Setting all **negative** elements resulting from the least squares computation to 0 is the simplest method for insuring **non**-negativity. Other than the multiplicative update algorithm, where an element set 0 stays 0, the ALS algorithm is flexible and can converge very fast. As with PMF there are numerous ways to compensate for input data uncertainties in NMF analysis. Penalty terms can be used to enforce constraints based upon prior knowledge or to predetermine characteristics of the computed solution. The matlab built-in function for **non**-**negative** **matrix** **factorization** (nnmf) which was used allows for input of pre-defined matrices and . However, for this thesis no such adjustments were made. The initial matrices and were set to be matrices of random values. For the matlab default value of 100 was used. The termination tolerances on change in size of the residual and relative change in the elements of and were also set to default setting. To deal with rotational ambiguity the NMF was run for 50 replicates. The ALS algorithm was chosen due to its fast converging properties. For the chosen number of factors 50 replicates appeared sufficient to produce a suitable local minimum of the cost function. This means nnmf resulted in the same factors for every test run with the same settings. For all 50 replicates the algorithm converged before was reached.

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Hence, a sound source separation technique is required to solve these above-mentioned problems. Sound source separation techniques with **Non**-**negative** **matrix** **factorization** (NMF) (2) have been proposed to solve the above-mentioned problems. When we utilize NMF, we need to specify the number of bases in advance. The Gamma Process **Non**-**negative** **Matrix** **Factorization** (GaP-NMF)(3) has solved the problem of NMF by introducing Gamma process to NMF and by extending NMF by Bayesian nonparametric method. GaP-NMF makes it possible to separate while inferring the number of unknown sound sources. However, there is also a problem with GaP-NMF. We expect to obtain a basis **matrix** as a set of the spectrum of a single tone so as to realize the multiple sound source separation. The basis **matrix** which is estimated by GaP-NMF is not a set of the spectrum of a single tone. Hence, in this study, we propose a method that introduces deep learning to the learning process of a basis **matrix** of GaP-NMF.

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Abstract—Earth Observation (EO) images clustering is a chal- lenging problem in data mining, where each image is represented by a high-dimensional feature vector. However, the feature vectors might not be appropriate to express the semantic content of images, which eventually lead to poor results in clustering and classification. To tackle this problem, we propose an interactive approach to generate compact and informative features from images content. To this end, we utilize a 3D interactive application to support user-images interactions. These interactions are used in the context of two novel **Non**-**negative** **Matrix** **Factorization** (NMF) algorithms to generate new features. We assess the quality of new features by applying k-means clustering on the generated features and compare the obtained clustering results with those achieved by original features. We perform experiments on a Synthetic Aperture Radar (SAR) image dataset represented by different state-of-the-art features and demonstrate the effectiveness of the proposed method. Moreover, we propose a divide-and-conquer approach to cluster a massive amount of images using a small subset of interactions.

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Abstract
This paper presents an algorithm for source localization using a beamforming-inspired spatial covariance model (SCM) and complex **non**-**negative** **matrix** **factorization** (CNMF). The spatial properties are modeled as the weighted sum of spatial kernels which encode the phase an the amplitude differences between microphones for every possible source location in a grid. The actual localization for each individual source in the multichan- nel mixture is estimated using complex-valued **non**-**negative** **matrix** **factorization** (CNMF) where each source spectrogram is modeled using a dictionary of spectral patterns learned a priori from training material.

