2011-04-09, 03:06 PM
Quality of data in many cases means only a first order approximation (linear fit, possibly log linear) can even be made.
It's not a complete model of everything, but the more you take into account, the more complicated your model gets.
And the problem there is: Your results only have a certain level of accuracy. The more variables you split that up into, the larger the uncertainty on each variable.
I'll give a simple example, that's not likely to come up, but demonstrates the principle.
You have a fit f(x) = kx.
As is fairly normal when you fit a line, you can calculate a range of values for k that fit the model well.
So you have [k-t, k+t] where you're 95% confident the true value of k lies.
If you do t/k, you can normalize the amount of error. If your value of k=100, and t=1, it's a fairly good fit (1% spread on either side). Conversely, if k=1 and t=1, it's a very loose fit (100% error possible easily).
You could, if you liked, include several constants:
f(x) = ix + jx
But looking at i and j independently, each still has a range of [i-t, i+t].
If i~=j, then you have a worse fit in every case compared to the original model - from i=j=50, a 2% spread on each side, from i=j=0.5, a 200% spread on each side.
Adding extra parameters to a model only works if you have proportionally more data available. You lose certainty every time you add variables.
When you look closely at most of nature's laws, you find yourself in a field of stochastic responses. Fluids are, ultimately, a huge number of individual particles bouncing off each other stochastically, and the best "correct" gas model would include all these collisions, and use them to predict what happens to the gas next. This approach has two problems:
1) hard to measure the position and trajectory of 10^20+ particles at once
2) hard to simulate 10^20 particles
The ideal gas law reduces 10^20 variables to approximately 10, and still gives "good" results. When you have a 19 orders of magnitude reduction in computational complexity, with a corresponding loss of accuracy that averages out to very little, you just have to accept that the simplified model is "good enough" in most cases.
If you want to find out what happens in a physical situation exactly, the perfect mathematical model is the world itself. Nothing else can ever be as completely accurate as the original.
It's not a complete model of everything, but the more you take into account, the more complicated your model gets.
And the problem there is: Your results only have a certain level of accuracy. The more variables you split that up into, the larger the uncertainty on each variable.
I'll give a simple example, that's not likely to come up, but demonstrates the principle.
You have a fit f(x) = kx.
As is fairly normal when you fit a line, you can calculate a range of values for k that fit the model well.
So you have [k-t, k+t] where you're 95% confident the true value of k lies.
If you do t/k, you can normalize the amount of error. If your value of k=100, and t=1, it's a fairly good fit (1% spread on either side). Conversely, if k=1 and t=1, it's a very loose fit (100% error possible easily).
You could, if you liked, include several constants:
f(x) = ix + jx
But looking at i and j independently, each still has a range of [i-t, i+t].
If i~=j, then you have a worse fit in every case compared to the original model - from i=j=50, a 2% spread on each side, from i=j=0.5, a 200% spread on each side.
Adding extra parameters to a model only works if you have proportionally more data available. You lose certainty every time you add variables.
When you look closely at most of nature's laws, you find yourself in a field of stochastic responses. Fluids are, ultimately, a huge number of individual particles bouncing off each other stochastically, and the best "correct" gas model would include all these collisions, and use them to predict what happens to the gas next. This approach has two problems:
1) hard to measure the position and trajectory of 10^20+ particles at once
2) hard to simulate 10^20 particles
The ideal gas law reduces 10^20 variables to approximately 10, and still gives "good" results. When you have a 19 orders of magnitude reduction in computational complexity, with a corresponding loss of accuracy that averages out to very little, you just have to accept that the simplified model is "good enough" in most cases.
If you want to find out what happens in a physical situation exactly, the perfect mathematical model is the world itself. Nothing else can ever be as completely accurate as the original.