agree); and level of education (high school, college, graduate school). It is important to note that with an ordinal feature, the difference between, say, small and medium, need not be the same as the difference between medium and large. We can order the categories, but the differences between consecutive categories may not be quantifiable. Even when they can be quantified, the differences between consecutive categories may vary. We give examples later in this section.

   Ordinal and nominal data are subtypes of categorical data. Another name for these types is qualitative. In contrast we also have quantitative features.

3. Quantitative feature: These data represent numeric amounts or quantities and so are called quantitative. Examples include: height measured to the nearest cm, price reported in USD, and distance measured to the nearest tenth of a km. Quantitative features can be further divided into discrete, meaning that only a small set of values are possible, and continuous, meaning that the quantity could in principal be reported to arbitrary precision. For example, the number of siblings takes on a discrete set of values (such as, 0, 1, 2,…, 8). On the other hand, height is measured in centimeters and can theoretically be reported to any number of decimal places so we consider it continuous. There is no hard and fast rule to determine whether a quantity is discrete or continuous.

Data Storage Types vs. Feature Types

Each column in a pandas data frame has its own storage type. These types can be integer, floating point, boolean, date-time format, category, and object (strings of varying length are stored as objects in python with pointers to the strings). It is essential to understand that a feature type is not the same as a pandas storage type. We use the term feature type to refer to the conceptual notion of the information, and the term storage type refers to the representation of the information in the computer.

Note

Pandas calls the storage type dtype, which is short for data type. We refrain from using the term data type here because it can be confused with both storage type and feature type.

A feature stored as an integer can represent nominal data, strings can be quantitative (e.g., “$100.00”), and, in practice, boolean values often represent nominal features that have only two possible values.

In order to determine a feature type, we often need to consult the dataset’s data dictionary or codebook. A data dictionary is a document included with the data that describes what each column in the data table represents. In the following example, we take a look at the storage types and feature types of the columns in the dogs data frame and see that the storage type may not be a good indicator of the kind of information contained in a field.

Next, we’ll give concrete examples of both storage and feature types.

10.1.1. Example: AKC Dog Breeds

Let’s take a look at the data table from the American Kennel Club. The subset of AKC data we are working with has 12 features and 172 breeds.

dogs = pd.read_csv('data/akc.csv')
dogs

	breed	group	score	longevity	...	size	weight	height	repetition
0	Border Collie	herding	3.64	12.52	...	medium	NaN	51.0	<5
1	Border Terrier	terrier	3.61	14.00	...	small	6.0	NaN	15-25
2	Brittany	sporting	3.54	12.92	...	medium	16.0	48.0	5-15
...	...	...	...	...	...	...	...	...	...
169	Wire Fox Terrier	terrier	NaN	13.17	...	small	8.0	38.0	25-40
170	Wirehaired Pointing Griffon	sporting	NaN	8.80	...	medium	NaN	56.0	25-40
171	Xoloitzcuintli	non-sporting	NaN	NaN	...	medium	NaN	42.0	NaN

172 rows × 12 columns

A cursory glance at the table shows us that breed, group and size appear to be strings, and the other columns numbers. The summary of the data frame, shown below, provides the index, name, count of non-null values, and dtype for each column.

dogs.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 172 entries, 0 to 171
Data columns (total 12 columns):
 #   Column          Non-Null Count  Dtype  
---  ------          --------------  -----  
 0   breed           172 non-null    object 
 1   group           172 non-null    object 
 2   score           87 non-null     float64
 3   longevity       135 non-null    float64
 4   ailments        148 non-null    float64
 5   purchase_price  146 non-null    float64
 6   grooming        112 non-null    float64
 7   children        112 non-null    float64
 8   size            172 non-null    object 
 9   weight          86 non-null     float64
 10  height          159 non-null    float64
 11  repetition      132 non-null    object 
dtypes: float64(8), object(4)
memory usage: 16.2+ KB

Several columns of this data frame have a numeric computational type, as signified by float64, which means that the column can contain non-integers. Also, note that pandas encodes string columns as the object dtype rather than using a string dtype. Notice that we guessed incorrectly that repetition is quantitative. Looking a bit more carefully at the data table, we see that repetition contains string values for ranges, such as “< 5”, “15-25” and “25-40”, so this feature is ordinal.

