I have many observations of measuring a physical parameter over time.

Each observation is basically a function p(t), t \in [t_1, t_2].

I also have the value \int_{t_1}^{t_2} q (t) dt = Q

I have many observations: \left\{ (p_1(t), Q_1), ( p_2(t), Q_2), \ldots (p_n(t), Q_n) \right\}.

I know that the connection p = f(q) is modeled very well by a quadratic polynomial.

I am after a parametric model for q = g(p).

The idea is top optimize the parameters of the model by linear least squares.

So id the model is q = g_{\boldsymbol{\theta}}(p) then I can integrate over the values by replacing each p value by q and minimize the difference with the value Q.

Nothing special here.

What I am after is a good parametric model to approximate the inverse of a quadratic function.

I can say that the range of values I am looking to work at is limited to q \in [0, 120], p \in [0, 900] (So only in positive quadrant) and I know the quadratic model is âsmilingâ.

Do you know a reasonable basis functions to approximate the inverse of a smiling quadratic function over the positive quadrant?

The usual suspects would be `sqrt`

or `log`

, but I am now aware of any basis built with them.