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Journal articles on the topic 'Negative squares'

Author: Grafiati

Published: 4 June 2025

Last updated: 8 July 2025

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1

Fendley, Paul, Kareljan Schoutens, and Hendrik van Eerten. "Hard squares with negative activity." Journal of Physics A: Mathematical and General 38, no. 2 (2004): 315–22. http://dx.doi.org/10.1088/0305-4470/38/2/002.

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2

Berg, C., J. P. R. Christensen, and P. H. Maserick. "Sequences with finitely many negative squares." Journal of Functional Analysis 79, no. 2 (1988): 260–87. http://dx.doi.org/10.1016/0022-1236(88)90014-6.

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3

Li, Yifeng, and Alioune Ngom. "Classification approach based on non-negative least squares." Neurocomputing 118 (October 2013): 41–57. http://dx.doi.org/10.1016/j.neucom.2013.02.012.

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4

Arsene, Gr, T. Constantinescu, and A. Gheondea. "Lifting of operators and prescribed numbers of negative squares." Michigan Mathematical Journal 34, no. 2 (1987): 201–16. http://dx.doi.org/10.1307/mmj/1029003552.

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5

Paatero, Pentti. "Least squares formulation of robust non-negative factor analysis." Chemometrics and Intelligent Laboratory Systems 37, no. 1 (1997): 23–35. http://dx.doi.org/10.1016/s0169-7439(96)00044-5.

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6

Behrndt, Jussi, and Carsten Trunk. "On the negative squares of indefinite Sturm–Liouville operators." Journal of Differential Equations 238, no. 2 (2007): 491–519. http://dx.doi.org/10.1016/j.jde.2007.01.026.

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7

Behrndt, Jussi, Roland Möws, and Carsten Trunk. "Eigenvalue estimates for operators with finitely many negative squares." Opuscula Mathematica 36, no. 6 (2016): 717. http://dx.doi.org/10.7494/opmath.2016.36.6.717.

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8

Rajkó, Róbert, and Yu Zheng. "Distance algorithm based procedure for non-negative least squares." Journal of Chemometrics 28, no. 9 (2014): 691–95. http://dx.doi.org/10.1002/cem.2625.

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9

Elphick, Clive, and William Linz. "Symmetry and asymmetry between positive and negative square energies of graphs." Electronic Journal of Linear Algebra 40 (May 13, 2024): 418–32. http://dx.doi.org/10.13001/ela.2024.8447.

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The positive and negative square energies of a graph, $s^+(G)$ and $s^-(G)$, are the sums of squares of the positive and negative eigenvalues of the adjacency matrix, respectively. The first results on square energies revealed symmetry between $s^+(G)$ and $s^-(G)$. This paper reviews examples of asymmetry between these parameters, for example using large random graphs and the ratios $s^+/s^-$ and $s^-/s^+$, as well as new examples of symmetry. Some questions previously asked about $s^{+}$ and $s^{-}$ are answered and several further avenues of research are suggested.

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10

Loconte, Lorenzo, Stefan Mengel, and Antonio Vergari. "Sum of Squares Circuits." Proceedings of the AAAI Conference on Artificial Intelligence 39, no. 18 (2025): 19077–85. https://doi.org/10.1609/aaai.v39i18.34100.

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Designing expressive generative models that support exact and efficient inference is a core question in probabilistic ML. Probabilistic circuits (PCs) offer a framework where this tractability-vs-expressiveness trade-off can be analyzed theoretically. Recently, squared PCs encoding subtractive mixtures via negative parameters have emerged as tractable models that can be exponentially more expressive than monotonic PCs, i.e., PCs with positive parameters only. In this paper we provide a more precise theoretical characterization of the expressiveness relationships among these models. First, we prove that squared PCs can be less expressive than monotonic ones. Second, we formalize a novel class of PCs – sum of squares PCs – that can be exponentially more expressive than both squared and monotonic PCs. Around sum of squares PCs, we build an expressiveness hierarchy that allows us to precisely unify and separate different tractable model classes such as Born Machines and PSD models, and other recently introduced tractable probabilistic models by using complex parameters. Finally, we empirically show the effectiveness of sum of squares circuits in performing distribution estimation.

