4D Visualization — Invoking Insight through 4+ Variable Visual Analytics

4D Function f(r,s,t) = r² * s² * t⁵ Mapped in Mathematica 6

My intrigue in modeling 4 variables using a 3 vector XYZ graph, using a single vector sliding scale controlling a 4th variable (Just like in the video shared above!), began while I prepared to enroll in my Immersive Data Science course. I was inspired while reading up on Python, and Calculus, (a class I flunked out of in my final semester of High School, deciding that the incredibly short-sighted premise that “you won’t always have a graphing calculator in your pocket to do this!!” as a rationale to force students to calculate tedious functions, no longer appealed to me.) specifically paying attention to partial derivatives and related plotting techniques, such as least squares formulas that use 3D modeling to guess the best fit function when we don’t have all the data we need to make a conclusion.

The often colorful, geometrically complex plots that map 3 variables in a 3D matrix I encountered while studying derivatives, were what inspired me. They were often functions of one another, or could be modeled as functions of one another to arrive at a line of best fit that would represent a function that closely mapped the correlational relationship of data. This got me thinking how one could probably model additional functions in these graphs through Javascript, or an existing GUI module such as we’ve seen in the above video, to take a function of x and y such as

f(x,y)=y∗x²

https://mathinsight.org/nondifferentiable_discontinuous_partial_derivatives

a step further into

f(x, y, z) = sqrt(z) * y *x²

The above function can then be mapped like in the video I shared, where we see one of the variables modified in the function moving all other variables across it’s axis through time following either a function or dataframe, at any specific moment providing a 3D slice of this 4D function.

I find this an appropriate place to mention the prevailing viewpoint in physics that time is the 4th dimension in the space-time continuum we experience. Visualization through time is quite likely the most obvious application of a 4D visualization — we already do plenty of 2D time lapse visualizations, such as the change in global median ages gif shown below —

https://www.vox.com/2016/5/20/11715620/global-aging

This same principle could be applied especially to exploratory analyses of data. Alongside machine learning algorithms, functions can be derived to find relationships of interest between variables that could lead to statistically significant results. Could analyses of multiple variables in a visual interface like this help data scientists find relationships between variables that would otherwise be overlooked? What about finding multi-correlational relationships, leading to more tightly fit functions that find causal relationships between multiple variables, where there was once only looser correlational relationships with plenty of unexplained variance??

I pose these questions at this early point in my data science path, not knowing the answers. I’m only now covering basic principles of data visualization on plain old 2D XY matrixes, hoping my queries lead me to the answers I seek, and towards more intriguing questions that might not have already been asked a couple hundred times by other curious data students in this exponentially growing field.

A cursory glance can show me that some of the experts already have answers for me. They are pointing to the reality that, often, less is more, and visually stunning designs and graphs might be eye catching and impressive-looking, but not necessarily the best way to convey specific, statistically significant findings to a broad audience. The mundane aspects of pruning and cleaning data for ease of analysis means often times data scientists are oriented around showcasing less info rather than more, especially when demonstrating findings. “Dimension Reduction” is the practice of finding meaningful variables through exploratory analysis, and paying close attention where it’s most likely one will find variables that show meaningful relationships with other variables. In defense of my view that intaking a ton of 4D visualizations may net novel interesting insights, dimension reduction means one can cram EVEN MORE meaningful data into a 4D visualization, with one dimension in a visualization representing a tightly, causally related set of variables that can be plotted for comparison with other sets of variables!

I explore these concepts here with You, because I want You to tell Us what direction to go from here. What could complex high density visualization do in our quest for meaningful insight? Could heat mapping within a 4D projection make a meaningful 5D visualization??

I am always eager to exchange thoughts. Share yours with mine:

Eric Likhtiger

Elikhtig@gmail.com

Lightsourceuniverse.org

http://curvetube.com/WebGL_Graph_Visualization/qHkjSxbnzAU.video

Operating at the intersection of Technology, Ecology, and Arts

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