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Correction to: Non-invasive assessment of plasma parameters inside an ion thruster combining optical emission spectroscopy and principal component analysis

The Original Article was published on 05 August 2021

Correction to: EPJ Techn Instrum 8, 13 (2021). https://doi.org/10.1140/epjti/s40485-021-00070-x.

In the publication of the article [1], the principle component analysis section contains mixed up indices. The corrected part of the affected section is shown below.

Corrected

First, the average spectrum of the entire reference data set \(\overline{S_{\mathrm{PCA}}}(\lambda)\) is subtracted from each spectrum \(S(\lambda)\). To derive the new coordinates, i.e., the PCA-axes, the covariances \(\sigma_{ij}\) of each of the n wavelength positions with every other wavelength position (including itself) are calculated using

$$ \sigma_{ij} = \frac{1}{m}\cdot\sum _{k=0}^{m}(x_{ki}-\overline{x_{i}}) \cdot(x_{kj}-\overline{x_{j}}) \quad \text{where } \overline{x_{i}}=\overline{x_{j}}=0. $$
(1)

Here, \(x_{ki}\) and \(x_{kj}\) are the values of wavelength positions i and j of the spectrum k. Since the average intensities of each wavelength position \(\overline{x_{i}}\) and \(\overline{x_{j}}\) were already subtracted before, they are now zero.

With these covariances the covariance matrix C is set up

$$ \mathbf{C}= \begin{pmatrix} \sigma_{00} & \sigma_{01} & \cdots & \sigma_{0n}\\ \sigma_{10} & \sigma_{11} & \cdots & \sigma_{1n}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{n0} & \sigma_{n1} & \cdots & \sigma_{nn} \end{pmatrix}. $$
(2)

All the changes that were requested are implemented in this correction and the original article [1] has been corrected.

References

  1. Nauschütt BT, Chen L, Holste K, Klar PJ. Non-invasive assessment of plasma parameters inside an ion thruster combining optical emission spectroscopy and principal component analysis. EPJ Tech Instrum. 2021;8(1). https://doi.org/10.1140/epjti/s40485-021-00070-x

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Correspondence to Benny T. Nauschütt.

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Nauschütt, B.T., Chen, L., Holste, K. et al. Correction to: Non-invasive assessment of plasma parameters inside an ion thruster combining optical emission spectroscopy and principal component analysis. EPJ Techn Instrum 8, 18 (2021). https://doi.org/10.1140/epjti/s40485-021-00075-6

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  • DOI: https://doi.org/10.1140/epjti/s40485-021-00075-6