- 1 Welcome to A Business Analyst’s Introduction to Business Analytics
**I Introductory Material**- 2 Becoming a Data-Driven Business Analyst
- 3 The Computing Environment
- 4 R: Basic Usage
- 5 R Packages: causact,tidyverse, etc.
**II DATA: Manipulation & Visualization**- 6 dplyr: Manipulating Data Frames
- 7 dplyr: Data Manipulation For Insight
- 8 ggplot2: Data Visualization Using The Grammar of Graphics
- 9 ggplot2: The Four Stages of Visualization
**III DATA STORIES: Modelling The Real World**- 10 Representing Uncertainty
- 11 Joint Distributions Tell You Everything
- 12 Graphical Models Tell Joint Distribution Stories
- 13 Bayesian Inference On Graphical Models
- 14 Generative DAGs As Prior Joint Distributions
- 15 Install Tensorflow, greta, and causact
- 16 greta: Bayesian Updating And Probabilistic Statements About Posteriors
- 17 causact: Quick Inference With Generative DAGs
- 18 The beta Distribution
- 19 Parameter Estimation
- 20 Posterior Predictive Checks
- 21 Decision Making
- 22 A Simple Linear Model
- 23 Linear Predictors and Inverse Link Functions
- 24 Multi-Level Modelling
- 25 Compelling Decisions and Actions Under Uncertainty

A large portion of thoughts and concepts in this chapter are inspired by Noah Ilinsky. See his talk here: (https://youtu.be/R-oiKt7bUU8)

Visualization makes data accessible to the human brain. Evolution has wired our eyes and brain to be very sophisticated in pattern recognition. This includes detecting patterns and violations of patterns in regards to position, color, size, shape, gaps, trends, etc. For example, in Figure 8.1, your brain will easily detect seven pattern violations.

To take advantage of the facile nature with which our eyes digest visual information, we will learn to map data - like a column of a data frame - to properties of visual markings made on the screen or on paper. These properties, called *aesthetics*, include things like position, shape, color, and transparency. By intelligently mapping data to aesthetics, we can not only see our data, but lead ourselves and our audience to visual insight.

Data presented in tables or even in statistical summaries are rarely as forthcoming with insight as is a good visualization. Anscombe (1973Anscombe, F. J. 1973. “Graphs in Statistical Analysis.” *The American Statistician* 27 (1): 17–21.) constructed four fictitious datasets to illustrate this point - each dataset consisting of `x-y`

value pairs: {`(x1,y1),(x2,y2),(x3,y3),(x4,y4)`

}. The four datasets, known as Anscombe’s quartet, have virtually indiscernible statistical properties. However, the distinguishing characteristics of each dataset is very evident when graphed. Due to the cogency of the arguments made by Anscombe, these datasets have been built-in to R. We can see the data in tabulated form using the following lines:

```
# install.packages("dplyr", dependencies = TRUE)
library("dplyr")
## retrieve the anscombe dataset
ansDF = anscombe %>% as_tibble()
## notice the x-values for the first three datasets are
## the same and scanning the y-values yields little insight
ansDF
```

```
## # A tibble: 11 x 8
## x1 x2 x3 x4 y1 y2 y3 y4
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 10 10 10 8 8.04 9.14 7.46 6.58
## 2 8 8 8 8 6.95 8.14 6.77 5.76
## 3 13 13 13 8 7.58 8.74 12.7 7.71
## 4 9 9 9 8 8.81 8.77 7.11 8.84
## 5 11 11 11 8 8.33 9.26 7.81 8.47
## 6 14 14 14 8 9.96 8.1 8.84 7.04
## 7 6 6 6 8 7.24 6.13 6.08 5.25
## 8 4 4 4 19 4.26 3.1 5.39 12.5
## 9 12 12 12 8 10.8 9.13 8.15 5.56
## 10 7 7 7 8 4.82 7.26 6.42 7.91
## 11 5 5 5 8 5.68 4.74 5.73 6.89
```

Notice how the `x`

-values for each of the first three datasets, (i.e. `x1`

, `x2`

, and `x3`

) are the same, yet patterns in the corresponding `y`

-values (i.e. `y1`

, `y2`

, and `y3`

, respectively) are not easily discernible. This tabulated form of data yields little insight.

