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What Does Multivariate Mean

How to Make a Multivariate Matrix and Why Everything Interesting is Multivariate by Nature
Brett Friedman
Multivariate adj. Having or involving a number of independent mathematical or statistical variables

A dry, but precise definition from Merriam-Webster defines multivariate as descriptive of anything with multiple independent variables. What things might that include? Anything. Everything. Most things at least. Concepts are like onions, they have layers; layers of global context and individual associations.

But "multivariate" is more than a common onion – it is a concept representative of our reality and a perspective representative of our modern epoch.

To understand what that means, you'll need to dive head first into the history of its usage. So let's go back one and a third centuries to 1889, before multi-variate was cool, when Francis Galton, Charles' Darwin's lesser known cousin, evolved the normal distribution from bi-variate to multi-variate.

For 80 years leading up to Galton's innovation, mathematicians as famed as Laplace and Gauss used uni-variate and bi-variate distributions for their many statistical needs. If you couldn't already guess, uni-variate describes single variable things and bi-variate describes things with two variables. But what I bet you didn't guess is what those things look like!

Is this a uni-variate or bi-variate matrix?

Surprise! That's a uni-variate matrix. You may have thought 'there are two images here, and bi describes things with two variables...' While that logic checks out, it goes wrong when you equate two images with two variables. There are in fact two images, but only one variables changes: the text. Thus, it is uni-variate.

To strengthen that point, take a look at this next image.

You could call this a bi-variate matrix. 2 variables interact. They go both ways... twice.
Way 1: Product 1 + Copy 1
Way 1: Product 2 + Copy 1
Way 2: Product 1 + Copy 2
Way 2: Product 2 + Copy 2

Up until 1889, that's as interesting as the world was. Simpler times called for simpler math. You could mix your cereal with milk, but try adding a glass, even half full, of orange juice and the pre-20th Century universe breaks down in an apocalyptic singularity.

That's when Galton said 'I love orange juice and no god will stop me from adding it to my breakfast!' And so just as he innovated the bi-variate distribution, he innovated the multi-variate distribution. It's been said, God was impressed. So impressed, he let Galton in on a little secret called Genetics.

Galton's protege Karl Pearson (the same Pearson on all your college textbooks), took those genetic secrets a step further and applied multi-variate maths to his 'Biometrics' inventing the Chi-Squared Test, Standard Deviation, as well as the Correlation and Regression Coefficients. More importantly, he passed down his brilliance to Sir Ronald Aylmer Fisher.

R.A. Fisher, as they called him, was the Beatles of 20th Century Statistics. He produced book after book of hits, from ANOVA to the Z-distribution, from the P-stat to the Meta-Analysis, he wrote and re-wrote it all.

Left to Right: Galton, Fisher, and Pearson. Dashing.

But before we fall head over heels for statistics, let's remember what's important here: Multi-variate reality. And what Fisher contributed to that reality was its formal usage in Anthropology, Botany, and Agriculture, "saving millions from starvation through rational crop breeding programs."

Yeah. Multivariate methodology just hits different.

Oh and if you needed more reason to hop off the bi-variate bandwagon and into the multi-variate Hover-Tesla, check out the extra infinity dimensions a multivariate matrix boasts.

So many variables!! Can I move closer, Doc?
Is that a secret message? Zoom and enhance.
I think we can go deeper Neo.
Ah, that's better.

The point: uni-variate methods test one variable, bi-variate methods test two variables, multi-variate methods test infinite variables.

Now Fisher didn't stop there, but we will for now. I want to move from what's basically ancient history (1930-1940) to more modern times...1958. That's when T.W. Anderson published and popularized his charming, mind-expanding "An Introduction to Multivariate Statistical Analysis". A textbook that took many young 60's hippies from broke to woke.

But enough history, let's talk today. After all, "Today is forever." And I still need to tell you why everything interesting is multi-variate by nature.

