Advice on Algorithm Generation?

Question

I am trying to determine the mathmatic specialty that takes simple input(x) sets and result(y) sets and tries to generate equations that replicate the behavior? For example:

Input Set Possabilities: 1,2,3,4,5,6,7,8,9,10.....(x)
Output set Possabilities: 4,17,123,213,217,318.... (y)

Any ideas on where I can go to learn more and what it is called??

Answer 1

♣ I see what u mean; u want to get knowledge out of computer’s womb; computers are designed to help us, not to substitute our brains!

Between input(x) and output(y) there must be a program composed by a human!

A womb will never give a birth to anything without our participation!



▬ OK! Then do it Timmy kid!

Answer 2

There are many, many ways to answer this question. There is no one technique or theory; it all depends on what kind of application you have in mind. I have a background in system theory; the problem of determining the system equations relating the input and the output is known as system identification. A simple model is the ARMA (autoregressive moving average) model, which models y[k] as



y[k] = a1 y[k-1] + a2 y[k-2] + ... + am y[k-m]

+ b0 x[k] + b1 x[k-1] + ... + bn x[k-n].



There are many techniques from differential equations and complex analysis for analysing and designing such systems.



More simply, you could look at the problem as a curve-fitting problem: find a curve, perhaps an nth degree polynomial, that passes through the given pairs (x1,y1), ..., (xn,yn). The Lagrange interpolation polynomial is a way of determining this curve. Other numerical techniques are the cubic spline and the least squares curve.



Depending on the application you have in mind, there are numerous other ways the input could be related to the output.

Answer 3

Try "Curve Fitting", "Least squares" and "Regression Analysis".



You can ALWAYS come up with a n degree polynomial that fits n+1 points (as long as no one input generates two different outputs). e.g. You can find a line (a linear/first degree polynomial) to fit two input->output pairs. You can always find a quadratic to fit three pairs. etc.