1594:
subjective and unsuitable for relatively high gas and liquid flow rates. Therefore, the machine learning methods are proposed by many authors. The methods are applied to differential pressure data collected during a calibration process. The method of elastic maps provided a 2D map, where the area of each regime is represented. The comparison with some other machine learning methods is presented in Table 1 for various pipe diameters and pressure.
24:
1532:
1563:
Most important applications of the method and free software are in bioinformatics for exploratory data analysis and visualisation of multidimensional data, for data visualisation in economics, social and political sciences, as an auxiliary tool for data mapping in geographic informational systems and
38:
data: a) Configuration of nodes and 2D Principal
Surface in the 3D PCA linear manifold. The dataset is curved and can not be mapped adequately on a 2D principal plane; b) The distribution in the internal 2D non-linear principal surface coordinates (ELMap2D) together with an estimation of the density
1593:
in a pipe. There are various regimes: Single phase water or air flow, Bubbly flow, Bubbly-slug flow, Slug flow, Slug-churn flow, Churn flow, Churn-annular flow, and
Annular flow. The simplest and most common method used to identify the flow regime is visual observation. This approach is, however,
969:
39:
of points; c) The same as b), but for the linear 2D PCA manifold (PCA2D). The “basal” breast cancer subtype is visualized more adequately with ELMap2D and some features of the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the
1769:
Wang, Y., Klijn, J.G., Zhang, Y., Sieuwerts, A.M., Look, M.P., Yang, F., Talantov, D., Timmermans, M., Meijer-van Gelder, M.E., Yu, J. et al.: Gene expression profiles to predict distant metastasis of lymph-node-negative primary breast cancer. Lancet 365, 671–679 (2005);
792:
430:
1026:
For example, on the 2D rectangular grid the elastic edges are just vertical and horizontal edges (pairs of closest vertices) and the bending ribs are the vertical or horizontal triplets of consecutive (closest) vertices.
800:
310:
1912:, Knowledge-Based Intelligent Information and Engineering Systems, B. Apolloni, R.J. Howlett and L. Jain (eds.), Lecture Notes in Computer Science, Vol. 4693, Springer: Berlin – Heidelberg, 2010, 635-641.
1760:, In: Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques, Olivas E.S. et al. Eds. Information Science Reference, IGI Global: Hershey, PA, USA, 2009. 28–59.
627:
436:
which is the energy of the springs with unit elasticity which connect each data point with its host node. It is possible to apply weighting factors to the terms of this sum, for example to reflect the
660:
1567:
The method is applied in quantitative biology for reconstructing the curved surface of a tree leaf from a stack of light microscopy images. This reconstruction is used for quantifying the
532:
177:
1402:
1276:
1124:
1084:
1180:
236:
112:
212:
147:
1345:
1309:
475:
1497:
1457:
993:
58:
embedded in the data space. This system approximates a low-dimensional manifold. The elastic coefficients of this system allow the switch from completely unstructured
1207:
1517:
1477:
1013:
74:
analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates. The method was developed by
1433:
1365:
1227:
1148:
651:
556:
325:
1807:
1936:
Measurement of gas and liquid flow rates in two-phase pipe flows by the application of machine learning techniques to differential pressure signals
1833:
964:{\displaystyle U_{G}={\frac {1}{2}}\mu \sum _{({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})\in G}\|{\bf {w}}_{i}-2{\bf {w}}_{j}+{\bf {w}}_{k}\|^{2}}
1854:
241:
1869:, In: B. Beliczynski et al. (Eds.), Lecture Notes in Computer Sciences, Vol. 4432, Springer: Berlin – Heidelberg 2007, 355–363.
1412:
51:
1896:
1578:
Recently, the method is adapted as a support tool in the decision process underlying the selection, optimization, and management of
1722:. The hybrid technology was developed for engineering applications. In this technology, elastic maps are used in combination with
561:
1819:
Michael Kass, Andrew Witkin, Demetri
Terzopoulos, Snakes: Active contour models, Int.J. Computer Vision, 1988 vol 1-4 pp.321-331
1535:
Application of principal curves build by the elastic maps method: Nonlinear quality of life index. Points represent data of the
1727:
1555:
incidence. Different forms and colors correspond to various geographical locations and years. Red bold line represents the
1923:
Identification of flow regime in vertical upward air–water pipe flow using differential pressure signals and elastic maps
787:{\displaystyle U_{E}={\frac {1}{2}}\lambda \sum _{({\bf {w}}_{i},{\bf {w}}_{j})\in E}\|{\bf {w}}_{i}-{\bf {w}}_{j}\|^{2}}
1972:
1723:
1230:
441:
75:
63:
1523:: one starts from a small number of nodes and gradually adds new nodes. Each epoch goes with its own number of nodes.
