Compare Functions: Which is Better?

[klinmy]

Hi, I would like know if it is possible to tell which function is better over another?

For instance, I have two functions:

(i) f(x,y)=exp(a*x) * exp(b*y)
(ii) f(x,y)=a(exp(x)) * b(exp(y))

I would like to know the characteristics of both the functions which make it better/less better over another. (e.g.: the effect of a and b towards the results; the complexity or etc.)
I've tried plotting the functions in 3D graphs, but didn't help much.. Please guide.

Thanks in advance!

 

[HallsofIvy]

I have no idea what you mean by "better". I can only think that it would depend upon what you wanted to use the function for. And since there are many different applications for functions, there cannot be one single answer to this question.

 

[klinmy]

I have no idea what you mean by "better". I can only think that it would depend upon what you wanted to use the function for. And since there are many different applications for functions, there cannot be one single answer to this question.

Thanks. I just wonder if it is possible to tell the differences between two functions when both of them do actually serve the same purpose. In this case, both functions are used to find the weights between nodes (graph), and x and y are two different properties to consider. So by looking at the functions, can we say that, e.g: output of function (ii) is easier to be predicted when a and b are tuned; function (i) is more complex than function (ii) and etc.

I hope this makes sense..
Thanks.

 

[Stephen Tashi]

As I interpret the question, it is this:

Suppose f(x) and g(x) are two different functions. Suppose we have a set of data points (x,y) and f and g agree with each other on those data points. (i.e. y = f(x) = g(x for all i). Is there a way to say which of f and g is the simpler representation of the data?

That is a sensible but imprecise question since, as HallsOfIvy indicates, any answer would depend on what you mean by "simpler". There is no universally accepted meaning for one function being simpler than another. However, there various specific definitions of "simpler" have been invented and there are specific results about how to compare functions once you establish which definition you are using. Thinking in terms of fitting equations to data, some people define "f is simpler than g" to mean that fitting f to g involves using the data to find fewer unknown constants.

 

[klinmy]

Thank you so much for rephrasing my question Stephen. It sounds so much clearer and more professional.

Sorry that I just noticed the mistake I made in function(ii), where the * is actually a +.

(i) f(x,y)=exp(a*x) * exp(b*y)
(ii) g(x,y)=a(exp(x)) + b(exp(y))

I tried running both the functions, and found that both generated the same shape of graph when the same a and b were used. So does this mean that both functions are actually the same?

Thanks.

 

[klinmy]

correction: similar shape with different interval between i.