Robust Statistics and Cluster Analysis in NDT for Multi-Parameter Signals

doi:10.21203/rs.3.rs-1785606/v1

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Robust Statistics and Cluster Analysis in NDT for Multi-Parameter Signals

https://doi.org/10.21203/rs.3.rs-1785606/v1

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Non-Destructive testing (NDT) often involves acquisition of complex multi-parameter data in which the target flaw or structure condition can be difficult to identify. The use of robust cluster analysis, based on the Mahalanobis distance, to assess such data is examined here. The addition of robust estimates for the covariance matrix allows this approach to work in an unsupervised learning environment. Being unsupervised makes the approach very versatile and largely eliminates the requirement for large training sets such as are used by other artificial intelligence (AI) techniques such as artificial neural nets (ANN) or support vector machines (SVM). This is particularly helpful when there are numerous different environments, such as fasteners holding together the various components of an aircraft wing, each of which would require its own training set. It also eliminates issues related to changes in the performance of the instrument(s) between training set and real application. The concept of the partial Mahalanobis distance, which only uses information from eigenvectors associated with the defect under investigation, is introduced. The use of only that fraction of the signal that is relevant to the detection of the defects significantly increases the resolving power of the method. Robust cluster analysis can provide automatic defect identification on sample sizes as small as 50 or 60.

Cluster analysis

Mahalanobis distance

NDT

robust statistics

Within the NDT community, there is interest in being able to analyze complex multi-parameter signals using automated analysis. Such complex signals can arise in many ways. It can be the result of several interfering effects being present in the data [1], resulting in overlapping signals of very different strength. It can arise because the signal is intrinsically complex and sections of it cannot be readily associated with the effects (defects) under investigation. This is typically the case in pulsed eddy current (PEC). In this situation, the signal is typically broken down into a few components using techniques like Principle Component Analysis (PCA) [2]–[7], modified Principal Components (MPCA) [8]–[12], Singular Value Decomposition (SVD)[13] or Independent Component Analysis (ICA)[6], [14], [15] or into more complex data using techniques like Wavelet Analysis (WA ) [7], [16]. Complex data can also arise when results from different techniques or different sensors are combined into a single data set, such as is commonly done using data fusion of images. Having reduced the signal(s) to a set of parameters, these are often then used in an artificial neural net (ANN)[17]–[20] or a support vector machine (SVM)[6], [21], [22] to classify the defects. Both of these approaches belong to a form of machine learning referred to as supervised learning in which the program is presented with a training set where the defect(s) have been classified manually. The program then learns an algorithm based on these examples and this algorithm would then be applied in the field. An advantage of this kind of approach is that it can provide instant feedback on the unknowns under investigation. However, the practical application of these approaches can be subject to a number of difficulties. In the first place, both of these approaches are sensitive to shifting or scaling of the input parameters. So for example, in eddy current testing, a change in the ambient temperature (say, 5°C) can result in a significant shift in the eddy current response, which is why inspections typically involve a calibration at the start and end of the process. If this wasn’t compensated for, then the learned algorithm may be ineffective. Another issue can be the range of sites of interest. For example, consider the inspection of an aircraft wing for small second (or deeper) layer cracks about fasteners. A number of different locations can immediately be identified. These include an isolated spot on a spar, an isolated spot on a rib, possibly several sites where a spar and a rib cross (3 layers as opposed to two). Then there is the possibility of different conductivity or thickness face sheets. The list of different sites expands quickly. Each type of site potentially requires its own model, which in turn requires samples to build the model. This quickly becomes unmanageable.

An alternative approach are techniques based on unsupervised learning such as cluster analysis or outlier detection [8], [12], [23]–[29]. These techniques have the advantage that they do not require a calibration set and, hence, do not suffer from issues that cause a difference in response between measurements carried out on such a set and the sample currently under inspection. They do require consistency during an individual inspection of samples. One drawback of such techniques is that they involve post-inspection analysis and, therefore, require a method of relating the analysis to the inspected items so that once defective items are identified in the analysis, the corresponding item can be tagged. Another issue is that such techniques simply identify outliers without consideration for why they are outside the cluster of the majority of data. Hence, full analysis typically requires either further data analysis by a human operator or further inspection by another technique. Despite these issues, the extreme flexibility of this approach makes it attractive to many situations.

