This chapter will show you how to use visualization and transformation to explore your data in a systematic way, a task that statisticians call Exploratory Data Analysis, or EDA for short. EDA is an interative cycle that involves:
1. Forming questions about your data.
1. Searching for answers by visualizing, transforming, and modeling your data.
1. Using what you discover to refine your questions about the data, or
EDA is not a formal process with a strict set of rules: you must be free to investigate every idea that occurs to you. Instead, EDA is a loose set of tactics that are more likely to lead to useful insights. This chapter will teach you a basic toolkit of these useful EDA techniques. Our discussion will lead to a model of data science itself, the model that I've built this book around.
In this chapter we'll combine what you've learned about dplyr and ggplot2 to iteratively ask questions, answer them with data, and then ask new questions.
Your goal during EDA is to develop your understanding of your data. The easiest way to do this is to use questions as tools to guide your investigation. When you ask a question, the question focuses your attention on a specific part of your dataset and helps you decide which graphs, models, or transforamtions to make.
EDA is fundamentally a creative process. And like most creative processes, the key to asking _quality_ questions is to generate a large _quantity_ of questions. It is difficult to ask revealing questions at the start of your analysis because you do not know what insights are contained in your dataset. On the other hand, each new question that you ask will expose you to a new aspect of your data and increase your chance of making a discovery. You can quickly drill down into the most interesting parts of your data---and develop a set of thought provoking questions---if you follow up each question with a new question based on what you find.
There is no rule about which questions you should ask to guide your research. However, two types of questions will always be useful for making discoveries within your data. You can loosely word these questions as:
The rest of this chapter will look at these two questions. I'll explain what variation and covariation are, and I'll show you several ways to answer each question. To make the discussion easier, let's define some terms:
For now, assume all the data you see in this book is be tidy. You'll encounter lots of other data in practice, so we'll come back to these ideas again in [tidy data] where you'll learn how to tidy messy data.
**Variation** is the tendency of the values of a variable to change from measurement to measurement. You can see variation easily in real life; if you measure any continuous variable twice---and precisely enough, you will get two different results. This is true even if you measure quantities that are constant, like the speed of light (below). Each of your measurements will include a small amount of error that varies from measurement to measurement.
caption = "The speed of light is a universal constant, but variation due to measurement error obscures its value. In 1879, Albert Michelson measured the speed of light 100 times and observed 30 different values (in km/sec)."
Discrete and categorical variables can also vary if you measure across different subjects (e.g. the eye colors of different people), or different times (e.g. the energy levels of an electron at different moments).
Every variable has its own pattern of variation, which can reveal interesting information. The best way to understand that pattern is to visualize the distribution of the values that you observe for the variable.
How you visualize the distribution of a variable will depend on whether the variable is categorical or continuous. A variable is **categorical** if it can only have a finite (or countably infinite) set of unique values. In R, categorical variables are usually saved as factors, integers, or character strings. To examine the distribution of a categorical variable, use a bar chart.
A variable is **continuous** if you can arrange its values in order _and_ an infinite number of unique values can exist between any two values of the variable. Numbers and date-times are two examples of continuous variables. To examine the distribution of a continuous variable, use a histogram.
A histogram divides the x axis into equally spaced intervals and then uses a bar to display how many observations fall into each interval. In the graph above, the tallest bar shows that almost 30,000 observations have a $carat$ value between 0.25 and 0.75, which are the left and right edges of the bar.
You can set the width of the intervals in a histogram with the `binwidth` argument, which is measured in the units of the $x$ variable. You should always explore a variety of binwidths when working with histograms, as different binwidths can reveal different patterns. For example, here is how the graph above looks when we zoom into just the diamonds with a binwidth of less than three and choose a smaller binwidth.
If you wish to overlay multiple histograms in the same plot, I recommend using `geom_freqpoly()` instead of `geom_histogram()`. `geom_freqpoly()` performs the same calculation as `geom_histogram()`, but instead of displaying the counts with bars, uses lines instead. It's much easier to understand overlapping lines than bars.
