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Is there a mathematical symbol that describes relationships/dependencies?
February 5, 2010 4:44 PM   Subscribe

Is there a mathematical symbol to describe relationships or dependencies between components and results?

Let's say that a relationship - perhaps a tenuous one, but whatever - exists between an outcome and several (possibly independent, likely interdependent) components A, B, and C. That is, outcome O is dependent on A, B, and C (which may be dependent to some extent on each other, but I'm not sure that's significant for this question). How does one describe this mathematically?

Seems to me that you could simply write O = f(A,B,C) and be done with it under the assumption that the magic happens in f(), but is there a more precise way to describe the relationship between (A,B,C) and O?

I was considering using the ≝ symbol but I don't know whether that's the proper usage.

I have to get this right because I'll be presenting this relationship to a bunch of academics who will chew me up if they spot something wrong.

So - any advice? Should I skip trying to describe this as a mathematical model and use my words?
posted by aberrant to Science & Nature (28 answers total)
 
...and the symbol I pasted doesn't really show. It's the equal sign with the word "def" above it.
posted by aberrant at 4:45 PM on February 5, 2010


A ⇒ B is like saying "A implies B" or "B follows from A".

A ∵ B is like saying "A is true because B is true".

A ∴ B is like saying "Because A is true, therefore B is true".
posted by Blazecock Pileon at 4:50 PM on February 5, 2010


Following your example, you might have something like:

A ^ B ^ C ⇒ O
posted by Blazecock Pileon at 4:52 PM on February 5, 2010


Are all of these variables meant to represent boolean values? That is, do A, B, and C have to be "true" in order for O to be "true"? Or is it more complicated than that?

If they're all booleans, then you could say O => A ^ B ^ C, which means "if O is true, then A, B, and C are all true". If O isn't dependent on anything else besides A, B, and C, then you can additionally say O = A ^ B ^ C, or O <=> A ^ B ^ C, which means "O is true if and only if A, B, and C are".
posted by equalpants at 5:01 PM on February 5, 2010


The symbol you're looking for is "not independent." It looks like dependent, but it has a line through it. You can make the symbol in latex like this.

In your example, O not-independent A,B,C.

Your wording makes it sound that A,B,C are causes of the outcome O. The only way I know how to write that are:
- to sketch the "causal model" with arrows from A, B, and C pointing at O
- I've never seen this, but it makes sense to me: A,B,C independent do(O), and A,B,C not-independent O.
posted by esprit de l'escalier at 5:10 PM on February 5, 2010


I would vote for O=f(A,B,C). That expression would tell me "O is a function of A, B, and C" which means "the value of O depends on the values of A, B, and C". It seems to me this is as much information as you have.

If you give more information about the nature of the relationship maybe we can offer something better.
posted by PercussivePaul at 5:14 PM on February 5, 2010 [1 favorite]


By the way let me add that O=f(A,B,C) is a pretty standard form that will be grasped right away by a broad audience, which is what you want in a presentation. As an engineer this is the sort of thing you often see.
posted by PercussivePaul at 5:16 PM on February 5, 2010 [1 favorite]


Yes. The symbol ⇒ represented by latex's \implies represents implication within a formal system, normally A ⇒ B is equivalent to either A is false or B is true. The turnstile character ⊦ represented by \vdash in latex usually means a particular fact is provable from the antecedent assumptions. The double turnstile or models character ⊨ represented by \vDash in latex usually means the conclusion is either modeled by the antecedent or holds in all models satisfying the antecedent. etc.

You've then got all manor of non-monotonic logics, modal logics, etc. In particularly, a non-monotonic logic is one of the most frequent "tenuous" deductions used by computer scientists, say reasoning by default. You would not use the the implies character for representing non-monotonic assertions.

Yes is the simple answer. :)
posted by jeffburdges at 5:24 PM on February 5, 2010


Thanks to all! Some clarification: A,B, and C are NOT boolean. They're actually rather abstract concepts that might be quantifiable. To give an example (not the one I'm using, but it's similar and common enough):

The concept of (security) risk (R) is traditionally thought of as an interrelationship among threat (T), vulnerability (V), and impact (I) (or consequence). There possibly are other (minor) factors, but these are the three main components.

Consider a methodology to quantify risk by quantifying each of these components. Then, R = f(T,V,I) ... or does it? Is there another way to describe this relationship symbolically?
posted by aberrant at 5:25 PM on February 5, 2010


this is as much information as you have.