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11

Verdooren, L. R. "Least squares estimators and non-negative estimators of variance components." Communications in Statistics - Theory and Methods 17, no. 4 (1988): 1027–51. http://dx.doi.org/10.1080/03610928808829671.

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12

Chen, Ying, Shiqing Zhang, and Xiaoming Zhao. "Facial Expression Recognition via Non-Negative Least-Squares Sparse Coding." Information 5, no. 2 (2014): 305–18. http://dx.doi.org/10.3390/info5020305.

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13

Kho, Jun Beom, Jaihie Kim, Ig-Jae Kim, and Andrew B. J. Teoh. "Cancelable fingerprint template design with randomized non-negative least squares." Pattern Recognition 91 (July 2019): 245–60. http://dx.doi.org/10.1016/j.patcog.2019.01.039.

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14

Zhigang, Chu, Shen Linbang, Yang Yang, and Wang Guangjian. "Non-negative least squares deconvolution method for mirror-ground beamforming." Journal of Vibration and Control 22, no. 16 (2016): 3470–78. http://dx.doi.org/10.1177/1077546314563778.

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15

Berg, Christian, and Zolt�n Sasv�ri. "Functions with a finite number of negative squares on semigroups." Monatshefte f�r Mathematik 107, no. 1 (1989): 9–34. http://dx.doi.org/10.1007/bf01470734.

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16

He, Kai, Yali Peng, Shigang Liu, and Jun Li. "Regularized Negative Label Relaxation Least Squares Regression for Face Recognition." Neural Processing Letters 51, no. 3 (2020): 2629–47. http://dx.doi.org/10.1007/s11063-020-10219-6.

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17

Bramble, James H., and Tong Sun. "A negative-norm least squares method for Reissner-Mindlin plates." Mathematics of Computation 67, no. 223 (1998): 901–17. http://dx.doi.org/10.1090/s0025-5718-98-00972-7.

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18

Sasv�ri, Zolt�n. "On bounded functions with a finite number of negative squares." Monatshefte f�r Mathematik 99, no. 3 (1985): 223–34. http://dx.doi.org/10.1007/bf01295156.

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19

Shaoqin, Gao, and Duan Huoyuan. "Negative norm least-squares methods for the incompressible magnetohydrodynamic equations." Acta Mathematica Scientia 28, no. 3 (2008): 675–84. http://dx.doi.org/10.1016/s0252-9602(08)60069-7.

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20

Zhang, Zhi-Min, Xiao-Qing Chen, Hong-Mei Lu, et al. "Mixture analysis using reverse searching and non-negative least squares." Chemometrics and Intelligent Laboratory Systems 137 (October 2014): 10–20. http://dx.doi.org/10.1016/j.chemolab.2014.06.002.

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21

Osaro, Aideyan D., Okeke J. Uche, and Ngutor Nenker. "Two-Stage Least Squares and Three-Stage Least Squares in Modelling Nigerian Economic Growth." International Journal of Development Mathematics (IJDM) 1, no. 2 (2024): 153–59. http://dx.doi.org/10.62054/ijdm/0102.11.

