2634:
307:
1602:
299:
367:
The only information is given by the ratios between components, so the information of a composition is preserved under multiplication by any positive constant. Therefore, the sample space of compositional data can always be assumed to be a standard simplex, i.e.
3321:, a town may be a compositional data point in a sample of towns; a town in which 35% of the people are Christians, 55% are Muslims, 6% are Jews, and the remaining 4% are others would correspond to the quadruple . A data set would correspond to a list of towns.
2941:
1429:
1412:
2257:
3328:, a rock composed of different minerals may be a compositional data point in a sample of rocks; a rock of which 10% is the first mineral, 30% is the second, and the remaining 60% is the third would correspond to the triple . A
86:
3474:
Olea, Ricardo A.; Martín-Fernández, Josep A.; Craddock, William H. (2021). "Multivariate classification of the crude oil petroleum systems in southeast Texas, USA, using conventional and compositional analysis of biomarkers".
3189:
2185:
The choice of denominator component is arbitrary, and could be any specified component. This transform is commonly used in chemistry with measurements such as pH. In addition, this is the transform most commonly used for
3072:
2503:
2053:
2637:
A representation of a tree in terms of its orthogonal components. l represents an internal node, an element of the orthonormal basis. This is a precursor to using the tree as a scaffold for the ilr transform
1092:
2630:
of clr transformed data. Another alternative is to construct log contrasts from a bifurcating tree. If we are given a bifurcating tree, we can construct a basis from the internal nodes in the tree.
1808:
3666:
2615:
2383:
2178:
2845:
1933:
433:
682:
1979:
1865:
3477:
In
Advances in Compositional Data Analysis—Festschrift in honor of Vera-Pawlowsky-Glahn, Filzmoser, P., Hron, K., Palarea-Albaladejo, J., Martín-Fernández, J.A., editors. Springer
361:
3372:, the proportions of people positively answering some different items can be expressed as percentages. As the total amount is identified as 100, the compositional vector of
2830:
392:
363:
represent values of different proportions. A, B, C, D and E are 5 different compositions within the simplex. A, B and C are all equivalent and D and E are equivalent.
3535:
Egozcue, Juan Jose; Pawlowsky-Glahn, Vera; Mateu-Figueras, Gloria; Barcelo-Vidal, Carles (2003), "Isometric logratio transformations for compositional data analysis",
2190:. The alr transform is not an isometry, meaning that distances on transformed values will not be equivalent to distances on the original compositions in the simplex.
2984:
3248:
3221:
3007:
1699:
1639:
2417:
1597:{\displaystyle \langle x,y\rangle ={\frac {1}{2D}}\sum _{i=1}^{D}\sum _{j=1}^{D}\log {\frac {x_{i}}{x_{j}}}\log {\frac {y_{i}}{y_{j}}}\qquad \forall x,y\in S^{D}}
3288:
3268:
2437:
1953:
1731:
1109:
717:
2201:
294:{\displaystyle {\mathcal {S}}^{D}=\left\{\mathbf {x} =\in \mathbb {R} ^{D}\,\left|\,x_{i}>0,i=1,2,\dots ,D;\sum _{i=1}^{D}x_{i}=\kappa \right.\right\}.\ }
3087:
1713:
Since the
Aitchison simplex forms a finite dimensional Hilbert space, it is possible to construct orthonormal bases in the simplex. Every composition
3017:
2986:
are the respective number of tips in the corresponding subtrees shown in the figure. It can be shown that the resulting basis is orthonormal
2454:
2004:
3623:
3526:
3506:
49:
3314:
molar concentrations. These compositions can be translated into weight per cent multiplying each component by the appropriated constant.
3380: − 1 components, assuming that the remaining component is the percentage needed for the whole vector to add to 100.
3350:, a partition of the sampling space into disjoint events is described by the probabilities assigned to such events. The vector of
752:
1741:
24:
are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is
3682:
2187:
78:
for short) can be represented by a real vector with positive components. The sample space of compositional data is a simplex:
3395:
2627:
3565:
Egozcue, Juan Jose; Pawlowsky-Glahn, Vera (2005), "Groups of parts and their balances in compositional data analysis",
1993:
that transform from the
Aitchison simplex to real space. All of these transforms satisfy linearity and as given below
2936:{\displaystyle a={\frac {\sqrt {s}}{\sqrt {r(r+s)}}}\quad {\text{and}}\quad b={\frac {-{\sqrt {r}}}{\sqrt {s(r+s)}}}}
2512:
2266:
2063:
74:
defined compositional data to be proportions of some whole in 1982. In particular, a compositional data point (or
1870:
401:
2623:
443:
3369:
1958:
1818:
3358:
parts. As they add to one, one probability can be suppressed and the composition is completely determined.
313:
3603:
3576:
3567:
3537:
3303:
1642:
3592:
3554:
2649:
33:
3306:
of each component. As the sum of all concentrations is not determined, the whole composition of
3339:, data obtained are typically transformed to relative abundances, rendering them compositional.
