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Mineral Species Frequency Distribution Conforms to a Large Number of Rare Events Model: Prediction of Earth’s Missing Minerals

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Abstract

A population model is introduced to describe the mineral species frequency distribution. Mineral species coupled with their localities conform to a large number of rare events (LNRE) distribution: 100 common mineral species occur at more than 1,000 localities, whereas \(34 \,\%\) of the approved 4,831 mineral species are found at only one or two localities. LNRE models formulated in terms of a structural type distribution allow the estimation of Earth’s undiscovered mineralogical diversity and the prediction of the percentage of observed mineral species that would differ if Earth’s history were replayed.

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References

  • Baayen RH (1993) Statistical models for word frequency distributions: a linguistic evaluation. Comput Humanit 26:347–363

    Article  Google Scholar 

  • Baayen RH (2001) Word frequency distributions, text, speech and language technology, vol 18. Kluwer Academic Publishers, Dordrecht

    Book  Google Scholar 

  • Baroni M, Evert S (2007) Words and echoes: assessing and mitigating the non-randomness problem in word frequency distribution modeling. In: Proceedings of the 45th annual meeting of the association for computational linguistics, Prague, pp 904–911

  • Baroni M, Evert S (2005) Testing the extrapolation quality of word frequency models. In: Danielsson P, Wagenmakers M (eds) Proceedings of corpus linguistics 2005, Birmingham, UK. The corpus linguistics conference series, vol 1

  • Bunge J, Barger K (2008) Parametric models for estimating the number of classes. Biom J 50(6):971–982

    Article  Google Scholar 

  • Bunge J, Fitzpatrick M (1993) Estimating the number of species: a review. J Am Stat Assoc 88(421):364–373

    Google Scholar 

  • Bunge J, Willis A, Walsh F (2014) Estimating the number of species in microbial diversity studies. Annu Rev Stat Appl 1:427–445

    Google Scholar 

  • Burnham KP, Overton WS (1978) Estimation of the size of a closed population when capture probabilities vary among animals. Biometrika 65(3):625–633

    Article  Google Scholar 

  • Burnham KP, Overton WS (1979) Robust estimation of population size when capture probabilities vary among animals. Ecology 60(5):927–936

    Article  Google Scholar 

  • Chao A (1984) Nonparametric estimation of the number of classes in a population. Scand J Stat 11(4):265–270

    Google Scholar 

  • Chao A, Bunge J (2002) Estimating the number of species in a stochastic abundance model. Biometrics 58(3):531–539

    Article  Google Scholar 

  • Chao A, Lee SM (1992) Estimating the number of classes via sample coverage. J Am Stat Assoc 87(417):210–217

    Article  Google Scholar 

  • Chao A, Ma MC, Yang MCK (1993) Stopping rules and estimation for recapture debugging with unequal failure rates. Biometrika 80:193–201

    Article  Google Scholar 

  • Chao A, Hwang WH, Chen YC, Kuo CY (2000) Estimating the number of shared species in two communities. Stat Sin 10:227–246

    Google Scholar 

  • Efron B, Thisted R (1976) Estimating the number of unseen species: how many words did Shakespeare know? Biometrica 63(3):435–447

    Google Scholar 

  • Efron B, Tibshirani RJ (1993) An introduction to the bootstrap, monographs on statistics and applied probability, vol 57. Chapman & Hall/CRC, London

    Book  Google Scholar 

  • Evert S (2004) A simple LNRE model for random character sequences. In: Proceedings of the 7èmes Journées Internationales d’Analyse Statistique des Données Textuelles, Louvain-la-Neuve, pp 411–422

  • Evert S, Baroni M (2007) zipfR: word frequency distributions in R. In: Proceedings of the 45th annual meeting of the association for computational linguistics, posters and demonstrations session, Prague, pp 29–32

  • Evert S, Baroni M (2008) Statistical models for word frequency distributions, package zipfR. http://zipfr.r-forge.r-project.org/materials/zipfR_0.6-5.pdf. Accessed 10 Nov 2008

  • Fisher RA, Corbet AS, Williams CB (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. J Anim Ecol 12(1):42–58

    Article  Google Scholar 

  • Good IJ (1953) The population frequencies of species and the estimation of population parameters. Biometrika 40:237–264