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Every Country strives to attain economic stability. In a bid to achieve this, several economic indices are explored to make this a reality. This study focused on analysis of two stage and three stage least squares (2SLS and 3SLS) in modeling Nigerian economic growth. Considering the limitations of ordinary least squares (OLS) in dealing with simultaneous equations, we explored the use of two stage and three stage least squares which are more robust methods for this purpose. A secondary data was obtained from Central Bank of Nigeria bullion and the National Bureau of Statistics bulletin to investigate the relationship among economic growth, capital inflow and external debt servicing. Models for GDP, external debt servicing and capital inflow were obtained using R and Microsoft excel statistical packages. The study shows with coefficients of 3.9159 and -6.1878, that foreign debt- servicing has negative impact on GDP in the 2SLS and 3SLS models respectively: which is not significant in the 2SLS but significant in the 3SLS model. Also, GDP has negative, though significant effect on capital inflow with coefficient of -0.010495 and -0.0139109 in the 2SLS and 3SLS models respectively: a pointer to the effects of insecurity in Nigerian economic growth. The study recommends that Government should deplete existing external debt stock, increase utilization of capital stock and labour force, reduce borrowing rate, resort to alternative ways of enhancing both current and recurrent expenditures and deal with insecurity.

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22

Grima, J. N., M. Bajada, S. Scerri, D. Attard, K. K. Dudek, and R. Gatt. "Maximizing negative thermal expansion via rigid unit modes: a geometry-based approach." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2179 (2015): 20150188. http://dx.doi.org/10.1098/rspa.2015.0188.

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Existent rigid unit mode (RUM) models based on rotating squares, which may explain the phenomenon of negative thermal expansion (NTE), are generalized so as to assess the NTE potential for novel systems made from rectangular or rhombic rigid units. Analytical models for the area coefficients of thermal expansion (CTE) of these innovative networks are derived in an attempt to determine the optimal geometrical parameters and connectivity for maximum NTE. It was found that all systems exhibit NTE, the extent of which is determined by the shape and connectivity of the elemental rigid units (side lengths ratio or internal angle). It was also found that some of the networks proposed here should exhibit significantly superior NTE properties when compared with the well-known network of squares, and that for optimal NTE characteristics, pencil-like rigid units should be used rather than square-shaped ones, as these permit larger pore sizes that are more conducive to NTE. All this compliments earlier work on the negative Poisson's ratio (auxetic) potential of such systems and may provide a route for the design of new materials exhibiting superior thermo-mechanical characteristics including specifically tailored CTEs or giant NTE characteristics.

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23

Timotheou, Stelios. "FAST NON-NEGATIVE LEAST-SQUARES LEARNING IN THE RANDOM NEURAL NETWORK." Probability in the Engineering and Informational Sciences 30, no. 3 (2016): 379–402. http://dx.doi.org/10.1017/s0269964816000061.

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The random neural network is a biologically inspired neural model where neurons interact by probabilistically exchanging positive and negative unit-amplitude signals that has superior learning capabilities compared to other artificial neural networks. This paper considers non-negative least squares supervised learning in this context, and develops an approach that achieves fast execution and excellent learning capacity. This speedup is a result of significant enhancements in the solution of the non-negative least-squares problem which regard (a) the development of analytical expressions for the evaluation of the gradient and objective functions and (b) a novel limited-memory quasi-Newton solution algorithm. Simulation results in the context of optimizing the performance of a disaster management problem using supervised learning verify the efficiency of the approach, achieving two orders of magnitude execution speedup and improved solution quality compared to state-of-the-art algorithms.

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24

De-Graft Acquah, Henry. "Comparing ols and rank-based estimation techniques for production analysis: An application to Ghanaian maize farms." Applied Studies in Agribusiness and Commerce 10, no. 4-5 (2016): 125–30. http://dx.doi.org/10.19041/apstract/2016/4-5/16.

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This paper introduces the rank-based estimation method to modelling the Cobb-Douglas production function as an alternative to the least squares approach. The intent is to demonstrate how a nonparametric regression based on a rank-based estimator can be used to estimate a Cobb-Douglas production function using data on maize production from Ghana. The nonparametric results are compared to common parametric specification using the ordinary least squares regression. Results of the study indicate that the estimated coefficients of the CobbDouglas Model using the Least squares method and the rank-based regression analysis are similar. Findings indicated that in both estimation techniques, land and Equipment had a significant and positive influence on output whilst agrochemicals had a significantly negative effect on output. Additionally, seeds which also had a negative influence on output was found to be significant in the robust rank-based estimation, but insignificant in the ordinary least square estimation. Both the least squares and rank-based regression suggest that the farmers were operating at an increasing returns to scale. In effect this paper demonstrate the usefulness of the rank-based estimation in production analysis. JEL CODE: Q18, D24, Q12, C1 and C67