371:
3619:
3522:
3502:
2951:
3653:
3611:
3584:
3546:
3434:
2440:
55:
on three variables graphically depicts the ratios of the three variables as positions in an
3226:
3199:
2992:
1648:
1612:
2393:
52:
1407:{\displaystyle \alpha \odot x=\left=C\qquad \forall x\in S^{D},\;\alpha \in \mathbb {R} }
3580:
3494:
3438:
3336:
3273:
3253:
2422:
2252:{\displaystyle \operatorname {clr} :S^{D}\rightarrow U,\quad U\subset \mathbb {R} ^{D}}
1938:
1716:
71:
2633:
696:
3676:
3558:
3390:
2451:
The isometric log ratio (ilr) transform is both an isomorphism and an isometry where
3596:
3405:
3362:
728:
45:
25:
2198:
The center log ratio (clr) transform is both an isomorphism and an isometry where
3516:
3451:
1955:
with respect to the given basis. They are called isometric log-ratio coordinates
3343:
3184:{\displaystyle b_{i}={\sqrt {\frac {rs}{r+s}}}\log {\frac {g(x_{R})}{g(x_{S})}}}
1990:
1702:
306:
56:
3588:
3550:
3347:
3318:
17:
3299:
2622:
There are multiple ways to construct orthonormal bases, including using the
3615:
3425:
Aitchison, John (1982). "The
Statistical Analysis of Compositional Data".
731:
in several different ways. The following vector space structure is called
3329:
59:
3079:
where each element in the ilr transformed data is of the following form
3067:{\displaystyle \operatorname {ilr} (x)=\operatorname {clr} (x)\Psi ^{T}}
2498:{\displaystyle \operatorname {ilr} :S^{D}\rightarrow \mathbb {R} ^{D-1}}
2048:{\displaystyle \operatorname {alr} :S^{D}\rightarrow \mathbb {R} ^{D-1}}
3400:
3325:
29:
32:. Measurements involving probabilities, proportions, percentages, and
3658:
3638:
310:
An illustration of the
Aitchison simplex. Here, there are 3 parts,
3645:
3427:
Journal of the Royal
Statistical Society. Series B (Methodological)
394:. In this context, normalization to the standard simplex is called
2632:
305:
3644:
Pawlowsky-Glahn, V.; Egozcue, J.J.; Tolosana-Delgado, R. (2007).
3332:
would contain one such triple for each rock in a sample of rocks.
3250:
are the set of values corresponding to the tips in the subtrees
3667:
Why, and How, Should
Geologists Use Compositional Data Analysis
2001:
The additive log ratio (alr) transform is an isomorphism where
3515:
van den
Boogaart, K. Gerald; Tolosana-Delgado, Raimon (2013),
3501:, Monographs on statistics and applied probability, Springer,
1609:
Endowed with those operations, the
Aitchison simplex forms a
1087:{\displaystyle x\oplus y=\left=C\qquad \forall x,y\in S^{D}}
449:
408:
93:
277:
1803:{\displaystyle x=\bigoplus _{i=1}^{D}x_{i}^{*}\odot e_{i}}
3009:
is built, the ilr transform can be calculated as follows
44:
Compositional data in three variables can be plotted via
2641:
Each vector in the basis would be determined as follows
3606:; Egozcue, Juan Jose; Tolosana-Delgado, Raimon (2015),
3462:
1867:
forms an orthonormal basis in the simplex. The values
405:
3276:
3256:
3229:
3202:
3090:
3020:
2995:
2954:
2848:
2837:
The elements within each vector are given as follows
2652:
2515:
2457:
2425:
2396:
2269:
2204:
2066:
2007:
1961:
1941:
1873:
1821:
1744:
1719:
1651:
1615:
1432:
1112:
755:
699:
446:
404:
374:
316:
89:
3354:
probabilities can be considered as a composition of
2439:. The inverse of this function is also known as the
1935:are the (orthonormal and Cartesian) coordinates of
3310:parts is needed and thus expressed as a vector of
3282:
3262:
3242:
3215:
3183:
3066:
3001:
2978:
2935:
2824:
2609:
2497:
2431:
2411:
2377:
2251:
2172:
2047:
1973:
1947:
1927:
1859:
1802:
1725:
1693:
1633:
1596:
1406:
1086:
711:
676:
427:
386:
355:
293:
2610:{\displaystyle \operatorname {ilr} (x)={\big }}
3646:"Lecture Notes on Compositional Data Analysis"
3499:The Statistical Analysis of Compositional Data
2602:
2536:
2378:{\displaystyle \operatorname {clr} (x)=\left}
2173:{\displaystyle \operatorname {alr} (x)=\left}
36:can all be thought of as compositional data.
8:
2597:
2572:
2560:
2541:
1445:
1433:
727:The simplex can be given the structure of a
3608:Modeling and Analysis of Compositional Data
3365:, for the classification of petroleum oils.
