Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Mosaic plots for data visualization
Introduction
This notebook gives descriptions and examples of using the function MosaicPlot of the paclet .
The function MosaicPlot summarizes the conditional probabilities of co-occurrence of the categorical values in a list of records of the same length or a datasets. The list of records is assumed to be a full array and the columns to represent categorical values. (Note, that if a column is numerical, but has a small number of different values, then it can be seen as categorical.)
I have read the descriptions of mosaic plots in the book “R in Action” by Robert Kabakoff, [2], and one of the references provided in the book ( by Steve Simon, [3]). I was impressed how informative mosaic plots are, and I figured they can be relatively easily implemented using (also known as “Tries”) [4,5].
Remark: The implementation of the paclet is based on the paclet .
I implemented MosaicPlot while working on a document analyzing the census income data from 1998, [6]. This is the reason that dataset is used in this document. A good alternative dataset provided by WL is ExampleData[{“Statistics”,”USCars1993”}].
Remark: The original version of this document the WL / Mathematica package provided by the project , see [1].
Load the paclet
In[1]:=
Needs["AntonAntonov`MosaicPlot`"]
Load the dataset
In[1]:=
dsCensusData=ResourceFunction["ImportCSVToDataset"]["~/Downloads/adult/dfAdult.csv"]
Data set
The description of the data set is given in the file “adult.names” of the data folder. The data folder provides two sets with the same type of data “adult.data” and “adult.test”; the former is used for training, the latter for testing.
The total number of records in the file “adult.data” is ; the total number of records in the file “adult.test” is .
32561
16281
Here is how the data looks like:
In[15]:=
Magnify[RandomSample[dsCensusData,12],0.6]
Out[15]=
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Since I did not understand the meaning of the column “fnlwgt” I dropped it from the data:
In[17]:=
dsCensusData=dsCensusData[All,KeyDrop[#,"fnlwgt"]&];
Here is the summary table of the data:
In[18]:=
ResourceFunction["RecordsSummary"][dsCensusData[]]
Out[18]=
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1 age | ||||||||||||
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2 workclass | ||||||||||||||
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3 education | ||||||||||||||
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4 education-num | ||||||||||||
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5 marital-status | ||||||||||||||
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6 occupation | ||||||||||||||
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7 relationship | ||||||||||||
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8 race | ||||||||||
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9 sex | ||||
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10 capital-gain | ||||||||||||
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11 capital-loss | ||||||||||||
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12 hours-per-week | ||||||||||||
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13 native-country | ||||||||||||||
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14 income | ||||
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On the summary table the numerical variables are described with min, max, and quartiles. The category variables are described with the tallies of their values. The tallies of values are ordered in decreasing order. The tallies of truncated values are summed under the value “(Other)”.
Note that:
-- only% of the labels are “>50K”;
-- of the records are for males;
-- “capital-gain” and “capital-loss” are very skewed.
-- only
24
--
2/3
-- “capital-gain” and “capital-loss” are very skewed.
Explanations
If we pick a categorical variable, say “sex”, we can visualize the frequencies of the appearance of the variable values with the following plot:
In[20]:=
 | MosaicPlot |
Out[20]=
The size of the rectangles depends on the frequencies of appearance of the values “Male” and “Female” in the data records. From the rectangle sizes we can see what we already knew from the data summary table: approximately of the records are about males.
2/3
We can subdivide every rectangle according to the frequencies of co-occurrence of ’s value with the values of a second categorical variable, say “relationship”:
r
r
In[22]:=
| MosaicPlot |
Out[22]=
The labels corresponding to the values of “relationship” are rotated for legibility. The "relationship" labels are placed according to the co-occurrence with the value "Male" of the variable "sex". The correspondent fractions of the pairs ("Female","Husband"), ("Female","Not-in-family"), etc., are deduced from the order of the "relationship" labels.
Using colored mosaic plots can help distinguishing which rectangles correspond to which values. Here is the last plot with rectangles colored across the "relationship" data variable:
In[26]:=
| MosaicPlot |
Out[26]=
From the visual representations of the “sex vs. relationship” mosaic plot we can see that large fraction of the males are husbands, none (or a very small fraction) of them are wives. We can also see that none (or a very small fraction) of the females are husbands, the largest fraction of them are “Not-in-family”, and the “Not-in-family” females are approximately three times more than the females that are wives.
