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Erratum to “Multivariate statistical analysis of atom probe tomography data” [Ultramicroscopy 110 (2010) 1362]
Ultramicroscopy 111 (2011) 1522
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Erratum
Erratum to ‘‘Multivariate statistical analysis of atom probe tomography data’’ [Ultramicroscopy 110 (2010) 1362] C.M. Parish n, M.K. Miller Oak Ridge National Laboratory, Oak Ridge, TN 37831-6064, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 February 2011 Accepted 12 February 2011 Available online 26 February 2011
A small error in the mathematics described in the previous paper (C.M. Parish and M.K. Miller: Multivariate statistical analysis of atom probe tomography data, Ultramicroscopy 110(11) (2010) 1362–1373) has been found. Although the results and conclusions are completely unchanged, it is important to correct the error. & 2011 Elsevier B.V. All rights reserved.
A forthcoming paper by Keenan, Smentkowski, Ulfig, Oltman, Larson, and Kelly pointed out an error in Ref. [1] Equation (7), where we described the multivariate statistical analysis (MVSA) ^ of a spectrum image dataset D as: model D ^ ¼ AST ¼ TPT , with T a A, P aS, D
ð7Þ
where A and S are score images and loading spectra from an arbitrary factor model, and T and P are the score images and loading spectra after re-orthogonalization using the ‘‘fPCA’’ procedure of Keenan [2]. (Superscript-T denotes a matrix transpose.) We then described an orthogonal factor rotation from the basis set TPT to a different basis set T~ P~ T with the same fit to the data but a different representation described by Eq. (8) [3,4] ^ ¼ TPT ¼ ðTRÞ R1 PT ¼ T~ P~ T D ð8Þ where R is a non-singular orthogonal rotation matrix such that R 1 ¼RT. As intended, T~ would be orthonormal and P~ nonorthogonal [3]. The fPCA procedure yielding Eq. (7) produces an orthonormal P and orthogonal T, but reference to [3,4] requires orthonormal T and orthogonal P. Accidentally omitted from our described experimental procedure was a step between Eqs. (7) and (8) where T and P were rescaled (leaving the model ^ unchanged). Due to an error, the resulting T and P were both D orthogonal, and neither orthonormal, so the rotations applied to yield Figs. 4 and 7 did not yield orthonormal T~ as intended. To accurately describe our procedure, a step between Eqs. (7) and (8) should be inserted, where each column i of T, Ti, was divided element-by-element by a scalar ki, and each column i of P, Pi, was multiplied element-by-element by ki. In this case, ki was equal to the sum of the squares of the elements of Ti. Eq. (8) was applied to
n
DOI of original article: 10.1016/j.ultramic.2010.07.006 Corresponding author. Tel.: +1 865 574 0092. E-mail address: [email protected] (C.M. Parish).
0304-3991/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2011.02.003
this orthogonal, but not properly orthonormalized, dataset, yielding Figs. 4 and 6 in the original paper. As a result, the rotations described as orthogonal in [1] were in not orthogonal, but rather ^ ¼ T~ P~ T , oblique. The MVSA reconstructions of the original data, D differ only by computational rounding error from the intended orthogonal rotations, and the conclusions of [1] are in no way affected. It can also be pointed out that the intended orthogonal rotations can be achieved by rescaling Eq. (7) for orthonormal T for the application of Eq. (8). The differences between the actual and intended procedure are minor (Online supplemental figs. S1– S3), with only the non-metallurgically interesting APT artifact components being noticeably affected.
Appendix A. Supplementary materials Supplementary materials associated with this article can be found in the online version at doi:10.1016/j.ultramic.2011.02.003.
References [1] C.M. Parish, M.K. Miller, Multivariate statistical analysis of atom probe tomography data, Ultramicroscopy 110 (11) (2010) 1362–1373. [2] M.R. Keenan, Multivariate analysis of spectral images composed of count data, in: H.F. Grahn, P. Geladi (Eds.), Techniques and Applications of Hyperspectral Image Analysis, John Wiley & Sons, Chichester, 2007, pp. 89–126. [3] M.R. Keenan, Exploiting spatial-domain simplicity in spectral image analysis, Surface and Interface Analysis 41 (2009) 79–87. [4] V.S. Smentkowski, S.G. Ostrowski, M.R. Keenan, A comparison of multivariate statistical analysis protocols for ToF-SIMS spectral images, Surface and Interface Analysis 41 (2009) 88–96.
