516:
82:
1120:
1381:
eye but extends beyond it; we believe in what we see through light microscopes because it agrees with what we see through magnifying glasses but extends beyond it; and similarly for electron microscopes. Such arguments are widely used in biology in extrapolating from animal studies to humans and from pilot studies to a broader population.
1380:
Extrapolation arguments are informal and unquantified arguments which assert that something is probably true beyond the range of values for which it is known to be true. For example, we believe in the reality of what we see through magnifying glasses because it agrees with what we see with the naked
590:
extrapolation is a method suitable for any distribution that has a tendency to be exponential, but with accelerating or decelerating factors. This method has been used successfully in providing forecast projections of the growth of HIV/AIDS in the UK since 1987 and variant CJD in the UK for a number
527:
A polynomial curve can be created through the entire known data or just near the end (two points for linear extrapolation, three points for quadratic extrapolation, etc.). The resulting curve can then be extended beyond the end of the known data. Polynomial extrapolation is typically done by means
1371:
log(N) times even with fast
Fourier transform (FFT). There exists an algorithm, it analytically calculates the contribution from the part of the extrapolated data. The calculation time can be omitted compared with the original convolution calculation. Hence with this algorithm the calculations of a
1206:
This divergence is a specific property of extrapolation methods and is only circumvented when the functional forms assumed by the extrapolation method (inadvertently or intentionally due to additional information) accurately represent the nature of the function being extrapolated. For particular
543:
High-order polynomial extrapolation must be used with due care. For the example data set and problem in the figure above, anything above order 1 (linear extrapolation) will possibly yield unusable values; an error estimate of the extrapolated value will grow with the degree of the polynomial
73:
to project, extend, or expand known experience into an area not known or previously experienced so as to arrive at a (usually conjectural) knowledge of the unknown (e.g. a driver extrapolates road conditions beyond his sight while driving). The extrapolation method can be applied in the
909:
133:
Linear extrapolation means creating a tangent line at the end of the known data and extending it beyond that limit. Linear extrapolation will only provide good results when used to extend the graph of an approximately linear function or not too far beyond the known data.
124:
of the process that created the existing data points. Some experts have proposed the use of causal forces in the evaluation of extrapolation methods. Crucial questions are, for example, if the data can be assumed to be continuous, smooth, possibly periodic, etc.
1366:
The extrapolated data often convolute to a kernel function. After data is extrapolated, the size of data is increased N times, here N is approximately 2–3. If this data needs to be convoluted to a known kernel function, the numerical calculations will increase
431:
1115:{\displaystyle {\begin{aligned}{\text{f}}(m,n,d_{1},d_{2})&={\text{round}}\left((n\cdot d_{1}-m)+(m\cdot d_{2})\right)\\&={\text{round}}\left((3\times 1.5-5\right)+(5\times 1.66))=8\end{aligned}}}
914:
1207:
problems, this additional information may be available, but in the general case, it is impossible to satisfy all possible function behaviors with a workably small set of potential behavior.
1339:
1130:
Typically, the quality of a particular method of extrapolation is limited by the assumptions about the function made by the method. If the method assumes the data are smooth, then a non-
713:
599:
Can be created with 3 points of a sequence and the "moment" or "index", this type of extrapolation have 100% accuracy in predictions in a big percentage of known series database (OEIS).
497:
903:
865:
1372:
convolution using the extrapolated data is nearly not increased. This is referred as the fast extrapolation. The fast extrapolation has been applied to CT image reconstruction.
827:
655:
220:
769:
1264:
266:
579:
will not rejoin itself, but may curve back relative to the X-axis. This type of extrapolation could be done with a conic sections template (on paper) or with a computer.
162:
109:
1137:
In terms of complex time series, some experts have discovered that extrapolation is more accurate when performed through the decomposition of causal forces.
274:
1281:
is mapped to the origin and vice versa. Care must be taken with this transform however, since the original function may have had "features", for example
1818:
Claude
Brezinski and Michela Redivo-Zaglia : "Extrapolation and Rational Approximation", Springer Nature, Switzerland, ISBN 9783030584177, (2020).
1735:
591:
of years. Another study has shown that extrapolation can produce the same quality of forecasting results as more complex forecasting strategies.
