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I stumbled over the second of the two algebraic functions in the example section: x^n/(x^n + (1-x)^n). as far as I see this is not a sigmoid function. It violates the definition of a sigmoid given in the section
Definition. Notably, the derivative is not non-negative and it has multiple inflection
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Perhaps what is meant is that the second derivative (curvature) has a local minimum and maximum? BTW, I do not agree that the function has global extrema, because as I learned them and as the article on them states, they are points in the domain of the function and are always also local extrema. This
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The
Properties section should be consistent with the Definition section. The Definition says that the derivative must be positive. The Properties section says that it must be either non-positive or non-negative. It's unnecessary anyway to restate properties that are explicit in the definition (other
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In particular see "Chapter 4: Artificial Neural
Networks" where the Boolean abilities of "perceptrons" are defined as well. I happened onto the tricky business of adding three folded planes together to make a "triangle" (and passing them through a second-layer sigmoid) because a neural net showed me
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Two of the above Z(t) but with reversed signs and slightly offset with different thresholds added together make a line, like a mountain range on a map, or a canyon. However, If you put three of these plateaus i.e. "folded sheets" (for a total of just 3 sigmoids) on the X-Y plane and get the signs of
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1) but the more gain you put in the more difficult the design becomes. For an OR you need a range of -0.25 to +2.25 (i.e. if inputs are "a" and "b" that vary from 0 to 1, add them and squash their sum back to approximately 0 or 1). The first hack starts out with the odd function y = 1*(x-0.5) + 0.5
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The article still seems to vacillate between identifying a sigmoid function as a general class of 'S'-shaped functions, and considering it to be specifically the logistic function. Not sure who can clear this up or on what authority. Seems like one avenue would be to research the history of the
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I've used the sigmoid function on and off, for a long time (about 8 years), and what I use is of course similar to what is presented here, but I would suggest adding two elements into the definition -- a "gain" or "sharpness" factor "k" or "g" -- and a "threshold" or "slider" term that allows the
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states "The logit and probit are both sigmoid functions ". I don't think both of these statements can be true because logit and probit are unbounded. I noticed the reference cited for the sentence claiming the "bounded" restriction is from
Lecture Notes in Computer Science in the article "From
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It would be interesting to add something like this. I fiddled with this notion with respect to what would be required for mother nature to build a squasher for making neuralogical ANDs and ORs, and was able to get to some pretty nice approximations -- as long as you stay within the interval.
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My guess is writers who distinguish between the two are (needlessly) splitting the hare (hair) and using two different names for the same function depending on where it is used. "Logistic" would seem to come from "logic" i.e. having 1 and 0 outcomes only; "Sigmoid" because of its shape as in
873:(just a straight line shifted to the right: yielding (0,0), (1,1) ). This clearly won't work. The trick then is to feedback a certain amount of x to give you some "gain", etc, etc. As I remember this works best if it goes through two iterations. I'm working from memory here... bill
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Given that you can make
Matterhorns to your heart's content anywhere on the plane, you can add them together and approximate any curve by "bleeding" one into another. This summation proves that sigmoids can be used to approximate any arbitrary curve, much like a 2-D Fourier
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this (!). I've not seen it documented anywhere, but I did see the results of it in a journal once. I'm sure someone who knows the literature better could cite the source. Proofs similar to the above are mentioned in
Mitchell. This stuff is easy to do in Excel. wvbailey
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has been deleting this edit, pointing to WP:EL (policy on external links) I appreciate his point, as in many topic of opinion, blogs are not an authoritative source. As this article is about math, I can't see the difference between the resource
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I am adding an example that includes "most" of the sigmoids having a closed-form equation. Hope that it is of interest and is not seen as COI. This is my very first
Knowledge edit, if I am doing something inappropriate please remove or modify.
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tanh for example has a derivative of 1-tanh^2. This is also confusing as f(...) can be mistaken for applying function f to (...) where in this case it means the result of multiplying function f with 1-f. dP/df = (P)*(1-P) would be clearer.
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So the article is inconsistent as to whether it must be upward sloping or whether it can alternatively be downward sloping, and (if downward sloping is precluded) as to whether it must have a positive or just a non-negative derivative.
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term "sigmoid function", and if there has been some change over time about how that term is applied, to expound briefly on the evolution of the term. Any 'history of math' scholars out there who may be able to shed some light?
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their k's right, add them together and pass them through a "second-layer" sigmoid you have a "triangle" that can be shrunk with higher values of k's make a single
Matterhorn stick up anywhere on the plane (or make a sink-hole).