Note

Why are decimal columns stored as the float64 dtype? In computer architecture, a floating-point number, or “float” for short, refers to a number that can have a decimal component. We won’t go in-depth into computer architecture in this book, but we will point out when it affects terminology, as in this case. The dtype float64 says that the column contains decimal numbers that each take up 64 bits of space when stored in computer memory.

Why are strings stored as the object dtype? Essentially, pandas uses optimized storage types for numeric data, like float64 or int64. However, it doesn’t have optimizations for Python objects like strings, dictionaries, or sets, so these are all stored as the object dtype. This means that the object dtype is ambiguous, but in most real-world cases we know whether object columns contain strings or some other Python type.

Next, let’s look at an example where the storage type differs from the feature type. At a first glance, we might guess ailments and children are quantitative features because they are stored as float64 dtypes. But, let’s look at the counts of their values.

display_df(dogs['ailments'].value_counts(), rows=8)

0.0    61
1.0    42
2.0    24
4.0    10
3.0     6
5.0     3
8.0     1
9.0     1
Name: ailments, dtype: int64

dogs['children'].value_counts()

1.0    67
2.0    35
3.0    10
Name: children, dtype: int64

Both ailments and children only take on a few integer values. What does a value of 3.0 for children or 9.0 for ailments mean? We need more information to figure this out. The name of the column and how the information is stored in the data frame is not enough. Instead, take a look the data dictionary, shown in the AKC Dog Breed Codebook table below.

Table 10.1 AKC Dog Breed Codebook

Feature	Description
breed	dog breed, e.g., Border Collie, Dalmatian, Vizsla
group	American Kennel Club grouping (herding, hound, non-sporting, sporting, terrier, toy, working)
score	AKC score
longevity	typical lifetime (years)
ailments	number of serious genetic ailments
purchase_price	average purchase price from puppyfind.com
grooming	grooming required once every: 1 = day, 2 = week, 3 = few weeks
children	suitability for children: 1 = high, 2 = medium, 3 = low
size	size: small, medium, large
weight	typical weight (kg)
height	typical height from the shoulder (cm)
repetition	number of repetitions to understand a new command: <5, 5-15, 15-25, 25-40, 40-80, >80

Although the data dictionary does not explicitly specify the feature types, the descriptions help us figure out that children represents the suitability of the breed for children, and a value of 1.0 corresponds to “high” suitability. We also figure out that ailments is a count of the number of serious genetic ailments that dogs of this breed tend to have. Based on the codebook, we treat children as a categorical feature, even though it is stored as a floating point number, and since low < medium < high, children is ordinal. Since ailments is a count, we treat it as a quantitative (numeric) feature type, and for some analyses we further define it as discrete numeric because there are only a few possible values that ailments can take on.

The codebook also confirms that the features: score, longevity, purchase_price, weight, and height are quantitative. It makes sense to compare the longevity for one breed to that of another by looking at the difference in their longevity values. For example, chihuahuas typically live about four years longer than dachshunds (16.5 to 12.6 years). It also makes sense to compare the weight of one breed to another as a ratio; for example, a dachshund is usually about five times heavier than a chihuahua (11 kg to 2 kg). Of these quantitative features, ailments is the only one that we consider to be discrete.

The data dictionary descriptions for breed , group, size and repetition suggest that these features are qualitative. Each of these variables have different, and yet commonly found, characteristics that are worth exploring a bit more. We do this by examining the counts of each unique value for the feature. We begin with breed.

dogs['breed'].value_counts()

Australian Cattle Dog         1
Staffordshire Bull Terrier    1
Dandie Dinmont Terrier        1
                             ..
Komondor                      1
Boykin Spaniel                1
Alaskan Malamute              1
Name: breed, Length: 172, dtype: int64

The breed feature has 172 unique values—that’s the same as the number of records in the data frame. We can think of breed as the primary key for the table. By design, each dog breed has one record, and this feature determines the dataset’s granularity. Although technically breed is a nominal feature, it doesn’t really make sense to analyze it. We would want to confirm that all values are unique and clean, and otherwise we would only use it to, say, label unusual values in a plot.

Next we examine group.

dogs['group'].value_counts()

sporting        28
terrier         28
working         27
hound           26
herding         25
non-sporting    19
toy             19
Name: group, dtype: int64

The group feature has seven unique values, and since these groupings do not have a natural ordering, we consider group a nominal feature.

