Reliability of a fitted curve? I would like to estimate the uncertainty or the reliability of a fitted curve. I intentionally don't name a precise mathematical quantity that I am looking for, since I don't know what it is.
Here $E$ (energy) is the dependent variable (response) and $V$ (volume) is the independent variable. I would like to find the Energy-Volume curve, $E(V)$, of some material. So I made some calculations with a quantum chemistry computer program to get the energy for some sample volumes (green circles in the plot). 
Then I fitted these data samples with the Birch–Murnaghan function:
$$
\mathbb{E}(E|V) = E_0 + \frac{9V_0B_0}{16}
\left\{
\left[\left(\frac{V_0}{V}\right)^\frac{2}{3}-1\right]^3B_0^\prime + 
\left[\left(\frac{V_0}{V}\right)^\frac{2}{3}-1\right]^2
\left[6-4\left(\frac{V_0}{V}\right)^\frac{2}{3}\right]\right\}\;,
$$
which depends on four parameters: $E_0, V_0, B_0, B_0'$. I also assume that this is the correct fitting function, so all errors just come from the noise of the samples. In what follows, the fitted function $(\hat{E})$ will be written as a function of $V$.
Here you can see the result (fitting with a least squares algorithm). The y-axis variable is $E$ and the x-axis variable is $V$. The blue line is the fit and the green circles are the sample points.

I now need some measure of the reliability (at best in dependence of the volume) of this fitted curve, $\hat{E}(V)$, because I need it to calculate further quantities like transition pressures or enthalpies. 
My intutition tells me that the fitted curve is most reliable in the middle, so I guess that the uncertainty (say uncertainty range) should increase near the end of the sample data, like in this sketch:

However, what it this kind of measure that I am looking for and how can I calculate it?
To be precise, there is actually only one error source here: The calculated samples are noisy due to computational limits. So if I would calculate a dense set of data samples they would form a bumpy curve.
My idea to find the desired uncertainty estimate is to calculate the following ''error'' based on the parameters as you learn it in school (propagation of uncertainty):
$$
\Delta E(V) = \sqrt{ \left(\frac{\partial E(V)}{\partial E_0} \Delta E_0\right)^2 + \left(\frac{\partial E(V)}{\partial V_0} \Delta V_0\right)^2 + \left(\frac{\partial E(V)}{\partial B_0} \Delta B_0\right)^2 + \left(\frac{\partial E(V)}{\partial B_0'} \Delta B_0'\right)^2}
$$
The $\Delta E_0, \Delta V_0, \Delta B_0$ and $\Delta B_0'$, are given by the fitting software.
Is that an acceptable approach or am I doing it wrong?
PS: I know that I could also just sum up the squares of the residuals between my data samples and the curve to get some kind of ''standard error'' but this is not volume dependent.
 A: This is an ordinary least squares problem!
Defining
$$x = V^{-2/3}, \ w = V_0^{1/3},$$
the model can be rewritten
$$\mathbb{E}(E|V) = \beta_0 + \beta_1 x + \beta_2 x^2 + \beta_3 x^3$$
where the coefficients $\beta=(\beta_i)^\prime$ are algebraically related to the original coefficients via
$$16 \beta = \pmatrix{16 E_0 + 54 B_0 w^3 - 9 B_0 B_0^\prime w^3\\
 -144 B_0 w^5 + 27 B_0 B_0^\prime w^5\\
 126 B_0 w^7 - 27 B_0 B_0^\prime w^7\\
 -36 B_0 w^9 + 9 B_0 B_0^\prime w^9}.$$
This is straightforward to solve algebraically or numerically: pick the solution for which $B_0, B_0^\prime$, and $w$ are positive.  The only reason to do this is to obtain estimates of $B_0, B_0^\prime, w$, and $E_0$ and to verify they are physically meaningful.  All analyses of the fit can be carried out in terms of $\beta$.
This approach is not only much simpler than nonlinear fitting, it is also more accurate: the variance-covariance matrix for $(E_0, B_0, B_0^\prime, V_0)$ returned by a nonlinear fit  is only a local quadratic approximation to the sampling distribution of these parameters, whereas (for normally distributed errors in measuring $E$, anyway) the OLS results are not approximations.
Confidence intervals, prediction intervals, etc. can be obtained in the usual way without needing to find these values: compute them in terms of the estimates $\hat\beta$ and their variance-covariance matrix.  (Even Excel could do this!)  Here is an example, followed by the (simple) R code that produced it.


#
# The data.
#
X <- data.frame(V=c(41, 43, 46, 48, 51, 53, 55.5, 58, 60, 62.5),
                E=c(-48.05, -48.5, -48.8, -49.03, -49.2, -49.3, -49.35, 
                    -49.34, -49.31, -49.27))
#
# OLS regression.
#
fit <- lm(E ~ I(V^(-2/3)) + I(V^(-4/3)) + I(V^(-6/3)), data=X)
summary(fit)
beta <- coef(fit)
#
# Prediction, including standard errors of prediction.
#
V0 <- seq(40, 65)
y <- predict(fit, se.fit=TRUE, newdata=data.frame(V=V0))
#
# Plot the data, the fit, and a three-SEP band.
#
plot(X$V, X$E, xlab="Volume", ylab="Energy", bty="n", xlim=c(40, 60))
polygon(c(V0, rev(V0)), c(y$fit + 3*y$se.fit, rev(y$fit - 3*y$se.fit)),
        border=NA, col="#f0f0f0")
curve(outer(x^(-2/3), 0:3, `^`) %*% beta, add=TRUE, col="Red", lwd=2)
points(X$V, X$E)


If you are interested in the joint distribution of the original parameter estimates, then it is easy to simulate from the OLS solution: simply generate multivariate Normal realizations of $\beta$ and convert those into the parameters. Here is a scatterplot matrix of 2000 such realizations.  The strong curvilinearity shows why the Delta method is likely to give poor results.

A: Cross-validation is a simple way to estimate reliability of your curve:
https://en.wikipedia.org/wiki/Cross-validation_(statistics)
Propagation of uncertainty with partial differentials is great is you really know $\Delta E_{0},\Delta V_{0},\Delta B_{0}$, and $\Delta B'$. However, the program you are using gives only fitting errors(?). Those will be too optimistic (unrealistically small).
You can calculate 1-fold validation error by leaving one of your points away from fitting and using the fitted curve to predict value of the point that was left away. Repeat this for all points so that each is left away once. Then, calculate validation error of your final curve (curve fitted with all points) as an average of prediction errors.
This will only tell you how sensitive your model is for any new data point. For example, it will not tell you how inaccurate your energy model is. However, this will be much more realistic error estimate mere fitting error.
Also, you can plot prediction errors as a function of volume if you want.