If O is an outcome of cause A, then you have some more information. For example, if the ground gets wet (O) when it rains (A), then you know two things:
- A and O are dependent (knowing one tells you something about the other)
- if someone wets the ground (say, with a sprinkler) then O and A are independent (knowing one tells you nothing about the other.)
posted by esprit de l'escalier at 5:25 PM on February 5, 2010


This is perhaps a bit too pedantic (I'm a mathematician), but to me O = f(A, B, C) would mean that if A, B, and C are known, then (at least in principle) one can determine O. This doesn't sound like the situation you have; it sounds like knowing A, B, and C gives you some information about O, but doesn't determine O uniquely.

So perhaps you should write something like O = f(A, B, C, ...) -- the implication being that O depends on A, B, C, and some other insignificant things you're not going to bother writing down.
posted by madcaptenor at 5:33 PM on February 5, 2010


It's a safe bet that your situation is covered by some sort of formal system. I'd imagine the turnstile character ⊦ is very likely the correct symbol here, as that denotes deduction within the rules of some system, not just the tautological formal implication. It's feels kinda asinine using turnstile if you can't even name the type of formal system of course, but hey. You might also use squiggly arrows or arrows with dots in the middle, those lack common formal meanings and more clearly convey the message to a general audience.

p.s. I'd only use O = f(A,B,C) if these have actual values, but you said they were assertions.
posted by jeffburdges at 5:35 PM on February 5, 2010


madcaptenor: I'm looking for pendantic answers - I'll be showing this to a bunch of physicists (and my background is not in physics, which puts me at instant disadvantage), and I'm sure they're just looking for a leg to chew. :)

Thanks for your suggestion.
posted by aberrant at 5:36 PM on February 5, 2010


jeffburdges: let's assume that A,B,C (or T,V,I) have actual values. That is, T is measured [0,5], V is [0,10), and I is measured in some dollar amount. Does that change things?
posted by aberrant at 5:38 PM on February 5, 2010


From your clarifying example, it definitely sounds like you're talking about a numerical relationship not a logical one. So O = f(A,B,C) sounds fine to me, since you're saying that in principle one could define your variables precisely and produce a formula to calculate O from A, B, and C, and that no other variables are needed. (If you're only approximating O you could use ≈) Your audience may be displeased that you're being too hand-wavey, but that has nothing to do with your notation.

I wouldn't use the ≝ symbol, because you aren't really saying how O is defined, just how it relates to some other known variables (namely, that it is completely determined by them). Really you're saying "There exists an f such that O = f(A,B,C)" or "O = f(A,B,C) for some f". O is presumably an existing concept with some meaning outside of your equation.

But I'm not a physicist so there may be some notational conventions in that community that I don't know about.
posted by serathen at 6:13 PM on February 5, 2010


I guess in the case of your specific example, since risk doesn't really have a precise natural definition, using ≝ would be reasonable. But even then a regular = ought to be okay, I think.
posted by serathen at 6:27 PM on February 5, 2010


I am a physicist, and after I read your question, the first thing that came to mind was the work of Bob Coecke and perhaps John Baez. Not really my field of expertise, but from what I gather they argue category theory provides a natural language for physics, and in particular the description of experiments. Coecke in particular makes extensive use of simple diagrams to describe physical processes, and I think his notation is probably the most suited to what you want to do, and it does have a rigorous foundation (although it is unlikely many physicists are familiar with it, unless they are at the Perimeter Institute).
posted by TheyCallItPeace at 7:20 PM on February 5, 2010


you could say:

The concept of (security) risk (R) is traditionally thought of as an interrelationship among threat (T), vulnerability (V), and impact (I) (or consequence). There possibly are other (minor) factors, but these are the three main components. To model this mathematically, we could introduce the random variables T, V, and I. Then define R to be a continuous, differentiable function of T, V, and I.

R := f(T,V,I).

The input data for the model is T, V, I, and the function f. Questions?

posted by water bear at 7:57 PM on February 5, 2010


Another mathematician, another vote for R = f(T,V,I).
posted by escabeche at 9:07 PM on February 5, 2010


let's assume that A,B,C (or T,V,I) have actual values. That is, T is measured [0,5], V is [0,10), and I is measured in some dollar amount. Does that change things?

I would go either with water bear's use of:

:=

to mean "defined as a function of", or with:

=

to mean "equal to".

