How to measure the fitting of a data curve to a model? Norm $\text{||model-data||}$ is the better way? I have a set of datas and a model to fit within the data.
The model is based on a couple of ODE's of form
$$\frac{dx_1}{dt}=f_1(x_1,x_2,t)\\
\frac{dx_2}{dt}=f_2(x_1,x_2,t)$$
and I have the data of $x_2$ to fit within the curve $(x_2,t)$ generated by model. Usually, I've calculated the vector norm $\text{||model-data||}$. However, it seems to me that this kind of evaluation cannot capture a lot of the behavior of datas, e.g. little "squeezeds".
I'd like to receive some good conseiuls about how to measure the fitting of a data curve to a model. In other words, how I say that some model is "better" than another?
Thank you so much.
 A: You can use different norms to obtain more information about the quality of your models.
$$error = data-model$$
$L_1$-norm provides information about the mean absolute error.
$$\|error\|_{1} = \sum\limits_{i=1}^n \, |error_i| = n\,\times mean(abs(error))$$
$L_\infty$-norm provides information about the error at the worst point.
$$\|error\|_{\infty} = \max\limits_{i} \, |error_i|$$
In a general interpretation for $L_p$ norms ($1\leq p <\infty$): when $p$ is small, it will measure if most of the data points are near the model's curve, the norm won't be sensible to small groups of points, even if their error values are high; when $p$ is large, it will measure if there are points with high error values, the norm won't be sensible to points near the model's curve.
You can compute a progression of $L_p$-norms to evaluate if the model is appropriate to your application.

Moreover, the mean of the $error$ vector (without taking the absolute value) can show you the bias of your data, if it is positive, most of the data points are over the curve, if it is negative, most of the data points are under the curve. We would expect a $0$-valued mean.
The standard deviation of the $error$ vector provides information about the variation on the $error$ vector. it can be measured through the $L_2$-norm, when the mean error is zero.

You can compute all these for the whole data and for subsets, for example, to identify if the behavior is different at the boundaries.
In short, there are many things you can do to evaluate if your fitting is good. However, I can not say what is the best procedure, it is highly dependent on your problem, you must choose based on its properties.
