DE102018216637A1 - Factor analysis method for the spectrometric analysis of a sample - Google Patents

Abstract

The invention relates to a method for factor analysis, in particular for real-time factor analysis, in the spectrometric examination of a sample, comprising: forming a data matrix (D) from a plurality of recorded spectra, and factoring the data matrix (D ) into a coefficient matrix (W) and into a base matrix (H) by non-negative matrix factorization, the base matrix (H) containing a number of base spectra (h _k ) corresponding to the recorded ones Spectra are assigned.

Description

Background of the invention

The invention relates to a method for factor analysis, in particular for real-time factor analysis, in the spectrometric examination of a sample. In factor analysis or in a multivariate statistical analysis, a data matrix, which may contain, for example, spectroscopic data or data records, is factored into a product of two matrices.

In the US 8,266,197 B1 A method for factor analysis or for multivariate statistical analysis of gas chromatography (GC) / mass spectroscopy (MS) data sets is described. In the method, a mass-to-charge spectrum of a sample, which is described by a data matrix D, is recorded. The data matrix D is factored by a principal component analysis (PCA) into a product of a matrix of orthogonal vectors and a matrix of orthonormal vectors. The two matrices are rotated with an orthogonal rotation matrix R to form two rotated matrices. The PCA solution is refined by applying an Alternating Least Squares (ALS) algorithm to the rotated matrix of orthogonal vectors by non-negative matrix factorization (NMF) determine.

In the US Pat. No. 7,725,517 B1 A method for spectral image analysis utilizing spatial simplicity is described. Here, a data matrix is factored into a "Spectral Shape" matrix and into a concentration matrix, which describes the spatial distribution of the pure components at each location of the sample. When performing a so-called "Multivariate Curve Resolution-Alternating Least Squares" (MCR-ALS) algorithm, "Zero Abundance" elements are selected in the concentration matrix to take into account that the analyzed samples are spatially "simple" in that respect in that they consist of comparatively discrete chemical phases.

In the article "Regularized Alternating Least Squares Algorithms for Non-Negative Matrix / Tensor Factorization", Springer, LNCS-4493, 793-802 (2007), by A. Cichocki & R. Zdunek, modified regularized ALS algorithms are proposed for the NMF , There, it is proposed to use a more general cost function using a weighting matrix instead of the cost function typically used in a NMF algorithm based on the Frobenius norm.

Object of the invention

The object of the invention is to provide a method for factor analysis, which allows a reduction in the amount of data recorded in the spectrometric examination data and which can be carried out in particular in real time.

Subject of the invention

This object is achieved by a method of the aforementioned type, comprising: forming a data matrix from a plurality of recorded spectra, and factoring the data matrix into a coefficient matrix and into a base matrix by non-negative matrix Factorization, where the base matrix contains a number of basis spectra associated with the recorded spectra. The number of base spectra of the basic matrix is typically significantly lower than the majority or the number of recorded spectra. For the purposes of this application, non-negative matrix factorization is also understood to mean more complex types of non-negative factorization, in particular non-negative tensor factorization.

In the method, non-negative matrix factorization is used to factorize the data matrix containing the acquired spectra, typically in the form of column vectors, into a coefficient matrix and a base matrix, the latter containing the Contains basic spectra, which are also referred to as hidden sources of the recorded spectra. In the non-negative matrix factorization, it is exploited that the majority of the recorded spectra are usually an overlay, more precisely a linear combination, of a lower number of base spectra obtained by means of non-negative matrix factorization reconstruct. The recorded spectra or data sets, which represent a recorded spectrum, have values that are always greater than zero, which is why they are well suited for non-negative matrix factorization, since this requires a non-negative data matrix. For non-negative matrix factorization, it is not necessary to know the base spectra, but rather they are generated during factorization without the need for additional assumptions about the base spectra. For non-negative matrix factorization, however, it is necessary to specify the number of base spectra.

The spectra or the corresponding data sets which form the columns of the data matrix are typically recorded at successive times, for example by one To monitor the process in real time and to detect a possible change in the composition of the sample. The sample can be a solid, but also a liquid or a gas. During the spectrometric examination of the sample, the ratio of the constituents of the sample can thus change, with the result that the spectra recorded at different times and / or different sample positions differ from one another. This may also be the case if only a predetermined number of constituents are present in the sample, each of which is assigned to a basic spectrum or a basic vector. However, the plurality of recorded spectra forms a linear combination of the base spectra, the linear coefficients being given by the coefficient matrix.