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25

Abiad, Aida, Leonardo De Lima, Dheer Noal Desai, Krystal Guo, Leslie Hogben, and José Madrid. "Positive and negative square energies of graphs." Electronic Journal of Linear Algebra 39 (June 8, 2023): 307–26. http://dx.doi.org/10.13001/ela.2023.7827.

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The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Let $s^+(G), s^-(G)$ denote the sum of the squares of the positive and negative eigenvalues of $G$, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if $G$ is a connected graph of order $n$, then $s^+(G)\geq n-1$ and $s^-(G) \geq n-1$. In this paper, we show partial results towards this conjecture. In particular, numerous structural results that may help in proving the conjecture are derived, including the effect of various graph operations. These are then used to establish the conjecture for several graph classes, including graphs with certain fraction of positive eigenvalues and unicyclic graphs.

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26

Alpay, D., and J. Rovnyak. "Loewner’s theorem for kernels having a finite number of negative squares." Proceedings of the American Mathematical Society 127, no. 4 (1999): 1109–17. http://dx.doi.org/10.1090/s0002-9939-99-04618-3.

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27

Kim, Dongmin, Suvrit Sra, and Inderjit S. Dhillon. "A non-monotonic method for large-scale non-negative least squares." Optimization Methods and Software 28, no. 5 (2013): 1012–39. http://dx.doi.org/10.1080/10556788.2012.656368.

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28

Copeland, Dylan M. "A negative-norm least-squares method for time-harmonic Maxwell equations." Journal of Mathematical Analysis and Applications 388, no. 1 (2012): 303–17. http://dx.doi.org/10.1016/j.jmaa.2011.09.004.

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29

Bl�mlinger, Martin. "Functions with a finite number of negative squares on compact groups." Archiv der Mathematik 67, no. 1 (1996): 70–75. http://dx.doi.org/10.1007/bf01196168.

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30

Khuri, Andrea I. "The Probability of a Negative Linear Combination of Independent Mean Squares." Biometrical Journal 36, no. 8 (1994): 899–910. http://dx.doi.org/10.1002/bimj.4710360802.

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31

Yang, YiChuan. "On the existence of directed rings and algebras with negative squares." Journal of Algebra 295, no. 2 (2006): 452–57. http://dx.doi.org/10.1016/j.jalgebra.2005.10.039.

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32

Moreno-Espinosa, Javier, José A. Nuñez-Mora, Jaime Beltrán-Godoy, and Leovardo Mata-Mata. "Effects of legal complexity on value-added tax (VAT) collection in Mexico from 1980 to 2019." PANORAMA ECONÓMICO 18, no. 38 (2023): 73–92. https://doi.org/10.29201/pe-ipn.v18i38.150.

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In this paper we investigate the dependence between Value Added Tax (VAT) and Tax Complexity. Three models are analyzed: Least Squares (LS), Generalized Least Squares (GLS), Iteratively Reweighted Least Squared (IRLS). In all the models the dependent variable is the VAT as proportion of GDP from 1980 to 2019 and the independent variables included the Structured, Interdependence, Entropy (As the Complexity measure of VAT Law). Dummies area included representing change of law in different years (1983, 1992, 1995, 2010). The Three models showed consistency in the negative sign and statistical significance of entropy.

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33

ŠIMIČEVIĆ, NEVEN. "IMPLICATIONS OF WEAK-INTERACTION SPACE DEFORMATION FOR NEUTRINO MASS MEASUREMENTS." International Journal of Modern Physics D 12, no. 09 (2003): 1615–26. http://dx.doi.org/10.1142/s021827180300392x.