  • Hazen RM, Grew ES, Downs RT, Golden J, Hystad G (2015) Mineral ecology: chance and necessity in the mineral diversity of terrestrial planets. Can Mineral 53(2). doi:10.3749/canmin.1400086

  • Heller G (1997) Estimation of the number of classes. S Afr Stat J 31:65–90

    Google Scholar 

  • Keating KA, Quinn JF, Ivie MA, Ivie LL (1998) Estimating the effectiveness of further sampling in species inventories. Ecol Appl 8(4):1239–1249

    Google Scholar 

  • Khmaladze EV (1987) The statistical analysis of large number of rare events. Tech. Rep. MS-R8804, Department of Mathematical Statistics, Center for Mathematics and Computer Science, CWI, Amsterdam, Netherlands

  • Khmaladze EV, Chitashvili RJ (1989) Statistical analysis of large number of rare events and related problems. Trans Tbilisi Math Inst 91:196–245

    Google Scholar 

  • Kyselý J (2010) Coverage probability of bootstrap confidence intervals in heavy-tailed frequency models, with application to precipitation data. Theor Appl Climatol 101:345–361

    Article  Google Scholar 

  • Ma C, Beckett JR, Rossman GR (2014) Monipite, MoNiP, a new phosphide mineral in a Ca-Al-rich inclusion from the Allende meteorite. Am Mineral 99(1):198–205

    Article  Google Scholar 

  • Miller RI, Wiegert RG (1989) Documenting completeness, species-area relations, and the species-abundance distribution of a regional flora. Ecology 70(1):16–22

    Article  Google Scholar 

  • Norris JL, Pollock KH (1998) Non-parametric MLE for Poisson species abundance models allowing for heterogeneity between species. Environ Ecol Stat 5(4):391–402

    Article  Google Scholar 

  • Shen TJ, Chao A, Lin CF (2003) Predicting the number of new species in further taxonomic sampling. Ecology 84(3):798–804

    Article  Google Scholar 

  • Sichel HS (1971) On a family of discrete distributions particularly suited to represent long-tailed frequency data. In: Proceedings of the third symposium on mathematical statistics, Pretoria, pp 51–97

  • Sichel HS (1975) On a distribution law for word frequencies. J Am Stat Assoc 70:542–547

    Google Scholar 

  • Sichel HS (1986) Word frequency distributions and type-token characteristics. Math Sci 11:45–72

    Google Scholar 

  • Soberón J, Llorente J (1993) The use of species accumulation functions for the prediction of species richness. Conserv Biol 7(3):480–488

    Article  Google Scholar 

  • Solow AR, Polasky S (1999) A quick estimator for taxonomic surveys. Ecology 80(8):2799–2803

    Article  Google Scholar 

  • Wang JP (2010) Estimating species richness by a Poisson-compound Gamma model. Biometrika 97(3):727–740

    Article  Google Scholar 

  • Wang JP (2011) SPECIES: an R package for species richness estimation. J Stat Softw 40(9):1–15

    Google Scholar 

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Acknowledgments

Joshua Golden, Edward Grew, and Dimitri Sverjensky provided valuable advice and discussions. We thank the Deep Carbon Observatory, the Keck Foundation, and a private foundation for support.

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089, Tucson, AZ, 85721-0089, USA

    Grethe Hystad

  2. Department of Geosciences, University of Arizona, 1040 E 4th Street, Tucson, AZ, 85721-0077, USA

    Robert T. Downs

  3. Geophysical Laboratory, Carnegie Institution, 5251 Broad Branch Road NW, Washington, DC, 20015, USA

    Robert M. Hazen

Authors
  1. Grethe Hystad
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  2. Robert T. Downs
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  3. Robert M. Hazen
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Correspondence to Grethe Hystad.

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Hystad, G., Downs, R.T. & Hazen, R.M. Mineral Species Frequency Distribution Conforms to a Large Number of Rare Events Model: Prediction of Earth’s Missing Minerals. Math Geosci 47, 647–661 (2015). https://doi.org/10.1007/s11004-015-9600-3

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  • DOI: https://doi.org/10.1007/s11004-015-9600-3

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