Also in mass spectrometry, for example in a residual gas analyzer having an ion trap for the spectrometric analysis of a gas, a different number of particles in the ion trap is at those spectra that are recorded at different times or during pulsed operation at different pulses detected. Accordingly, also in this case, the recorded spectra on a different concentration or a different ratio of the particles or the constituents of the sample, which are due to a limited number of particle types or constituents of the spectrometrically examined sample in the form of the gas, which form the base spectra of the base matrix. The method described above can, in addition to the residual gas analysis and in other types of mass spectrometry, for example in the Proton Transfer Reaction (PTR) -Massenspektrometrie, the Selective Reagent Ionization (SRI) -Massenspektrometrie, or in the time-of-flight secondary ions Mass spectrometry (ToF-SIMS).

In gas chromatography, fragments of different molecules are detected, which form the constituents of the spectrometrically examined gas. If the fragments have different ratios in the recorded spectra, those fragments which belong to a single type of molecule can be determined and the base spectrum of a respective molecule can be reconstructed.

For certain samples to be examined and / or for certain spectrometric methods, an assumption may be made about the number of base spectra or basic vectors to be contained in the base matrix. For example, in a coating process, especially a sputtering process in which a substance, e.g. of carbon, on a substrate, e.g. Upon Ru, it can be assumed that the sample has exactly these two constituents and that it is therefore sufficient to specify a number of two basis spectra for the base matrix which correspond to the spectra of the two constituents.

In a development, when factoring the data matrix, the number of base spectra of the base matrix is increased until a reconstruction error of the factorization of the data matrix is below a threshold value. In the event that the number of base spectra underlying the majority of the recorded spectra is not known, a number of basic spectra are first given for the factorization. In the (numerically performed) non-negative matrix factorization, the error (so-called reconstruction error) in the approximation of the data matrix by the product of the coefficient matrix and the base matrix is too large, i. if this is above a predetermined threshold, it can be assumed that the number of base spectra has been selected too small. In this case, the number of base spectra or column vectors of the base matrix is increased and factorization is performed again. In the event that the reconstruction error is very small and / or that one of the base spectra contributes only a very small proportion to the data matrix when multiplied by the coefficient matrix, the number of base spectra in the factorization possibly reduced.

bestimmt. Hierbei steht D_ij für die Datenmatrix (NxK Matrix, Spektren in Zeilen angeordnet), N für die Anzahl der aufgenommenen Spektren, K für die Anzahl der gemessenen Energiekanäle pro aufgenommenem Spektrum, und d_j, d_k sind die j-te bzw. k-te Komponente des mittleren (über alle N aufgenommenen Spektren gemittelten) Spektrums. Für Eigenwerte bzw. für Singulärwerte, deren Betrag unter einem vorgegebenen Schwellwert liegt, wird typischerweise kein Basis-Spektrum bei der Faktorisierung der Daten-Matrix benötigt. Das untere Limit bzw. der Schätzwert für die Anzahl der Basis-Spektren entspricht der Anzahl der Eigen- bzw. Singulärwerte, die größer sind als der Schwellwert.In another variant, a lower limit for the number of base spectra of the base matrix for factoring the data matrix is determined based on a principal component analysis of the data matrix or a covariance matrix calculated from the data matrix. In principal component analysis (PCA) or, if applicable, in its generalization, the singular value decomposition (SVD), the eigenvalues or singular values of the data matrix D or the thereof derived, ie from the data matrix D, KxK covariance matrix cov (D): $$cOs{\left(\underset{\_}{D}\right)}_{jk}=\frac{1}{N-1}{\displaystyle \underset{i=1}{\overset{N}{\Sigma }}\left(D_{ij}-D_j\right)\left(D_{ik}-d_k\right)}$$

certainly. Here, D _ij stands for the data matrix (NxK matrix, spectra arranged in rows), N for the number of recorded spectra, K for the number of measured energy channels per recorded spectrum, and d _j , d _k are the j-th and k, respectively -th component of the mean spectrum (averaged over all N spectra). For eigenvalues or for singular values whose amount is under one given threshold value, typically no base spectrum is required in the factorization of the data matrix. The lower limit or the estimate for the number of base spectra corresponds to the number of eigenvalues or singular values that are greater than the threshold value.