In this paper, we consider the use of the Minimum Covariance Determinant (MCD) [30]–[32]and clustering based on the Mahalanobis distance (MD). The novel concept of a partial Mahalanobis distance, which considers only those dimensions relevant to the defect under investigation, is introduced as a method to reduce interference from sources that are unrelated to the defects under consideration. The use of only that fraction of the signal that is relevant to the detection of the defects significantly increases the resolving power of the method. Specific examples are taken from the analysis of pulsed eddy current data looking for defects in second layers emanating from fasteners [8]. This is representative of the second case mentioned above, where one is looking for a small effect amongst much larger variations in highly correlated data.

Cluster analysis is based on the assumption that measurements on locations without defects (blanks) will give, more or less the same response, and thus the scores, which represent signal features that describe this response, will be clustered. Measurements on locations with defects are expected to give a different response (see Fig. 1). Serious outliers can easily be found qualitatively by visual inspection of 3D scattergraphs of the scores. However, this becomes more challenging for less serious outliers that lie close to the cluster. In order to establish a quantitative approach, which could lead to a Probability of Detection (POD) curve, a boundary around the blank cluster needs to be described, such that everything inside the boundary is considered to be defect free and anything outside the boundary is considered a defect (crack). One way of defining this boundary is using the Mahalanobis distance (MD) [33], which can be thought of as the distance a point is from the centroid of the cluster in standard deviations, i.e. it takes into account that the cluster is, by assumption, an ellipsoid with different dimensions in arbitrarily oriented directions. Conveniently, the square of the MD is a chi-square for normally distributed data.

In order to obtain the Mahalanobis distance, it is necessary to first subtract out the cluster centre from the raw scores to produce the centred raw scores, S, where S is an m dimensional column vector, m being the number of scores or features used to describe a response. The Mahalanobis Distance Squared (MDS) is then calculated as [33]

$$MDS={S}^{T}{C}^{-1}S$$

where C is the m x m covariance matrix of the blanks. It can be useful to do an eigen-decomposition of C.

$$C= \text{V}E{V}^{T}$$

Where the columns of V are the eigenvectors of C and E is a diagonal m x m matrix whose diagonal elements are the eigenvalues of C. The inverse of C can be obtained by taking the inverse of E in Eq. 2, so

$$MDS= {S}^{T}V{E}^{-1}{V}^{T}S= \left({S}^{T}V\right){E}^{-1}\left({V}^{T}S\right)={\left({V}^{T}S\right)}^{T}{E}^{-1}\left({V}^{T}S\right)={\stackrel{´}{S}}^{T}{E}^{-1}\stackrel{´}{S}$$

where

$$\stackrel{\text{´}}{\text{S}}={V}^{T}\text{S}$$

which is just the scores rotated to the eigenvector basis from the original one. The Mahalanobis Distance Squared is then calculated using the formula

MDS$=\sum \frac{{{\stackrel{´}{S}}_{i}}^{2}}{{e}_{i}}$ (5)

where ${e}_{i}$is the i^th eigenvalue of the covariance matrix and represents the variance associated with the i^th eigenvector. Since the variance is the square of the standard deviation, the terms in the sum in Eq. 5 are the squares of scores divided by their standard deviation. If the ${\stackrel{´}{s}}_{i}$ are normally distributed, then it is obvious that the MDS should be distributed as a chi-square with as many degrees of freedom (dof) as there are terms in the sum. The value of this representation is that the terms in the computation of MDS are now independent of each other. This means that a subset of the dimensions can be taken to obtain a partial Mahalanobis Distance Squared (PMDS), which will also be chi-square distributed. The PMDS would be preferable, if it was “known” that some of the rotated scores described variation that had nothing to do with the defect under investigation. Such variation would increase the MDS and help to mask outliers due to defects. For example, in [8], the authors found that repeat variation was mostly confined to the first eigenvector. By its nature, repeat variation has no information about what class a data point might belong to. Eliminating this dimension would remove a large source of masking variation.