1679:
1388:
arguments, extrapolation arguments may be strong or weak depending on such factors as how far the extrapolation goes beyond the known range.
57:, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to
1140:
Even for proper assumptions about the function, the extrapolation can diverge severely from the function. The classic example is truncated
1815:
Avram Sidi: "Practical
Extrapolation Methods: Theory and Applications", Cambridge University Press, ISBN 0-521-66159-5 (2003).
1564:
1498:
17:
1640:
1745:
1765:
1402:
85:
Example illustration of the extrapolation problem, consisting of assigning a meaningful value at the blue box, at
661:
1275:
443:
1680:"Reconstruction from truncated projections using mixed extrapolations of exponential and quadratic functions"
1442:
871:
833:
1761:
519:
Lagrange extrapolations of the sequence 1,2,3. Extrapolating by 4 leads to a polynomial of minimal degree (
515:
499:). It is possible to include more than two points, and averaging the slope of the linear interpolant, by
1316:. In effect, a set of data from a small region is used to extrapolate a function onto a larger region.
775:
608:
167:
39:
719:
563:
can be created using five points near the end of the known data. If the conic section created is an
1581:
1226:
1710:
1427:
1286:
225:
1515:
1485:
1452:
1331:
1282:
75:
31:
65:
and a higher risk of producing meaningless results. Extrapolation may also mean extension of a
1831:
1576:
1510:
1324:
1301:
1181:
1149:
529:
61:, which produces estimates between known observations, but extrapolation is subject to greater
1809:
1351:
1320:
1313:
1293:
545:
35:
1457:
1355:
437:
140:
1481:
8:
1422:
1417:
1305:
500:
88:
1335:
1660:
1594:
1528:
533:
426:{\displaystyle y(x_{*})=y_{k-1}+{\frac {x_{*}-x_{k-1}}{x_{k}-x_{k-1}}}(y_{k}-y_{k-1}).}
1199: = 0, but will produce extrapolations that eventually diverge away from the
1741:
1702:
1656:
1278:
504:
69:, assuming similar methods will be applicable. Extrapolation may also apply to human
1777:
1694:
1664:
1652:
1598:
1586:
1532:
1520:
1437:
1407:
1274:
with the part of the complex plane outside of the unit circle. In particular, the
1216:
540:
that fits the data. The resulting polynomial may be used to extrapolate the data.
1641:"Decomposition by Causal Forces: A Procedure for Forecasting Complex Time Series"
1432:
1131:
1447:
1385:
503:-like techniques, on the data points chosen to be included. This is similar to
1619:
1565:"Forecasting by Extrapolation: Conclusions from Twenty-Five Years of Research"
1825:
1638:
1620:"Probnet: Geometric Extrapolation of Integer Sequences with error prediction"
1462:
1267:
1220:
560:
537:
58:
1781:
1706:
1524:
1347:
1309:
1297:
1141:
587:
1698:
1172: = 0 however, the extrapolation moves arbitrarily away from the
571:, when extrapolated it will loop back and rejoin itself. An extrapolated
1590:
1412:
1397:
1271:
1195: = 0 will produce better agreement over a larger interval near
62:
46:
81:
66:
70:
54:
1292:
Another problem of extrapolation is loosely related to the problem of
1343:
576:
1499:"Causal Forces: Structuring Knowledge for Time-series Extrapolation"
572:
1737:
Across the
Boundaries: Extrapolation in Biology and Social Science
1677:
1156: = 0, we may estimate that the function behaves as sin(
564:
594:
120:
A sound choice of which extrapolation method to apply relies on
568:
1496:
1551:
1289:, at infinity that were not evident from the sampled data.
1766:"Arguments whose strength depends on continuous variation"
1639:
J. Scott
Armstrong; Fred Collopy; J. Thomas Yokum (2004).
1168: = 0, this is an excellent estimate. Away from
27:
Method for estimating new data outside known data points
1327:
features that were not evident from the initial data.
1219:, a problem of extrapolation may be converted into an
602:
Example of extrapolation with error prediction :
1229:
912:
874:
836:
778:
722:
664:
611:
446:
277:
228:
170:
143:
91:
1562:
1258:
1114:
897:
859:
821:
763:
707:
649:
491:
425:
260:
214:
156:
103:
1203:-axis even faster than the linear approximation.