Now that you can visualize variation, what should you look for in your plots? And what type of follow-up questions should you ask? I've put together a list below of the most useful types of information that you will find in your graphs, along with some follow up questions for each type of information. The key to asking good follow up questions will be to rely on your **curiosity** (What do you want to learn more about?) as well as your **skepticism** (How could this be misleading?).
In both bar charts and histograms, tall bars reveal common values of a variable. Shorter bars reveal rarer values. Places that do not have bars reveal values that were not seen in your data. To turn this information into useful questions, look for anything unexpected:
* Which values are the most common? Why?
* Which values are the rare? Why? Does that match your expectations?
* Can you see any unusual patterns? What might explain them?
As an example, the histogram below suggests several interesting questions:
* Why are there more diamonds at whole carats and common fractions of carats?
* Why are there more diamonds slightly to the right of each peak than there
* How are the observations in separate clusters different from each other?
* How can you explain or describe the clusters?
* Why might the appearance of clusters be misleading?
The histogram shows the length (in minutes) of 272 eruptions of the Old Faithful Geyser in Yellowstone National Park. Eruption times appear to be clustered in to two groups: there are short eruptions (of around 2 minutes) and long eruption (4-5 minutes), but little in between.
Many of the questions above will prompt you to explore a relationship *between* variables, for example, to see if the values of one variable can explain the behavior of another variable.
### Unusual values
Outliers are observations that are unusual; data points that are don't seem to fit the pattern. Sometimes outliers are data entry errors; other times outliers suggest important new science. When you have a lot of data, outliers are sometimes difficult to see in a histogram.
For example, take this distribution of the `x` variable from the diamonds dataset. The only evidence of outliers is the unusually wide limits on the x-axis.
```{r}
ggplot(diamonds) +
geom_histogram(aes(x = y), binwidth = 0.5)
```
This is because there are so many observations in the common bins that the rare bins are so short that you can't see them (although maybe if you stare intently at 0 you'll spot something). To make it easy to see the unusual vaues, we need to zoom into to small values of the y-axis with `coord_cartesian()`:
```{r}
ggplot(diamonds) +
geom_histogram(aes(x = y), binwidth = 0.5) +
coord_cartesian(ylim = c(0, 50))
```
(`coord_cartesian()` also has an `xlim()` argument for when you need to zoom into the x-axis. ggplot2 also has `xlim()` and `ylim()` functions that work slightly differently: they throw away the data outside the limits.)
This allows us to see that there are three unusual values: 0, ~30, and ~60. We pluck them out with dplyr:
```{r}
unusual <- diamonds %>%
filter(y < 3 | y > 20) %>%
arrange(y)
unusual
```
The y variable measures one of the three dimensions of these diamonds in mm. We know that diamonds can't have a 0 measurement. So these must be invalid measurements. We might also suspect that measureents of 32mm and 59mm are implausible: those diamonds are over an inch long, but don't cost hundreds of thousands of dollars!
When you discover an outlier it's a good idea to trace it back as far as possible. You'll be in a much stronger analytical position if you can figure out why it happened. If you can't figure it out, and want to just move on with your analysis, it's a good idea to replace it with a missing value, which we'll discuss in the next section.
### Exercises
1. Explore the distribution of each of the `x`, `y`, and `z` variables
in `diamonds`. What do you learn? Think about a diamond and how you
might decide which dimension is the length, width, and depth.
1. Explore the distribution of `price`. Do you discover anything unusual
or surprising? (Hint: carefully think about the `binwidth` and make sure
ggplot2 subscribes to the philosophy that missing values should never silently go missing. However, it's not obvious where you should plot missing values, so ggplot2 doesn't display in the plot, but does warn that they're been removed.