1928:{\displaystyle x_{i}^{*},i=1,2,\ldots ,D-1}
428:{\displaystyle \scriptstyle {\mathcal {C}}}
1392:
3657:
3275:
3255:
3234:
3228:
3207:
3201:
3169:
3148:
3135:
3104:
3095:
3089:
3058:
3019:
2994:
2953:
2903:
2897:
2885:
2855:
2847:
2815:
2809:
2785:
2775:
2751:
2741:
2717:
2707:
2683:
2681:
2657:
2651:
2601:
2600:
2585:
2554:
2535:
2534:
2514:
2483:
2479:
2478:
2468:
2456:
2424:
2395:
2348:
2342:
2305:
2299:
2268:
2243:
2239:
2238:
2215:
2203:
2157:
2141:
2135:
2112:
2102:
2096:
2065:
2033:
2029:
2028:
2018:
2006:
1960:
1940:
1883:
1878:
1872:
1845:
1826:
1820:
1794:
1781:
1776:
1766:
1755:
1743:
1718:
1676:
1657:
1650:
1614:
1588:
1560:
1550:
1544:
1530:
1520:
1514:
1502:
1491:
1481:
1470:
1451:
1431:
1400:
1399:
1383:
1360:
1355:
1336:
1331:
1304:
1299:
1289:
1278:
1267:
1262:
1256:
1238:
1233:
1223:
1212:
1201:
1196:
1190:
1178:
1173:
1163:
1152:
1141:
1136:
1130:
1111:
1078:
1049:
1039:
1020:
1010:
983:
973:
963:
952:
940:
930:
923:
905:
895:
885:
874:
862:
852:
845:
833:
823:
813:
802:
790:
780:
773:
754:
698:
654:
644:
633:
622:
616:
598:
588:
577:
566:
560:
548:
538:
527:
516:
510:
493:
474:
461:
448:
447:
445:
420:
416:
407:
406:
403:
373:
347:
334:
321:
315:
264:
254:
243:
194:
189:
183:
177:
173:
172:
159:
140:
127:
112:
98:
92:
91:
88:
693:is the number of parts (components) and
3417:
677:{\displaystyle {\mathcal {C}}=\left,\ }
1974:{\displaystyle (\operatorname {ilr} )}
3376:components can be defined using only
1860:{\displaystyle e_{1},\ldots ,e_{D-1}}
7:
3639:CoDaWeb – Compositional Data Website
3518:Analyzing Compositional Data with R
3302:, compositions can be expressed as
1989:There are three well-characterized
3463:Egozcue & Pawlowsky-Glahn 2005
3439:10.1111/j.2517-6161.1982.tb01195.x
3055:
2996:
1569:
1370:
1059:
739:and has the following operations:
14:
356:{\displaystyle x_{1},x_{2},x_{3}}
1100:Powering (scalar multiplication)
113:
2890:
2884:
2230:
2188:multinomial logistic regression
1568:
1369:
1058:
3175:
3162:
3154:
3141:
3051:
3045:
3033:
3027:
2927:
2915:
2878:
2866:
2819:
2816:
2678:
2669:
2624:Gram–Schmidt orthogonalization
2528:
2522:
2474:
2406:
2400:
2364:
2358:
2321:
2315:
2282:
2276:
2221:
2079:
2073:
2024:
1968:
1962:
1628:
1616:
1366:
1324:
1055:
1003:
743:Perturbation (vector addition)
706:
700:
499:
454:
421:
413:
165:
120:
1:
1733:can be decomposed as follows
3396:Response surface methodology
2628:singular-value decomposition
2447:Isometric logratio transform
1997:Additive log ratio transform
2825:{\displaystyle e_{\ell }=C}
3699:
3337:high throughput sequencing
2194:Center log ratio transform
1645:. The uniform composition
3589:10.1007/s11004-005-7381-9
3401:Applications of simplices
2419:is the geometric mean of
387:{\displaystyle \kappa =1}
3551:10.1023/A:1023818214614
2979:{\displaystyle k,r,s,t}
1641:-dimensional Euclidean
66:Simplicial sample space
3683:Statistical data types
3284:
3264:
3244:
3217:
3185:
3068:
3003:
2980:
2937:
2826:
2638:
2611:
2499:
2433:
2413:
2379:
2253:
2174:
2049:
1985:Linear transformations
1975:
1949:
1929:
1861:
1804:
1771:
1727:
1695:
1635:
1598:
1507:
1486:
1408:
1294:
1228:
1168:
1088:
968:
890:
818:
719:denotes a row vector.