The threshold, which determines the lower limit of the necessary number of basis vectors, may be an absolute value, but typically the threshold is determined based on the eigenvalues, such as the magnitude of the largest eigenvalue, using a numerical variant for the evaluation of the Scree plot, the Kaiser-Guttmann criterion, the mean or the median of the eigenvalues, etc. If a lower bound is determined for the start value in this way, the procedure described above for determining the number of base spectra can be abbreviated and thus computing time can be saved. In particular, the reconstruction error in the factorization of the data matrix may already be below the threshold value when the starting value of the number of base spectra is used, so that an increase in the number of base spectra of the basic matrix can be dispensed with ,

In a further variant, for the purpose of updating the base spectra, the formation of the data matrix and the factorization of the data matrix are performed several times in succession. The update can be carried out, for example, at constant intervals or time intervals. The update is particularly favorable, if necessary to adjust the number of base spectra, which may change over time, for example, if during the spectrometric examination additional constituents are formed or if certain constituents disappear during the spectrometric examination, because they decay or are no longer present or appear new (eg when creating ToF-SIMS depth profiles of layer systems).

In an advantageous variant, the method additionally comprises determining at least one further coefficient matrix for at least one further plurality of recorded spectra based on the determined base spectra, and storing the plurality of further recorded spectra in the form of the further coefficient matrix.

When recording spectra, the fundamental problem is that a large amount of data is generated and must be stored. These data are e.g. in the case of spectra frequently recorded in time series, they are at least temporarily redundant with respect to the base spectra. To advance the amount of data to be stored, i. Already during the measurement, it can be exploited that the basic spectra or the basic vectors of the basic matrix are known, so that it is sufficient to store the further coefficient matrix or the linear factors from which the further majority of the spectra results by multiplication with the base matrix.

For this purpose it is not necessary to carry out a non-negative matrix factorization again, since the calculation of the further coefficient matrix requires only one matrix multiplication with the inverse basic matrix. Since the coefficient matrix typically has a significantly smaller dimension than the data matrix, the amount of data to be stored can be significantly reduced in this way. For determining the further coefficient matrix, typically a number of (further) recorded spectra are combined in the further data matrix which corresponds to the plurality of recorded spectra of the data matrix for which the non-negative matrix factorization was performed ,

In a further variant, the method comprises: comparing the base spectra with known base spectra, in particular for identifying constituents of the sample. In this variant, the base spectra generated during the factorization of the data matrix are compared or correlated with known base spectra, which can be stored, for example, in a database. The known base spectra stored in the database typically each correspond to a constituent of the sample. The constituent may be, for example, a chemical element, a chemical compound, a type of molecule, etc. In the comparison, a measure of the correlation between the base spectra generated in the factorization and the known base spectra can be determined. In particular, the constituents of the sample can be identified on the basis of the comparison.

In a further variant, the spectra are recorded by a spectrometric method which is selected from the group comprising: Electron Probe Microanalysis (EPMA), scanning electron spectroscopy (SEM) with connected energy-dispersive X-ray spectrometer (EDX), wavelength dispersive X-ray spectrometer (US Pat. WDX), X-ray fluorescence (XRF), Electron energy loss spectroscopy (EELS), Particle-induced X-ray emission (PIXE), Auger electron spectroscopy (AES), Gamma ray spectroscopy, Secondary ion mass spectrometry (SIMS), X-ray photoelectron spectroscopy (XPS), IR spectroscopy, Raman spectroscopy, Magnetic Resonance Imaging (MRI) scans, Computerized Axial Tomography (CAT) scans, IR reflectometry, mass spectrometry, especially with an ion trap, multidimensional chromatographic / spectroscopic techniques, hyperspectral imaging, and remote imaging sensors.

It is understood that non-negative matrix factorization can be used not only for factor analysis of recorded spectra but also for factor analysis of other data or data sets. For example, non-negative matrix factorization may be used for factor analysis of image data, e.g. to identify common mistakes when examining a sample or measuring device. For example, to determine errors or aberrations instead of a fixed base, for example in the form of the Zernike polynomials, the basis vectors of the base matrix can be used as the basis for determining the errors or the size of the errors. Also for the inspection of an object, for example in the form of a wafer used in photolithography, more precisely for the detection of common errors, which are due to the photomask used and their amplitude varies, the method described above or the non-negative Matrix factorization can be applied.

Further features and advantages of the invention will become apparent from the following description of embodiments of the invention, with reference to the figures of the drawing, which show details essential to the invention, and from the claims. The individual features can be realized individually for themselves or for several in any combination in a variant of the invention.

list of figures

• 1 eine schematische Darstellung einer Mehrzahl von XPS-Spektren bei einem Sputter-Prozess von C auf Ru,
• 2 eine schematische Darstellung von zwei Basis-Spektren, die durch Nicht-Negative-Matrix-Faktorisierung aus einer Daten-Matrix mit den XPS-Spektren von 1 berechnet wurden,
• 3 eine schematische Darstellung der zeitabhängigen Veränderung der Amplituden der Basis-Spektren, sowie
• 4 eine schematische Darstellung des bei der Faktorisierung aufgetretenen Rekonstruktions-Fehlers.

