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Topic: How to define the complexity of a curve? (Read 1198 times) |
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cuckoo
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How to define the complexity of a curve?
« on: Apr 8th, 2009, 4:59am » |
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How to define the complexity of a curve, a function or a sequence?
For example, the sequence generated by function f(x)=x is much less complex than a randomly generated sequence.
And function f(x)=e^x*log(x+1) seems contains more uncertainty than f(x)=x.
So how to quantify a function's 'entropy', or 'uncertainty', or 'irregularty', or 'information complexity' (and so on)?
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towr
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Re: How to define the complexity of a curve?
« Reply #1 on: Apr 8th, 2009, 5:39am » |
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on Apr 8th, 2009, 4:59am, cuckoo wrote:| And function f(x)=e^x*log(x+1) seems contains more uncertainty than f(x)=x. |
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It doesn't seem to contain any uncertainty from what I can see. It's as deterministic as hell.
Quote:| So how to quantify a function's 'entropy', or 'uncertainty', or 'irregularty', or 'information complexity' (and so on)? |
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How do they even apply to functions?
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cuckoo
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Re: How to define the complexity of a curve?
« Reply #2 on: Apr 8th, 2009, 6:05am » |
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Kolmogrov complexity, defined as length of the shortest program to compute a function, is a theoretic measure defines what I calls "complexity, uncertainly, ireggularty, entropy".
But I want a practical, computable way to measure a function's complexity. 
on Apr 8th, 2009, 5:39am, towr wrote:
It doesn't seem to contain any uncertainty from what I can see. It's as deterministic as hell.
How do they even apply to functions? |
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| « Last Edit: Apr 8th, 2009, 6:06am by cuckoo » |
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towr
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Re: How to define the complexity of a curve?
« Reply #3 on: Apr 8th, 2009, 6:52am » |
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on Apr 8th, 2009, 6:05am, cuckoo wrote:Kolmogrov complexity, defined as length of the shortest program to compute a function, is a theoretic measure defines what I calls "complexity, uncertainly, ireggularty, entropy".
But I want a practical, computable way to measure a function's complexity. |
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Well, how about picking a basic set of operands, and then use the length of the shortest expression using those operands that computes the function. You could even give them a weight.
Of course a simple recurrence sequence like
an = 2 an-12-1
is chaotic. Which is not born out by it's Kolmogorov complexity (or rather any approximation thereof).
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