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The negative values for the squares of both electron and muon neutrino masses obtained in recent experiments are explained as a possible consequence of a change in metric within the weak-interaction volume in the energy–momentum representation. Using a model inspired by a combination of the general theory of relativity and the theory of deformation for continuous media, it is shown that the negative value of the square of the neutrino mass can be obtained without violating allowed physical limits. The consequence is that the negative value is not necessarily unphysical.

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34

Elia, Nova, and Marselina Marselina. "Tingkat Pengangguran Berdasarkan Jumlah Penduduk, Pendapatan Perkapita, dan Investasi Asing di Indonesia Tahun 1996-2020." Studi Ekonomi dan Kebijakan Publik 1, no. 2 (2023): 123–35. http://dx.doi.org/10.35912/sekp.v1i2.1391.

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Abstract: Purpose: This study aims to analyze and prove the effect of population, per capita income, and foreign investment on the open unemployment rate in Indonesia, including population, per capita income, and foreign investment. Methodology: The method used in this research is multiple linear regression analysis with the least squares or ordinary least square calculation model, namely by minimizing the sum of the squares of the regression error or error. Results: The results of this study are that the number of people has a statistically significant positive effect; GDP Per Capita has a statistically significant negative effect; and foreign investment has a statistically significant negative effect on the Open Unemployment Rate. Conclusion: All independent variables have a significant effect on the open unemployment rate in Indonesia. Keywords: 1. Foreign Investment 2. Income Per Capita 3. Indonesia 4. Level Open Unemployment 5. Population

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35

Yi-zhuo LIANG, 梁一卓, 刘玲 Ling LIU, 彭力 Li PENG, et al. "Research on Weighted Bayesian Inversion Algorithm with Non-negative Least Squares Constraint." ACTA PHOTONICA SINICA 49, no. 10 (2020): 1029001. http://dx.doi.org/10.3788/gzxb20204910.1029001.

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36

Bruzual, Ramón, Marisela Domínguez, and Boris Lora. "Representation of generalized Toeplitz kernels with a finite number of negative squares." Acta Scientiarum Mathematicarum 78, no. 1-2 (2012): 111–28. http://dx.doi.org/10.1007/bf03651306.

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37

Ziyatdinov, Andrey, Joelle Mbatchou, Anthony Marcketta, et al. "Joint testing of rare variant burden scores using non-negative least squares." American Journal of Human Genetics 111, no. 10 (2024): 2139–49. http://dx.doi.org/10.1016/j.ajhg.2024.08.021.

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38

Magron, Victor, Mohab Safey El Din, and Markus Schweighofer. "Algorithms for weighted sum of squares decomposition of non-negative univariate polynomials." Journal of Symbolic Computation 93 (July 2019): 200–220. http://dx.doi.org/10.1016/j.jsc.2018.06.005.

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39

Nalci, Alican, Igor Fedorov, Maher Al-Shoukairi, Thomas T. Liu, and Bhaskar D. Rao. "Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem." IEEE Transactions on Signal Processing 66, no. 12 (2018): 3124–39. http://dx.doi.org/10.1109/tsp.2018.2824286.

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40

Zhang, Hongping, Peiliang Xu, Wenhui Han, Maorong Ge, and Chuang Shi. "Eliminating negative VTEC in global ionosphere maps using inequality-constrained least squares." Advances in Space Research 51, no. 6 (2013): 988–1000. http://dx.doi.org/10.1016/j.asr.2012.06.026.

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41

Bombara, Giuseppe, Marco Cococcioni, Beatrice Lazzerini, and Francesco Marcelloni. "S-NNLS: An efficient non-negative least squares algorithm for sequential data." International Journal for Numerical Methods in Biomedical Engineering 27, no. 5 (2011): 770–73. http://dx.doi.org/10.1002/cnm.1331.

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42

Hwang, Dosik, and Yiping P. Du. "Improved myelin water quantification using spatially regularized non-negative least squares algorithm." Journal of Magnetic Resonance Imaging 30, no. 1 (2009): 203–8. http://dx.doi.org/10.1002/jmri.21783.