Either would be clear.
posted by Blazecock Pileon at 11:06 PM on February 5, 2010


Building on water bear's elaborated example, I think it's probably important to specify whether "O = f(A,B,C)" is your actual assertion about the real world (that R is completely determined by A,B, and C) or if it's a simplified model you're talking about ("There possibly are other (minor) factors" which you're ignoring for now) or if you're defining a new thing, O, which you're now going to talk about and hopefully explain how it's related to some existing thing the audience is interested in.
posted by hattifattener at 12:14 AM on February 6, 2010


Consider a methodology to quantify risk by quantifying each of these components. Then, R = f(T,V,I) ... or does it? Is there another way to describe this relationship symbolically?

Well, I think that the relationship is multiplicative: if you double the probability of attack, or the probability of a vulnerability existing, or the cost of an attack, you double the expected-value of the damage (the risk).

So R is proportional to T*V*I. The proportionality sign suggests that there may also be other variables that you haven't explicitly mentioned.
posted by James Scott-Brown at 2:27 AM on February 6, 2010


I think going to an "intuitive" proportionality model like the one James S-C just presented is a recipe for total disaster before an audience of physicists. They will start demanding plots showing the relationship, asking where the error bars on your data points are and proposing that you try to fit to T^k_1 * V^k_2 * I^k_3 and show that the Chi-squared is minimal at k_i = 1.

I'd vote for not using mathematical notation if your understanding of the problem is not mathematical and you're not actually going to do any mathematical "processing" in your presentation.
posted by themel at 3:56 AM on February 6, 2010


"So - any advice? Should I skip trying to describe this as a mathematical model and use my words?"

Yes. If you can't explain the specific mathematical relationship in plain English, it's probably not yet ready to be rendered as formal notation.
posted by polymodus at 4:30 AM on February 6, 2010


Another thought: If you want to show them something, maybe a flowchart or diagram would be sufficiently expressive and without the technical baggage that comes with even the simplest-looking operator and relation symbols.

Above all, I think don't use notation to show off or be clever. You're talking to physicists, not mathematicians.
posted by polymodus at 4:37 AM on February 6, 2010


Are your objects—"outcomes", "components," and "results"—numbers, or other mathematical objects? Or are you using the ordinary real-life senses of those words?

I've never heard those words used together in a mathematical context, so I'm guessing they're not mathematical at all. If that's true, don't use fake math notation; use words or pictures. If your audience is as judgmental as you say, I doubt they would would look kindly on the use of mathematical notation for something it isn't meant to express.

But I could be misinterpreting your intention. Could you give an example of what kind of thing you'd use this notation to say?
posted by k. at 7:33 AM on February 6, 2010


aberrant: I think that you're best off avoiding mathematical notation.

If you feel the need to include some notation in your presentation I'd third-or-fourth-the-vote for the "R = f(T,V,I)", but I'd hedge my bets by using some kind of wavy equals sign. And, I'd introduce it in some slide that has bullet points like this:

- WHAT WE ARE INTERESTED IN: describe your O here, in words

- WHAT WE KNOW IMPACTS O: here you list

- WHAT WE KNOW DOESN'T IMPACT O: here you list any obvious other factors the audience might throw up.

- TO CONCLUDE: O appears to be impacted materially by A, B, and C (and not by the list of other possible impactors in the previous point). The exact nature of that relationship is still very unclear, (which is what this talk is about). (And, if you must, O ~ f(A,B,C), but we have no clue how to characterize f, hence the deliberately left-undefined squiggly equals).

I can't judge your audience but I think if you hit those points you won't need the gratuitous equation.

FWIW basically it sounds like what you're looking for is an equation involving so-called "random variables" (that's a precise term-of-art). They're a very useful tool but the kind of thing best avoided unless you understand them well, as they have lots of subtle gotchas (and, even if YOU know them inside and out, you'd be surprised how many working engineers and so on don't grasp them in their full nuance).
posted by hoople at 10:16 AM on February 6, 2010


≺ and ≼ ("precedes"; LaTeX \prec and \preceq) are often used for defining custom relations. So you would just say something like:

ab indicates the causal relation "b depends on a"

And then usually you'd go on to explain the properties of the relation. Is it transitive, well-founded, etc? There's also ≼.

(If your browser font is small, ≺ might look like <>
posted by qxntpqbbbqxl at 11:03 AM on February 6, 2010


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