Error propagation in least squares

In summary, the conversation discusses the calculation of error or uncertainty in the slope of a linear relationship obtained by plotting temperature measurements against another quantity. The speaker is looking for a method to factor in the uncertainty of their measurements, specifically with regards to software like IGOR Pro which calculates these values automatically. Suggestions are made for a Chi Squared Analysis and a book on probability and statistics for further guidance. The speaker also shares a link to a resource that suggests taking the average of the worst possible slopes as the error in slope, but asks if there are any other methods for estimating error.
  • #1
refrigerator
15
0
I am doing a calculation involving taking three or more temperature measurements and then plotting them against another quantity (dependent). I get a relationship that is pretty linear, so I take the line of best fit to obtain an equation with a slope and an intercept.

Now, my question is: how do you calculate error/uncertainty in the resulting slope? I have looked around the Internet and found ways to calculate error purely on the distribution of points, but I am rather looking for error caused by uncertainties in my measurements. For instance, if my temperature readings are good to 0.1K, how would that factor into the uncertainty of the slope? (I have previously used software like IGOR Pro that I think calculated those values for me, but I want to know how it is done.)

Should I, for example, take the worst cases (highest and lowest possible slopes based on measurement uncertainty) and take the difference as the error? Or is this a bit pessimistic? (A formula would be great, I could understand it from there.)

Thank you.
 
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  • #2
How do you know that you have the best fit? I recommend a Chi Sqaured Analysis. See "Probability and Statistics for Engineering and the Sciences" by Jay L. Devore. That should point you in the right direction.

Hope this helps.
 
  • #3
Hmm, I currently do not have access to any books, so an electronic resource would be preferable. But I will make sure to take a look at that book as soon as I get access to a library. Looks like it has some useful information.

I am looking for error propagation. I would like to determine a fit through the least squares technique, and then determine probable error in slope based on uncertainty in my readings. I have done further searching, and found this:

http://www.ghiweb.com/cap/Lab114115/App%20B%20-%20graphing/appB%20Slope.htm

Here, they take the average of the worst possible slopes as the error in slope. This is sort of similar to what I originally thought I might do. I suppose this is a reasonable definition, but before I move on does anyone know of any other ways of estimating error?

Thank you!
 
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1. What is error propagation in least squares?

Error propagation in least squares is a method used in data analysis to estimate the uncertainty or error in the result of a least squares analysis. It takes into account the uncertainties in the input data and calculates the propagated uncertainty in the final result.

2. How is error propagation calculated in least squares?

Error propagation in least squares is calculated using the partial derivatives of the least squares equation with respect to each variable. These derivatives are then multiplied by the corresponding uncertainties in the input data and squared. The sum of these squared values is then used to calculate the propagated uncertainty.

3. Why is error propagation important in least squares?

Error propagation is important in least squares because it gives a measure of the uncertainty in the final result. This allows for a more accurate interpretation of the data and helps in making informed decisions based on the results of the analysis.

4. Can error propagation be reduced in least squares?

Yes, error propagation can be reduced in least squares by increasing the number of data points or by improving the accuracy of the input data. It can also be reduced by using alternative methods, such as weighted least squares, that take into account the uncertainties in the input data.

5. Are there any limitations to error propagation in least squares?

One limitation of error propagation in least squares is that it assumes that the uncertainties in the input data are independent and normally distributed. If this assumption is not met, the propagated uncertainty may not accurately reflect the true uncertainty in the final result. Additionally, error propagation does not account for any systematic errors in the data, which may also affect the accuracy of the results.

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