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I've found this page quite useful in my work in the statistics of vaccines research, especially the illustration of the six normalized functions, but agree it needs major clean up. Separate pages would be preferable for
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I have seen it used as a "hack" when a fast S-shaped function was needed, avoiding the (computer) evaluation of exp(x). Its derivative is flat at 0 and 1, and it is symmetrical with respect to the midpoint (meaning,
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Do you really mean "local minimum" and "local maximum"? The example function given clearly doesn't have any local minima or maxima (but it does have a global minimum of 0 and a global maximum of 1) -- Somebody
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is not even the special case of the logistic function mentioned in the text. How the reader could know what function the formula applies to. This part of the text is very confusing.
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As it stands now, the article apparently flips between talking about sigmoid functions as members of a class of functions with "S"-shaped graph in general and talking about the
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I see that someone changed the image size recently in order to avoid resolution problems. Maybe the image should be replaced after all with the almost identical vector image
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Natural to
Artificial Neural Computation" so I suspect this definition is likely to be domain-specific. Should the "bounded" restriction be removed or qualified here?
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Somewhere I actually worked out the math for this ... a problem arises because, to be useful, the AND etc needs some "gain" in the middle (i.e. a slope : -->
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You see the failure of "the law of excluded middle" (LoEM) -- no matter how huge the k, the value of the function at X = "thr" = 0.5. This violates the LoEM.
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The curve can be "flipped around" by changing the sign of k; thus the sigmoid can be made to act like a
Boolean NOT if "thr" is 0.5 and k is positive,
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In general, a sigmoid function is real-valued and differentiable, having either a non-negative or non-positive first derivative which is bell shaped.
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A sigmoid function is a bounded differentiable real function that is defined for all real input values and has a positive derivative at each point.
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Sigmoidal and sigmoid seem to me as they are the same thing; maybe there is some very slight difference but it's not pointed out by the article. --
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of the unit . The sigmoid function has the useful property that its derivative is easily expressed in terms of its output..." (Mitchell 1997:96-97)
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Isn't it better to redirect this page to the logistic function page? Or restore this page to its former glory? The current page is kinda pathetic.
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function has none. The image of the function has supremum of 1 and infimum of 0, though (ie. the asymptotes of this function are y=0 and y=1).
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This page is missing a separation of symmetrical and asummetrical sigmoid functions , e.g. the Gompertz function is an asymmetric sigmoid
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Maybe it's for complex argument values? One is led to think of reals only because the plot is 2d, but maybe the text doesn't assume that.
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Finally, I promise to come back and clean up this page as soon as I've finished my research on the statistics of vaccines research :).
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You assure that f(0)=0, f(1)=1 and that f'(0)=f'(1)=0. By playing around with a and b you can get different shapes to suit you
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points to a very useful implementation of the model in Excel. I've found the author moved the site so the link redirects to
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is a shape parameter governing how fast the curve approaches the asymptotes for a given slope at the inflection point. When
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properties that can be derived from the definition should be mentioned), but it's unhelpful at least to be inconsistent.
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Are private websites in images allowed, like the one in this page? Is it not considered a kind of subtle advertisement?
545:{\displaystyle y={\mbox{tanh}}(x-d)\,{\Bigg (}1-\exp {\bigg (}-{\bigg (}{\frac {x-d}{s}}{\bigg )}^{4}{\bigg )}{\Bigg )},}
407:{\displaystyle y={\mbox{tanh}}(x-d)\,{\Bigg (}1-\exp {\bigg (}-{\bigg (}{\frac {x-d}{s}}{\bigg )}^{2}{\bigg )}{\Bigg )},}
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Dunning, AJ; et al. (28 Dec 2015). "Some extensions in continuous methods for immunological correlates of protection".
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You can build e.g. an OR gate by adding X1 and X2, subtracting "thr" = 0.5 and then squashing the sum with the sigmoid:
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I'd like to add a gallery of sigmoid-like curves to this article. The hemoglobin example is a nice one. Any others? --
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considered "S shaped"? If this is a typo, I'm not sure what other function it was supposed to be. --
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So this is really confusing and I'd suggest to remove the example and would do so if nobody objects.
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Similarly, in a plane, the value of Z(t) will be 0.5 all along a line (it looks like a folded plane)
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My text by Mitchell, which I listed on the article page (the only reference, BTW), equates the two:
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The "gain" at X = "thr", is the derivative of course, but it is 1/4 the value of k (as I remember)
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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The article claims "A sigmoid function is a bounded, differentiable, real function " whereas
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Given that you can build an OR and a NOT you now can approximate any Boolean function.