We look at size next.

dogs['size'].value_counts()

medium    60
small     58
large     54
Name: size, dtype: int64

The size feature has a natural ordering: small < medium < large so it is ordinal. We don’t know how the category “small” is determined, but we do know that a small breed is in some sense smaller than a medium-size breed, which is smaller than a large one. We have an ordering, but differences and ratios don’t make sense conceptually for this feature.

Nominal features, in comparison, do not provide meaning in even the direction of the differences. A dog breed in the group sporting and a breed in toy differ from each other in several ways so group values are not easily reduced to an ordering.

The repetition feature is an example of a quantitative variable that has been collapsed into categories and become ordinal. The codebook tells us that repetition is the number of times a new command needs to be repeated before the dog understands it. The numeric values have been placed into categories: <5, 5-15, 15-25, 25-40, 40-80, >80.

dogs['repetition'].value_counts()

25-40     39
15-25     29
40-80     22
5-15      21
80-100    11
<5        10
Name: repetition, dtype: int64

Notice that these categories have different widths. The first is fewer than 5 repetitions, while others are 10, 15, and 40 repetitions wide. The ordering is clear, but the gaps from one category to the next are not the same magnitude.

Now that we have double checked the values in the variables against the descriptions in the codebook, we can augment the data dictionary to include this additional information about the feature types. Our dictionary is shown in the Revised AKC Dog Breed Codebook table below.

Table 10.2 Revised AKC Dog Breed Codebook

Feature	Description	Feature Type	Storage Type
breed	dog breed, e.g., Border Collie, Dalmatian, Vizsla	primary key	string
group	AKC group (herding, hound, non-sporting, sporting, terrier, toy, working)	qualitative - nominal	string
score	AKC score	quantitative	floating point
longevity	typical lifetime (years)	quantitative	floating point
ailments	number of serious genetic ailments (0, 1, …, 9)	quantitative - discrete	floating point
purchase_price	average purchase price from puppyfind.com	quantitative	floating point
grooming	groom once every: 1 = day, 2 = week, 3 = few weeks	qualitative - ordinal	floating point
children	suitability for children: 1 = high, 2 = medium, 3 = low	qualitative - ordinal	floating point
size	size: small, medium, large	qualitative - ordinal	string
weight	typical weight (kg)	quantitative	floating point
height	typical height from the shoulder (cm)	quantitative	floating point
repetition	number of repetitions to understand a new command: <5, 5-15, 15-25, 25-40, 40-80, >80	qualitative - ordinal	string

10.1.2. Transforming Qualitative Features

We discussed transformations in the Wrangling Dataframes chapter, but there are a few additional transformations related to the categories of qualitative features that we may want to perform. We may want to:

• Relabel categories

• Collapse categories

• Convert a quantitative feature into ordinal

We’ll explain when we may want to make these transformations and give examples.

Relabel Categories. Summary statistics, like the mean and the median, make sense for quantitative data, but typically not for qualitative data. For example, the average price for toy breeds makes sense ($687), but the “average” of children suitability doesn’t. However, pandas will happily compute the mean of the values in the children column if we ask it to.

# Don't use this value in actual data analysis!
dogs["children"].mean()

1.4910714285714286

Note

This is a key difference between storage types and feature types—storage types say what operations we can write code to compute, while feature types say what operations make sense for the data.

Instead, we want to consider the distribution of ones, twos, and threes of children for toy breeds.

toy_dogs = dogs.query('group == "toy"')
sns.countplot(data=toy_dogs, x='children')

Fig. 10.1 Bar plot of the ordinal feature, children. The y-axis shows the counts associated with each category (1=low, 2=medium, 3= high suitability for children).

We can transform children by replacing the numbers with their string descriptions. Changing 1, 2, 3 into low, medium, and high makes it easier to recognize that children is categorical. With strings, we would not be tempted to compute a mean, the categories would be connected to their meaning, and labels for plots would have reasonable values by default. Why would we not always want to have categorical data represented by strings? Strings generally take up more computer memory to store, which can greatly increase the size of a dataset if it contains many categorical features.