One might think that statistical transformations can be used to yield more insight. So let’s try this using a statistical transformation of the data that summarize’s the relationship between respective `x`

’s and `y`

’s using the equation of a line, i.e. a linear regression. The `lm`

function in R can be used to get linear regression output. For this book, basic linear regression knowledge is assumed - for an introduction or refresher on linear regression, please consult OpenIntro’s introductory statistics textbook: https://www.openintro.org/stat/textbook.php Notice (below) that the output for both the slope coefficient (\(\approx 0.5\)) and the y-intercept (\(\approx 3\)) is nearly identical for all four datasets and one might (wrongly) assume the datasets to be quite similar as a result:

```
model1 = lm(y1 ~ x1, data = ansDF) ##predict y1 using x1
model2 = lm(y2 ~ x2, data = ansDF) ##predict y2 using x2
model3 = lm(y3 ~ x3, data = ansDF) ##predict y3 using x3
model4 = lm(y4 ~ x4, data = ansDF) ##predict y4 using x4
##show results of regression (i.e. intercept and slope)
coef(model1)
```

```
## (Intercept) x1
## 3.0000909 0.5000909
```

```
## (Intercept) x2
## 3.000909 0.500000
```

```
## (Intercept) x3
## 3.0024545 0.4997273
```

```
## (Intercept) x4
## 3.0017273 0.4999091
```

Despite the statistical transformation (i.e. regression output) yielding nearly identical insights, visualizing the data, as shown in Figure 8.2, tells a much richer story; the type of story we want to tell as we use data visualization for both exploratory analysis and managerial persuasion.

Going forward, I will ask you to change the way you look at plots such as those shown above. Specifically, I request that you think of plots as representing a mapping of data to visual properties which you see on a screen or paper. To this end, take notice of the visual markings in each of the four plots. For each plot, one `X`

column and one `Y`

column is extracted - these are your *data*. For example,

```
## # A tibble: 11 x 2
## x2 y2
## <dbl> <dbl>
## 1 10 9.14
## 2 8 8.14
## 3 13 8.74
## 4 9 8.77
## 5 11 9.26
## 6 14 8.1
## 7 6 6.13
## 8 4 3.1
## 9 12 9.13
## 10 7 7.26
## 11 5 4.74
```

yields 11 rows (or observations) of \((x,y)\) pairs for the upper-right plot of Figure 8.2. For each observation, the \(x\)-value is mapped to horizontal position and the \(y\)-value is mapped to vertical position:
\[
\begin{aligned}
\textrm{x} &\rightarrow \textrm{horizontal postion}\\
\textrm{y} &\rightarrow \textrm{vertical postion}\\
\end{aligned}
\]
To display the mapped visual aesthetics, a *geom* or visual marker is used - in our example this *geom* was chosen to be points:
\[
\textrm{geom} = \textrm{points}
\]
An alternative *aesthetic* mapping and *geom* selection would be the following:

\[ \begin{aligned} x &\rightarrow \textrm{horizontal position}\\ y &\rightarrow \textrm{fill color}\\ \textrm{geom} &= \textrm{rectangular bar} \end{aligned} \]

While Figure 8.2 (upper right-hand plot) is far superior in revealing the curvilinear relationship between \(x2\) and \(y2\) than Figure 8.3, they both are representations of the exact same data. For example, notice the maximum \(y2\) value of Figure 8.2 is now represented by the lightest shading in Figure 8.3; both are viusal representation of the exact same data point \((x2,y2) = (11,9.26)\) with the first representation being the more usful of the two.

The key lessons to takeaway from this exploration of Anscombe’s quartet are the following:

- Tabulated data is a struggle to read and obscures discovery of patterns or relationships.
- Statistical transformations tell stories, but the true story of the underlying data might be missed if not visualized.
- Graphing data enables us to readily see patterns that tabulation or statistical tests fail to help us with.
- Mapping of data to aesthetic properties and visualization using different geoms impact the effectiveness of a visualization’s ability to reveal underlying patterns in the data.

In this section, we show how to specify the mapping of data to aesthetic properties and visual markings using the `ggplot2`

package (Wickham 2009Wickham, Hadley. 2009. *Ggplot2 Elegant Graphics for Data Analysis*. Springer-Verlag New York. http://ggplot2.org.) that is part of the `tidyverse`

package group (Wickham 2017Wickham, Hadley. 2017. *Tidyverse: Easily Install and Load the ’Tidyverse’*. https://CRAN.R-project.org/package=tidyverse.). The mapping is accomplished via a set of rules known as the *grammar of graphics*.