1439:
strategy is used. This strategy starts with a rigid grids (small length, small bending and large elasticity modules
483:
17:
1711:
1951:, Series Intelligent Systems Reference Library, Volume 99, Springer International Publishing, Switzerland 2016.
28:
1519:). The training goes in several epochs, each epoch with its own grid rigidness. Another adaptive strategy is
1790:
1715:
1540:
1127:
1015:
are the stretching and bending moduli respectively. The stretching energy is sometimes referred to as the
67:
43:
s algorithm. Data are available for public competition. Software is available for free non-commercial use.
152:
1897:
Semi-automated 3D leaf reconstruction and analysis of trichome patterning from light microscopic images
1370:
1244:
1092:
1034:
1967:
1922:
1161:
217:
214:(if there are several closest nodes then one takes the node with the smallest number). The data set
93:
1734:
1719:
1579:
186:
121:
70:, this system effectively approximates non-linear principal manifolds. This approach is based on a
1883:
1803:
437:
59:
1317:
1281:
447:
1850:
1575:
and their patterning, which is a marker of the capability of a plant to resist to pathogenes.
1482:
1442:
978:
1935:
1586:
1548:
1435:. For improving the approximation various additional methods are proposed. For example, the
1185:
1910:
Portfolio optimization through elastic maps: Some evidence from the
Italian stock exchange
1708:
1590:
1544:
1502:
1462:
998:
115:
55:
425:{\displaystyle D={\frac {1}{2}}\sum _{j=1}^{k}\sum _{s\in K_{j}}\|s-{\bf {w}}_{j}\|^{2}}
23:
1867:
Detection of Gene
Expressions in Microarrays by Applying Iteratively Elastic Neural Net
1830:
Principal manifolds and graphs in practice: from molecular biology to dynamical systems
1536:
1418:
1350:
1212:
1133:
636:
541:
1961:
32:
1585:
The method of elastic maps has been systematically tested and compared with several
66:(for high bending and low stretching modules). With some intermediate values of the
1786:
1552:
1229:
is a linear problem with the sparse matrix of coefficients. Therefore, similar to
1948:
1909:
1130:
of the elastic map, i.e. its location is such that it minimizes the total energy
1771:
16:"Elastic net" redirects here. For the statistical regularization technique, see
1531:
1949:
Computational
Intelligence Paradigms in Economic and Financial Decision Making
35:
480:
On the set of nodes an additional structure is defined. Some pairs of nodes,
1572:
71:
1539:
171 countries in 4-dimensional space formed by the values of 4 indicators:
1789:- Multidimensional Data Visualization Tool (free for non-commercial use).
1589:
methods on the applied problem of identification of the flow regime of a
1568:
1849:, LNCSE 58, Springer: Berlin – Heidelberg – New York, 2007.
1846:
1234:
1866:
305:{\displaystyle K_{j}=\{s\ |\ {\bf {w}}_{j}{\mbox{ is a host of }}s\}}
1879:
1847:
Principal
Manifolds for Data Visualisation and Dimension Reduction
1829:
1757:
1737:(SOMs) in applications to economic and financial decision-making.
1733:
The textbook provides a systematic comparison of elastic maps and
1895:
H. Failmezger, B. Jaegle, A. Schrader, M. HĂĽlskamp, A. Tresch.,
79:
62:(zero elasticity) to the estimators located closely to linear
1886:, Badie, B., Berg-Schlosser, D., Morlino, L. A. (Eds.), 2011.
1938:, International Journal of Multiphase Flow 67(2014), 106-117
1167:
223:
164:
99:
1925:, International Journal of Multiphase Flow 61 (2014) 62-72.
622:{\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})}
1845:
A.N. Gorban, B. Kegl, D. Wunsch, A. Zinovyev (Eds.),
290:
54:. By their construction, they are a system of elastic
1505:
1485:
1465:
1445:
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1353:
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1284:
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1215:
1188:
1164:
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663:
639:
564:
544:
486:
450:
328:
244:
220:
189:
155:
124:
96:
27:
Linear PCA versus nonlinear
Principal Manifolds for
1880:
Data visualization in political and social sciences
1899:, PLoS Computational Biology, 2013, 9(4):e1003029.
1511:
1491:
1479:coefficients) and finishes with soft grids (small
1471:
1451:
1427:
1396:
1359:
1339:
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786:
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469:
424:
304:
230:
206:
171:
141:
106:
1884:"International Encyclopedia of Political Science"
1019:, while the bending energy is referred to as the
1598:TABLE 1. Flow regime identification accuracy (%)
118:. Elastic map is represented by a set of nodes
1564:for visualisation of data of various nature.