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Erratum
Erratum to ‘‘Multivariate statistical analysis of atom probe tomography data’’ [Ultramicroscopy 110 (2010) 1362] C.M. Parish n, M.K. Miller Oak Ridge National Laboratory, Oak Ridge, TN 37831-6064, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 February 2011 Accepted 12 February 2011 Available online 26 February 2011
A small error in the mathematics described in the previous paper (C.M. Parish and M.K. Miller: Multivariate statistical analysis of atom probe tomography data, Ultramicroscopy 110(11) (2010) 1362–1373) has been found. Although the results and conclusions are completely unchanged, it is important to correct the error. & 2011 Elsevier B.V. All rights reserved.
A forthcoming paper by Keenan, Smentkowski, Ulfig, Oltman, Larson, and Kelly pointed out an error in Ref. [1] Equation (7), where we described the multivariate statistical analysis (MVSA) ^ of a spectrum image dataset D as: model D ^ ¼ AST ¼ TPT , with T a A, P aS, D
ð7Þ
where A and S are score images and loading spectra from an arbitrary factor model, and T and P are the score images and loading spectra after re-orthogonalization using the ‘‘fPCA’’ procedure of Keenan [2]. (Superscript-T denotes a matrix transpose.) We then described an orthogonal factor rotation from the basis set TPT to a different basis set T~ P~ T with the same fit to the data but a different representation described by Eq. (8) [3,4] ^ ¼ TPT ¼ ðTRÞ R1 PT ¼ T~ P~ T D ð8Þ where R is a non-singular orthogonal rotation matrix such that R 1 ¼RT. As intended, T~ would be orthonormal and P~ nonorthogonal [3]. The fPCA procedure yielding Eq. (7) produces an orthonormal P and orthogonal T, but reference to [3,4] requires orthonormal T and orthogonal P. Accidentally omitted from our described experimental procedure was a step between Eqs. (7) and (8) where T and P were rescaled (leaving the model ^ unchanged). Due to an error, the resulting T and P were both D orthogonal, and neither orthonormal, so the rotations applied to yield Figs. 4 and 7 did not yield orthonormal T~ as intended. To accurately describe our procedure, a step between Eqs. (7) and (8) should be inserted, where each column i of T, Ti, was divided element-by-element by a scalar ki, and each column i of P, Pi, was multiplied element-by-element by ki. In this case, ki was equal to the sum of the squares of the elements of Ti. Eq. (8) was applied to
n
DOI of original article: 10.1016/j.ultramic.2010.07.006 Corresponding author. Tel.: +1 865 574 0092. E-mail address: [email protected] (C.M. Parish).
0304-3991/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2011.02.003
this orthogonal, but not properly orthonormalized, dataset, yielding Figs. 4 and 6 in the original paper. As a result, the rotations described as orthogonal in [1] were in not orthogonal, but rather ^ ¼ T~ P~ T , oblique. The MVSA reconstructions of the original data, D differ only by computational rounding error from the intended orthogonal rotations, and the conclusions of [1] are in no way affected. It can also be pointed out that the intended orthogonal rotations can be achieved by rescaling Eq. (7) for orthonormal T for the application of Eq. (8). The differences between the actual and intended procedure are minor (Online supplemental figs. S1– S3), with only the non-metallurgically interesting APT artifact components being noticeably affected.
Appendix A. Supplementary materials Supplementary materials associated with this article can be found in the online version at doi:10.1016/j.ultramic.2011.02.003.
References [1] C.M. Parish, M.K. Miller, Multivariate statistical analysis of atom probe tomography data, Ultramicroscopy 110 (11) (2010) 1362–1373. [2] M.R. Keenan, Multivariate analysis of spectral images composed of count data, in: H.F. Grahn, P. Geladi (Eds.), Techniques and Applications of Hyperspectral Image Analysis, John Wiley & Sons, Chichester, 2007, pp. 89–126. [3] M.R. Keenan, Exploiting spatial-domain simplicity in spectral image analysis, Surface and Interface Analysis 41 (2009) 79–87. [4] V.S. Smentkowski, S.G. Ostrowski, M.R. Keenan, A comparison of multivariate statistical analysis protocols for ToF-SIMS spectral images, Surface and Interface Analysis 41 (2009) 88–96.