In language, rules of grammar are used to convey meaning when words are combined. For example, Figure 8.4 is similar to a meme circulating on Facebook that shows how English grammar, in this case spacing and the use of a hyphen, changes the meaning of words.

Through these rules, readers can correctly comprehend the meaning an author wishes to convey. Just like with words, graphics also have an underlying grammar which can be leveraged to accurately describe a graphic or visual. This grammar, formalized in the lengthy and terse work of Wilkinson (2006Wilkinson, Leland. 2006. *The Grammar of Graphics*. Springer Science & Business Media.) has thankfully been made much more accessible to R-users via Hadley Wickham’s excellent `ggplot2`

package. Once we learn to use this grammar properly, good graphics become easier to both describe and create.

English grammatical rules specify that a *complete sentence* satisfies three conditions:

- It begins with a capital letter.
- It includes an ending punctuation mark like a period(.) or question mark(?).
- It contains a main clause with a subject and verb.

Analogously, there are conditions required by the `ggplot2`

package’s implementation of the grammar of graphics to specify a *complete plot*:

- It begins with a dataset.
- It includes a geometric object, called a
*geom*, along with that*geom’s*minimal required set of*aesthetic mappings*which specify how data is to be transformed into a visual display.

We can use the `starwars`

dataset from the `dplyr`

package to illustrate these two conditions. The `starwars`

dataset is part of the `ggplot2`

package, it can be accessed like any other data frame:

```
## # A tibble: 87 x 14
## name height mass hair_color skin_color eye_color birth_year sex gender
## <chr> <int> <dbl> <chr> <chr> <chr> <dbl> <chr> <chr>
## 1 Luke~ 172 77 blond fair blue 19 male mascu~
## 2 C-3PO 167 75 <NA> gold yellow 112 none mascu~
## 3 R2-D2 96 32 <NA> white, bl~ red 33 none mascu~
## 4 Dart~ 202 136 none white yellow 41.9 male mascu~
## 5 Leia~ 150 49 brown light brown 19 fema~ femin~
## 6 Owen~ 178 120 brown, gr~ light blue 52 male mascu~
## 7 Beru~ 165 75 brown light blue 47 fema~ femin~
## 8 R5-D4 97 32 <NA> white, red red NA none mascu~
## 9 Bigg~ 183 84 black light brown 24 male mascu~
## 10 Obi-~ 182 77 auburn, w~ fair blue-gray 57 male mascu~
## # ... with 77 more rows, and 5 more variables: homeworld <chr>, species <chr>,
## # films <list>, vehicles <list>, starships <list>
```

To create a visual, we pass this dataframe as the argument value for the `data`

argument in the `ggplot`

function call:

```
# install.packages("ggplot2")
library("dplyr") ##load for starwars dataset
library("ggplot2") ##load for plotting
ggplot(data = starwars) + ## specify dataset. Use "+" to continue with more plot specifications
geom_point(mapping = aes(x = height, y = mass)) ## pick type of geom to display and map it graphical parameters
```

The initial `ggplot()`

function call initiates the creation of a plot using a given dataset. The `geom_point()`

function adds a layer of visual markings, i.e. point geoms, to the plot where:

- each point represents one row of the
`starwars`

dataframe, - the x-position of each point is determined by the value of the
`height`

variable, - the y-position of each point is determined by the value of the
`mass`

variable, and - intelligent defaults are set for every other decision required to make the corresponding plot.

See https://ggplot2.tidyverse.org/reference/#section-layer-geoms for complete list of available geometric objects.

Add another aesthetic mapping for a useful variation. To discover more aesthetics that can be controlled when using `geom_point()`

, use `?geom_point`

to open the `Help`

pane in the lower-right of RStudio. Scrolling down, you will discover the aesthetics shown in Figure 8.6 can be controlled when using this layer.

Figure 8.6: Aesthetics that can be controlled when using geom_point().

One way to control an aesthetic is to map it to the data by specifying the mapping within the `aes()`

function call.

The other way is to map the aesthetic to a constant outside of the `aes()`

function, but within the `geom`

function, like this:

```
ggplot(data = starwars) +
geom_point(mapping = aes(x = height, y = mass), shape = 15, color = "red")
```

where color and shape are specified outside of the `aes()`

function because they are mapped to constant values and not a column of the dataframe.