713:
678:
649:
593:
543:
429:
388:
364:
357:
295:
259:
3650:Universitat de Girona
3616:10.1002/9781119003144
3604:Pawlowsky-Glahn, Vera
3285:
3265:
3245:
3243:{\displaystyle x_{S}}
3218:
3216:{\displaystyle x_{R}}
3186:
3069:
3004:
3002:{\displaystyle \Psi }
2981:
2938:
2827:
2636:
2612:
2500:
2434:
2414:
2380:
2254:
2175:
2050:
1976:
1950:
1930:
1862:
1805:
1751:
1728:
1696:
1694:{\displaystyle \left}
1636:
1634:{\displaystyle (D-1)}
1599:
1487:
1466:
1409:
1274:
1208:
1148:
1089:
948:
870:
798:
714:
679:
629:
573:
523:
430:
389:
358:
309:
296:
239:
26:represented by points
3568:Mathematical Geology
3538:Mathematical Geology
3304:molar concentrations
3274:
3254:
3227:
3200:
3088:
3018:
2993:
2952:
2846:
2650:
2513:
2455:
2423:
2412:{\displaystyle g(x)}
2394:
2267:
2202:
2064:
2055:. This is given by
2005:
1959:
1939:
1871:
1819:
1742:
1717:
1649:
1613:
1430:
1110:
753:
697:
444:
402:
372:
314:
87:
3581:2005MatGe..37..795E
1888:
1786:
1643:inner product space
1365:
1341:
1309:
1272:
1243:
1206:
1183:
1146:
3280:
3260:
3240:
3213:
3181:
3064:
2999:
2976:
2933:
2822:
2814:
2807:
2780:
2773:
2746:
2739:
2712:
2705:
2639:
2607:
2495:
2429:
2409:
2375:
2249:
2170:
2045:
1971:
1945:
1925:
1874:
1857:
1800:
1772:
1723:
1691:
1631:
1594:
1404:
1351:
1327:
1295:
1258:
1229:
1192:
1169:
1132:
1084:
733:Aitchison geometry
723:Aitchison geometry
709:
674:
425:
424:
398:and is denoted by
384:
365:
353:
291:
22:compositional data
3625:978-1-119-00314-4
3528:978-3-642-36809-7
3508:978-94-010-8324-9
3283:{\displaystyle S}
3263:{\displaystyle R}
3179:
3127:
3126:
2931:
2930:
2908:
2888:
2882:
2881:
2861:
2786:
2784:
2752:
2750:
2718:
2716:
2684:
2682:
2432:{\displaystyle x}
2368:
2325:
2163:
2118:
1948:{\displaystyle x}
1726:{\displaystyle x}
1709:Orthonormal bases
1684:
1665:
1566:
1536:
1464:
1311:
1245:
1185:
990:
912:
840:
737:Aitchison simplex
673:
661:
605:
555:
290:
3690:
3663:
3661:
3628:
3599:
3561:
3531:
3511:
3481:
3480:
3471:
3465:
3460:
3454:
3449:
3443:
3442:
3422:
3289:
3287:
3286:
3281:
3269:
3267:
3266:
3261:
3249:
3247:
3246:
3241:
3239:
3238:
3222:
3220:
3219:
3214:
3212:
3211:
3190:
3188:
3187:
3182:
3180:
3178:
3174:
3173:
3157:
3153:
3152:
3136:
3128:
3125:
3114:
3106:
3105:
3100:
3099:
3073:
3071:
3070:
3065:
3063:
3062:
3008:
3006:
3005:
3000:
2985:
2983:
2982:
2977:
2942:
2940:
2939:
2934:
2932:
2911:
2910:
2909:
2904:
2898:
2889:
2886:
2883:
2862:
2857:
2856:
2831:
2829:
2828:
2823:
2813:
2808:
2803:
2779:
2774:
2769:
2745:
2740:
2735:
2711:
2706:
2701:
2662:
2661:
2616:
2614:
2613:
2608:
2606:
2605:
2596:
2595:
2559:
2558:
2540:
2539:
2504:
2502:
2501:
2496:
2494:
2493:
2482:
2473:
2472:
2441:softmax function
2438:
2436:
2435:
2430:
2418:
2416:
2415:
2410:
2384:
2382:
2381:
2376:
2374:
2370:
2369:
2367:
2353:
2352:
2343:
2326:
2324:
2310:
2309:
2300:
2258:
2256:
2255:
2250:
2248:
2247:
2242:
2220:
2219:
2179:
2177:
2176:
2171:
2169:
2165:
2164:
2162:
2161:
2152:
2151:
2136:
2119:
2117:
2116:
2107:
2106:
2097:
2054:
2052:
2051:
2046:
2044:
2043:
2032:
2023:
2022:
1980:
1978:
1977:
1972:
1954:
1952:
1951:
1946:
1934:
1932:
1931:
1926:
1887:
1882:
1866:
1864:
1863:
1858:
1856:
1855:
1831:
1830:
1809:
1807:
1806:
1801:
1799:
1798:
1785:
1780:
1770:
1765:
1732:
1730:
1729:
1724:
1700:
1698:
1697:
1692:
1690:
1686:
1685:
1677:
1666:
1658:
1640:
1638:
1637:
1632:
1603:
1601:
1600:
1595:
1593:
1592:
1567:
1565:
1564:
1555:
1554:
1545:
1537:
1535:
1534:
1525:
1524:
1515:
1506:
1501:
1485:
1480:
1465:
1463:
1452:
1413:
1411:
1410:
1405:
1403:
1388:
1387:
1364:
1359:
1340:
1335:
1317:
1313:
1312:
1310:
1308:
1303:
1293:
1288:
1271:
1266:
1257:
1246:
1244:
1242:
1237:
1227:
1222:
1205:
1200:
1191:
1186:
1184:
1182:
1177:
1167:
1162:
1145:
1140:
1131:
1093:
1091:
1090:
1085:
1083:
1082:
1054:
1053:
1044:
1043:
1025:
1024:
1015:
1014:
996:
992:
991:
989:
988:
987:
978:
977:
967:
962:
946:
945:
944:
935:
934:
924:
913:
911:
910:
909:
900:
899:
889:
884:
868:
867:
866:
857:
856:
846:
841:
839:
838:
837:
828:
827:
817:
812:
796:
795:
794:
785:
784:
774:
718:
716:
715:
712:{\displaystyle }
710:
683:
681:
680:
675:
671:
667:
663:
662:
660:
659:
658:
648:
643:
627:
626:
617:
606:
604:
603:
602:
592:
587:
571:
570:
561:
556:
554:
553:
552:
542:
537:
521:
520:
511:
498:
497:
479:
478:
466:
465:
453:
452:
434:
432:
431:
426:
412:
411:
393:
391:
390:
385:
362:
360:
359:
354:
352:
351:
339:
338:
326:
325:
300:
298:
297:
292:
288:
284:
280:
279:
276:
269:
268:
258:
253:
199:
198:
182:
181:
176:
164:
163:
145:
144:
132:
131:
116:
103:
102:
97:
96:
3698:
3697:
3693:
3692:
3691:
3689:
3688:
3687:
3673:
3672:
3643:
3635:
3626:
3602:
3564:
3534:
3529:
3514:
3509:
3493:
3490:
3485:
3484:
3473:
3472:
3468:
3461:
3457:
3450:
3446:
3424:
3423:
3419:
3414:
3387:
3295:
3272:
3271:
3252:
3251:
3230:
3225:
3224:
3203:
3198:
3197:
3165:
3158:
3144:
3137:
3115:
3107:
3091:
3086:
3085:
3054:
3016:
3015:
2991:
2990:
2989:Once the basis
2950:
2949:
2899:
2844:
2843:
2787:
2753:
2719:
2685:
2653:
2648:
2647:
2581:
2550:
2511:
2510:
2477:
2464:
2453:
2452:
2449:
2421:
2420:
2392:
2391:
2354:
2344:
2311:
2301:
2292:
2288:
2265:
2264:
2237:
2211:
2200:
2199:
2196:
2153:
2137:
2108:
2098:
2089:
2085:
2062:
2061:
2027:
2014:
2003:
2002:
1999:
1987:
1957:
1956:
1937:
1936:
1869:
1868:
1841:
1822:
1817:
1816:
1790:
1740:
1739:
1715:
1714:
1711:
1656:
1652:
1647:
1646:
1611:
1610:
1584:
1556:
1546:
1526:
1516:
1456:
1428:
1427:
1379:
1273:
1207:
1147:
1129:
1125:
1108:
1107:
1074:
1045:
1035:
1016:
1006:
979:
969:
947:
936:
926:
925:
901:
891:
869:
858:
848:
847:
829:
819:
797:
786:
776:
775:
772:
768:
751:
750:
725:
695:
694:
650:
628:
618:
594:
572:
562:
544:
522:
512:
509:
505:
489:
470:
457:
442:
441:
400:
399:
370:
369:
343:
330:
317:
312:
311:
260:
190:
188:
184:
171:
155:
136:
123:
111:
107:
90:
85:
84:
68:
48:. The use of a
42:
12:
11:
5:
3696:
3694:
3686:
3685:
3675:
3674:
3671:
3670:
3664:
3641:
3634:
3633:External links
3631:
3630:
3629:
3624:
3600:
3575:(7): 795–828,
3562:
3545:(3): 279–300,
3532:
3527:
3512:
3507:
3489:
3486:
3483:
3482:
3466:
3455:
3452:Egozcue et al.
3444:
3433:(2): 139–177.
