Abstract
A good reference for the material in this section is Chap. 15 of Numerical Recipes [1]. Frequently we are given a set of data points \((x_i, y_i), i = 1, 2, \cdots , N\), with corresponding error bars, \(\sigma _i\), through which we would like to fit to a smooth function \(f(x)\). The function could be straight line (the simplest case), a higher order polynomial, or a more complicated function.
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Notes
- 1.
\(\chi ^2\) should be thought of as a single variable rather than the square of something called \(\chi \). This notation is standard.
- 2.
Although this result is only valid if the fitting model is linear in the parameters, it is usually taken to be a reasonable approximation for non-linear models as well.
- 3.
It is conventional to include the factor of \(1/2\).
- 4.
I find the use of the word “null” in the quote to be confusing. It is, however, common usage in the statistics literature. The brackets round it are mine.
References
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, 2nd edn. (Cambridge University Press, Cambridge, 1992)
C.M. Bishop, Pattern Recognition and Machine Learning (Springer, New York, 2006)
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Young, P. (2015). Fitting Data to a Model. In: Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid to Ask. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-19051-8_3
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