Embodiments are illustrated in the schematic drawing and will be explained in the following description. It shows

• 1 a schematic representation of a plurality of XPS spectra in a sputtering process from C to Ru,
• 2 a schematic representation of two basic spectra obtained by non-negative matrix factorization from a data matrix with the XPS spectra of 1 were calculated
• 3 a schematic representation of the time-dependent change in the amplitudes of the base spectra, as well
• 4 a schematic representation of the occurred in the factorization reconstruction error.

In 1 For example, a number of twenty-five XPS spectra d _j (j = 1, ..., 25) are shown taken at successive times during a sputtering process of carbon on a sample of Ru. As is customary in X-ray photoelectron spectroscopy, is in 1 The measured intensity I (in arbitrary units) is plotted against the binding energy E _B (in eV), which is determined from the kinetic energy of the photoelectrons. During the sputtering process, the composition of the Ru sample changes, resulting in the 1 discernible differences in the recorded XPS spectra d _j result. In the 1 XPS spectra shown are summarized in a data matrix D for the factor analysis, wherein the columns of the data matrix D correspond to a respective recorded spectrum d _j and in the rows of the data matrix D, the intensity values I for a typically large number are contained by different binding energies or so-called energy channels E _{i of} a respective recorded spectrum d _j .

wobei die Basis-Matrix H die unbekannten Quellen (hier: die Basis-Spektren h_k (max{k}<= max{j})) als Spalten-Vektoren enthält und W eine Koeffizienten-Matrix darstellt. Die Matrix E bezeichnet den Rekonstruktions-Fehler. Die beiden Matrizen W und H erfüllen das Nicht-Negativitäts-Kriterium, d.h. die Einträge bzw. die Norm der beiden Matrizen W und H ist jeweils > 0 oder = 0. Für den Rekonstruktions-Fehler E ist dieses Kriterium im Allgemeinen nicht zwingend erforderlich.In order to obtain the in the XPS spectra d _j of 1 The data matrix D is factored using non-negative matrix factorization (hereinafter: NMF). The following applies: $$\underset{\_}{\text{D}}=\underset{\_}{\text{W}}\underset{\_}{\text{H}}+\underset{\_}{\text{e}}.$$

where the base matrix H contains the unknown sources (here: the base spectra _hk (max {k} <= max {j})) as column vectors and W represents a coefficient matrix. The matrix E denotes the reconstruction error. The two matrices W and H satisfy the non-negativity criterion, ie the entries or the norm of the two matrices W and H are> 0 or = 0 in each case. For the reconstruction error E, this criterion is generally not absolutely necessary.

minimieren, wobei || || die Frobenius-Norm bezeichnet. Für die Minimierung der Funktion F können verschiedene Algorithmen verwendet werden, die aus der Literatur zur NMF bekannt sind, beispielsweise der Algorithmus von Lee und Seung.In the NMF, various cost functions or reconstruction errors E can be defined which describe the deviation between the data matrix D and the product WH. For the following calculation, the quadratic error (Frobenius norm) was used to determine the deviation or the reconstruction error E. The factorization problem with the Frobenius norm can be defined as follows: For given D find two non-negative matrices which are the function $$\text{F}\left(\underset{\_}{\text{W}}.\underset{\_}{\text{H}}\right)={\Vert \underset{\_}{\text{D}}-\underset{\_}{\text{W}}\underset{\_}{\text{H}}\Vert }^2$$

minimize, where || || the Frobenius norm. For the minimization of the function F, various algorithms known from the literature on NMF can be used, for example the algorithm of Lee and Seung.

In the example shown it is assumed that the sample consists of the two constituents C and Ru; Accordingly, the NMF searches for two base spectra _hk (k = 1.2), ie the base matrix H has only two columns, which are the hidden sources of the in 1 shown XPS spectra d _j acts.

