# Calculating error in coefficients determined from fitting a curve to data?

In summary, the conversation discusses the process of calculating the error or uncertainty associated with each coefficient in a data set. The data is modeled by a function and the coefficients have been determined, but the task now is to determine the error for each coefficient. One approach is to use a formula for the coefficients and calculate the error using the partial derivatives and residuals of the data.
I have a set of data points (x0, y0), (x1, y1), ... (xi, yi)

With each yi there is an associated error ei.

The data is modeled by the function:

$$y = a\exp(-bln^2(c/x))$$

I have determined values for the coefficients a, b, c and I know the residuals produced from the values of the coefficients I've calculated and the set of data. What I'm trying to do now is to calculate the error, or uncertainty if you prefer, associated with each coefficient. How do I go about this?

A rough approach could be this one:

if you calculated the coefficients a, b, and c starting from your data, it means you have at your disposal a formula for theese coefficients. For example

$a = a (x_1,\dots,x_n,y_1,\dots,y_n)$

Then you could write

$\Delta a = \sum_{k=1}^n\frac{\partial a}{\partial y_k}\epsilon_k$

## 1. What is the purpose of calculating error in coefficients when fitting a curve to data?

The purpose of calculating error in coefficients is to determine the accuracy and reliability of the fitted curve. It helps to quantify the uncertainty in the calculated coefficients and provides a measure of how well the curve fits the data.

## 2. How is error in coefficients calculated?

Error in coefficients can be calculated using various methods such as least squares, maximum likelihood, or Bayesian inference. These methods use statistical techniques to estimate the uncertainty in the fitted coefficients.

## 3. Can error in coefficients be negative?

No, error in coefficients cannot be negative. It represents the uncertainty or variation in the calculated coefficient and therefore, it is always a positive value.

## 4. What is the significance of error bars in a curve fitting plot?

Error bars in a curve fitting plot represent the range of values within which the true value of the coefficient is likely to fall. They provide a visual representation of the uncertainty in the fitted coefficients.

## 5. How does the number of data points affect the error in coefficients?

The number of data points used to fit a curve has a direct impact on the error in coefficients. As the number of data points increases, the error in coefficients decreases, indicating a more accurate and reliable fit.

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