The Mahalanobis distance approach requires that one know both the centroid and the covariance matrix of the blank cluster. The covariance matrix can be thought of as being made up of two components. The first is repeat variation, which describes variation that occurs when repeat measurements are performed on the same fastener, i.e. unavoidable variation due to the probe and the way the inspection is carried out. For example, inconsistent probe tilt can cause a variation in the signal that has nothing to do with nature of the fastener being inspected. Proper probe design and technique can minimize this type of variation. The other type of variation is due to uncontrolled or unknown variation in the blank fasteners themselves. For example, if the distance between the fastener and a nearby edge varies, this can cause substantial variation in the response to a fastener [34]. Alternatively, damage to the fastener or the hole it is in can cause excessive variations [35]. The ability to find cracks using any analysis, cluster or otherwise, requires that both of these sources of variation be small relative to the changes due to a crack in at least some dimensions. So, for example, in conventional eddy current, ideally, it is arranged that the signal of interest occurs at 90° to the lift-off signal [36]. This way large variation in lift-off won’t obscure the ability to detect cracks. In the samples described in [8], it was found that, typically, 97% of the repeat variance was associated with a single eigenvector. This eigenvector was very highly correlated with the first eigenvector of the blank covariance matrix of those tests. Using a PMDS that ignored the first eigenvector helped to make the notches more visible.

In the lab, where which fasteners are blanks and which are not is known, it is easy to find the center (centroid) of the blank cluster and calculate its covariance matrix. On an unknown sample, like a real aircraft wing, this is a much more challenging proposition, even when it is expected that the vast majority of the fasteners are blanks. The effect of outliers on the centroid is usually mild. However, the effect of even one large outlier on the covariance matrix calculated in the standard fashion can be huge. In this case, the value of e_i in [2] is much larger than it should be and so the MD is smaller than it should be. This causes cracks to be “masked”, i.e. undetected. Ref. [12] used robust statistics to attempt to overcome this problem. The accuracy of the robust method in determining the covariance matrix is thus very important to obtaining useful MDs.

Robust statistics is a large field of mathematics, which attempts to provide reliable estimates of statistical parameters in contaminated data sets. One such algorithm is the Minimum Covariance Determinant (MCD) [32], which is available through scikit-learn [37] in python. By default, this algorithm separates the data into a core cluster (50% + m + 1), where the density of points is the greatest. In most schemes, the actual estimation of the covariance matrix is carried out in a two-step process. In the first step, the core cluster is identified based on the data that is closest together and, hence, gives the covariance matrix with the smallest determinant. The MD of all the points is then calculated and a new covariance matrix is calculated using some fraction of the data (greater than 50% and less than or equal to 1) with the smallest MDs. The actual estimate tends to underestimate the true value of the covariance matrix. To illustrate this, a series of simulations based on real data collected for samples described in [8] was carried out. The true 5 x 5 covariance matrix for the blanks from a sample set [8] was determined and decomposed using the Cholesky method [38], to give a lower triangular matrix. This matrix was then used to generate 50 sets of pseudo data (5 scores) with the same covariance matrix [38]. The MCD algorithm was run using different data fractions ranging from 0.58 to 1.0.

The eigenvectors of the robust covariance matrix were an excellent match to the eigenvectors of the true covariance matrix at all data fractions. The dot product between the true and robust eigenvectors typically only differed from 1 by less than 1%.

The ratio of the average robust eigenvalues to the true eigenvalues at different data fractions is shown in Fig. 2. The true eigenvalues are sorted in descending value ranging from 0.89 (1) to 1.74E-4 (5). As is well known, the eigenvalues calculated from the robust covariance matrix underestimate the true values [32]. This underestimation decreases slightly as the fraction of the data used by the MCD increases. Consequently, MDs calculated using the robust covariance matrix will tend to be high. For all data fractions, the standard deviation of the eigenvalues is what would be predicted on the basis of the chi-square distribution and 49 degrees of freedom, which is approximately 20%.