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the function is the 'absolute sigmoid function' shown in the illustration, and when
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as it talks about the logistic function specifically. (And is unsourced anyway.) –
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I wonder if it would be useful to list the following function among the sigmoids:
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example, which doesn't show an explicit formula, but a family of them. These are
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External link to Logistic Function implementation in Excel should be maintained
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it approximates the error function.. This should be considered for inclusion.
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A sigmoid curve is produced by a mathematical function having an "S" shape
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The sigmoidal shape of hemoglobin's oxygen-dissociation curve results from
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I also think the double sigmoid function is wrong. What about this one?
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I'm pretty sure that not all sigmoid functions have the derivative:
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The neat thing about this more expanded definition is the following:
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the following is called a "sigmoidal function" in another article:
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but then in the very next sentence the section "Properties" says
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as a specific example. Frankly, I would just remove the section
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An external link that has been in this page for a good while
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Right you are, I don't know what I was thinking. Fixed. (The
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redirect to this article? Or are they different things?
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the function is the 'square root sigmoid function'; when
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the function approximates the arctangent function, when
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function to be "slid" back and forth across the X-axis:
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551:
549:
548:
543:
538:
537:
531:
530:
524:
523:
518:
517:
510:
505:
494:
492:
491:
482:
481:
463:
462:
439:
435:
413:
411:
410:
405:
400:
399:
393:
392:
386:
385:
380:
379:
372:
367:
356:
354:
353:
344:
343:
325:
324:
301:
297:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
2297:
2296:
2292:
2291:
2290:
2288:
2287:
2286:
2267:
2266:
2244:
2225:nomination page
2211:
2199:
2152:
2150:
2125:
2095:
2090:
2089:
2088:
2078:
2067:
2057:
2031:
2030:
2026:
1978:
1977:
1952:
1951:
1926:
1925:
1900:
1899:
1874:
1873:
1854:
1853:
1819:
1807:
1789:
1764:
1763:
1736:
1735:
1723:
1696:
1671:
1631:
1556:
1522:
1521:
1515:
1499:220.225.131.157
1492:
1454:
1419:"σ(y) = 1/(1+e)
1390:
1316:
1315:
1244:
1243:
1174:
1166:
1159:
1158:
1153:
1079:
1078:
1057:
1041:
1033:
1032:
957:
949:
943:
915:
892:
891:
790:
789:
739:
723:
700:
699:
694:
590:Y(t) = 1/(1 + e
581:
561:
511:
495:
422:
421:
373:
357:
284:
283:
273:
230:
220:Coffee2theorems
202:
194:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
2295:
2293:
2285:
2284:
2279:
2269:
2268:
2243:
2240:
2221:
2220:
2210:
2207:
2198:
2195:
2194:
2193:
2192:
2191:
2124:
2121:
2094:
2091:
2087:
2086:
2023:
2022:
2018:
1991:
1988:
1985:
1965:
1962:
1959:
1939:
1936:
1933:
1913:
1910:
1907:
1887:
1884:
1881:
1861:
1834:
1830:
1826:
1822:
1816:
1811:
1806:
1802:
1798:
1795:
1792:
1788:
1783:
1780:
1777:
1774:
1771:
1746:
1743:
1722:
1719:
1695:
1692:
1670:
1667:
1651:
1650:
1641:
1640:
1630:
1627:
1626:
1625:
1580:
1577:
1574:
1569:
1566:
1563:
1559:
1555:
1552:
1549:
1545:
1541:
1538:
1535:
1532:
1529:
1514:
1511:
1490:
1478:218.82.217.162
1453:
1450:
1449:
1448:
1432:
1431:
1430:
1429:
1422:
1421:
1420:
1412:
1411:
1389:
1386:
1385:
1384:
1374:130.234.198.85
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1327:
1323:
1302:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1255:
1251:
1228:128.111.110.55
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1180:
1177:
1172:
1169:
1152:
1149:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1064:
1060:
1056:
1053:
1048:
1044:
1040:
996:
993:
990:
987:
984:
981:
978:
973:
966:
963:
960:
955:
952:
946:
942:
939:
936:
933:
930:
927:
922:
918:
914:
911:
908:
905:
902:
899:
886:
885:
833:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
774:
771:
768:
765:
762:
759:
756:
751:
746:
742:
738:
735:
730:
726:
722:
719:
716:
713:
710:
707:
693:
690:
679:
678:
661:
660:
655:
654:
649:
648:
647:
646:
643:
640:
633:
632:
628:
627:
626:
625:
619:
618:
614:
613:
612:
611:
605:
604:
601:
594:
593:
592:
591:
580:
577:
560:
557:
555:
553:
552:
541:
536:
529:
522:
516:
508:
504:
501:
498:
490:
485:
480:
475:
472:
469:
466:
461:
454:
451:
448:
445:
442:
432:
429:
415:
414:
403:
398:
391:
384:
378:
370:
366:
363:
360:
352:
347:
342:
337:
334:
331:
328:
323:
316:
313:
310:
307:
304:
294:
291:
277:
272:
269:
256:
255:
229:
226:
211:82.103.198.180
201:
200:local extrema?