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distributed in space containing patients from both c-means clusters. This GMM cluster was excluded because it would not represent any speci ﬁc subtype ( Figure 3C ). Patients inside the c-means GMM intersection had higher c-means membership likelihoods (roughly . .70) to belong to their own cluster than those outside the intersection (p , .01, Wilcoxon rank-sum test) ( Figure 3D ). We chose the cluster cores using the likeli- hoods of c-means with a cutoff of .70. As a result, 2 core subtypes were de ﬁned after ﬁltering out 50 ambiguous patients Figure 2. Inter-item correlations, relationship between factors, sociodemographic information, and clinical information. The 4-factor structure, derived from the Pharmacotherapy Monitoring and Outcome Survey sample with initial measure of Positive and **Negative** Syndrome Scale (PANSS) scores, was adopted as the reference on which the international sample was projected to derive the factor loadings. (A, B) Heat maps show interitem correlations for the original PANSS subscales (A) and the current orthonormal projective **non**-**negative** **matrix** **factorization** (OPNMF) 4-factor representation of psychopathology after controlling for symptom severity (total PANSS score) (B). Correlation strength is color-coded (light yellow to red: positive correlations; cyan to blue: **negative** correlations). (C) Box plot shows the bootstrap results (repeated 10,000 times) for the Pearson correlations among the 4-factor loadings. Bootstrap samples were drawn with replacement from the original international sample, and then the correlation analysis was done on them. The red line depicts the median, the green diamond depicts the mean, and the whiskers represent the 5th and 95th percentiles. (D, E) Graphs show effects of sociodemographic and clinical features on the 4-factor loadings. (D). Scatter plots show 4-way analysis of variance results of the signi ﬁcant **negative** association between age (adjusted for gender, illness duration, and total PANSS score) and the cognitive loading (p = .033) as well as the signiﬁcant positive associations between the symptom severity (total PANSS score) and the 4-factor loadings (**negative**: p = 4.98 3 10 2105 ; positive: p = 2.52 3 10 –51 ; affective: p = 1.75 3 10 2133 ; cognitive: p =

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model. DNA methylation profiles as measured by microarrays involve assays containing thousands to millions of cells, each of which may have its own profile. Consequently, the resulting measurements are averages over numerous cells and take values in [0, 1]. However, it is assumed that the cell populations consist of few homogeneous subpop- ulations having a common methylation profile, which are also shared across different samples. These subpopulations are referred to as cell types (see below for a specific example). Accordingly, the measurements can be modelled as mixtures of cell type- specific profiles, where the mixture proportions differ from sample to sample. Letting D ∈ [0, 1] m×n denote the **matrix** of methylation profiles for m CpGs and n samples, we suppose that D ≈ T A, where T :,k ∈ {0, 1} m represents the methylation profile of the k-th cell type and A ki equals the proportion, i.e. A ki ≥ 0 and P r k=1 A ki = 1, of the k-th cell type in the i-th sample, k = 1, . . . , r, i = 1, . . . , n. The total number of cell types r is sometimes known; otherwise, it has to be determined in a data-driven way. Decomposing D in the above manner constitutes an important preliminary step before studying possible associations between phenotype and methylation, since one needs to adjust for heterogeneity of the samples w.r.t. their cell type composition. If one of T or A is (approximately) known, the missing factor can be determined by solving a constrained least squares problem. While it may be possible to obtain T and/or A via experimental techniques if sufficient prior knowledge about the composition of the cell populations is available, this tends to require considerable effort, and it is thus desirable to recover both T and A. At this point, the proposed **matrix** **factorization** comes into play.

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For MMP-7 two low energy docking poses were found with significantly diverse binding modes. As visible in Figure 5a (the **non** zinc-binding mode), the ligand interacts with four amino acids of the receptor by the establishment of hydrogen bonds to Ala182, Ala184, Gly244 and Asp245. In this pose the ligand does not coordinate to the catalytic zinc(II) ion. The aliphatic linker between the scaffold and the carboxylic acid populates the hydrophobic S1’ channel and the phthalimide blocks the groove at the active site. The benzyl residue can be engaged in π-interactions with Tyr172 and Phe185. In the second pose, depicted in Figure 5c (the zinc-binding mode), the benzyl moiety populates the S 1 ’ pocket and interacts with Tyr241 via a face to edge aromatic interaction.

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Applications of the **Factorization** method include inverse electromag- netic problems [47, 48] and inverse problems in elasticity [17] as well as inverse elliptic problems [47, 27], especially applications in impedance to- mography [33, 35, 38] and optical tomography [39]. Let us also point out the applications of the method to scattering problems for periodic structures [7, 3], which will be considered in more detail in Chapter II. A striking feature of the method is its sound theoretical foundation together with its simple implementation. On the other hand, the assumptions nec- essary for the construction of a **Factorization** method are probably among the most restrictive ones compared to the methods cited above and there are many situations where it is not known yet whether or not the Factor- ization method applies, see [49]. Especially, the Linear Sampling method, closely related to the **Factorization** method through its methodology, has been shown to be applicable in much more situations than the Factoriza- tion method.

