variation.Rmd

# Exploratory Data Analysis (EDA)

```{r include = FALSE}
library(ggplot2)
library(dplyr)
library(broom)
knitr::opts_chunk$set(fig.height = 2)
```


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 involves iteratively

1. forming questions about your data 
2. searching for answers by visualizing and transforming your data
3. using what you discover to refine your questions about the data, or to choose new questions to investigate

There is no formal way to do Exploratory Data Analysis because you must be free to investigate every idea that occurs to you. However, some tactics will reliably lead to 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. 

## Questions

> "There are no routine statistical questions, only questionable statistical routines."---Sir David Cox

> "Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise."---John Tukey

Your goal during EDA is to develop a complete understanding of your data set and the information that it contains. 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 data set and helps you decide which graphs or models to make.

During EDA, the _quantity_ of questions that you ask matters more than the quality of the 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 data set. 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

1. What type of **variation** occurs **within** my variables? and

2. What type of **covariation** occurs **between** my variables?

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: 

* A _variable_ is a quantity, quality, or property that you can measure. 

* A _value_ is the state of a variable when you measure it. The value of a variable may change from measurement to measurement.
  
* An _observation_ is a set of measurements that you make under similar conditions (you usually make all of the measurements in an observation at the same time and on the same object). An observation will contain several values, each associated with a different variable. I'll sometimes refer to an observation as a data point.

* _tabular data_ is a set of values, each associated with a variable and an observation. Tabular data is _tidy_ if each value is placed in its own "cell", each variable in its own column, and each observation in its own row. 

    Throughout the rest of this chapter I will use the word data to mean tidy tabular data. Other types of "unstructured" data exist, but you would not use the methods in this chapter on unstructured data until you first reorganized the unstructured data into tidy tabular data.

   

## Variation

> "What type of variation occurs within my variables?"

**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 should be constant, like the speed of light (below). Each of your measurements will include a small amount of error that varies from measurement to measurement.

```{r include = FALSE}
options(digits = 7)
```

```{r, variation, echo = FALSE}

mat <- as.data.frame(matrix(morley$Speed + 299000, ncol = 10))

knitr::kable(mat, 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).*", col.names = rep("", ncol(mat)))
```

```{r include = FALSE}
options(digits = 3)
```

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.

### Visualizing distributions

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.

```{r}
ggplot(data = diamonds) +
  geom_bar(mapping = aes(x = cut))
```

The height of the bars displays how many observations occurred with each x value. You can compute these values manually with `table()`. 

```{r}
table(diamonds$cut)
```

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.

```{r message = FALSE}
ggplot(data = diamonds) +
  geom_histogram(aes(x = carat), binwidth = 0.5)
```

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 $x$ axis. 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 with a binwidth of 0.01.

```{r message = FALSE}
ggplot(data = diamonds) +
  geom_histogram(aes(x = carat), binwidth = 0.01)
```

If you wish to overlay multiple histograms in the same plot, I recommend using `geom_freqpoly()` or `geom_density2d()` instead of `geom_histogram()`. `geom_freqpoly()` makes a frequency polygon, a line that connects the tops of the bars that would appear in a histogram. Like `geom_histogram()`, `geom_freqpoly()` accepts a binwidth argument.

`geom_density()` plots a one dimensional kernel density estimate of a variable's distribution. The result is a smooth version of the information contained in a histogram or a frequency polygon. You can control the smoothness of the density with `adjust`. `geom_density()` displays _density_---not count---on the y axis; the area under each curve will be normalized to one, no matter how many total observations occur in the subgroup, which makes it easier to compare subgroups. 

```{r message = FALSE, fig.show='hold', fig.width=3}
zoom <- coord_cartesian(xlim = c(55, 70))

ggplot(data = diamonds) +
  geom_freqpoly(aes(x = depth, color = cut), binwidth = 0.2) +
  zoom

ggplot(data = diamonds) +
  geom_density(aes(x = depth, color = cut), adjust = 3) +
  zoom
```

### Asking questions about variation

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?).

* *Typical values*
   
    In both bar charts and histograms, tall bars reveal common values of a variable. Shorter bars reveal less common or rare values. Places that do not have bars reveal seemingly impossible values. To turn this information into a useful question, look for anything unexpected:
    
    + Which values are the most common? Why might that be?
    
    + Which values are the most rare? Why might that be?
    
    + Is there an unusual pattern in the distribution? Why might that be?
    
    + Do the typical values change if you look at subgroups of the data?
    
    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 are slightly to the left of each peak?
    