Other times you want to understand what makes observations with missing values different to observations with recorded values. For example, in `nycflights13::flights`, missing value in the `dep_time` variable indicate that the flight was cancelled. So you might want to compare the scheduled departure times for cancelled and non-cancelled times. You can do by making a new variable with `is.na()`.
However this plot isn't great because there are many more non-cancelled flights than cancelled flights. In the next section we'll explore some techniques for making improving this comparison.
### Exercises
1. Recall what the `na.rm = TRUE` argument does in `mean()` and `sum()`.
Why is that a similar operation for `geom_point()`?
If variation describes the behavior _within_ a variable, covariation describes the behavior _between_ variables. **Covariation** is the tendency for the values of two or more variables to vary together in a correlated way. The best way to spot covariation is to visualize the relationship between two or more variables. How you do that should again depend on the type of variables involved.
It's common to want to explore the distribution of a continuous variable broken down by a categorical, as in the previous histogram. The default appearance of `geom_freqpoly()` is not that useful for that sort of comparison because the height is the count. That means if one of the groups is much smaller than the others, it's hard to see the differences in shape. For example, lets explore how the price of a diamond varies with its quality:
To make the comparison easier we need to swap what is displayed on the y-axis. Instead of display count, we'll display __density__, which is the count standardised so that the area under each frequency polygon is one.
There's something rather surprising about this plot - it appears that fair diamonds (the lowest quality) have the highest average cut! But maybe that's because frequency polygons are a little hard to interpret - there's a lot going on in this plot.
Another alternative to display the distribution of a continuous variable broken down by a categorical variable is the boxplot. A **boxplot** is a type of visual shorthand for a distribution of values that is popular among statisticians. Each boxplot consists of:
We see much less information about the distribution, but the boxplots are much more compact so we can more easily compare them (and fit more on one plot). It supports the counterintuive finding that better quality diamonds are cheaper on average! In the exercises, you'll be challenged to figure out why.
`cut` is an ordered factor: fair is worse than good, which is wrose than very good and so on. Most factors are unordered, so it's fair game to reorder to display the results better. For example, take the `class` variable in the `mpg` dataset. You might be interested to know how hwy mileage varies across classes:
Covariation will appear as a systematic change in the medians or IQRs of the boxplots. To make the trend easier to see, wrap the $x$ variable with `reorder()`. The code below reorders the x axis based on the median hwy value of each group.
If you wish to add more information to your boxplots, use `geom_violin()`. In a violin plot, the width of the "box" displays a kernel density estimate of the shape of the distribution.
1. Install the ggstance pacakge, and create a horizontal boxplot.
1. One problem with boxplots is that they were developed in an era of
much smaller datasets and tend to display an prohibitively large
number of "outlying values". One approach to remedy this problem is
the letter value plot. Install the lvplot package, and try using
`geom_lvplot()` to display the distribution of price vs cut. What
do you learn? How do you interpret the plots?
1. Compare and contrast `geom_violin()` with a facetted `geom_histogram()`,
or coloured `geom_freqpoly()`. What are the pros and cons of each
method?
### Visualizing two categorical variables
There are two basic techniques for visulaising covariation between categorical variables. One is to count the number of observations at each location and display the count with the size of a point. That's the job of `geom_count()`:
```{r}
ggplot(data = diamonds) +
geom_count(mapping = aes(x = cut, y = color))
```
The size of each circle in the plot displays how many observations occurred at each combination of values. Covariation will appear as a strong correlation between specific x values and specific y values. As with bar charts, you can calculate the specific values with `count()`.
```{r}
diamonds %>% count(color, cut)
```
This allows you to reproduce `geom_count()` by hand, or instead of mapping count to `size`, you could instead use `geom_raster()` and map count to `fill`:
```{r}
diamonds %>%
count(color, cut) %>%
ggplot(mapping = aes(x = color, y = cut)) +
geom_raster(aes(fill = n))
```
If the categorical variables are unordered, you might want to use the seriation package to simultaneously reorder the rows and columns in order to more clearly reveal interesting patterns.