Let us make another mosaic plot for a different kind of relationship, “sex vs. education”:
By comparing the sizes of the rectangles corresponding to the values “Bachelors”, “Doctorate”, “Masters”, and “Some-college” on the “sex vs. education” mosaic plot we can see that the fraction of men that have finished college is larger than the fraction of women that have finished college.
We can further subdivide the rectangles according the co-occurrence frequencies with a third categorical variable. We are going to choose that third variable to be “income”, the values of which can be seen as outcomes or consequents of the values of the first two variables of the mosaic plot.
(The exact numbers of these observations can be seen tooltip table shown when hovering with the mouse over the rectangles.)
Instead of having the consequent (or outcome) variable to be the last variable in the mosaic plot, it is also useful to start with the consequent variable to get a perspective of how the attributes breakdown for it. Here is an example of a mosaic plot for “income vs. relationship vs. sex” (using a different color scheme):
It might be useful to make a mosaic plot for a subset of the records. Here is an example of a mosaic plot with splitting across four columns made only for people who have bachelor, master, or doctorate degrees:
Similar to the previous mosaic plot is this plot of “sex vs. education vs. marital-status vs. income”:
Options
MosaicPlot takes the following options:
In addition, MosaicPlot takes all the options of Graphics. (Because MosaicPlot is implemented with Graphics.)
The options are explained in the sub-sections below.
Visualizing categorical columns + a numerical column (“ExpandLastColumn”)
Visualizing categorical columns + a numerical column (“ExpandLastColumn”)
If the last data column is numerical then MosaicPlot can use it as pre-computed contingency statistics. This functionality is specified with the option “ExpandLastColumn”True.
In order to explain the functionality we are going to use following interpretation. If the last of column of the data is numerical then we can treat the data as a contracted version of a longer list of records made only of the categorical columns. For example, consider the following table with observations of people’s hair and eyes color:
The table above can be considered as a contracted version of this table:
Setting the option “ExpandLastColumn” to True gives a mosaic plot corresponding to that latter, observations-expanded table:
The last data column (which is numerical) does not need to be made of integers:
Controlling the size of the gap between the rectangles (“Gap” and “GapFactor”)
Controlling the size of the gap between the rectangles (“Gap” and “GapFactor”)
Contingency values labels (“LabelRotation” and “LabelStyle”)
Contingency values labels (“LabelRotation” and “LabelStyle”)
The labels derived from the distinct values (levels) of each column of the data can be rotated and given style options.
The option “LabelRotation” takes directional specification for Text (the fourth argument of Text). The option “LabelStyle” takes options and arguments for the function Style.
Labels for categorical variables (“ColumnNames” and “ColumnNamesOffset”)
Labels for categorical variables (“ColumnNames” and “ColumnNamesOffset”)
The names of the data columns (data’s variables) are specified with the option “ColumnNames”. (The list of names given to “ColumnNames” can be formatted with Style.) The distance of the column names from the rectangles is specified with the option “ColumnNamesOffset”.
Start of the rectangle splitting (“FirstAxis”)
Start of the rectangle splitting (“FirstAxis”)
The starting axis of the data splitting is specified by “FirstAxis”.
Tooltips with exact contingency statistics (“Tooltips”)
Tooltips with exact contingency statistics (“Tooltips”)
MosaicPlot has an interactive feature using Tooltip that gives a table with the exact co-occurrence (contingency) values when hovering with the mouse over the rectangles. The option “Tooltips” takes the values True or False.
Visualizing non-existing contingencies (“ZeroProbability”)
Visualizing non-existing contingencies (“ZeroProbability”)
The non-existing contingencies have to be represented in the mosaic plot. MosaicPlot uses very thin rectangles for them and the size of these rectangles is controlled with the option “ZeroProbability”.
Coloring of the rectangles (ColorRules)
Coloring of the rectangles (ColorRules)
The rectangles can be colored using the option ColorRules which specifies how the colors of the rectangles are determined from the indices of the data columns.
If coloring for only one column index is specified the value of ColorRules can be of the form
The colors are used with Blend in order to color the rectangles according to the order of the unique values of the specified data columns.
The grid of plots below shows mosaic plots of the same data with different values for the option ColorRules (given as plot labels).
The default value for ColorRules is Automatic. When Automatic is given to ColorRules, MosaicPlot finds the data column with the largest number of unique values and colors them according to their order using ColorData[7,”ColorList”].
References