The consequences of the underestimation shown in Fig. 2 on the partial Mahalanobis distance squared (PMDS) are shown in Fig. 3. To generate this plot for each curve (data fraction), 50 random data points of 5 scores each were generated as before using the Cholesky matrix. The PMDS using the first 3 rotated scores was calculated for all 50 points and sorted from smallest to largest. Each point was then plotted vs the corresponding point from the chi-square distribution. So, if PDMS(j) is the j^th ranked point, then the j^th point in the graph is ($\left({\chi }_{1-\left(j-0.5\right)/n}^{2}\left(3\right), PMDS\left(j\right)\right)$. The dashed black line in Fig. 3 has a slope of 1 and so represents data having an exact chi-square distribution. The red line labelled as having a data fraction of 1, was generated not using the MCD, but the empirical covariance matrix. Both it and the line representing the MCD calculation using all of the data have a perfect chi-square distribution within expected statistical scatter. The curves for the remaining data fractions are all approximately straight lines but with steeper slopes indicating that they are overestimating the true MD. It would be straightforward to divide them by their slopes to scale them back to the chi-square value, but the real question is “How does this overestimate affect the detection of outliers?” If the MD of the outliers also scales up in proportion to this overestimate, then there is no significant effect. This question is related to the power of the test, i.e. the probability of a Type II error (i.e. a false negative), where an outlier is missed.

In order to investigate the ability of a robust MD method to detect an outlier, another series of simulations was done. As before a Cholesky decomposition was performed on a 5 x 5 covariance matrix calculated from real life data. This matrix was used to generate 50 pseudo-data as before. However, this time an outlier was added to the data. The outlier was calculated based on the eigenvalues and eigenvectors of the true covariance matrix. For example, in Fig. 4, the outlier, in terms of the eigenvector basis was $C=\left(\text{0,0},m\sqrt{{e}_{3}},-m\sqrt{{e}_{3}},0\right)$, where m is a scaling factor and ${e}_{i}$ is the i^th eigenvalue. This specific choice produces a point that can be thought of as $m\sqrt{2}$ standard deviations from the cluster center. This point was then rotated back into the original basis set $\stackrel{´}{C}=C{V}^{T}$ and added to the previously generated data. A robust covariance matrix was then calculated using a data fraction of 0.7 and the PMDS calculated using Vectors 3 and 4. Two different methods were then used to determine whether the outlier was “detected”. In the first, the outlier was “detected if it had one of the 3 largest PMDSs (i.e. was in the top 95% of the data). In the second approach, the point was considered detected if it was greater than a cut-off of 9.33, which was the average value of the 95 percentile of the 0.7 curve in Fig. 3. The first approach had the advantage of ensuring there would never be more than 5% false calls, but also commited the analysis to that level of false calls. It also required no simulations or other methods to determine a cut-off and so was very flexible. At each value of m, one hundred such simulations were performed and the number detected is shown in Fig. 4. A number of other series of simulations were carried out using other outlier directions and they all produced essentially identical results, as would be expected when the scaling is in terms of standard deviations. Finally, it is noted that the choice of data fraction at or below 90% had little effect on this result as might be expected from Fig. 3. The use of 70% maintains the high breakdown capability of the MCD algorithm.

The results of the analysis above suggest that a data point needs to be approximately 3 standard deviations away from the cluster center (in terms of the true covariance matrix) in order to have a 90% chance of being detected. The question then is “How big is this in terms of crack size?”. This question obviously depends on the inspection being made. However, as an example, data from Sample 13 of [8] is shown in Fig. 5. The data has been rotated by the eigenvectors of the true covariance matrix of the blanks and scaled with the square root of the corresponding eigenvalue. The results suggest that, at least for this case, a threshold of three standard deviations should result in substantial detection of the notches.

Another important consideration is: How well does this approach work as a function of cluster size? In order to investigate this a series of graphs similar to Fig. 4 were generated for blank clusters of 30, 40 50, 60, 80, and 100 samples. A logit fit to the data for each graph allowed the a₉₀, the size at which a defect would have a 90% chance of being detected, to be determined. The result is shown in Fig. 6. Down to a cluster size of 60, cluster size has little effect on a₉₀. However, below this a₉₀ starts to increase sharply. As might be expected, this result depends somewhat on the number of parameters contributing to the covariance matrix. More parameters require larger data sets to achieve constant performance, while fewer parameters require less.