Today, multivariate methodology is ubiquitously applied in academia, increasingly exploited in business, and near completely ignored in everyday life. We've seen it academically throughout history; flash-forward 60 years and it's in every discipline under the moon as well as currently shifting the paradigm from uni-disciplinary studies to the burgeoning multi-disciplinary approach.

In business, 80's academic econometricians switched from PHDs to MBAs and moved into management, treating people like variables. It worked. Jump forward another 20 years and Mad Marketers start hand-making ads with dozens of colors, messages, and call to action variants for the shiny new Internet.

Moving into the new 20's, Airlines use multi-variate methods for pricing, financiers use it for risk models, and 'Data Scientists' a.k.a. statisticians, are some of the most highly demanded workers on the market in all indsutries.

https://www.bls.gov/emp/images/growing_occupations.png

So it's clear the collective conscious is onto something here. Avid academics and edgy businessmen are typical early adopters, but there's an untapped blue ocean waving hello. Tide goes in, tide goes out. Can't explain that.

Or can we?

Let's leave the laptop and jump into your life for a moment.

Just imagine you're sitting at home, perfectly relaxed, with a hot mug of your favorite drink. As you look closely, you get to thinking about this article, about how much multi-variate analysis makes sense for every facet of investigation, academic or otherwise, and that all multi-variate really means is 'involving multiple variables'.

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You think, 'technically, that mug is multi-variate. That handle can be red, it can be square, it can not exist at all. It's shape can be more cylindrical or octagonal, it's mouth covered, it's sides smooth.'

You see it morph a million times. Then you take your thought experiment a step further. You played with physical characteristics of the mug, but your perspective is a variable too. So you stand up, walk around the mug. Get a 360 degree view. The fact is, all those angles exist no matter which spot you stand in.

Love tea. It's multivariate too.

You can only ever observe one angle at a time; You need either data or people to provide alternative angles. Multivariate analysis is like looking at all angles at once, shifting a few variables and observing how the system changes. And ironically, this allows you to understand each single variable at a deeper level than if you studied it alone.

Now imagine treating your job, your relationships, and yourself like variables. Your possibilities, like your variables, become endless.

Just one more thing. A direct answer. Why is everything interesting multi-variate by nature? Because complexity is cool and any thing multi-variate is naturally complex.

Quick example. The Beatles wrote pop songs 60 years ago that still slap today. But there are numerous nameless pop artists from the past 20 years that no one remembers. The difference is their complexity. The Beatles were multi-variate. They had layers and layers of variable complexity: they treated their instruments, melodies, harmonies, time signatures, chord structures and more as variables and changed them all until they got something good.

Today's artists optimize for catchiness and immediate gratification using simple song structure. As a result, you get bored after 10-15 listens. There's no more to learn about those songs, no more to experience.

Pop Song Chord Structure Comparison. Left courtesy of Adam McCann. Right courtesy of Nailguitar.com.

That principle applies to nearly all aspects of life. The more you learn about any topic, the more you know you don't know, the more you want to know, and so on. Multi-variate analysis allows you to peel back many layers of complexity at once. To find complexity and dive into it. To learn more with less. The Beatles knew exactly which chord structures worked, which chords worked, which notes at what times.

That being said, it's hard to claim they intentionally used statistical methods to make music. But there you have it, they did it naturally.

Science, money, art. The most interesting things we've built for ourselves are all multi-variate by nature.

Now you know what multivariate means, you know how to visualize it, and you know why it's naturally part of everything interesting. This takes me to my final point. We are living in a multi-variate world. It's time to stop approaching things from a 1D or 2D 'Flatland' and start applying multi-variate perspective to our lives. Take it from Edward Tufte, the legendary data visualizer:

"Escaping this flatland is the essential task of envisioning information for all the interesting worlds (physical, biological, imaginary, human) that we seek to understand are inevitably and happily multivariate in nature."

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How to Run a Multivariate Test

The Beginner's Guide

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How to Run a Multivariate Test
The Beginner's Guide

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