    ```{r echo = FALSE, message = FALSE, warning = FALSE, fig.height = 2}
    ggplot(data = diamonds) +
      geom_histogram(aes(x = carat), binwidth = 0.01) + xlim(0, 3)
    ```
    
* *Range of values*

    The range of values, or spread, of the distribution reveals how certain you can be when you make predictions about a variable. If the variable only takes a narrow set of values, like below, you are unlikely to be far off if you make a prediction about a future observation. Even if the observation takes a value at the distant extreme of the distribution, the value will not be far from your guess. 
    
    ```{r echo = FALSE, message = FALSE, fig.height = 2}
    mpg$hwy2 <- mpg$hwy / 10 + 22
    ggplot(mpg) + geom_histogram(aes(x = hwy2), binwidth = 1) + xlim(10, 45)
    ```
    
    If the variable takes on a wide set of values, like below, the possibility that your guess will be far off the mark is much greater. The extreme possibilities are farther away. 
    
    ```{r echo = FALSE, message = FALSE, fig.height = 2}
    ggplot(mpg) + geom_histogram(aes(x = hwy), binwidth = 1) + xlim(10, 45)
    ```    
    
    As a quick rule, wide distributions imply less certainty when making predictions about a variable; narrow distributions imply more certainty. A distribution with only a single repeated value implies complete certainty: your variable is a constant. Ask yourself
    
    + Do your data show a surprising amount of certainty or uncertainty? Why?
    
    + Does the range of the distribution change if you look at individual subgroups of the data?
    
* *Outliers*

    Outliers are data points that do not seem to fit the overall pattern of variation, like the diamond on the far right of the histogram below. This diamond has a y dimension of `r diamonds$y[which(diamonds$y > 50)]` mm, which is much larger than the other diamonds.
    
    ```{r echo = FALSE, message = FALSE, fig.height = 2}
    ggplot(diamonds[24000:24500, ]) + geom_histogram(aes(x = y), binwidth = 0.25)
    ```   
    
    An outlier is a signal that something unique happened to the observation. Whenever you spot an outlier, ask yourself
    
    + What can explain the unusual value?
    
    If you can figure out what happened, a discovery might follow. In the case above, the unique event was a measurement error.

* *Clusters*

    Clusters of similar values suggest that subgroups exist in your data. To understand the subgroups, ask:
    
    + How are the observations within each cluster similar to each other?
    
    + 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 below displays two distinct clusters. It shows the length in minutes of 272 eruptions of the Old Faithful Geyser in Yellowstone National Park; Old Faithful appears to oscillate between short and long eruptions.

    ```{r echo = FALSE, message = FALSE, fig.height = 2}
    ggplot(faithful) + geom_histogram(aes(x = eruptions))
    ```  

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. Questions about relationships are examples of the second general question that I proposed for EDA. Let's look at that question now.

## Covariation

> "What type of covariation occurs between my variables?"

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.

### Visualizing two categorical variables

Visualize covariation between categorical variables with `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 specifc x values and specific y values. As with bar charts, you can calculate the specific values with `table()`.

```{r}
table(diamonds$color, diamonds$cut)
```

### Visualizing one categorical variable and one continuous variable

Visualize covariation between continuous and categorical variables with boxplots. A **boxplot** is a type of visual shorthand for a distribution of values that is popular among statisticians. Each boxplot consists of:

* A box that stretches from the 25th percentile of the distribution to the 75th percentile, a distance known as the Inter-Quartile Range (IQR). In the middle of the box is a line that displays the median, i.e. 50th percentile, of the distribution. These three lines give you a sense of the spread of the distribution and whether or not the distribution is symmetric about the median or skewed to one side. 

* Visual points that display observations that fall more than 1.5 times the IQR from either edge of the box. These outlying points have a strong chance of being outliers, so they are included in the boxplot for inspection. 

* A line (or whisker) that extends from each end of the box and goes to the farthest non-outlier point in the distribution.

```{r, echo = FALSE}
knitr::include_graphics("images/EDA-boxplot.pdf")
```

The chart below shows several boxplots, one for each level of the class variable in the mpg data set. Each boxplot represents the distribution of hwy values for points with the given level of class. To make boxplots, use `geom_boxplot()`.

```{r fig.height = 3}
ggplot(data = mpg) +
  geom_boxplot(aes(x = class, y = hwy))
```

Covariation will appear as a systematic change in the medians or IQR's 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.

```{r fig.height = 3}
ggplot(data = mpg) +
  geom_boxplot(aes(x = reorder(class, hwy, FUN = median), y = hwy))
```