In fact, the situation is actually somewhat worse as the cluster size decreases, as it is often the a_90/95 that is the quantity of interest for aviation applications [39], [40]. This latter quantity will increase as the scatter in the covariance matrix increases. As can be seen in Fig. 3, a plot of the MDs versus chi-square yields an almost straight line. If the MDs had a chi-square distribution, the slope of this line would be one. The slopes in Fig. 3 are averages over 100 repeat simulations. If instead the 95 percentile was plotted in the same way, it also produces approximately a straight line. The closer the slope of that line is to that of the average, the more reproducible the results are and, consequently, the closer the a_90/95 would be to the a₉₀. Figure 7 shows how the slopes of these two lines vary as a function of sample size for a data fraction of 70%. It can be seen that at sample sizes of 60 or greater, there is little effect on the average slope value but below this value, the relative slope begins to rise. This is reflected in similar behaviour for the a₉₀ in Fig. 6. However, for all the sample sizes investigated, the slope associated with the 95 percentile increases as the sample size decreases. The increase is slow at first, but increases rapidly as the sample size decreases. This would result in a rapidly increasing a_90/95 at the smaller sample sizes. Hence, there is an emphasis on attaining large data sets to improve the estimate of a_90/95.An inspection of the underside of the outer wing of a Twin Otter was recently carried out and serves as an example of the sample sizes of interest. Running along a spar, there were typically 13 fasteners between ribs. Consequently, sample sizes well over 100 were easily obtainable for each spar or stringer. In fact, the data for several stringers could be combined making even larger data sets. Going along a rib, there were typically only 4 fasteners between spars for approximately 29 fasteners per rib. Still, combining the results for several ribs could push the numbers up. The challenge came when considering fasteners at the intersection of a rib and a spar. In one case, there were only 25 such fasteners in a wing panel that had a different conductivity from the other wing panels. In this case, a qualitative assessment based on a 3D scattergraph was required.

The size requirement of this approach depends on the number of parameters that are used to describe the data. As a guideline, Rousseeuw and Hubert [31] state that as a minimum the number of samples needs to be greater than five times the number of dimensions. In the example shown in Fig. 6, this would translate to a minimum of 25 data points. However, as Fig. 6 shows, in order to achieve a reasonable a₉₀, one needs at least 50 samples and preferably more. Intuitively this makes sense, since the more parameters there are, the bigger the covariance matrix becomes and the more coefficients need to be evaluated. If more unknowns are to be evaluated, then more samples are required to achieve the same level of uncertainty.

In principle, one would want to reduce the number of dimensions used in the PMDS to just those that have relevant information. Simply adding dimensions with no or little contribution from the defect signal increases the magnitude of the cut-off with no corresponding increase in signal. For example, the 90% cut-off of a chi-square with one degree of freedom is 2.71 (standard deviations). With two degrees of freedom (DOF) it increases to 4.61 and, at 3 DOF, it is 6.25. Simulations using a sample size of 50 and the same data as Fig. 6, put the a₉₀ at 2.3, 3.3 and 3.9 standard deviations for 1, 2 and 3 dimensions respectively. This is approximately linear with the square root of the chi-square limits as might be expected. On the other hand, reducing the number of dimensions below what is represented in the defect signal can result in a significant increase in the number of misses. For example, in Fig. 5 (right), if only the scores associated with Score 3 were used, one would miss all the defects above the grey circle. Using just Score 4 would result in the loss of all the points to the left of the circle, i.e. most of the notches. Fortunately, the question as to what dimensions to include or exclude can be established empirically because the rotated scores (Eq. 4), are independent of each other in terms of their contribution to the PDMS as was shown in Eq. 5. One can add (or subtract) a dimension to the MD and if the defects have no contribution to that dimension, the largest outliers will only increase marginally. On the other hand, if one adds or removes a dimension with a large contribution from defects then the PMDS associated with that point can be expected to change significantly. Hence, the choice of dimensions to include in the PDMS can be established by examination of the data. In the case where the inspection is expected to possess very few or no defects then it makes sense to establish which dimensions are likely to be useful using a test piece with a selection of representative flaws. It is important, however, that such a test sample contain enough unflawed samples that a reasonably accurate empirical covariance matrix can be established. Test pieces tend to be notoriously defect rich.