198:
193:
190:
189:
188:
181:
178:error function
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2294:
2283:
2280:
2278:
2275:
2274:
2272:
2265:
2264:
2260:
2256:
2252:
2249:
2241:
2239:
2238:
2234:
2230:
2226:
2219:
2216:
2215:
2214:
2208:
2206:
2203:
2196:
2190:
2186:
2182:
2177:
2176:
2175:
2171:
2167:
2163:
2162:Deacon Vorbis
2158:
2149:
2148:
2147:
2146:
2142:
2138:
2134:
2130:
2122:
2120:
2118:
2114:
2110:
2106:
2102:
2092:
2082:
2075:
2062:
2054:
2051:
2047:
2043:
2039:
2035:
2028:
2025:
2021:
2017:
2016:
2012:
2008:
2003:
1989:
1986:
1983:
1963:
1960:
1957:
1937:
1934:
1931:
1911:
1908:
1905:
1885:
1882:
1879:
1859:
1850:
1832:
1828:
1824:
1814:
1804:
1796:
1793:
1786:
1781:
1775:
1769:
1761:
1758:
1741:
1733:
1729:
1720:
1718:
1716:
1712:
1708:
1707:209.93.31.116
1704:
1693:
1691:
1690:
1686:
1682:
1678:
1676:
1668:
1666:
1665:
1661:
1657:
1649:
1646:
1645:
1644:
1639:
1636:
1635:
1634:
1628:
1624:
1620:
1616:
1612:
1611:
1610:
1609:
1606:
1605:
1600:
1596:
1591:
1578:
1575:
1567:
1564:
1561:
1557:
1553:
1550:
1543:
1539:
1536:
1530:
1518:
1512:
1510:
1508:
1504:
1500:
1496:
1488:
1487:
1483:
1479:
1475:
1472:
1469:
1465:
1461:
1458:
1451:
1447:
1443:
1439:
1434:
1433:
1427:
1423:
1418:
1417:
1416:
1415:
1414:
1413:
1409:
1408:
1407:
1406:
1402:
1398:
1395:
1387:
1383:
1379:
1375:
1353:
1350:
1344:
1341:
1338:
1335:
1332:
1325:
1321:
1314:
1313:
1311:
1309:
1305:
1281:
1278:
1272:
1269:
1266:
1263:
1260:
1253:
1249:
1239:
1237:
1233:
1229:
1225:
1218:
1205:
1199:
1196:
1193:
1187:
1184:
1178:
1175:
1170:
1167:
1156:
1150:
1148:
1147:
1143:
1139:
1127:
1108:
1105:
1102:
1099:
1093:
1090:
1087:
1084:
1062:
1058:
1054:
1051:
1046:
1042:
1038:
1030:
1021:
1020:
1016:
1012:
1007:
991:
988:
985:
979:
976:
971:
964:
961:
958:
953:
950:
944:
937:
934:
931:
925:
920:
916:
912:
909:
903:
897:
889:
884:
880:
876:
869:
868:
867:
865:
861:
857:
853:
849:
828:
822:
819:
816:
813:
807:
804:
801:
795:
785:
769:
766:
763:
757:
754:
749:
744:
740:
736:
733:
728:
724:
720:
717:
711:
705:
697:
691:
689:
688:
685:
677:
676:0-07-042807-7
674:
670:
666:
665:
664:
657:
656:
651:
650:
644:
641:
638:
637:
635:
634:
630:
629:
623:
622:
621:
620:
616:
615:
609:
608:
607:
606:
602:
599:
598:
597:
589:
588:
587:
586:
585:
578:
576:
575:
572:
567:
558:
556:
539:
520:
506:
502:
499:
496:
483:
473:
470:
467:
464:
449:
446:
443:
430:
427:
420:
419:
418:
417:or this one:
401:
382:
368:
364:
361:
358:
345:
335:
332:
329:
326:
311:
308:
305:
292:
289:
282:
281:
280:
276:
270:
265:
260:
254:
250:
246:
241:
240:
239:
238:
235:
227:
225:
224:
221:
216:
215:
212:
206:
199:
197:
191:
187:
184:
179:
175:
174:
173:
172:
169:
165:
157:
142:
138:
137:High-priority
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
62:High‑priority
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2253:
2245:
2222:
2212:
2204:
2200:
2156:
2126:
2099:— Preceding
2096:
2061:cite journal
2037:
2033:
2027:
2019:
2004:
1851:
1762:
1759:
1731:
1727:
1724:
1701:— Preceding
1697:
1679:
1672:
1652:
1647:
1642:
1637:
1632:
1603:
1592:
1519:
1516:
1489:
1455:
1425:
1391:
1241:
1219:
1157:
1154:
1130:smootherstep
1022:
1008:
890:
887:
786:
698:
695:
680:
668:
662:
595:
582:
562:
554:
416:
278:
274:
231:
217:
207:
203:
195:
183:Jorge Stolfi
168:65.147.0.105
161:
136:
96:
40:WikiProjects
2105:Raffamaiden
1604:victor falk
1493:—Preceding
1392:Please see
1222:—Preceding
1138:81.6.34.172
846:—Preceding
234:HappyCamper
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2271:Categories
2020:References
1304:Jfmiller28
1025:smoothstep
1011:Juancentro
659:transform.
2007:Adunning2
1656:Duoduoduo
571:Hagman-de
2202:points.
2113:contribs
2101:unsigned
2070:|first1=
2053:26707389
1703:unsigned
1495:unsigned
1438:Wvbailey
1224:unsigned
875:Wvbailey
860:contribs
848:unsigned
684:Wvbailey
245:Dasomath
228:Examples
158:Untitled
2255:Raoouul
2137:Tea2min
2040:(107).
1464:MrOllie
139:on the
2170:videos
2166:carbon
1852:where
1681:Olbran
1031:, and
852:Pasmao
36:scale.
559:Image
2259:talk
2233:talk
2185:talk
2181:Y2PK
2157:Done
2141:talk
2109:talk
2081:link
2074:help
2050:PMID
2011:talk
1711:talk
1685:talk
1660:talk
1619:talk
1503:talk
1482:talk
1470:and
1442:talk
1401:talk
1378:talk
1308:talk
1232:talk
1142:talk
1126:GLSL
1015:talk
879:talk
856:talk
673:ISBN
436:tanh
298:tanh
271:Sign
249:talk
131:High
2227:. —
2042:doi
1990:3.4
1964:2.9
1938:1.5
1615:Kri
569:?--
471:exp
333:exp
2273::
2261:)
2235:)
2187:)
2172:)
2168:•
2143:)
2115:)
2111:•
2065::
2063:}}
2059:{{
2048:.
2038:15
2036:.
2013:)
1745:∞
1742:±
1713:)
1687:)
1662:)
1621:)
1576:−
1562:−
1505:)
1484:)
1444:)
1403:)
1380:)
1351:−
1336:−
1279:−
1264:−
1234:)
1197:−
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1136:.
1103:−
1094:∗
1088:∗
1052:−
1017:)
980:∈
962:−
935:−
926:−
881:)
862:)
858:•
820:−
805:−
758:∈
734:−
500:−
484:−
474:
468:−
447:−
362:−
346:−
336:
330:−
309:−
251:)
2257:(
2231:(
2183:(
2164:(
2139:(
2107:(
2083:)
2076:)
2072:(
2055:.
2044::
2009:(
1987:=
1984:k
1961:=
1958:k
1935:=
1932:k
1912:2
1909:=
1906:k
1886:1
1883:=
1880:k
1860:k
1833:k
1829:/
1825:1
1821:)
1815:k
1810:|
1805:x
1801:|
1797:+
1794:1
1791:(
1787:x
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1779:)
1776:x
1773:(
1770:f
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1683:(
1658:(
1617:(
1579:1
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1568:x
1565:a
1558:e
1554:+
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1540:2
1537:=
1534:)
1531:x
1528:(
1520:σ
1501:(
1480:(
1440:(
1399:(
1376:(
1360:)
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1326:/
1322:1
1306:(
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1203:)
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1188:P
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1179:t
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1140:(
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42::
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