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43

ACCARDI, L., and R. ROSCHIN. "RENORMALIZED SQUARES OF BOSON FIELDS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 08, no. 02 (2005): 307–26. http://dx.doi.org/10.1142/s0219025705001998.

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The standard renormalization procedure consists of introducing a cutoff and then trying to remove it by some limiting procedure. In Ref. 2 a new renormalization technique was introduced based on the idea of renormalizing a closed set of commutation relations and then finding a nontrivial representation for them. In Ref. 5 it was proved that, in the case of quadratic fields the new renormalization procedure leads to quadratic field operator which is gamma distributed in the quadratic vacuum (as one would intuitively expect from the "square" of a white noise) and to Meixner or Pascal distributed Poisson fields. It is natural to ask if the same result can be obtained with the usual cutoff and take-limit procedure. In this paper we prove that the answer to this question is negative. More precisely, we show that, independently of the choice of the cutoff (cf. Sec. 7), if a quadratic field admits a limit in the sense of mixed moments, then this limit will be Gaussian distributed in the vacuum and consequently the associated Poisson fields will have a Poisson distribution.

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44

Deng, Xinchen, Ramie Ali-Adeeb, Jeffrey L. Andrews, et al. "Monitor Ionizing Radiation-Induced Cellular Responses with Raman Spectroscopy, Non-Negative Matrix Factorization, and Non-Negative Least Squares." Applied Spectroscopy 74, no. 6 (2020): 701–11. http://dx.doi.org/10.1177/0003702820906221.

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Radiation therapy (RT) is one of the most commonly prescribed cancer treatments. New tools that can accurately monitor and evaluate individual patient responses would be a major advantage and lend to the implementation of personalized treatment plans. In this study, Raman spectroscopy (RS) was applied to examine radiation-induced cellular responses in H460, MCF7, and LNCaP cancer cell lines across different dose levels and times post-irradiation. Previous Raman data analysis was conducted using principal component analysis (PCA), which showed the ability to extract biological information of glycogen. In the current studies, the use of non-negative matrix factorization (NMF) allowed for the discovery of multiplexed biological information, specifically uncovering glycogen-like and lipid-like component bases. The corresponding scores of glycogen and previously unidentified lipids revealed the content variations of these two chemicals in the cellular data. The NMF decomposed glycogen and lipid-like bases were able to separate the cancer cell lines into radiosensitive and radioresistant groups. A further lipid phenotype investigation was also attempted by applying non-negative least squares (NNLS) to the lipid-like bases decomposed individually from three cell lines. Qualitative differences found in lipid weights for each lipid-like basis suggest the lipid phenotype differences in the three tested cancer cell lines. Collectively, this study demonstrates that the application of NMF and NNLS on RS data analysis to monitor ionizing radiation-induced cellular responses can yield multiplexed biological information on bio-response to RT not revealed by conventional chemometric approaches.

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45

Jaszczak, Agnieszka, Ewelina Pochodyła, and Beata Płoszaj-Witkowska. "Transformation of Green Areas in Central Squares after Revitalization: Evidence from Cittaslow Towns in Northeast Poland." Land 11, no. 4 (2022): 470. http://dx.doi.org/10.3390/land11040470.