3416:
3415:
3413:
3410:
3409:
3408:
3403:
3398:
3393:
3386:
3383:
3382:
3381:
3366:
3359:
3340:
3333:
3322:
3315:
3294:
3291:
3279:
3259:
3237:
3233:
3210:
3206:
3194:
3193:
3192:
3191:
3177:
3172:
3168:
3164:
3161:
3156:
3151:
3147:
3143:
3140:
3134:
3131:
3124:
3121:
3118:
3113:
3110:
3103:
3098:
3094:
3077:
3076:
3075:
3074:
3061:
3057:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
2998:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2946:
2945:
2944:
2943:
2929:
2926:
2923:
2920:
2917:
2914:
2907:
2902:
2896:
2893:
2880:
2877:
2874:
2871:
2868:
2865:
2860:
2854:
2851:
2835:
2834:
2833:
2832:
2821:
2818:
2812:
2806:
2802:
2799:
2796:
2793:
2790:
2783:
2778:
2772:
2768:
2765:
2762:
2759:
2756:
2749:
2744:
2738:
2734:
2731:
2728:
2725:
2722:
2715:
2710:
2704:
2700:
2697:
2694:
2691:
2688:
2680:
2677:
2674:
2671:
2668:
2665:
2660:
2656:
2620:
2619:
2618:
2617:
2604:
2599:
2594:
2591:
2588:
2584:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2557:
2553:
2549:
2546:
2543:
2538:
2533:
2530:
2527:
2524:
2521:
2518:
2492:
2489:
2486:
2481:
2476:
2471:
2467:
2463:
2460:
2448:
2445:
2428:
2408:
2405:
2402:
2399:
2388:
2387:
2386:
2385:
2373:
2366:
2363:
2360:
2357:
2351:
2347:
2341:
2338:
2335:
2332:
2329:
2323:
2320:
2317:
2314:
2308:
2304:
2298:
2295:
2291:
2287:
2284:
2281:
2278:
2275:
2272:
2246:
2241:
2236:
2233:
2229:
2226:
2223:
2218:
2214:
2210:
2207:
2195:
2192:
2183:
2182:
2181:
2180:
2168:
2160:
2156:
2150:
2147:
2144:
2140:
2134:
2131:
2128:
2125:
2122:
2115:
2111:
2105:
2101:
2095:
2092:
2088:
2084:
2081:
2078:
2075:
2072:
2069:
2042:
2039:
2036:
2031:
2026:
2021:
2017:
2013:
2010:
1998:
1995:
1986:
1983:
1970:
1967:
1964:
1944:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1886:
1881:
1877:
1854:
1851:
1848:
1844:
1840:
1837:
1834:
1829:
1825:
1813:
1812:
1811:
1810:
1797:
1793:
1789:
1784:
1779:
1775:
1769:
1764:
1761:
1758:
1754:
1750:
1747:
1722:
1710:
1707:
1689:
1683:
1680:
1675:
1672:
1669:
1664:
1661:
1655:
1630:
1627:
1624:
1621:
1618:
1607:
1606:
1605:
1604:
1591:
1587:
1583:
1580:
1577:
1574:
1571:
1563:
1559:
1553:
1549:
1543:
1540:
1533:
1529:
1523:
1519:
1513:
1510:
1505:
1500:
1497:
1494:
1490:
1484:
1479:
1476:
1473:
1469:
1462:
1459:
1455:
1450:
1447:
1444:
1441:
1438:
1435:
1422:
1421:
1417:
1416:
1415:
1414:
1402:
1398:
1395:
1391:
1386:
1382:
1378:
1375:
1372:
1368:
1363:
1358:
1354:
1350:
1347:
1344:
1339:
1334:
1330:
1326:
1323:
1320:
1316:
1307:
1302:
1298:
1292:
1287:
1284:
1281:
1277:
1270:
1265:
1261:
1255:
1252:
1249:
1241:
1236:
1232:
1226:
1221:
1218:
1215:
1211:
1204:
1199:
1195:
1189:
1181:
1176:
1172:
1166:
1161:
1158:
1155:
1151:
1144:
1139:
1135:
1128:
1124:
1121:
1118:
1115:
1102:
1101:
1097:
1096:
1095:
1094:
1081:
1077:
1073:
1070:
1067:
1064:
1061:
1057:
1052:
1048:
1042:
1038:
1034:
1031:
1028:
1023:
1019:
1013:
1009:
1005:
1002:
999:
995:
986:
982:
976:
972:
966:
961:
958:
955:
951:
943:
939:
933:
929:
922:
919:
916:
908:
904:
898:
894:
888:
883:
880:
877:
873:
865:
861:
855:
851:
844:
836:
832:
826:
822:
816:
811:
808:
805:
801:
793:
789:
783:
779:
771:
767:
764:
761:
758:
745:
744:
724:
721:
708:
705:
702:
687:
686:
685:
684:
670:
666:
657:
653:
647:
642:
639:
636:
632:
625:
621:
615:
612:
609:
601:
597:
591:
586:
583:
580:
576:
569:
565:
559:
551:
547:
541:
536:
533:
530:
526:
519:
515:
508:
504:
501:
496:
492:
488:
485:
482:
477:
473:
469:
464:
460:
456:
451:
423:
419:
415:
410:
383:
380:
377:
350:
346:
342:
337:
333:
329:
324:
320:
304:
303:
302:
301:
287:
283:
278:
275:
272:
267:
263:
257:
252:
249:
246:
242:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
197:
193:
187:
180:
175:
170:
167:
162:
158:
154:
151:
148:
143:
139:
135:
130:
126:
122:
119:
115:
110:
106:
101:
95:
72:John Aitchison
67:
64:
41:
38:
13:
10:
9:
6:
4:
3:
2:
3695:
3684:
3681:
3680:
3678:
3668:
3665:
3660:
3655:
3651:
3647:
3642:
3640:
3637:
3636:
3632:
3627:
3621:
3617:
3613:
3609:
3605:
3601:
3598:
3594:
3590:
3586:
3582:
3578:
3574:
3570:
3569:
3563:
3560:
3556:
3552:
3548:
3544:
3540:
3539:
3533:
3530:
3524:
3520:
3519:
3513:
3510:
3504:
3500:
3496:
3495:Aitchison, J.
