zbMATH — the first resource for mathematics

Differential models for evolutionary compositions. (English) Zbl 1323.37051
Summary: General systems are frequently decomposable into parts and these parts can evolve in time or space, a frequent occurrence in the field of Geosciences. In most cases, fitting models to forecast future states of the system is a goal of the analysis. Modelling interactions between parts may also be of common interest. The system can be analysed from different points of view; the traditional one consists in modelling each part of the system in time. Alternatively, modelling the evolution of the parts as proportions is proposed herein and attention is centred on the compositional evolution. The compositions are expressed in orthogonal coordinates (ilr) and then modelled using first-order differential equations with constant coefficients. Simple models are shown to be very flexible, including many of the standard growth curve models. The models are fitted using regression techniques on the integrated coordinates. The use and interpretation of these differential models is illustrated with several examples: a simulated example; urban waste in Catalonia (Spain); oil production and reserves; and growth of a luzonite crystal.

37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
86A17 Global dynamics, earthquake problems (MSC2010)
Full Text: DOI
[1] Aitchison J (1986) The statistical analysis of compositional data. Monographs on statistics and applied probability. Chapman & Hall Ltd., London, p 416 (Reprinted in 2003 with additional material by The Blackburn Press)
[2] Aitchison, J; Barceló-Vidal, C; Egozcue, JJ; Pawlowsky-Glahn, V; Bayer, U (ed.); Burge, H (ed.); Skala, M (ed.), A concise guide for the algebraic-geometric structure of the simplex, the sample space for compositional data analysis, 387-392, (2002), Berlin
[3] Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6): 716-723 · Zbl 0314.62039
[4] Billheimer, D; Guttorp, P; Fagan, W, Statistical interpretation of species composition, J Am Stat Assoc, 96, 1205-1214, (2001) · Zbl 1073.62573
[5] Borgheresi, M; Buccianti, A; Benedetto, F; Vaughan, DJ, Application of compositional techniques in the field of crystal chemistry: a case study of luzonite, a sn-bearing mineral, Math Geosci, 45, 183-206, (2013)
[6] Brewer, D; Barenco, M; Callard, R; Hubank, M; Stark, J, Fitting ordinary differential equations to short time course data, Philos Trans R Soc A, 366, 519-544, (2008) · Zbl 1153.37444
[7] Brockwell PJ, Davis RA (1987) Time series: theory and methods. Springer, New York, p 584 · Zbl 0604.62083
[8] Caithamer, P, Regression and time series analysis of the world oil peak of production: another look, Math Geosci, 40, 653-670, (2008) · Zbl 1184.62215
[9] Carnahan B, Luther HA, Wilkes JO (1969) Applied numerical methods. Wiley and Sons, New York, p 604
[10] Davison AC, Hinkley DV (1997) Bootstrap methods and their application. Cambridge University Press, Cambridge, p 582 · Zbl 0886.62001
[11] Deffeyes KS (2005) Beyond oil. Hill and Wang, New York, p 224
[12] Egozcue, JJ, Reply to on the harker variation diagrams by J. A. cortés, Math Geosci, 41, 829-834, (2009) · Zbl 1178.86018
[13] Egozcue JJ, Barceló-Vidal C, Martín-Fernández JA, Jarauta-Bragulat E, Díaz-Barrero JL, Mateu-Figueras G (2011a) Elements of simplicial linear algebra and geometry. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, Chichester
[14] Egozcue JJ, Jarauta-Bragulat E, Díaz-Barrero JL (2011b) Calculus of simplex-valued functions. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, Chichester
[15] Egozcue JJ, Pawlowsky-Glahn V (2005) Groups of parts and their balances in compositional data analysis. Math Geol 37(7):795-828 · Zbl 1177.86018
[16] Egozcue, JJ; Pawlowsky-Glahn, V; Buccianti, A (ed.); Mateu-Figueras, G (ed.); Pawlowsky-Glahn, V (ed.), Simplicial geometry for compositional data, 67-77, (2006), London · Zbl 1155.86002
[17] Egozcue JJ, Pawlowsky-Glahn V (2011) Basic concepts and procedures. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, Chichester
[18] Egozcue JJ, Pawlowsky-Glahn V, Díaz-Barrero JL (2008) Otros espacios euclídeos. La Gaceta Real Soc Mat Española 11(2):263-267 · Zbl 1153.37444
[19] Egozcue, JJ; Pawlowsky-Glahn, V; Mateu-Figueras, G; Barceló-Vidal, C, Isometric logratio transformations for compositional data analysis, Math Geol, 35, 279-300, (2003) · Zbl 1302.86024
[20] Egozcue JJ, Pawlowsky-Glahn V, Tolosana-Delgado R, Ortego MI, van den Boogaart KG (2013) Bayes spaces: use of improper distributions and exponential families. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A, Mat (RACSAM) 107:475-486. doi:10.1007/s13398-012-0082-6 · Zbl 1280.62030
[21] Euler, L, Réchérches générales sur la mortalité et la multiplication du gendre humain, Hist Acad R Sci B-Lett Berl, 16, 144-164, (1760)
[22] Hubbert, MK, Energy from fossil fuels, Science, 109, 103-109, (1949)
[23] Jarauta-Bragulat E, Egozcue JJ (2011) Approaching predator-prey Lotka-Volterra equations by simplicial linear diferential equations. In: Proceedings of the 4th International Workshop on Compositional Data Analysis, CoDaWork-2011, Sant Feliu de Guixols, Girona
[24] Kermack, WO; McKendrick, AG, A contribution to the mathematical theory of epidemics, Proc R Soc London, 115, 700-721, (1927) · JFM 53.0517.01
[25] Lotka AJ (1925) Elements of physical biology. Williams and Wilkins, Baltimore, p 460 · JFM 51.0416.06
[26] Mateu-Figueras G, Pawlowsky-Glahn V, Egozcue JJ (2011) The principle of working on coordinates. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, Chichester · Zbl 1155.86002
[27] Odum EP (1971) Fundamentals of ecology. Saunders, W. Washington Square, Philadelphia
[28] Patzek, TW; Croft, GD, A global coal production forecast with multi-hubbert cycle analysis, Energy, 35, 3109-3122, (2010)
[29] Pawlowsky-Glahn V, Egozcue JJ (2001) Geometric approach to statistical analysis on the simplex. Stoch Environ Res Risk Assess (SERRA) 15(5):384-398 · Zbl 0987.62001
[30] Pontriaguin LS (1969) Équations différentielles ordinaires. MIR, Moscow, p 399
[31] Tolosana-Delgado R, Pawlowsky-Glahn V, Egozcue JJ (2008) Indicator kriging without order relation violations. Math Geosci 40(3):327-347 · Zbl 1158.86005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.