# How many variables to describe a system?

## Main Question or Discussion Point

I was arguing with a friend over the whole 'big data' trend and how people are looking to get answers to questions by amassing many samples over many variables and hoping to mash it together somehow. (Under the assumption not only of lots of correlations to search over but also that distributions over many simultaneous parameters are a requirement)

Not being a physicist I was wondering if there actually exist physical systems with equations which rely on lots of variables. Has anyone ever found anything like this? That is to say, found a joint system and set of descriptive equations such that the equations require at least, say, 10 independent variables? Or maybe some kind of proof that there will exist systems that cannot be described by less than a certain number of variables?

The counter-example he gave was of QTLs =. quantitative trait loci like height for example which are known to depend on lots of genes but no-one really knows how. But then dependency on lots of genes in an unknown manner might be funneled through some common mechanism such that the actual number of important variables is much smaller than expected.

Sorry if it's a bit philosophical...

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phinds
Gold Member
2019 Award
Well, the Standard Model is not an equation but it has more that 10 variables (about 20 actually) needed to describe everything (e.g. mass of an electron). They are not thought of as variables so much as constants, but they have arbitrary values that cannot be derived from anything.

Doug Huffman
Gold Member
Consider a system of one particle; intrinsic properties and extrinsic properties.

@phinds
But those would all be constants like you said, right? Not variables which can in our universe take on many different values?
I'm thinking more some function which takes various values depending on some set of input variables (which themselves can have different values).

A toy height model from before would, for example, be a set of many binary genes (allele 0 or 1 in the simple case) such that the total height of the person is the 'sum' of their values. But this is just a toy model, no-one really has a list of such genes and how they work together in any real world example that I know of so it could be something completely different...

@Doug Huffman
Same question I guess, by intrinsic/extrinsic do you mean constant values for all such particles in our universe or something like spin which has a few values? If the latter, how many of those are there in a simple one-particle system?

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Not being a physicist I was wondering if there actually exist physical systems with equations which rely on lots of variables.
Yes, definitely.
Has anyone ever found anything like this?
Absolutely.
That is to say, found a joint system and set of descriptive equations such that the equations require at least, say, 10 independent variables?
At the moment I can't think of a particular enlightening "big" equation, but it sure is easy to do one; e.g. the total mass of 10 unspecified atoms will depend on 10 independent variables, since different types of atoms have different mass.

EDIT: Or consider 10 resistors in series in an electronic circuit; the total resistance will depend on 10 independent variables; Rtotal = R1 + R2 + ... + R10.

EDIT 2: Even better, consider 10 potentiometers (= variable resistors), connected in series. The total resistance will depend on how you set these 10 independent potentiometers.

Or maybe some kind of proof that there will exist systems that cannot be described by less than a certain number of variables?
Proofs are for mathematics :). Physics relies on experimental evidence and models.

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@DennisN
I like the potentionmeter example, I think it's actually a good way to represent how people think of QTL traits, especially if the underlying model is additive with no epistasis (loosely, epistasis is a nonlinear relationship between genes).
Could your example be considered a limiting case when all the components (potentiometers) exist on the same axis (resistivity)? In this case it's possible to combine the potentiometers together into one 'meta-potentiometer' with a single resistance value, right? If so I'd like to rephrase the original question - can you think of a real-world, fully described system with 10+ orthogonal variables (e.g. each one takes values on a different (orthogonal) axis)?

In this case it's possible to combine the potentiometers together into one 'meta-potentiometer' with a single resistance value, right?
Not before the potentiometers are actually set (and the individual resistances are known) (you can not reduce the general equation to a more simple one.). Only after; when the potentiometers are set, you have a total resistance, but it still depends on ten variables, since you can change the potentiometers whenever and however you like.

If so I'd like to rephrase the original question - can you think of a real-world, fully described system with 10+ orthogonal variables (e.g. each one takes values on a different (orthogonal) axis)?
Yes, it is very easy :D (it is generally harder to reduce a description of a system to few variables than many). Consider 5 stereo radios (with 10 outputs), tuned to 5 different radio channels. Connect them to a mixer which mixes the signals into two outgoing stereo signals. Now, try to describe the output waveforms of these two signals. They will depend (at least) on the 10 different incoming signals and how the 10 potentiometers are set on the mixer.

And here was me thinking I'd figured something out lol :)
Thanks

Your radio example sounds suspiciously like our brain... 5 types of inputting classic sensory information (touch, taste, smell, sight, sound) which are received in distinct anatomical areas and eventually sort-of converge on common processing areas from which they output via defined neuronal pathways. Which is nowhere near being solved. Back to radios, if you were given a system like you described and could only control inputs and observe outputs do you think you'd have any hope of figuring stuff out?

Back to radios, if you were given a system like you described and could only control inputs and observe outputs do you think you'd have any hope of figuring stuff out?
It depends on what you mean by "stuff" :D. I have never done anything like that with radios in the real world, but it depends on what you mean by "control inputs". In my example I assumed I could not control what the radio stations were broadcasting, so with "inputs" do you mean the 10 potentiometers?

EDIT: What I think of is that I can figure some "stuff" out. But I won't be able to predict the waveforms, since I can't control the broadcasts.

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In the least invasive case, 'input' means original broadcast from the 'environmental channel', 'output' means final waveform produced by the mixer. More invasive cases allow expanding the measurements to include 'stuff in the middle', like, say, mixer input or potentiometer resistance. In either case the task is to infer system wiring, rules and create a generative/predictive model.

Funny that we should be talking about radios and biology, have you read this? Kinda funny/infamous in certain circles :)