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Conclusions: A good agreement has been found between the SRTs and slope and those of other matrix tests. Since sentences are difficult to memorize, the Italian matrix test is suitable for repeated measurements.
In general, the simplified matrix type test can be used for speech-recognition measurements not only in noise but also in quiet. For German, Neumann et al. [13] showed that the simplified German matrix test is a valid audiometric test for quantifying speech perception in quiet in children from age 4. These findings were confirmed for a group of hearing-impaired children measured without and with hearing devices (hearing aids or cochlear implants).
I should conduct an activity which expects to operate with a weighted, directed graph. In particular, the graph has to represent the Italian railway network.I managed to obtain the network on a specified layer, on QGIS, and I can export it in various formats like .shp, .shx, .geojsonl.json etc.The question regards, from any of these files, how to get an adjacency, in order to create the graph I need (or simply create the graph, from which I may extract the matrix).
Mélanges are heterogeneous geological deposits and represent the most widespread bimrock (block-in-matrix) formations. This paper presents the efforts undertaken to characterise an Italian mélange composed of a clayey-marly matrix enclosing strong calcareous blocks. Due to its low uniaxial compressive strength, this geomaterial can be classified as a soft rock. The weak nature of the marly matrix and its water sensitivity, as well as the presence of rock inclusions, made the collection and preparation of intact specimens extremely complex and time-consuming operations. The difficulties encountered during these phases are described in detail; the various non-conventional procedures considered and developed to overcome these problems are also presented. The potential of the solutions proposed lies in the fact that they can be conveniently applied to other soft rocks with a block-in-matrix internal arrangement, such as the Italian mélange. To characterise the Italian mélange, point load and uniaxial compressive tests were carried out. From the results of these tests, a conversion factor equal to 14 is proposed to correct the point load strength index in order to estimate the uniaxial compressive strength of soft rocks, such as the mélange under study. Moreover, to estimate local strains and the deformability of the geomaterial, the non-destructive digital image correlation technique was applied.
In other words, we apply matrix completion to identify a counterfactual value (or expected value, based on the information coming from the observed portion of the matrix) for the average importance level of each creative skill across professions. Then we get from the results of our analysis that such counterfactual value is typically in good agreement with the actual average importance level. However, for some professions, the observed level deviates positively/negatively from the counterfactual (i.e., the counterfactual provides an underestimate/overestimate), hence such professions are associated with a surplus/deficit in creativity levels. In this regard, the methodological approach used in this work resembles the one employed in the analysis of total factor productivity, in which a set of actual outputs is compared versus the corresponding set of predicted outputs according to given inputs and a specific production function.Footnote 2 In this case, the emergence of outliers may be explained, e.g., by the absence of relevant inputs in the model or by its lack of additional nonlinear terms.
Our work builds on four main streams of existing literature dealing respectively with: (1) the relationship between ICT and soft skills; (2) skill relatedness, (3) creativity; (4) matrix completion. In the work, these issues are connected as follows. Our analysis allows the assessment of patterns and peculiarities of creative soft skills in the Italian labor market, as the endowments and formation of creativity among Italian workers could mitigate the negative effects of digitalization and job automation on low-skilled professions. More specifically, our analysis relies on the application of matrix completion to an occupation matrix whose entries represent, e.g., average importance levels of various soft skills (including ones associated with creativity) in different jobs. As detailed in the work, such an occupation matrix is quite well approximated by a low-rank matrix (an essential assumption for its successful applicationFootnote 4), which could be justified by the presence of skill relatedness across industries.
In the following, we apply Matrix Completion (MC) to predict average importance levels of skills associated with creativity (or, shortly, creativity levels) for the various professions in our occupation matrix, exploiting the similarity patterns detected automatically by that machine learning technique from its application to the specific dataset.