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In order to control adduct formation in the positive ion- mode, the effect of tissue washing prior to **matrix** deposition was investigated. The use of physiological saline solution as a sodium donor increased the total number of lipid-related sig- nals to 67, but did not reduce the abundance of protonated molecular ions. Moreover, **matrix** cluster formation turned out to be significantly enhanced, even after **matrix** recrystalli- zation (Fig. 1 (f, g)). On the other side, a tissue washing step using ammonium formate as a proton donor enhanced chem- ical noise throughout the spectrum and clearly reduced its signal-to-noise ratio. However, it successfully suppressed the formation of sodiated as well as potassiated molecular ions and turned out to be essential for signal assignment of the 48 lipid-related protonated molecular ions (Fig. 1(h) and ESM, Tables S1 , S2 , and S3 ).

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Nimmt man zu den positiven Zahlen die negativen dazu, dann ist die Subtraktion unbe- schränkt ausführbar.0. Nachdem sein Gehalt überwiesen wurde, lautet der Kontostand auf 1491[r]

The following result is an extension of Lemma 2 of [5] to the case of nonzero feedthrough **matrix** D and covers both the continuous- as well as the discrete-time case. Corollary 3.2 For a given n-th order system G = (A, B, C, D) and the observer based controller K (11), suppose F is a state feedback gain and L is a state es- timator gain, such that A + BF and A + LC are stable. Then the frequency-weighted controllability and observ- ability grammians for Enns’ method [2] applied to the frequency-weighted left coprime **factorization** based con- troller reduction problem with weights defined in (3) can be computed by solving Lyapunov equations of order n. For a continuous-time system these equations are

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not analyzed in the present paper, it is fairly straightforward to understand how the logic would work in such a model: a **non**-negativity constraint on quantity would make it more tempting for firms to choose a high price. Thus, in a model where firms choose prices and there is a **non**-negativity constraint on quantity, the boldness eﬀect would tend to make uninformed firms harmful to consumers, since it creates an incentive to set a higher price. Also in such a model, the boldness eﬀect would be stronger when the production costs are relatively low. Hence, one should expect that for suﬃciently low values of the marginal cost parameter, the boldness eﬀect would be stronger than the eﬀect present in Vives (1984) also under Bertrand competition, with the result being that, for a subset of the parameter space, informed firms is good for expected total surplus.

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Here T r denotes the trace of square **matrix**, Σ t denotes the integrated covariance **matrix** at
time t and H t is time t **matrix** of conditional covariance forcasts.
Results are reported in Table 3. Based on these results the VHAR-KCV model outper- forms the VHAR at all forecasting horizons. In the large study by Symitsi et al. (2018) out of twelve models under consideration the VHAR model was shown to be the best model for forecasting covariance **matrix**. Thus, the VHAR-KCV is already significant improvement. This improvement maybe due to the fact that the VHAR-KCV model with kernel covari- ance estimator simply puts higher weight to the more recent data, whereas the VHAR with realized covariance estimator puts equal weight to all data points.

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There are multiple reasons for that notable performance. The algorithm uses on the fly aggregation. However, there are no hash tables and no overhead for hashing and managing keys. Further, the cache efficiency is greatly improved, for example the result **matrix** C is computed row by row, this leads to reduction of write cache misses. This is also evident from the chart in figure 4.2. The graphic depicts the cache misses of the different algorithms for the case of matrix5 0. One can recognize that there is a correlation between the number of cache misses and the execution times of the algorithms, respectively the speedups. The fastest algorithm causes at least cache misses. Furthermore, it is interesting that the cache misses of gustavson are about 16 times less than those of join_aggregate. This coincide almost exactly with the speedup coefficient. Nevertheless, we can also see that cache misses are not the only factor influencing the performance. Although outerP_hash2 has less cache misses than outerP_hash, both algorithm have practically the same execution time. This indicates that performance is dependent on multiple parameters with cache misses being just one of them. Factors as instruction count, branch mispredictions and migrations play also crucial role in end running time. We will investigate this issue further in the chapter Cache Efficiency.