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In recent years, unfavorable changes in the development of city squares can be observed in Poland due to poorly conducted revitalization. These changes include, among others, the elimination of greenery or its significant reduction and the introduction of impervious surfaces to the places formerly designated by forms of greenery. This article highlights the need to redesign the central squares in small towns as part of their revitalization, including three levels of approach: functional, ecological, and aesthetic. The research aimed to analyze the degree of changes in the development of green areas in central squares due to the implementation of revitalization projects. Central squares in four selected towns of Cittaslow from the Warmia and Mazury region (northeastern Poland) were selected as the research area. The methods of comparative analysis were used, considering the comparison of functions, space, and composition, as well as the “research by design” method, which was adopted. A comparative analysis of changes in the arrangement of squares and analysis of changes in greenery structure was performed. The results showed that, in two examples, the changes related to removing greenery from the squares were not on a large scale due to the compensation with new forms. However, the vegetation was removed in two squares without introducing new forms or compensating for the greenery to a minimum extent. This allowed us to answer whether the structure and various forms submitted in the greenery projects in the squares, so-called landscape architecture objects, correspond to the current ecological design requirements. From the perspective of landscape architecture and ecological design, it was positive to introduce a diversified vegetation structure and supplement it with medium and tall forms, as evidenced by the development of the square in Biskupiec. However, the results also showed a negative aspect of the revitalization conducted by removing the greenery and replacing it with an impermeable surface, for example, in the square in Bartoszyce. The final stage was to propose general guidelines for revitalizing squares and design proposals considering introducing plant forms in the squares in the four analyzed towns.

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46

Yamaguchi, Kyoko, Salma Abdelbaky, Lianbo Yu, Christopher C. Oakes, Lynne V. Abruzzo, and Kevin R. Coombes. "PLASMA: Partial LeAst Squares for Multiomics Analysis." Cancers 17, no. 2 (2025): 287. https://doi.org/10.3390/cancers17020287.

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Background/Objectives: Recent growth in the number and applications of high-throughput “omics” technologies has created a need for better methods to integrate multiomics data. Much progress has been made in developing unsupervised methods, but supervised methods have lagged behind. Methods: Here we present the first algorithm, PLASMA, that can learn to predict time-to-event outcomes from multiomics data sets, even when some samples have only been assayed on a subset of the omics data sets. PLASMA uses two layers of existing partial least squares algorithms to first select components that covary with the outcome and then construct a joint Cox proportional hazards model. Results: We apply PLASMA to the stomach adenocarcinoma (STAD) data from The Cancer Genome Atlas. We validate the model both by splitting the STAD data into training and test sets and by applying them to the subset of esophageal cancer (ESCA) containing adenocarcinomas. We use the other half of the ESCA data, which contains squamous cell carcinomas dissimilar to STAD, as a negative comparison. Our model successfully separates both the STAD test set (p = 2.73 × 10−8) and the independent ESCA adenocarcinoma data (p = 0.025) into high-risk and low-risk patients. It does not separate the negative comparison data set (ESCA squamous cell carcinomas, p = 0.57). The performance of the unified multiomics model is superior to that of individually trained models and is also superior to an unsupervised method (Multi-Omics Factor Analysis; MOFA), which finds latent factors to be used as putative predictors in a post hoc survival analysis. Conclusions: Many of the factors that contribute strongly to the PLASMA model can be justified from the biological literature.

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47

Zhang, T., W. Zhu, and R. McGraw. "Joint cluster and non-negative least squares analysis for aerosol mass spectrum data." Journal of Physics: Conference Series 125 (July 1, 2008): 012026. http://dx.doi.org/10.1088/1742-6596/125/1/012026.

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Paatero, Pentti. "A weighted non-negative least squares algorithm for three-way ‘PARAFAC’ factor analysis." Chemometrics and Intelligent Laboratory Systems 38, no. 2 (1997): 223–42. http://dx.doi.org/10.1016/s0169-7439(97)00031-2.

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Liu, Hongwei, Xiangli Li, and Xiuyun Zheng. "Solving non-negative matrix factorization by alternating least squares with a modified strategy." Data Mining and Knowledge Discovery 26, no. 3 (2012): 435–51. http://dx.doi.org/10.1007/s10618-012-0265-y.

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Song, Yingchun, Jianjun Zhu, and Zhiwei Li. "The least-squares estimation of adjustment model constrained by some non-negative parameters." Survey Review 42, no. 315 (2010): 62–71. http://dx.doi.org/10.1179/003962610x12572516251367.

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