Collapse Categories. Let’s create a new column, called play, to represent the groups of dogs whose “purpose” is to play (or not). (This is a fictitious distinction used for demonstration purposes). This group consists of the toy and non-sporting breeds. The new feature, play, is a transformation of group that collapses categories: toy and non-sporting are combined into one category, and the remaining categories are placed in a second, non-play category. The boolean (bool) storage type is useful to indicate the presence or absence of this characteristic.

with_play = dogs.assign(
    play=(dogs["group"] == "toy") | (dogs["group"] == "non-sporting"))
with_play

	breed	group	score	longevity	...	weight	height	repetition	play
0	Border Collie	herding	3.64	12.52	...	NaN	51.0	<5	False
1	Border Terrier	terrier	3.61	14.00	...	6.0	NaN	15-25	False
2	Brittany	sporting	3.54	12.92	...	16.0	48.0	5-15	False
...	...	...	...	...	...	...	...	...	...
169	Wire Fox Terrier	terrier	NaN	13.17	...	8.0	38.0	25-40	False
170	Wirehaired Pointing Griffon	sporting	NaN	8.80	...	NaN	56.0	25-40	False
171	Xoloitzcuintli	non-sporting	NaN	NaN	...	NaN	42.0	NaN	True

172 rows × 13 columns

Representing a two-category qualitative feature as a boolean has a few advantages. For example, the mean of play makes sense because it returns the fraction of True values. When booleans are used for numeric calcuations, True becomes 1 and False becomes 0.

with_play['play'].mean()

0.22093023255813954

This storage type gives us a shortcut to compute counts and averages of boolean values. Later in the book, we’ll see that it’s also a handy encoding for modeling.

Convert Quantitative to Ordinal. Finally, another transformation that we sometimes find useful is to convert numeric values into categories. For example, we might collapse the values in ailments into categories: 0, 1, 2, 3, 4+. In other words, we turn ailments from a quantitative feature into an ordinal feature with the mapping 0→0, 1→1, 2→2, 3→3, and any value 4 or larger → 4+. Why might we want to make this transformation? Since so few breeds have more than three genetic ailments, we think the simplification will be clearer and adequate for our investigation.

Note

As of this writing (late 2021), pandas also implements a category dtype which is designed to work with qualitative data. However, this storage type is not yet widely adopted by the visualization and modeling libraries, which limits its usefulness. For that reason, we do not transform our qualitative variables into the category dtype. We expect that future readers may want to use the category dtype as more libraries support it.

10.1.3. The Importance of Feature Types

Feature types guide us in our data analysis. They help specify the operations, visualizations, and models we can meaningfully apply to the data. The Plots for Feature Types table below gives a mapping of the various plots that are typically good options for each feature type. Whether the variable(s) are quantitative or qualitative generally determines the set of viable plots to make, although there are exceptions. Other factors that enter into the decision are the number of observations, and whether the data takes on only a few distinct values. For example, we might make a bar chart, rather than a histogram, for a discrete quantitative variable.

Table 10.3 Plots for Feature Types

Feature Type	Dimension	Plot
Quantitative	One Feature	Rug plot, histogram, density curve, box-and-whisker plot, violin plot
Qualitative	One Feature	Bar plot, dot chart, line plot, pie chart
Quantitative	Two Features	Scatter plot, smooth curve, contour plot, heat map, quantile-quantile plot
Qualitative	Two Features	Side-by-side bar plots, mosaic plot, overlaid lines
Mixed	Two Features	Overlaid density curves, side-by-side box-and-whisker plots, overlaid smooth curves, quantile-quantile plot

The feature type also helps us decide the kind of summary statistics to calculate. With qualitative data, we usually don’t compute means or standard deviations, and instead compute the count, fraction, or percentage of records in each category. With a quantitative feature, we compute the mean or median as a measure of center, and, respectively, the standard deviation or inner quartile range (75th percentile - 25th percentile) as a measure of spread. In addition to the quartiles, we may find other percentiles informative.

Note

The nth percentile is that value q such that n% of the data values fall at or below it. The value q might not be unique, and there are several approaches to select a unique value from the possibilities. With enough data, there should be little difference between these definitions.

To compute percentiles in Python, we prefer using:

# Uses our definition of percentile
np.percentile(data, interpolation='lower')

When exploring the data, we need to know how to interpret the shapes that our plots reveal. We also need to recognize certain kinds of features and understand what they tell us about the data. The next three sections give guidance with this interpretation. We also introduce many of the types of plots listed in Table 10.3 through the examples, and those that are not introduced here are covered in the Data Visualization chapter.

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10. Exploratory Data Analysis

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10.2. What to Look For in a Distribution