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function, allowed to depend on the observed data through S T . We further assume that,
as T tends to infinity, ˆ φ T converges to a **non**-stochastic limiting shrinkage function φ.
The goal is to find the ‘optimal’ φ.
In order to accomplish this goal, we had to invoke some heavy-duty machinery from a research field called random **matrix** theory (RMT), which dates back to the seminal works of Wigner ( 1955 ) and Marˇcenko and Pastur ( 1967 ). This field studies large-sample, or asymptotic, properties of various features of sample covariance matrices, with a major focus on the sample eigenvalues. Under a rather lengthy set of regularity conditions, it can be shown that the sample eigenvalues are **non**-stochastic in the limit. More particularly, as the dimension N and the sample size T tend to infinity together, with their concentration ratio N/T converging to a limit c ∈ (0, 1) ∪ (1, ∞), 7 the empirical distribution of the

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eingesetzt wird. Strom kann so von Energiehändlern nicht nur für elektrische Anwendungen, sondern auch zum Be- treiben von Heizanlagen verwendet werden. Jedoch wird dies unter anderem wegen seines geringen energetischen Wirkungsgrades oft kritisch betrachtet. Für Stromprodu- zenten stellen **negative** Preise zusätzliche Kosten dar, die letztlich ihr operatives Ergebnis gefährden können, wenn sie nicht durch Gegengeschäfte (Hedging) ﬁ nanziell abge- sichert oder an den Endkonsumenten transferiert werden. **Negative** Strompreise sind nach wie vor eine ungewöhn- liche Markterscheinung, jedoch steigt ihre Häuﬁ gkeit. Als Abwehrmaßnahmen vor Negativpreisphänomenen für die (Energie-)Anbieterseite sind neben technischen Lö- sungen auch regulatorische Maßnahmen denkbar. Erste Eingriffe in Form einer festgesetzten Preisuntergrenze von -500 Euro/MWh 22 wurden von der EEX bereits imple-

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Wirklich in die Physik Einzug gehalten hat die **negative** En- tropie dann wohl durch Schrödinger. In seinem Büchlein „Was ist Leben?“ aus dem Jahr 1944, das weitgehend frei von Mathematik ist, schreibt er: „Was ist denn dieses kost- bare Etwas in unserer Nahrung, das uns vor dem Tode be- wahrt? Das ist leicht zu beantworten. Jeder Vorgang, jedes

lerie zu Galerie, von schöner Landschaft zu schöner Landschaft. Es wäre also ein sehr positiver Mensch. Und es konnte einen Menschen geben, der ungefähr dasselbe durchmacht, aber mit einem solchen Charakter, dass er jedem einzelnen Bilde tief hingegeben ist, mit Enthusiasmus an jedes einzelne Bild sich verliert, so dass er unmittelbar, wenn er davor steht, sich selber ganz vergisst und ganz in dem lebt, was er sieht; und so bei dem nächsten Bilde, bei dem dritten und so weiter. So geht er mit einer Seele, die an jede Einzelheit hingegeben ist, durch das Ganze hindurch; aber weil er so jeder Einzelheit hingegeben ist, verwischt jeder nächste Eindruck den vorhergehenden, und wenn er zurückkommt, hat er doch nur ein Chaos in seiner See- le. Das wäre ein Mensch, entgegengesetzt in gewisser Beziehung dem ersten, dem positiven; er wäre ein sehr negativer Mensch. So könnten wir in der mannigfaltigsten Weise Beispiele finden für positive und **negative** Menschen. Wir könnten als einen ne- gativen Menschen denjenigen bezeichnen, der so viel gelernt hat, dass sein Urteil unsicher geworden ist gegenüber jeder Tat- sache; dass er nicht weiß, was wahr ist und was falsch und zu einem Zweifler dem Leben und dem Wissen gegenüber wird. Er wäre ein negativer Mensch. - Ein anderer könnte dieselben und ebenso viele Eindrücke haben; aber er geht so durch das Leben, dass er die Eindrücke verarbeitet und die Fülle der Eindrücke einzureihen weiß in die Fülle seiner von ihm aufgesammelten Weisheit. Er wäre ein positiver Mensch im besten Sinne des Wortes.

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I hope to have shown that new and interesting research results in the classical topic of orthogonal polynomials can be obtained using computer algebra algorithms. The most important computer algebra algorithms utilized are the algorithms of linear algebra, polynomial **factorization** and the solution of polynomial systems, e. g. by Gr¨ obner bases.

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