For more information on what shapes and colors are available, execute the following R function `vignette(“ggplot2-specs”)`

to open up details in the `Help`

pane of RStudio.`

Multiple layers of `geoms`

can be put on one plot. For example, we might want to name the points:

```
ggplot(data = starwars) +
geom_point(mapping = aes(x = height, y = mass)) +
geom_text(mapping = aes(x = height, y = mass, label = name), check_overlap = TRUE)
```

To make the function call more concise, we can avoid the redundancy of mapping `x`

-`y`

positions for every geom by specifying mapping defaults in the initial `ggplot()`

function. Any geom layers will use these defaults unless overridden with an `aes()`

call from within that geom. Additionally, we can omit the `data`

and `mapping`

argument names by specifying those argument values in the order that the function expects. This yields the same plot with a little less typing:

Using `check_overlap = TRUE`

omits data labels that would otherwise overwrite and obscure a previously drawn data label. Experiment leaving this argumnet out of the `geom_text()`

function to see the ugliness that happens when every label is printed.

We will see a few more geom’s throughout the course, most prominently featured will be `geom_col(), geom_density(), geom_histogram(), and geom_segment`

. Each of these has a minimum set of aesthetic mappings which must be specified in order to produce a plot:

`geom` |
Required Aesthetics | Notes |
---|---|---|

`geom_col()` |
`x,y` |
Map `x` to a discrete variable and `y` to a continuous variable |

`geom_density()` |
`x` |
Map `x` to a continuous variable |

`geom_histogram()` |
`x` |
Map `x` to a contiunuous variable and bin similar `x` values together |

`geom_linerange()` |
`x,y,yend` |
Map `x` to a discrete variable and `y,yend` to two related continuous variables |

To generate minimally working examples of these other plot types, the following subset of data from the built-in `mpg`

dataset will be used:

```
## create mpgDF data frame
mpgDF = mpg %>%
group_by(manufacturer) %>%
summarize(cityMPG = mean(cty),
hwyMPG = mean(hwy),
numCarModels = n_distinct(model),
) %>%
filter(numCarModels >= 2)
mpgDF ## view contents of data frame
```

```
## # A tibble: 9 x 4
## manufacturer cityMPG hwyMPG numCarModels
## <chr> <dbl> <dbl> <int>
## 1 audi 17.6 26.4 3
## 2 chevrolet 15 21.9 4
## 3 dodge 13.1 17.9 4
## 4 ford 14 19.4 4
## 5 hyundai 18.6 26.9 2
## 6 nissan 18.1 24.6 3
## 7 subaru 19.3 25.6 2
## 8 toyota 18.5 24.9 6
## 9 volkswagen 20.9 29.2 4
```

Recall from the `dplyr`

chapter that the chaining operator, `%>%`

makes the object to its left the first argument of the function to its right. Since the first argument to the ggplot function is assumed to be the `data`

argument (see https://ggplot2.tidyverse.org/reference/ggplot.html), `mpgDF %>% ggplot()`

passes the `mpgDF`

data frame as the `data`

argument value used in the `ggplot()`

function; `ggplot(mpgDF)`

and `ggplot(data = mpgDF)`

are other equivalent ways call the function.

See https://serialmentor.com/dataviz/histograms-density-plots.html for more information on both density plots and histograms.

Above, we learned that specifying a *complete plot* required a dataset, a geom, and a minimal set of mappings. Behind the scenes, other grammatical elements were chosen by default; in reality, you can make all of these other decisions explicit. Some of these other elements included a coordinate system, a statistical transformation, and scales:

**Coordinate Systems**A coordinate system (coord for short) determines how data coordinates are mapped to the plane of a graphic. The default coordinate system is a two-axis system, think x- and y-coordinates, called the cartesian coordinate system.

**Statistical Transformations:**A statistical transformation is a way of manipulating or transforming data prior to its display. It usually is used to summarize data in a meaningful way such as when creating a histogram of one variable or summarizing the relationship of two variables using a linear regression line. These transformations are optional, but can prove useful as shortcuts to get from data to useful visuals.**Scales:**Whereas aesthetic mappings relate data to attributes that you can visually perceive (e.g. color, symbol shapes, fill, etc.), scales dictate how the mapping from data to attribute is performed. For example, a scale might determine which colors are mapped to which values in the data.

We will learn more about these other elements on an as needed basis. For now, we recognize the grammar for what it is, namely a strong foundation for understanding and describing a wide range of graphics.