The previous examples were based on a single outlier, but the results do not change significantly if more outliers are added. Figure 8 shows the results obtained when 5 outliers were added in the plane defined by Eigenvectors 3 and 4. The five defect points are all the same distance from the origin but randomly oriented. Two curves based on the 5 defects are shown. In one (open circles), a point was considered detected if it had one of the top 7 PMDS, which implies only two false calls out of 50 if all 5 of the defects are found. However, the number of false calls goes up as the detected fraction goes down. In the second example (full circles) a cut-off was used such that a 5% false call rate overall was obtained across all the simulations. This second curve is very close to that obtained earlier with only one defect.

The distribution of PMDS scores for the simulation in Fig. 8, corresponding to defects at 3 standard deviations, is shown in Fig. 9. The 100 simulations were ranked on the basis of the highest PMDS for a blank. The black curve is from the run with the 5th highest blank PMDS score (95th percentile) and the blue curve from the 5th smallest (5th percentile). Over most of their range, the two curves have very similar slopes, suggesting that all such curves are very similar. Obviously the false call rate is very high (12%) for the 95th percentile curve. On the other hand, there are no false calls for the 5th percentile curve. One positive feature of using the cut-off approach is that while it may result in a high false call rate a small number of times, in neither of the two cases examined here did it miss any of the defects. The cut-off approach also works better as the number of defects to be found increases. When it is expected that there are only going to be one or two defects, if any, the rank approach is easy to implement and requires no simulations. However, as the number of expected defects increases one needs increasingly complex decision trees to determine how far down in rank to go to find defects. The deeper one goes, the fewer defects escape, but the more false positives are obtained. The cut-off approach does not suffer this issue. To obtain a cut-off, one can use simulations as was done here but a simplistic alternative is to plot the PMDS against the corresponding chi-square for the same percentile as is done in Fig. 9. The appropriate cut-off tends to be where the curve breaks from linearity. For the 95th percentile, this occurs at about a chi-square value of 3.3 and for the 5 percentile curve it occurs about a chi-square value of 5. Both curves give a cut-off of approximately 7, which is close to the value of 6.8 determined from simulations. This approach can be done automatically with little effort.

It should be emphasized that the point of cluster analysis or outlier detection, is just that, to detect outliers for further investigation. While the primary interest here has been in analyzing complex signals from relatively small data files, the approach can be very useful for automatic scanning of large data files of even simple data, such as eddy current signals from heat exchangers. In this case, the approach can use large amounts of “normal” data to automatically identify regions of interest and bring them to the attention of human analysts for investigation, saving the human analyst many hours of time scanning through the large data files. The less common the defects are and the larger the data sets the better the cluster analysis will work. This is different from humans who have a tendency to be less vigilant when scanning large amounts of data with little expectation of finding anything, the proverbial “needle in a haystack”.

Robust cluster analysis based on the Mahalanobis distance and the MCD, has been shown to be a practical method to find outliers in complex data. It has the advantage of being a very versatile technique. It is capable of unsupervised learning, which significantly lowers the requirement for training data and eliminates issues surrounding changes in the data between training set and field data. The concept of a partial Mahalanbois distance, which only uses information from eigenvectors associated with the defect under investigation, was introduced. This approach eliminates dimensions with little or no relevance to detecting the defects of interest. When coupled with a robust estimator of the covariance matrix such as the MCD it can identify outliers with a modest false positive rate (5%) on even rather small data sets (> 60) that might occur in an NDT application. The primary advantage of the partial approach is that it allows the investigator to eliminate dimensions with little or no content due to the defects under investigation. Including such dimensions serves only to mask the presence of outliers and increase the a₉₀. The selection of the dimensions to use can benefit from experience with a training set, but because no numerical model is generated, the experience gathered is readily transferred to any similar situation without concerns about whether the data has been scaled or rotated.

Acknowledgements

Funding for this project was received from DTAES, DND Canada.

Ethics approval and consent to participate

Not applicable

Consent for publication

Not applicable

Availability of data and material

The programs (in python) used to generate the data from the simulations described in this work are available from the authors on reasonable request

Competing interests

Thomas Krause is an editor in this journal

Funding

The authors acknowledge the support of DTAES, DND Canada for funding for this project.

Authors' contributions

Ross Underhill preformed the simulations described in this paper. Both authors discussed the results and both authors wrote and reviewed this paper.

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No competing interests reported.

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Robust Statistics and Cluster Analysis in NDT for Multi-Parameter Signals

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