`geom_boxplot()` works best when the categorical variable is mapped to the x aesthetic. You can invert the axes with `coord_flip()`.

```{r}
ggplot(data = mpg) +
  geom_boxplot(aes(x = reorder(class, hwy, FUN = median), y = hwy)) +
  coord_flip()
```

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.

```{r}
ggplot(data = mpg) +
  geom_violin(aes(x = reorder(class, hwy, FUN = median), y = hwy)) +
  coord_flip()
```


### Vizualizing two continuous variables

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.

```{r}
ggplot(data = diamonds) +
  geom_point(aes(x = carat, y = price))
```

Scatterplots become less useful as the size of your data set 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()`.

```{r fig.show='hold', fig.width=3}
ggplot(data = diamonds) +
  geom_bin2d(aes(x = carat, y = price))

# install.packages("hexbin")
ggplot(data = diamonds) +
  geom_hex(aes(x = carat, y = price))
```

`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 data set is not big.


```{r}
ggplot(data = faithful, aes(x = eruptions, y = waiting)) +
  geom_point() +
  geom_density2d()
```

### Asking questions about covariation

When you explore plots of covariation, look for the following sources of insight:

* *Outliers* 

    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.
    
    ```{r echo = FALSE}
    ggplot(data = diamonds) +
      geom_point(aes(x = x, y = y)) +
      coord_cartesian(xlim = c(3, 12), ylim = c(3, 12))
    ```


* *Clusters*

    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.

    ```{r echo = FALSE, fig.height = 3}
    ggplot(data = iris, aes(y = Sepal.Length, x = Sepal.Width)) +
      geom_jitter() +
      geom_density2d(h= c(1,1)) +
      geom_rug(position = "jitter")
    ```

* *Patterns*

    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.
    
    ```{r echo = FALSE, message = FALSE, fig.height = 2}
    ggplot(faithful) + geom_point(aes(x = eruptions, y = waiting))
    ``` 

    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 of the second.

### Visualizing three or more variables

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.

```{r eval = FALSE}
library(plotly)
plot_ly(data = iris, x = Sepal.Length, y = Sepal.Width, z = Petal.Width, 
        color = Species, type = "scatter3d", mode = "markers")
```

```{r, echo = FALSE}
knitr::include_graphics("images/EDA-plotly.png")
```

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.

This doesn't mean that you should ignore complex interactions in your data. You can explore multivariate relationships in several ways. You can

* visualize each combination of variables in a multivariate relationship, two at a time

* use aesthetics and facetting to add additional variables to a 2D plot

* use a clustering algorithm to spot clusters in multivariate space

* use a modeling algorithm to spot patterns and outliers in multivariate space

## Clusters

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.

### Hierarchical clustering

Hierarchical clustering uses a simple algorithm to locate groups of points that are near each other in n-dimensional space:

1. Identify the two points that are closest to each other
2. Combine these points into a cluster
3. Treat the new cluster as a point
4. Repeat until all of the points are grouped into a single cluster

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 data set. 

```{r, echo = FALSE}
knitr::include_graphics("images/EDA-hclust.pdf")
```

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.

```{r}
small_iris <- sample_n(iris, 50)
  
iris_hclust <- small_iris %>% 
  select(Sepal.Length, Sepal.Width, Petal.Length, Petal.Width) %>% 
  dist() %>% 
  hclust(method = "complete")
```

Use `plot()` to visualize the results as a dendrogram. Each observation in the data set 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.

```{r fig.height = 4}
plot(iris_hclust, labels = small_iris$Species)
```

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 data set. To visualize the results, map the output of `cutree()` to an aesthetic.

```{r}
(clusters <- cutree(iris_hclust, 3))

ggplot(small_iris, aes(x = Sepal.Width, y = Sepal.Length)) +
  geom_point(aes(color = factor(clusters)))
```

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 effects the outcome of the algorithm.

```{r, echo = FALSE}
knitr::include_graphics("images/EDA-linkage.pdf")
```

* *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.


```{r fig.height = 4}
small_iris %>% 
  select(Sepal.Length, Sepal.Width, Petal.Length, Petal.Width) %>% 
  dist() %>% 
  hclust(method = "single") %>% 
  plot(labels = small_iris$Species)
```


### K means clustering

K means clustering provides a simulation based alternative to hierarchical clustering. It identifies the "best" way to group your data into a pre-defined number of clusters. The figure below visualizes (in two dimensional space) the k means algorith:

1. Randomly assign each data point to one of $k$ groups
2. Compute the centroid of each group
3. Reassign each point to the group whose centroid it is nearest to
4. Repeat steps 2 and 3 until group memberships cease to change

```{r, echo = FALSE}
knitr::include_graphics("images/EDA-kmeans.pdf")
```

