Seeing Your Data
Antaeus relies almost exclusively on scatter plots to visualize data. Here is the reason for this.
The most fundamental concept in mathematics above the level of basic set theory is that of a function. All of mathematics is built up using functions. A function is a set of rules that establishes a correspondence between one set of numbers (X) and another set of numbers (Y).
Consider this simple data table with three variables (named X, Y and Z) and 4 records:
| X | Y | Z |
| 4 | 4 | 7 |
| 6 | 5 | 1 |
| 7 | 8 | 1 |
| 10 | 5 | 3 |
This table specifies three sets of numbers:
X = {4, 6, 7, 10}, Y = {4, 5, 8, 5}, and Z = {7, 1, 1, 3}
It also defines six different functions:
Y(X), Y(Z), X(Y), X(Z), Z(X) and Z(Y).
Because this data table contains four records, each of these functions consists of four rules. For example, the function Y(X) consists of the four rules:
4
4, 6
5, 7
8 and 10
5
And the function Z(Y) consists of the four rules:
4
7, 5
1, 8
1 and 5
3
Each of these functions is visually represented by a scatter plot. For instance, Y(X) can be visualized by the following scatter plot.
Any pair of columns of numeric values in a data table represents two functions, one mapping the first set of numbers into the second and one mapping the second set of numbers into the first. Corresponding to these functions are two scatter plots with the "X" and "Y" axes interchanged. In Antaeus, the "Switch" button in each of the virtual scatter plot matrix diagrams toggles this state.
Every possible scatter plot is a mathematical function that can be described by a finite number of rules, each of which establishes a mapping between two numbers. Conversely, every possible mathematical function is a scatter plot.
You may not be used to thinking of a function as a set of one-to-one mapping rules as illustrated above. The simpler definition of a function as a formula�a single rule in the form of a mathematical equation�is the one most often used. These are the functions of physics and engineering. They completely define the relationship between X and Y. The "scatter plot" of such functions would coincide with the plot of their equations, a smooth curve.
But this simpler definition is inadequate because, in reality, it's likely not possible to create a formula that describes the emergent properties of actual phenomena. Outside of physics, formula-functions are almost always approximating abstractions—or reifications.
If data cannot be reduced to an equation, a human brain, one with expert knowledge of the phenomena represented by the data, can try to discern the relationship between X and Y by the visual appearance of the function—the scatter plot—assuming that "where there's smoke, there's fire". The other alternative is to stay within the realm of mathematics by reducing the data to statistics and analyzing these abstractions rather than the actual data. The conservative choice might be to do both.
The sets of numbers, X and Y, that define a function (scatter plot) are visualized individually in Antaeus as quantile plots. These show the precise distribution of single set of numbers. A histogram is an approximation to a quantile plot that summarizes the distribution of the set of numbers. A sunflower plot is an approximation to a scatter plot that summarizes the distribution of the set of data points (X, Y).
Seeing the data means seeing the scatter plots and quantile plots of the data. No other method allows for the visual assessment of the mathematical function represented by the mapping of one set of numbers into another.