3492:
3491:
3487:
3478:
3470:
3467:
3464:
3459:
3456:
3453:
3448:
3445:
3440:
3436:
3432:
3428:
3421:
3418:
3411:
3407:
3404:
3402:
3399:
3397:
3394:
3392:
3391:Mixture model
3389:
3388:
3384:
3379:
3375:
3371:
3367:
3364:
3360:
3357:
3353:
3349:
3345:
3341:
3338:
3334:
3331:
3327:
3323:
3320:
3316:
3313:
3309:
3305:
3301:
3297:
3296:
3292:
3290:
3277:
3257:
3235:
3231:
3208:
3204:
3170:
3166:
3159:
3149:
3145:
3138:
3132:
3129:
3122:
3119:
3116:
3111:
3108:
3101:
3096:
3092:
3084:
3083:
3082:
3081:
3080:
3059:
3048:
3042:
3039:
3036:
3030:
3024:
3021:
3014:
3013:
3012:
3011:
3010:
2987:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2924:
2921:
2918:
2912:
2905:
2900:
2894:
2891:
2875:
2872:
2869:
2863:
2858:
2852:
2849:
2842:
2841:
2840:
2839:
2838:
2810:
2804:
2800:
2797:
2794:
2791:
2788:
2781:
2776:
2770:
2766:
2763:
2760:
2757:
2754:
2747:
2742:
2736:
2732:
2729:
2726:
2723:
2720:
2713:
2708:
2702:
2698:
2695:
2692:
2689:
2686:
2675:
2672:
2666:
2663:
2658:
2654:
2646:
2645:
2644:
2643:
2642:
2635:
2631:
2629:
2625:
2592:
2589:
2586:
2582:
2578:
2575:
2569:
2566:
2563:
2555:
2551:
2547:
2544:
2531:
2525:
2519:
2516:
2509:
2508:
2507:
2506:
2505:
2490:
2487:
2484:
2469:
2465:
2461:
2458:
2446:
2444:
2442:
2426:
2403:
2397:
2371:
2361:
2355:
2349:
2345:
2339:
2336:
2333:
2330:
2327:
2318:
2312:
2306:
2302:
2296:
2293:
2289:
2285:
2279:
2273:
2270:
2263:
2262:
2261:
2260:
2259:
2244:
2234:
2231:
2227:
2224:
2216:
2212:
2208:
2205:
2193:
2191:
2189:
2166:
2158:
2154:
2148:
2145:
2142:
2138:
2132:
2129:
2126:
2123:
2120:
2113:
2109:
2103:
2099:
2093:
2090:
2086:
2082:
2076:
2070:
2067:
2060:
2059:
2058:
2057:
2056:
2040:
2037:
2034:
2019:
2015:
2011:
2008:
1996:
1994:
1992:
1984:
1982:
1965:
1942:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1884:
1879:
1875:
1852:
1849:
1846:
1842:
1838:
1835:
1832:
1827:
1823:
1795:
1791:
1787:
1782:
1777:
1773:
1767:
1762:
1759:
1756:
1752:
1748:
1745:
1738:
1737:
1736:
1735:
1734:
1720:
1708:
1706:
1704:
1687:
1681:
1678:
1673:
1670:
1667:
1662:
1659:
1653:
1644:
1625:
1622:
1619:
1589:
1585:
1581:
1578:
1575:
1572:
1561:
1557:
1551:
1547:
1541:
1538:
1531:
1527:
1521:
1517:
1511:
1508:
1503:
1498:
1495:
1492:
1488:
1482:
1477:
1474:
1471:
1467:
1460:
1457:
1453:
1448:
1442:
1439:
1436:
1426:
1425:
1424:
1423:
1420:Inner product
1419:
1418:
1396:
1393:
1389:
1384:
1380:
1376:
1373:
1361:
1356:
1352:
1348:
1345:
1342:
1337:
1332:
1328:
1321:
1318:
1314:
1305:
1300:
1296:
1290:
1285:
1282:
1279:
1275:
1268:
1263:
1259:
1253:
1250:
1247:
1239:
1234:
1230:
1224:
1219:
1216:
1213:
1209:
1202:
1197:
1193:
1187:
1179:
1174:
1170:
1164:
1159:
1156:
1153:
1149:
1142:
1137:
1133:
1126:
1122:
1119:
1116:
1113:
1106:
1105:
1104:
1103:
1099:
1098:
1079:
1075:
1071:
1068:
1065:
1062:
1050:
1046:
1040:
1036:
1032:
1029:
1026:
1021:
1017:
1011:
1007:
1000:
997:
993:
984:
980:
974:
970:
964:
959:
956:
953:
949:
941:
937:
931:
927:
920:
917:
914:
906:
902:
896:
892:
886:
881:
878:
875:
871:
863:
859:
853:
849:
842:
834:
830:
824:
820:
814:
809:
806:
803:
799:
791:
787:
781:
777:
769:
765:
762:
759:
756:
749:
748:
747:
746:
742:
741:
740:
738:
734:
730:
722:
720:
703:
692:
668:
664:
655:
651:
645:
640:
637:
634:
630:
623:
619:
613:
610:
607:
599:
595:
589:
584:
581:
578:
574:
567:
563:
557:
549:
545:
539:
534:
531:
528:
524:
517:
513:
506:
502:
494:
490:
486:
483:
480:
475:
471:
467:
462:
458:
440:
439:
438:
437:
436:
417:
397:
381:
378:
375:
348:
344:
340:
335:
331:
327:
322:
318:
308:
285:
281:
273:
270:
265:
261:
255:
250:
247:
244:
240:
236:
233:
230:
227:
224:
221:
218:
215:
212:
209:
206:
203:
200:
195:
191:
185:
178:
168:
160:
156:
152:
149:
146:
141:
137:
133:
128:
124:
117:
108:
104:
99:
83:
82:
81:
80:
79:
77:
73:
65:
63:
61:
58:
54:
51:
47:
46:ternary plots
39:
37:
35:
31:
27:
23:
19:
3649:
3607:
3572:
3566:
3542:
3536:
3521:, Springer,
3517:
3498:
3476:
3469:
3458:
3447:
3430:
3426:
3420:
3406:Ternary plot
3377:
3373:
3363:chemometrics
3355:
3351:
3311:
3307:
3195:
3078:
2988:
2947:
2836:
2640:
2621:
2450:
2389:
2197:
2184:
2000:
1991:isomorphisms
1988:
1814:
1712:
1608:
736:
732:
729:vector space
726:
690:
688:
395:
366:
75:
70:In general,
69:
43:
40:Ternary plot
21:
15:
3344:probability
1703:zero vector
76:composition
57:equilateral
50:barycentric
3669:(wikibook)
3488:References
3479:: 303−327.
3348:statistics
3319:demography
18:statistics
3659:10256/297
3610:, Wiley,
3559:122844634
3497:(2011) ,
3300:chemistry
3133:
3056:Ψ
3043:
3025:
2997:Ψ
2901:−
2805:⏟
2795:…
2771:⏟
2761:…
2737:⏟
2727:…
2703:⏟
2693:…
2676:
2659:ℓ
2598:⟩
2590:−
2573:⟨
2567:…
2561:⟩
2542:⟨
2520:
2488:−
2475:→
2340:
2331:⋯
2297:
2274:
2235:⊂
2222:→
2146:−
2133:
2124:⋯
2094:
2071:
2038:−
2025:→
1920:−
1911:…
1885:∗
1850:−
1836:…
1788:⊙
1783:∗
1753:⨁
1671:…
1623:−
1582:∈
1570:∀
1542:
1512:
1489:∑
1468:∑
1446:⟩
1434:⟨
1397:∈
1394:α
1377:∈
1371:∀
1362:α
1346:…
1338:α
1306:α
1276:∑
1269:α
1251:…
1240:α
1210:∑
1203:α
1180:α
1150:∑
1143:α
1117:⊙
1114:α
1072:∈
1060:∀
1030:…
950:∑
918:…
872:∑
800:∑
760:⊕
704:⋅
631:∑
611:…
575:∑
525:∑
484:…
418:⋅
376:κ
274:κ
241:∑
228:…
169:∈
150:…
3677:Category
3597:53061345
3385:See also
3330:data set
3293:Examples
60:triangle
3577:Bibcode
3326:geology
1701:is the
735:or the
396:closure
30:simplex
3622:
3595:
3557:
3525:
3505:
3370:survey
3196:where
2948:where
2390:Where
1815:where
689:where
672:
289:
3593:S2CID
3555:S2CID
3412:Notes
3368:In a
28:on a
3620:ISBN
3523:ISBN
3503:ISBN
3346:and
3270:and
3223:and
201:>
53:plot
3654:hdl
3612:doi
3585:doi
3547:doi
3435:doi
3361:In
3342:In
3335:In
3324:In
3317:In
3298:In
3130:log
3040:clr
3022:ilr
2887:and
2673:exp
2626:or
2517:ilr
2459:ilr
2337:log
2294:log
2271:clr
2206:clr
2130:log
2091:log
2068:alr
2009:alr
1966:ilr
1539:log
1509:log
34:ppm
16:In
3679::
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3648:.
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94:S
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