In order to apply MC to our occupation matrix (or, for a successive analysis, to its suitable submatrix, see Sect. 5.1), we generate artificially partially observed matrices from it, by artificially obscuring randomly 10%, 25% and 50% of the entries in the 25 columns associated with the creative skills, focusing each time on the prediction capability of matrix completion on each single row (occupation). The procedure is repeated several times (details are in the Appendix). This sampling procedure is justified by the fact that we want to apply MC to predict entries of the occupation matrix that are related to creativity (i.e., that belong to one of the 25 selected columns). The three choices \(10\%\), \(25\%\) and \(50\%\) are made in order to assess the accuracy (and robustness) of MC for quite different percentages of obscured entries. Finally, the motivation itself behind artificially obscuring known elements of the occupation matrix is that, in this way, it is possible to validate/test the predictions obtained by MC (i.e., a ground truth is available for comparison purposes).
where \(\Omega ^\mathrmtr\) is a training set of positions (i, j) corresponding to the known entries of the partially observed matrix \(\mathbfM \in \mathbb R^m \times n\), \(\mathbfZ \in \mathbb R^m \times n\) is the completed matrix, \(\Vert \mathbfZ\Vert _*\) is its nuclear norm, and \(\uplambda \ge 0\) is a regularization constant. Then, we solve it by applying the Soft Impute algorithm (Mazumder et al. 2009). This is proved therein to converge to an optimal solution of the optimization problem (1). The optimization problem itself is solved by the Soft Impute algorithm several times, for different choices of the set of obscured entries. For each such repetition, the best value of \(\uplambda \) is found by minimizing a suitable error on a validation subset of missing entries, whereas the final performance is evaluated on the remaining test set of other missing entries. Technical details about the choice of the various training/validation/test sets and of the regularization parameter of the optimization problem (1) are reported in the Appendix. Additional details about the optimization problem (1) and the Soft Impute algorithm are reported also in Metulini et al. (2022).
As a preliminary check for the applicability of MC, we compute the singular value decomposition of our occupation matrix. Figure 1 shows that its singular values decay quite fast to 0, hence that the matrix can be well approximated by a low-rank one. As reported in the Appendix, this is a necessary condition for an effective application of MC, which is satisfied by the dataset under analysis.
In order to better illustrate the approach used in our analysis, we consider a specific instance of the optimization problem (1), then we visualize its results. Figure 2 shows the original matrix, the locations of the observed and the missing entries in the original matrix (where the green cells denote the elements in the training set, the blue cells refer to the validation set and the red ones are related to the test set), the reconstructed matrix, and the error in absolute value for each cell. As the second subfigure shows, all the obscured entries are at the intersection between specific rows (one of which is associated with the test set) and the set of columns associated with creativity.
Visual representation of a the original matrix, b the locations of its observed (green) and missing entries (blue: validation; red: test), c the reconstructed matrix for a specific repetition, and d the absolute value of the prediction error for the same repetition (color figure online)
Then, Fig. 3 presents a visual representation of the (empirical) mean and standard deviation of the Root Mean Square Error (RMSE) of MC prediction on the test set (see the Appendix for details on its definition) per profession, where each row (profession) refers to the mean and standard deviation computed with respect to the repetitions having as test set elements belonging only to that specific row of the original matrix. As the figure shows, both the mean and standard deviation per profession are typically small (taking into account that the entries of the original matrix are numbers between 0 and 100).
Figure 4 reports more detailed results for all the repetitions for which the elements of the test set refer to a specific profession. In particular, the figure shows how the RMSE of MC prediction (evaluated, respectively, on the training set, validation set, and test set) changes as a function of the regularization parameter \(\uplambda \). The sizes of the three sets are also reported in the figure. As the figure reveals, the optimal value of \(\uplambda \) (which is computed based on the RMSE of MC prediction on the validation set) is associated with a quite small RMSE of MC prediction for both the validation and test sets. Moreover, for each of the two cases, the variability of the RMSE curve with respect to changing the training set (i.e., performing another repetition of the analysis, for the same choice of the test set) is very small, whereas the variability of the RMSE curve of MC prediction for the training set itself is negligible. It is worth mentioning that, for the specific case reported in Fig. 4, the performance of the method on the test set is slightly better than the one on the validation set. This is likely due to the fact that the elements of the two sets are sampled from different portions of the occupation matrix. Moreover, the test set has a much smaller cardinality (since its elements come from a specific row of the occupation matrix). In any case, the behaviour of the RMSE curve as a function of the regularization parameter \(\uplambda \) is similar for both sets, being also the minima of these two curves achieved for almost the same values of \(\uplambda \). Although Figs. 3 and 4 refer only to the percentage \(25\%\) of missing entries in the selected columns associated with creativity, similar results have been obtained for the other two percentages.
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