3 ms·

You don't! These techniques are for getting a perfect fit.

unlikelymordant 4y ago
The polynomial will go through your samples exactly, but you might have arbitrarily high error between the samples ( runges phenomenon), if your data is noisy at all it is much better to fit a low order polynomial to your data that minimises error. If your curve has no noise and you are free to choose the sample locations , minimax is the best way to guarantee minimum error between samples.
- Someone 4y ago
  Indeed. Computing the exact polynomial going through all noisy points also doesn’t make sense because adding or removing a data point wildly changes the polynomial. So, you can’t claim any of your polynomials really describes your problem. Even if your data isn’t noisy and the point added lies exactly on the previous polynomial (making the problem degenerate) chances are float rounding and numerical stability of your code can have that effect (https://arachnoid.com/polysolve/#x_limits https://arachnoid.com/polysolve/#x_limits)
jll29 4y ago
The parent asked regarding overfitting, because fitting a polynomial to a finite set of given data points can be viewed as an instance of a machine learning problem (inducing a regressor or classifier). What is called "control point"in the OP would be called the "training data". Now whether polynomials are a good idea for any given problem is a matter of the nature of the data, but overfitting in this context means that the polynomial induced via the parameters A, B, C and D can generalize to further (unseen) data points or not. In the gaming context this is not a relevant question because, as you rightly say, the problem is "just to fit" what we have. But in scenarios where the polynomial is an intensional representation for a function specified extensionally, i.e. via as a set of finite data points, and where we gradually become aware of additional data points, we can use the polynomial to forecast further data points - the interpolated ones. Recent example paper: https://arxiv.org/pdf/1808.03216.pdf https://arxiv.org/pdf/1808.03216.pdf

The polynomial will go through your samples exactly, but you might have arbitrarily high error between the samples ( runges phenomenon), if your data is noisy at all it is much better to fit a low order polynomial to your data that minimises error. If your curve has no noise and you are free to choose the sample locations , minimax is the best way to guarantee minimum error between samples.

Someone 4y ago
Indeed. Computing the exact polynomial going through all noisy points also doesn’t make sense because adding or removing a data point wildly changes the polynomial. So, you can’t claim any of your polynomials really describes your problem. Even if your data isn’t noisy and the point added lies exactly on the previous polynomial (making the problem degenerate) chances are float rounding and numerical stability of your code can have that effect (https://arachnoid.com/polysolve/#x_limits https://arachnoid.com/polysolve/#x_limits)

jll29 4y ago

The parent asked regarding overfitting, because fitting a polynomial to a finite set of given data points can be viewed as an instance of a machine learning problem (inducing a regressor or classifier). What is called "control point"in the OP would be called the "training data". Now whether polynomials are a good idea for any given problem is a matter of the nature of the data, but overfitting in this context means that the polynomial induced via the parameters A, B, C and D can generalize to further (unseen) data points or not. In the gaming context this is not a relevant question because, as you rightly say, the problem is "just to fit" what we have. But in scenarios where the polynomial is an intensional representation for a function specified extensionally, i.e. via as a set of finite data points, and where we gradually become aware of additional data points, we can use the polynomial to forecast further data points - the interpolated ones. Recent example paper: https://arxiv.org/pdf/1808.03216.pdf https://arxiv.org/pdf/1808.03216.pdf