2 shows the basic spectra h ₁ , h ₂ (the columns) of the matrix H for the elements C and Ru determined by means of the NMF. 3 shows the time-dependent change in the amplitudes of the two basic spectra h ₁ , h ₂ . In 3 It can clearly be seen that over the entire duration of the measurement series, the amplitude of the first basic spectrum h ₁ increases sharply and the amplitude of the second basic spectrum h ₂ decreases (slightly). 4 shows the reconstruction error E, which has occurred in the numerical implementation of the NMF of the data matrix D in the matrices W and H, depending on the binding energy E _B or a respective data line (discrete) binding energy E _B For the implementation of the NMF, a threshold value for the reconstruction error E is specified and the above-described minimization process of the NMF is terminated as soon as the function F (W, H) is less than the threshold value of the reconstruction error E.

The measure, which is compared with the threshold value, can be determined in different ways from the reconstruction error E at the different binding energies E _B. For example, this may be the maximum absolute error over all binding energies E _B or the median of the absolute errors over all binding energies E _B. The measure which is compared with the threshold value can also be determined on the basis of a weighting function of the binding energies E _B which takes into account a background noise. Even a non-normalized x ² value, ie the simple sum over the quadratic deviations, can serve as a measure for the reconstruction error E.

In the event that it is not known in advance how large the number of constituents or base spectra h _{k of} the base matrix H is, a (starting) value for the number of base spectra h _{k can be} specified and be checked whether the reconstruction error E in the implementation of the NMF is below a predetermined threshold. If the threshold value for the reconstruction error E can not be undershot in the case of the NMF, this is a sign that the selected number of base spectra h _{i has been selected} too low. In this case, the number of base spectra h _i or the number of columns of the base matrix H can be increased by a discrete value, for example by one, until the reconstruction error E in the factorization of the data matrix NMF is below the threshold.

The starting value for the number of basis spectra h _{k of} the base matrix H for the NMF can also be determined on the basis of a principal component analysis of the data matrix D or the K × K covariance matrix cov (D) formed from the data matrix: $$cOv{\left(\underset{\_}{D}\right)}_{jk}=\frac{1}{N-1}{\displaystyle \underset{i=1}{\overset{N}{\Sigma }}\left(D_{ij}-d_j\right)\left(D_{ik}-d_k\right)}$$

The lower limit or a first estimated value for the number of base spectrums h _k corresponds to the number of eigenvalues determined in the main component analysis, which are greater than a predefined threshold value. Typically, the threshold value is determined from the amounts of the eigenvalues, for example, depending on the magnitude of the largest eigenvalue (eg, 1/10 of the largest eigenvalue), the mean of the eigenvalues, the median of the eigenvalues, using a numerical variant of the scree test (with a predefined value) Minimum percentage to explain the variance in the measured spectra), the Kaiser-Guttmann criterion, etc. In the scree test, the eigenvalues are plotted against the number by size, or the size of their contribution to explaining the variance, and so graphically A numerical variant can, for example, analyze a change in the derivative of this curve or the curvature of that curve, thereby determining the minimum number of eigenvectors, or alternatively the cumulative variance with compared to the default value and thus the Mind number of estates necessary to achieve the required eigenvalues.

In the event that the number of constituents of the sample during the recording of the plurality of spectra d _j changes, it may be useful to perform the factorization of the data matrix described above by the NMF multiple times, for example, in equidistant intervals. In particular in combination with the adaptation of the number of base spectra described above h _i is ensured in this way that the number of base spectra h _i matches the number of constituents in the sample.

Is a set of current base spectrums h _k and it is assumed that in the factorization of the data matrix D a constant number of basis spectra h _i can occur in the base matrix H, when recording more XPS spectra d _k To reduce the amount of data to be stored proceed as follows:

wobei W die weitere Koeffizienten-Matrix bezeichnet, die sich bei gegebener weiterer Daten-Matrix D und gegebener Basis-Matrix H ergibt. An Stelle der vollständigen weiteren XPS-Spektren d_i bzw. der weiteren Daten-Matrix wird lediglich die weitere Koeffizienten-Matrix W abgespeichert, was zu einer signifikanten Reduzierung der zu speichernden Datenmenge führt. Die weiteren aufgenommenen XPS-Spektren werden in Form der zugehörigen Koeffizienten-Matrizen W blockweise gespeichert, wobei die Anzahl der in einem Block gespeicherten XPS-Spektren d_k der Mehrzahl der in der Daten-Matrix D vorhandenen XPS-Spektren (im hier beschriebenen Beispiel: 25) entspricht.From a further plurality of XPS spectra d _i (In the present example: i = 1, ..., 25), a further data matrix D is formed. From the further data matrix D of the XPS spectra d _i is based on the already known base matrix H or the base spectra h ₁ . h ₂ another coefficient matrix W is determined by solving the following equation for W: $$\underset{\_}{\text{D}}=\underset{\_}{\text{W}}\underset{\_}{\text{H}}.$$

where W denotes the further coefficient matrix that results for given further data matrix D and given base matrix H. Instead of the complete XPS spectra d _i or the further data matrix, only the further coefficient matrix W is stored, which leads to a significant reduction in the amount of data to be stored. The further recorded XPS spectra are stored as blocks in the form of the associated coefficient matrices W, the number of XPS spectra stored in a block d _k the majority of the XPS spectra present in the data matrix D (in the example described here: 25).