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.

```{r}
iris_kmeans <- small_iris %>% 
  select(Sepal.Length, Sepal.Width, Petal.Length, Petal.Width) %>% 
  kmeans(centers = 3, nstart = 20, iter.max = 50)

iris_kmeans$cluster
```

Unlike `hclust()`, the k means algorithm does not porvide an intuitive visual interface. Instead, `kmeans()` returns a kmeans class object. Subset the object with `$cluster` to access a list of cluster assignments for your data set, e.g. `iris_kmeans$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.

```{r}
ggplot(small_iris, aes(x = Sepal.Width, y = Sepal.Length)) +
  geom_point(aes(color = factor(iris_kmeans$cluster)))

small_iris %>% 
  group_by(iris_kmeans$cluster) %>% 
  summarise(n_obs = n(), avg_width = mean(Sepal.Width), avg_length = mean(Sepal.Length))
```


### Asking questions about clustering

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?

* How stable are the clusters if you re-run the algorithm?

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. 

## Models

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.

```{r echo = FALSE}
diamonds2 <- filter(diamonds, x > 3, y > 3, y < 12)
diamond_mod <- lm(y ~ x, data = diamonds2)
resids <- augment(diamond_mod)
diamonds3 <- bind_rows(filter(resids, abs(.resid) > 0.5),  
                       sample_n(filter(resids, abs(.resid) <= 0.5), 1000)) %>% 
  select(x, y)

ggplot(diamonds3, aes(x = x, y = y)) +
  geom_point() +
  geom_smooth(method = lm, se = FALSE)
```

The model describes the relationship between x and y as 

$$\hat{y} = 0.13 + 0.98 x$$

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 dimenisonal plot. 

```{r echo = FALSE, fig.width = 3, fig.show='hold'}
diamond_mod <- lm(y ~ x, data = diamonds3)
resids <- augment(diamond_mod)

ggplot(resids) +
  geom_histogram(aes(x = .resid), binwidth = 0.1)

ggplot(resids) +
  geom_point(aes(x = x, y = .resid))
```

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.

## Exploring further

> Every data set contains more variables and observations than it displays. 

You now know how to explore the variables displayed in your data set, 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.

### To make new variables

Use dplyr's `mutate()` function to calculate new variables from your existing variables.

```{r}
diamonds %>% 
  mutate(volume = x * y * z) %>% 
  head()
```

The window functions from Chapter 3 are particularly useful for calculating new variables. To calculate a variable from two or more variables, use basic operators or the `map2()`, `map3()`, and `map_n()` functions from purrr. You will learn more about purrr in Chapter ?.

If you are statistically trained, you can use R to extract potential variables with more sophisticated algorithms. R provides `prcomp()` for Principle Components Analysis and `factanal()` for factor analysis. The psych and SEM packages also provide further tools for working with latent variables. 

### To make new observations

If your data set contains subgroups, you can derive from your data a new data set of observations that describe the subgroups. To do this, first use dplyr's `group_by()` function to group the data into subgroups. Then use dplyr's `summarise()` function to calculate group level statistics. The measures of location, rank and spread listed in Chapter 3 are particularly useful for describing subgroups.

```{r}
mpg %>% 
  group_by(class) %>% 
  summarise(n_obs = n(), avg_hwy = mean(hwy), sd_hwy = sd(hwy))
```

## A last word on variables, values, and observations

Variables, values, and observations provide a basis for Exploratory Data Analysis: _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 admidst 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.

## Exploratory Data Analysis and Data Science

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 data set, 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. 

```{r, echo = FALSE}
knitr::include_graphics("images/EDA-data-science-1.png")
```

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.

```{r, echo = FALSE}
knitr::include_graphics("images/EDA-data-science-2.png")
```

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.

```{r, echo = FALSE}
knitr::include_graphics("images/EDA-data-science-3.png")
```

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.

```{r, echo = FALSE}
knitr::include_graphics("images/EDA-data-science-4.png")
```

This model od data science forms a roadmap for the rest of the book. 

* Part 1 of the book covered the central tasks of the model above, Exploratory Data Analysis. 

* Part 2 will cover the logistical tasks of working with data in a computer language: importing and tidying the data, skills I call Data Wrangling. 

* Part 3 will teach you some of the most efficient ways to program in R with data. 

* Part 4 will discuss models and how to apply them. 

* Part 5 will teach you R Markdown, the most popular format for reporting and reproducing the results of an R analysis.

```{r, echo = FALSE}
knitr::include_graphics("images/EDA-data-science-5.png")
```