ガシアンラインプロファイルの統計的に有意なレベルを計算する方法


1

一部の論文では、ガウスモデルを使用してラインプロファイルに適合しています。たとえば、this paper、イチジク1は、3つのガウス線プロファイル、O VII、O VIII、およびNe IXを描画します。しかし、データはノイズが多いようで、ガウスの高さ(ガウスラインの下部から上部、特にO VIII、およびNe IX)は、残差データの3 *(1シグマエラーバーの高さ)を超えていません。しかし、彼らは本文で(3ページの左下に)次のように述べています。「3本の線の重心波長は、それぞれ赤方偏移ゼロのO VII、O VIII、およびNe IXと一致しています。これらの線は、6.5で統計的に有意です。3.1、4.5シグマレベル。」だから私は重要性の計算アルゴリズムに戸惑っている。私の記憶では、sigma = sqrt(残差データの分散)とガウスの高さIntensity = exp(a *(x-b)/ c ^ 2)、有意性= a / sigma。次に、私の計算に問題がありますか?

2

It is difficult to say, there isn't a description in the paper as to how they arrive at those numbers. Given that the equivalent widths of the lines have signal to noise levels that mean they are only greater than zero at significance levels of 2-3, then it isn't clear these are meaningful numbers at all.

However, here is how it might have been done.

You fit a continuum model to the spectrum, get a minimum chi-squared $\chi^2_1$ (or maximum [log] likelihood value); you then fit a continuum plus absorption line model and get a new, lower value, of chi-squared $\chi^2_2$ (or higher value of max log likelihood)

You then get the difference in chi-squared $\Delta \chi^2 = \chi^2_1 - \chi^2_2$ (or form the likelihood ratio) of the two models, which should be distributed as chi-squared with the number of degrees of freedom being equal to the number of new parameters in the more advanced model (i.e. 3 for a line centre, width and depth). You then check a chi-squared table for the corresponding p-value and significance.

In other words, if $\Delta \chi^2 = 6.25$, that corresponds to p=0.1, or if $\Delta \chi^2 = 11.35$ for p=0.01 etc.

In your comments you say:

First i use powerlaw to fit the continue get reduced chi-square1 = 1.00277 / dof=179. Then i fit with powerlaw+gauss, the reduce chi-square2 = 0.88091 / dof=176. The Δχ2 = 24.455. Is that right?

Yes. A chisquared test for 3 d.o.f. yields p=0.00002012

To convert to a number of Gaussian sigma, it is the stuff of standard statistics that $$ n\sigma = \sqrt{2} {\rm erf}^{-1} (1 - p)$$ which in the case above suggests a significance of $4.26\sigma$.