1823:
1152:. For instance, taking only data from near the
1678:Shuangren Zhao; Kang Yang; Xintie Yang (2011).
1475:
1187:Taking more terms in the power series of sin(
595:Geometric Extrapolation with error prediction
1266:. This transform exchanges the part of the
1184:. I.e., the error increases without bound.
30:For the journal of speculative fiction, see
708:{\displaystyle {f_{1}(x,y)={\frac {x}{y}}}}
268:, linear extrapolation gives the function:
1806:Extrapolation Methods. Theory and Practice
1375:
1580:
1514:
1497:J. Scott Armstrong; Fred Collopy (1993).
492:{\displaystyle x_{k-1}<x_{*}<x_{k}}
137:If the two data points nearest the point
1760:
1350:that are divergent outside the original
1342:as extrapolation methods that lead to a
514:
80:
1687:Journal of X-Ray Science and Technology
1210:
14:
1824:
1617:
898:{\displaystyle n={\text{sequence}}(3)}
860:{\displaystyle m={\text{sequence}}(5)}
1733:
34:. For the John McLaughlin album, see
1645:International Journal of Forecasting
1304:is expanded at one of its points of
1740:. Oxford: Oxford University Press.
1354:. In this case, one often obtains
1340:Levin-type sequence transformations
24:
1223:problem by the change of variable
822:{\displaystyle {d_{2}=f_{1}(5,3)}}
650:{\displaystyle {\text{sequence}}=}
544:extrapolation. This is related to
25:
1843:
215:{\displaystyle (x_{k-1},y_{k-1})}
1657:10.1016/j.ijforecast.2004.05.001
1403:Minimum polynomial extrapolation
764:{\displaystyle d_{1}=f_{1}(3,2)}
38:. For the Apple TV+ series, see
582:
1754:
1727:
1671:
1632:
1611:
1556:
1545:
1490:
1259:{\displaystyle {\hat {z}}=1/z}
1236:
1099:
1096:
1084:
1058:
1030:
1011:
1005:
980:
960:
922:
892:
886:
854:
848:
815:
803:
758:
746:
688:
676:
644:
620:
417:
385:
294:
281:
255:
229:
209:
171:
13:
1:
1799:
1443:Extrapolation domain analysis
1134:will be poorly extrapolated.
510:
261:{\displaystyle (x_{k},y_{k})}
532:or using Newton's method of
7:
1563:J. Scott Armstrong (1984).
1391:
111:, given the red data points
10:
1848:
1164:. In the neighborhood of
1125:
40:Extrapolations (TV series)
29:
1552:AIDSCJDUK.info Main Index
128:
115:
1468:
1428:Richardson extrapolation
1332:sequence transformations
554:
1453:Interior reconstruction
1376:Extrapolation arguments
1361:
1150:trigonometric functions
1144:representations of sin(
436:(which is identical to
164:to be extrapolated are
76:interior reconstruction
32:Extrapolation (journal)
1812:, North-Holland, 1991.
1782:10.22329/il.v33i1.3610
1734:Steel, Daniel (2007).