The storage of a multiplicity of further coefficient matrices W can, if appropriate, take place as long as desired, or only until an update of the basic spectra h _k he follows. It is understood that the respective current base matrix H or the respective current base spectra h _k must be stored together with the respectively associated additional coefficient matrices W to the respective recorded further XPS spectra d _k to be able to reconstruct.

With sufficient speed of processing the data, for example in an analyzer provided for this purpose, e.g. in the form of a computer or the like, which is in particular coupled to a spectrometer used in the spectrometric examination of the sample, for example a mass spectrometer, optionally with an ion trap, the factor analysis can take place in real time. However, it should be understood that, alternatively, the factor analysis may be performed or completed only after completion of the spectrometric examination of the sample. Also in this case, in the manner described above, the amount of data to be stored can be significantly reduced.

The basis spectra determined in the manner described above h _i can be compared with known base spectra, for example, stored in a database and each associated with a constituent of the sample. For example, the in 2 illustrated first base spectrum h ₁ Carbon are assigned as constituents, while the second base spectrum h ₂ Ru is assigned as a constituent. It is understood that conclusions about the precision of the NMF can be made based on the comparison or on the correlation with the base spectra stored in the database.

The factor analysis described above can be carried out not only in the spectrometric examination of a sample by XPS, but also in a variety of other spectrometric methods, which are for example selected from the group comprising: EPMA, SEM, in particular with EDX, WDX, XRF, EELS, PIXE, AES, gamma ray spectroscopy, SIMS, XPS, IR spectroscopy, Raman spectroscopy, MRI scans, CAT, IR reflectometry, mass spectrometry, especially with an ion trap, multidimensional chromatographic / spectroscopic techniques and hyperspectral imaging.

QUOTES INCLUDE IN THE DESCRIPTION

This list of the documents listed by the applicant has been generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Cited patent literature

• US 8266197 B1 [0002]
• US 7725517 B1 [0003]

Claims (7)

Method for factor analysis, in particular for real-time factor analysis, in the spectrometric examination of a sample, comprising: taking a plurality of spectra (d _j ) of the sample, forming a data matrix (D) from the plurality of recorded spectra ( d _j ), and factorizing the data matrix (D) into a coefficient matrix (W) and into a base matrix (H) by non-negative matrix factorization, the base matrix (H) having a number of Contains base spectra (h _k ) associated with the recorded spectra (d _j ). Method according to Claim 1 in which, when factoring the data matrix (D), the number of base spectra (h _k ) of the base matrix (H) is increased until a reconstruction error (E) of the factorization of the data matrix (D ) is below a threshold. Method according to Claim 1 or 2 in which a lower limit for the number of base spectra (h _k ) of the base matrix (H) for the factorization of the data matrix (D) is based on a principal component analysis of the data matrix (D) or one of the data matrix. Matrix (D) calculated covariance matrix is determined. Method according to one of the preceding claims, in which, in order to update the basic spectra (h _k ), the formation of the data matrix (D) and the factorization of the data matrix (D) are performed several times in succession. Method according to one of the preceding claims, further comprising: determining a further coefficient matrix (W) for at least one further plurality of recorded spectra (d _i ) based on the determined base spectra (h _k ), and storing the further plurality of recorded spectra (d _i ) in the form of the further coefficient matrix (W). Method according to one of the preceding claims, further comprising: comparing the base spectra (h _k ) with known base spectra, in particular for identifying constituents of the sample. Method according to one of the preceding claims, in which the spectra (d _j ) are recorded by a spectrometric method which is selected from the group comprising: EPMA, SEM, in particular with EDX, WDX, XRF, EELS, PIXE, AES, gamma ray Spectroscopy, SIMS, XPS, IR spectroscopy, Raman spectroscopy, MRI scans, CAT, IR reflectometry, mass spectrometry, especially with an ion trap, multidimensional chromatographic / spectroscopic techniques, hyperspectral imaging.

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