527:{\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j})}
8:
1865:M. ChacĂłn, M. LĂ©vano, H. Allende, H. Nowak,
1781:
1779:
1391:
1374:
1334:
1321:
1298:
1285:
1265:
1248:
1113:
1096:
1031:The total energy of the elastic map is thus
952:
897:
775:
740:
464:
451:
413:
389:
299:
258:
1209:, minimization of the quadratic functional
1752:
1750:
1600:of different machine learning algorithms
1504:
1484:
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95:
1808:Institut des Hautes Études Scientifiques
1596:
1530:
22:
1834:International Journal of Neural Systems
1746:
114:be a data set in a finite-dimensional
7:
1810:), Bures-Sur-Yvette, ĂŽle-de-France.
172:{\displaystyle s\in {\mathcal {S}}}
1413:expectation-maximization algorithm
1158:For a given splitting of dataset
1154:Expectation-maximization algorithm
149:in the same space. Each datapoint
52:nonlinear dimensionality reduction
14:
1397:{\displaystyle \{{\bf {w}}_{j}\}}
1271:{\displaystyle \{{\bf {w}}_{j}\}}
1119:{\displaystyle \{{\bf {w}}_{j}\}}
1836:, Vol. 20, No. 3 (2010) 219–232.
1380:
1254:
1102:
1079:{\displaystyle U=D+U_{E}+U_{G}.}
940:
923:
903:
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858:
841:
763:
746:
718:
701:
605:
588:
571:
510:
493:
401:
278:
193:
128:
1730:(ICA) and backpropagation ANN.
82:and A.A. Pitenko in 1996–1998.
1758:Principal Graphs and Manifolds
1756:A. N. Gorban, A. Y. Zinovyev,
1728:Independent Component Analysis
1415:guarantees a local minimum of
1237:, a splitting method is used:
1175:{\displaystyle {\mathcal {S}}}
886:
835:
729:
695:
616:
565:
521:
487:
268:
231:{\displaystyle {\mathcal {S}}}
107:{\displaystyle {\mathcal {S}}}
1:
444:of any subset of data points
207:{\displaystyle {\bf {w}}_{j}}
142:{\displaystyle {\bf {w}}_{j}}
1724:Principal Component Analysis
1559:, approximating the dataset.
1231:principal component analysis
633:. Call this set of triplets
442:probability density function
1828:A. N. Gorban, A. Zinovyev,
1989:
1934:H. Shaban, S. Tavoularis,
1921:H. Shaban, S. Tavoularis,
1712:artificial neural networks
1089:The position of the nodes
558:. Some triplets of nodes,
183:, namely the closest node
18:Elastic net regularization
15:
1707:Here, ANN stands for the
1340:{\displaystyle \{K_{j}\}}
1304:{\displaystyle \{K_{j}\}}
657:The stretching energy is
538:. Call this set of pairs
470:{\displaystyle \{s_{i}\}}
1541:gross product per capita
1492:{\displaystyle \lambda }
1452:{\displaystyle \lambda }
1407:If no change, terminate.
988:{\displaystyle \lambda }
292: is a host of
238:is divided into classes
68:elasticity coefficients
1716:support vector machine
1560:
1513:
1493:
1473:
1453:
1429:
1398:
1361:
1341:
1305:
1272:
1223:
1203:
1176:
1144:
1128:mechanical equilibrium
1120:
1080:
1009:
989:
965:
797:The bending energy is
788:
647:
623:
552:
528:
471:
426:
365:
306:
232:
208:
173:
143:
108:
44:
1714:, SVM stands for the
1534:
1514:
1494:
1474:
1454:
1430:
1399:
1362:
1342:
1306:
1273:
1224:
1204:
1202:{\displaystyle K_{j}}
1177:
1145:
1126:is determined by the
1121:
1081:
1010:
990:
966:
789:
648:
624:
553:
529:
472:
427:
345:
307:
233:
209:
174:
144:
109:
86:Energy of elastic map
26:
1735:self-organizing maps
1720:self-organizing maps
1580:financial portfolios
1512:{\displaystyle \mu }
1503:
1483:
1472:{\displaystyle \mu }
1463:
1443:
1419:
1371:
1351:
1318:
1282:
1245:
1213:
1186:
1162:
1134:
1093:
1035:
1008:{\displaystyle \mu }
999:
979:
801:
661:
637:
562:
542:
484:
448:
326:
319:D is the distortion
317:approximation energy
242:
218:
187:
153:
122:
94:
1973:Dimension reduction
1804:ViDaExpert overview
1601:
534:, are connected by
50:provide a tool for
1597:
1571:distances between
1561:
1509:
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1469:
1449:
1425:
1394:
1357:
1337:
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1219:
1199:
1172:
1140:
1116:
1076:
1005:
985:
961:
896:
784:
739:
643:
619:
548:
524:
467:
438:standard deviation
422:
388:
302:
294:
228:
204:
169:
139:
104:
60:k-means clustering
45:
1855:978-3-540-73749-0
1705:
1704:
1428:{\displaystyle U}
1360:{\displaystyle U}
1222:{\displaystyle U}
1143:{\displaystyle U}
830:
825:
690:
685:
646:{\displaystyle G}
551:{\displaystyle E}
366:
343:
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266:
1980:
1952:
1945:
1939:
1932:
1926:
1919:
1913:
1906:
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1843:
1837:
1826:
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1817:
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1783:
1774:
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1754:
1616:Higher pressure
1613:Larger diameter
1602:
1587:machine learning
1549:infant mortality
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1498:
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1709:backpropagation
1599:
1591:gas-liquid flow
1557:principal curve
1545:life expectancy
1529:
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872:
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804:
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743:
715:
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664:
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539:
507:
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398:
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324:
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245:
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151:
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125:
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119:
116:Euclidean space
92:
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88:
21:
12:
11:
5:
1986:
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1838:
1821:
1812:
1795:
1791:Institut Curie
1775:
1762:
1745:
1744:
1742:
1739:
1718:, SOM for the
1703:
1702:
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1291:
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1262:
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1218:
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1192:
1169:
1155:
1152:
1139:
1115:
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13:
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1924:
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1915:
1911:
1905:
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1898:
1892:
1889:
1885:
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1878:A. Zinovyev,
1875:
1872:
1868:
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1802:A. Zinovyev,
1799:
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1785:A. Zinovyev,
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1030:
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1022:
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982:
956:
946:
934:
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917:
914:
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869:
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769:
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724:
712:
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691:
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674:
669:
665:
656:
655:
654:
640:
632:
611:
599:
594:
582:
577:
545:
537:
536:elastic edges
516:
504:
499:
478:
459:
455:
443:
439:
417:
407:
395:
392:
382:
378:
374:
371:
367:
361:
356:
353:
350:
346:
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332:
329:
322:
321:
320:
318:
313:
296:
284:
261:
255:
250:
246:
199:
182:
159:
156:
134:
117:
85:
83:
81:
80:A.Y. Zinovyev
77:
73:
69:
65:
64:PCA manifolds
61:
57:
53:
49:
42:
37:
34:
33:breast cancer
30:
29:visualization
25:
19:
1943:
1930:
1917:
1904:
1891:
1874:
1861:
1841:
1824:
1815:
1798:
1765:
1732:
1706:
1689:SOM (large)
1672:SOM (small)
1621:Elastic map
1607:Calibration
1584:
1577:
1566:
1562:
1556:
1553:tuberculosis
1527:Applications
1520:
1436:
1410:
1157:
1088:
1025:
1020:
1016:
974:
631:bending ribs
630:
535:
479:
435:
316:
314:
180:
89:
48:Elastic maps
47:
46:
40:
1968:Data mining
1882:, In: SAGE
1772:Data online
1521:growing net
1182:in classes
76:A.N. Gorban
41:elastic map
1962:Categories
1947:M. Resta,
1908:M. Resta,
1787:ViDaExpert
1741:References
1314:For given
1241:For given
1021:thin plate
72:mechanical
36:microarray
1573:trichomes
1507:μ
1487:λ
1467:μ
1447:λ
1437:softening
1367:and find
1347:minimize
1003:μ
983:λ
953:‖
915:−
898:‖
890:∈
832:∑
828:μ
776:‖
758:−
741:‖
733:∈
692:∑
688:λ
414:‖
396:−
390:‖
375:∈
368:∑
347:∑
181:host node
160:∈
1806:, IHES (
1793:, Paris.
1610:Testing
1569:geodesic
1017:membrane
1726:(PCA),
1235:k-means
629:, form
440:of the
56:springs
1853:
1023:term.
975:where
273:
265:
179:has a
1701:84.1
1698:82.1
1695:94.6
1684:88.6
1681:83.6
1678:94.2
1675:94.9
1667:70.5
1664:61.7
1661:88.5
1650:70.5
1647:76.2
1644:89.2
1641:99.1
1627:98.2
1411:This
1278:find
1851:ISBN
1692:100
1658:100
1655:SVM
1638:ANN
1633:100
1630:100
1624:100
1499:and
1459:and
995:and
315:The
90:Let
1233:or
31:of
1964::
1832:,
1778:^
1749:^
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1537:UN
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