### Exercises
1. How could you rescale the count dataset above to more clearly see
the differences across colours or across cuts?
1. Use `geom_raster()` together with dplyr to explore how average flight
Visualize covariation between two continuous variables with a scatterplot, i.e. `geom_point()`. Covariation will appear as a structure or pattern in the data points. For example, an exponential relationship seems to exist between the carat size and price of a diamond.
Scatterplots become less useful as the size of your dataset grows, because points begin to pile up into areas of uniform black (as above). You can make patterns clear again with `geom_bin2d()`, `geom_hex()`, or `geom_density2d()`.
`geom_bin2d()` and `geom_hex()` divide the coordinate plane into two dimensional bins and then use a fill color to display how many points fall into each bin. `geom_bin2d()` creates rectangular bins. `geom_hex()` creates hexagonal bins. You will need to install the hexbin package to use `geom_hex()`.
`geom_density2d()` fits a 2D kernel density estimation to the data and then uses contour lines to highlight areas of high density. It is very useful for overlaying on raw data even when your dataset is not big.
Two dimensional plots reveal outliers that are not visible in one dimensional plots. For example, some points in the plot below have an unusual combination of $x$ and $y$ values, which makes the points outliers even though their $x$ and $y$ values appear normal when examined separately.
Two dimensional plots can also reveal clusters that may not be visible in one dimensional plots. For example, the two dimensional pattern in the plot below reveals two clusters, a separation that is not visible in the distribution of either variable by itself, as verified with a rug geom.
Patterns in your data provide clues about relationships. If a systematic relationship exists between two variables it will appear as a pattern in the data. If you spot a pattern, ask yourself:
+ Could this pattern be due to coincidence (i.e. random chance)?
+ How can you describe the relationship implied by the pattern?
+ How strong is the relationship implied by the pattern?
+ What other variables might affect the relationship?
+ Does the relationship change if you look at individual subgroups of the data?
A scatterplot of Old Faithful eruption lengths versus the wait time between eruptions shows a pattern: longer wait times are associated with longer eruptions. The scatterplot also displays the two clusters that we noticed above.
Patterns provide one of the most useful tools for data scientists because they reveal covariation. If you think of variation as a phenomenon that creates uncertainty, covariation is a phenomenon that reduces it. If two variables covary, you can use the values of one variable to make better predictions about the values of the second. If the covariation is due to a causal relationship (a special case), then you can use the value of one variable to control the value
In general, outliers, clusters, and patterns become easier to spot as you look at the interaction of more and more variables. However, as you include more variables in your plot, data becomes harder to visualize.
You can extend scatterplots into three dimensions with the plotly, rgl, rglwidget, and threejs packages (among others). Each creates a "three dimensional," graph that you can rotate with your mouse. Below is an example from plotly, displayed as a static image.
You can extend this approach into n-dimensional hyperspace with the ggobi package, but you will soon notice a weakness of multidimensional graphs. You can only visualize multidimensional space by projecting it onto your two dimensional retinas. In the case of 3D graphics, you can combine 2D projections with rotation to create an intuitive illusion of 3D space, but the illusion ceases to be intuitive as soon as you add a fourth dimension.
Cluster algorithms are automated tools that seek out clusters in n-dimensional space for you. Base R provides two easy to use clustering algorithms: hierarchical clustering and k means clustering.
You can visualize the results of the algorithm as a dendrogram, and you can use the dendrogram to divide your data into any number of clusters. The figure below demonstrates how the algorithm would proceed in a two dimensional dataset.
To use hierarchical clustering in R, begin by selecting the numeric columns from your data; you can only apply hierarchical clustering to numeric data. Then apply the `dist()` function to the data and pass the results to `hclust()`. `dist()` computes the distances between your points in the n dimensional space defined by your numeric vectors. `hclust()` performs the clustering algorithm.