1525:10.1002/for.3980120205
1503:Journal of Forecasting
1296:, where (typically) a
1260:
1116:
899:
861:
823:
765:
709:
651:
530:Lagrange interpolation
524:
493:
427:
262:
216:
158:
112:
105:
1699:10.3233/XST-2011-0284
1356:rational approximants
1352:radius of convergence
1321:analytic continuation
1314:radius of convergence
1294:analytic continuation
1261:
1117:
900:
862:
824:
766:
710:
652:
518:
494:
428:
263:
217:
159:
157:{\displaystyle x_{*}}
106:
84:
36:Extrapolation (album)
1808:by C. Brezinski and
1591:10.1287/inte.14.6.52
1458:Extreme value theory
1330:Also, one may use
1300:representation of a
1227:
1211:In the complex plane
910:
872:
834:
776:
720:
662:
609:
444:
438:linear interpolation
275:
226:
168:
141:
89:
18:Linear extrapolation
1423:Regression analysis
1418:Prediction interval
1323:can be thwarted by
104:{\displaystyle x=7}
1256:
1112:
1110:
895:
857:
819:
761:
705:
647:
546:Runge's phenomenon
534:finite differences
525:
489:
423:
258:
212:
154:
113:
101:
1336:Padé approximants
1279:point at infinity
1239:
1180:) remains in the
1051:
973:
920:
884:
846:
702:
615:
505:linear prediction
383:
122:a prior knowledge
16:(Redirected from
1839:
1810:M. Redivo Zaglia
1793:
1792:
1790:
1788:
1758:
1752:
1751:
1731:
1725:
1724:
1722:
1721:
1715:
1709:. Archived from
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1675:
1669:
1668:
1636:
1630:
1629:
1627:
1626:
1615:
1609:
1608:
1606:
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1560:
1554:
1549:
1543:
1542:
1540:
1539:
1518:
1494:
1488:
1479:
1438:Trend estimation
1408:Multigrid method
1370:
1276:compactification
1265:
1263:
1262:
1257:
1252:
1241:
1240:
1232:
1217:complex analysis
1176:-axis while sin(
1121:
1119:
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1113:
1111:
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1029:
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1784:
1762:Franklin, James
1759:
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1728:
1719:
1717:
1713:
1682:
1676:
1672:
1637:
1633:
1624:
1622:
1618:V. Nos (2021).
1616:
1612:
1603:
1601:
1582:10.1.1.715.6481
1561:
1557:
1550:
1546:
1537:
1535:
1495:
1491:
1486:Merriam–Webster
1480:
1476:
1471:
1433:Static analysis
1394:
1378:
1368:
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1248:
1231:
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1132:smooth function
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1794:
1770:Informal Logic
1753:
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1726:
1670:
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1610:
1555:
1544:
1509:(2): 103–115.
1489:
1473:
1472:
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1467:
1466:
1465:
1460:
1455:
1450:
1448:Dead reckoning
1445:
1440:
1435:
1430:
1425:
1420:
1415:
1410:
1405:
1400:
1393:
1390:
1386:slippery slope
1377:
1374:
1363:
1360:
1312:with a larger
1255:
1251:
1247:
1244:
1238:
1235:
1212:
1209:
1160:) ~
1148:) and related
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9:
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3:
2:
1844:
1833:
1832:Extrapolation
1830:
1829:
1827:
1817:
1814:
1811:
1807:
1804:
1803:
1783:
1779:
1775:
1771:
1767:
1763:
1757:
1749:
1747:9780195331448
1743:
1739:
1738:
1730:
1716:on 2017-09-29
1712:
1708:
1704:
1700:
1696:
1693:(2): 155–72.
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1508:
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1493:
1487:
1483:
1482:Extrapolation
1478:
1474:
1464:
1463:Interpolation
1461:
1459:
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1454:
1451:
1449:
1446:
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1341:
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1322:
1317:
1315:
1311:
1308:to produce a
1307:
1303:
1299:
1295:
1290:
1288:
1287:singularities
1284:
1280:
1277:
1273:
1269:
1268:complex plane
1253:
1249:
1245:
1242:
1233:
1222:
1221:interpolation
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1208:
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1711:the original
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1613:
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1516:10.1.1.42.40
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1298:power series
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1484:, entry at
1413:Overfitting
1398:Forecasting
1306:convergence
1272:unit circle
1270:inside the
63:uncertainty
47:mathematics
1800:References
1720:2014-06-03
1625:2023-03-14
1604:2012-01-10
1569:Interfaces
1538:2012-01-10
1285:and other
511:Polynomial
501:regression
71:experience
55:estimation
1651:: 25–36.
1577:CiteSeerX
1511:CiteSeerX
1344:summation
1237:^
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1091:×
1071:−
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1018:⋅
1000:−
987:⋅
577:hyperbola
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1826:Category
1764:(2013).
1707:21606580
1392:See also
1325:function
1302:function
1182:interval
883:sequence
845:sequence
614:sequence
573:parabola
1787:29 June
1665:8816023
1599:5805521
1533:3233162
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1126:Quality
565:ellipse
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129:Linear
116:Method
67:method
1714:(PDF)
1683:(PDF)
1661:S2CID
1595:S2CID
1529:S2CID
1469:Notes
1384:Like
1334:like
1283:poles
1050:round
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555:Conic
1789:2021
1742:ISBN
1703:PMID
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