So, i have a dataset which behaviour could be approached by the power law:
$$ N(x)= a - bx^{-1/c} $$
So i am doing this to get the regression:
nlm = NonlinearModelFit[data, a - b*x^(-1/c), {{a, 0}, {b, 0}, {c,2}}, x, MaxIterations -> \[Infinity]]
So i get the following result:
Then i get these standard deviations for a, b, c:
Now i am using the gaussian law for error propagation to calculate the standard deviation of N(x):
\[Sqrt]((D[a - b*x^(-1/c), a]*11426.108699236138`)^2 + (D[a - b*x^(-1/c), b]*1.434690881054167`*^8)^2 + (D[a - b*x^(-1/c),c]*0.03626373539623972`)^2)
a = 3.2763042912759576`*^6;
b = 1.4018942844796562`*^9;
c = 2.320026692762461`;
Then i am getting the resulting function:
When i plot the resulting function sigma N(x), i am getting this graph:
So the error term is declines for x going to infinity.
Now my goal is to get the standard deviation for:
$$ x \rightarrow \infty $$
So i have done this:
So i am not that experienced in error calculations, and i am not sure if my action was right. I would be thankful if somebody could have a look on it.
NonlinearModelFit
you are fitting $N(x)=a−b x^{−1/c} + \epsilon$ where $\epsilon \sim N(0,\sigma^2)$. In other words, how you include the residual error is critical. Do the observed residuals support such an error structure? Are you looking for an estimate of the limit as $x\rightarrow \infty$ of $SE(\hat{a}-\hat{b} x^{-1/\hat{c}})$ (standard error of the mean prediction) or $SE(\hat{a}-\hat{b} x^{-1/\hat{c}}+\epsilon)$ (standard error of a single prediction)? $\endgroup$MeanPredictionBands
andSinglePredictionBands
. From those equations you can extract the associated standard errors. $\endgroup$