Use `plot()` to visualize the results as a dendrogram. Each observation in the dataset will appear at the bottom of the dendrogram labeled by its rowname. You can use the labels argument to set the labels to something more informative.
To see how near two data points are to each other, trace the paths of the data points up through the tree until they intersect. The y value of the intersection displays how far apart the points are in n-dimensional space. Points that are close to each other will intersect at a small y value, points that are far from each other will intersect at a large y value. Groups of points that are near each other will look like "leaves" that all grow on the same "branch." The ordering of the x axis in the dendrogram is somewhat arbitrary (think of the tree as a mobile, each horizontal branch can spin around meaninglessly).
You can split your data into any number of clusters by drawing a horizontal line across the tree. Each vertical branch that the line crosses will represent a cluster that contains all of the points downstream from the branch. Move the line up the y axis to intersect fewer branches (and create fewer clusters), move the line down the y axis to intersect more branches and (create more clusters).
`cutree()` provides a useful way to split data points into clusters. Give cutree the output of `hclust()` as well as the number of clusters that you want to split the data into. `cutree()` will return a vector of cluster labels for your dataset. To visualize the results, map the output of `cutree()` to an aesthetic.
You can modify the hierarchical clustering algorithm by setting the method argument of hclust to one of "complete", "single", "average", or "centroid". The method determines how to measure the distance between two clusters or a lone point and a cluster, a measurement that affects the outcome of the algorithm.
* *complete* - Measures the greatest distance between any two points in the separate clusters. Tends to create distinct clusters and subclusters.
* *single* - Measures the smallest distance between any two points in the separate clusters. Tends to add points one at a time to existing clusters, creating ambiguously defined clusters.
* *average* - Measures the average distance between all combinations of points in the separate clusters. Tends to add points one at a time to existing clusters.
* *centroid* - Measures the distance between the average location of the points in each cluster.
K means clustering provides a simulation based alternative to hierarchical clustering. It identifies the "best" way to group your data into a predefined number of clusters. The figure below visualizes (in two dimensional space) the k means algorithm:
Use `kmeans()` to perform k means clustering with R. As with hierarchical clustering, you can only apply k means clustering to numerical data. Pass your numerical data to the `kmeans()` function, then set `center` to the number of clusters to search for ($k$) and `nstart` to the number of simulations to run. Since the results of k means clustering depend on the initial assignment of points to groups, which is random, R will run `nstart` simulations and then return the best results (as measured by the minimum sum of squared distances between each point and the centroid of the group it is assigned to). Finally, set the maximum number of iterations to let each simulation run in case the simulation cannot quickly find a stable grouping.
Unlike `hclust()`, the k means algorithm does not provide an intuitive visual interface. Instead, `kmeans()` returns a kmeans class object. Subset the object with `$cluster` to access a list of cluster assignments for your dataset, e.g. `iris_kmeans$cluster`. You can visualize the results by mapping them to an aesthetic, or you can apply the results by passing them to dplyr's `group_by()` function.
Ask the same questions about clusters that you find with `hclust()` and `kmeans()` that you would ask about clusters that you find with a graph. Ask yourself:
* Do the clusters seem to identify real differences between your points? How can you tell?
* Are the points within each cluster similar in some way?
* Are the points in separate clusters different in some way?
* Might there be a mismatch between the number of clusters that you found and the number that exist in real life? Are only a couple of the clusters meaningful? Are there more clusters in the data than you found?
Keep in mind that both algorithms _will always_ return a set of clusters, whether your data appears clustered or not. As a result, you should always be skeptical about the results. They can be quite insightful, but there is no reason to treat them as a fact without doing further research.
A model is a type of summary that describes the relationships in your data. You can use a model to reveal patterns and outliers that only appear in n-dimensional space. To see how this works, consider the simple linear model below. I've fit it to a two dimensional pattern so we can visualize the results.
which is the equation of the blue model line in the graph above. Even if we did not have the graph, we could use the model coefficients in the equation above to determine that a positive relationship exists between $y$ and $x$ such that a one unit increase in $x$ is associated with an approximately one unit increase in $y$. We could use a model statistic, such as adjusted $r^{2}$ to determine that the relationship is very strong (here adjusted $r^{2} = 0.99$).
Finally, we could spot outliers in our data by examining the residuals of the model, which are the distances between the actual $y$ values of our data points and the $y$ values that the model would predict for the data points. Observations that are outliers in n-dimensional space will have residuals that are outliers in one dimensional space. You can find these outliers by plotting a histogram of the residuals or by visualizing the residuals against any variable in a two dimensional plot.
You can easily use these techniques with n-dimensional relationships that cannot be visualized easily. When you spot a pattern or outlier, ask yourself the same questions that you would ask when you spot a pattern or outlier in a graph. Then visualize the residuals of your model in various ways. If a pattern exists in the residuals, it suggests that your model does not accurately describe the pattern in your data.
I'll postpone teaching you how to fit and interpret models with R until Part 4. Although models are something simple, descriptions of patterns, they are tied into the logic of statistical inference: if a model describes your data accurately _and_ your data is similar to the world at large, then your model should describe the world at large accurately. This chain of reasoning provides a basis for using models to make inferences and predictions. As a result, there is more to learn about models than we can examine here.
You now know how to explore the variables displayed in your dataset, but you should know that these are not the only variables in your data. Nor are the observations that are displayed in your data the only observations. You can use the values in your data to compute new variables or to measure new (group-level) observations. These new variables and observations provide a further source of insights that you can explore with visualizations, clustering algorithms, and models.
Variables, values, and observations provide a basis for EDA: _if a relationship exists between two_ variables, _then the relationship will exist between the_ values _of those variables when those values are measured in the same_ observation. As a result, relationships between variables will appear as patterns in your data.
Within any particular observation, the exact form of the relationship between variables may be obscured by mediating factors, measurement error, or random noise; which means that the patterns in your data will appear as signals obscured by noise.
Due to a quirk of the human cognitive system, the easiest way to spot signal amidst noise is to visualize your data. The concepts of variables, values, and observations have a role to play here as well. To visualize your data, represent each observation with its own geometric object, such as a point. Then map each variable to an aesthetic property of the point, setting specific values of the variable to specific levels of the aesthetic. You could also compute group-level statistics from your data (i.e. new observations) and map them to geoms, something that `geom_bar()`, `geom_boxplot()` and other geoms do for you automatically.
As a term, "data science" has been used in different ways by many people. This fluidity is necessary for a term that describes a wide breadth of activity, as data science does. Nonetheless, you can use the principles in this chapter to build a general model of data science. The model requires one limit to the definition of data science: data science must rely in some way on human judgement applied to data.
To judge or interpret the information in a dataset, you must first comprehend that information, which is difficult to do. The easiest way to comprehend data is to visualize, transform, and model it, a process that we have referred to as Exploratory Data Analysis.
Once you comprehend the information in your data, you can use it to make inferences from your data. Often this involves making deductions from a model. This is what you do when you conduct a hypothesis test, make a prediction (with or without a confidence interval), or score cases in a database.
But all of this work will involve a computer; you cannot do it in your head, nor on paper with a pencil. To work efficiently, you will need to know how to program in a computer language, such as R. You will also need to know how to import data to use with the computer language, and how to tidy the data into the format that works best for that computer language.
Finally, if your work is meaningful at all, it will have an audience, which means that you will need to share your work in a way that your audience can understand. Your audience might be fellow scientists who will want to reproduce the work, non-scientists who will want to understand your findings in plain terms, or yourself (in the future) who will be thankful if you make your work easy to re-learn and recreate. To satisfy these audiences, you may choose to communicate your results in a report or to bundle your work into some type of useful format, like an R package or a Shiny app.
