×

zbMATH — the first resource for mathematics

Modelling spatio-temporal variation in sparse rainfall data using a hierarchical Bayesian regression model. (English) Zbl 1426.62353
Summary: Rainfall is a critical component of climate governing vegetation growth and production, forage availability and quality for herbivores. However, reliable rainfall measurements are not always available, making it necessary to predict rainfall values for particular locations through time. Predicting rainfall in space and time can be a complex and challenging task, especially where the rain gauge network is sparse and measurements are not recorded consistently for all rain gauges, leading to many missing values. Here, we develop a flexible Bayesian model for predicting rainfall in space and time and apply it to Narok County, situated in southwestern Kenya, using data collected at 23 rain gauges from 1965 to 2015. Narok County encompasses the Maasai Mara ecosystem, the northern-most section of the Mara-Serengeti ecosystem, famous for its diverse and abundant large mammal populations and spectacular migration of enormous herds of wildebeest, zebra and Thomson’s gazelle. The model incorporates geographical and meteorological predictor variables, including elevation, distance to Lake Victoria and minimum temperature. Salient features of our model are the use of non-stationary covariance structures and the facility to handle excess zeros and many missing observations. We assess the efficiency of the model by comparing it empirically with the established Gaussian process, Kriging, simple linear and Bayesian linear models. We use the model to predict total monthly rainfall and its standard error for all \(5 \times 5\) km grid cells in Narok County for each year from 1965 to 2015. Using the Monte Carlo integration method, we estimate seasonal and annual rainfall and their standard errors for 29 sub-regions of Narok for each of the 51 years spanning 1965–2015. The non-stationary model can handle data from a sparse network of observations with many missing values and performs at least as well as or better than four established and widely used models on the Narok rainfall data set. The model produces rainfall predictions consistent with expectation and in good agreement with blended station and satellite rainfall values. The predictions are precise enough for most practical purposes. The model is very general and applicable to other variables besides rainfall.
MSC:
62P12 Applications of statistics to environmental and related topics
86A32 Geostatistics
62F15 Bayesian inference
Software:
spTimer; fields
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Allcroft, DJ; Glasbey, CA, A latent Gaussian Markov random-field model for spatiotemporal rainfall disaggregation, Journal of Royal Statistical Society, Series C, 52, 487-498, (2003) · Zbl 1111.62362
[2] Bakar, KS; Sahu, SK, spTimer: Spatio-Temporal Bayesian Modelling Using R, Journal of Statistical Software, 63, 1-32, (2015)
[3] Banerjee, S.; Gelfand, AE; Finley, AO; Sang, H., Gaussian predictive process models for large spatial data sets, Journal of Royal Statistical Society, Series B, 70, 825-848, (2008) · Zbl 05563371
[4] Bartzke, GS; Ogutu, JO; Mukhopadhyay, S.; Mtui, D.; Dublin, HT; Piepho, HP, Rainfall trends and variation in the Maasai Mara ecosystem and their implications for animal population and biodiversity dynamics, PLoS ONE, 13, e0202814, (2018)
[5] Boutton, T. W., Tieszen, L. L., and Imbamba, S. K. (1988a),“Biomass dynamics of grassland vegetation in Kenya,” AfricanJournal of Ecology, 26, 89-101.
[6] Boutton, T. W., Tieszen, L. L., and Imbamba, S. K. (1988b),“Seasonal changes in the nutrient content of East Africangrassland vegetation,” African Journal of Ecology, 26(2), 103-115.
[7] Briggs, WM; Levine, RA, Wavelets and field forecast verification, Monthly Weather Review, 125, 1329-1341, (1997)
[8] Brown, PE; Diggle, PJ; Lord, ME; Young, PC, Space-time calibration of radar rainfall data, Journal of Royal Statistical Society, Series C, 50, 221-241, (2001)
[9] Chai, T., and Draxler, R. R. (2014), “Root mean square error (RMSE) or mean absolute error (MAE)?—Arguments against avoiding RMSE in the literature,” Geoscientific Model Development, 7, 1247-1250.
[10] Coe, MJ; Cumming, DH; Phillipson, J., Biomass and production of large African herbivores in relation to rainfall and primary production, Oecologia, 22, 341-354, (1976)
[11] Cressie, N. A. C., and Wikle, C. K. (2011), Statistics for Spatio-Temporal Data, New York: Wiley. · Zbl 1273.62017
[12] Deshmukh, IK, A common relationship between precipitation and grassland peak biomass for east and southern Africa, African Journal of Ecology, 22, 181-186, (1984)
[13] Dublin, HT; Ogutu, JO, Population regulation of African buffalo in the Mara-Serengeti ecosystem, Wildlife Research, 42, 382-393, (2015)
[14] East, R., Rainfall, soil nutrient status and biomass of large African savanna mammals, African Journal of Ecology, 22, 245-270, (1984)
[15] Fritz, H.; Duncan, P., On the carrying capacity for large ungulates of African savanna ecosystems, Proceedings of Royal Society of London B, 256, 77-82, (1994)
[16] Furrer, R., Nychka, D., and Sain, S. (2013), Fields: Tools for Spatial Data, University Corporation for Atmospheric Research. R package version 8.4.1. http://cran.r-project.org/web/packages/fields. Accessed 1 Nov 2017.
[17] Georgiadis, NJ; McNaughton, SJ, Elemental and fibre contents of savanna grasses: variation with grazing, soil type, season and species, Journal of Applied Ecology, 27, 623-634, (1990)
[18] Hofer, H., and Campbell, K. (1995), People and Wildlife: Spatial Dynamics and Zones of Interaction, Chicago and London: The University of Chicago Press.
[19] Lorenz, E.; Jenkner, C.; Sauerbrei, W.; Becher, H., Modeling variables with a spike at zero: Examples and practical recommendations, American Journal of Epidemiology, 185, 650-660, (2017)
[20] McNaughton, SJ, Ecology of a grazing ecosystem: The Serengeti, Ecological monographs, 55, 259-294, (1985)
[21] Mills, E. D. (2013), Adjusting for covariates in zero-inflated gamma and zero-inflated log-normal models for semicontinuous data,, Technical report, University of Iowa, Iowa Research Online, https://ir.uiowa.edu/etd/2583. Accessed 1 Nov 2017.
[22] Mukhopadhyay, S.; Sahu, SK, A Bayesian spatio-temporal model to estimate long term exposure to outdoor air pollution at coarser administrative geographies in England and Wales, Journal of Royal Statistical Society, Series A, 181, 465-486, (2018)
[23] Norton-Griffiths, M.; Herlocker, D.; Pennycuick, L., The patterns of rainfall in the Serengeti ecosystem, Tanzania, African Journal of Ecology, 13, 347-374, (1975)
[24] Ogutu, JO; Owen-Smith, N., Oscillations in large mammal populations: Are they related to predation or rainfall?, African Journal of Ecology, 43, 332-339, (2005)
[25] Ogutu, JO; Piepho, HP; Dublin, HT, Responses of phenology, synchrony and fecundity of breeding by African ungulates to interannual variation in rainfall, Wildlife Research, 40, 698-717, (2014)
[26] Ogutu, JO; Piepho, HP; Dublin, HT, Reproductive seasonality in African ungulates in relation to rainfall, Wildlife Research, 41, 323-342, (2015)
[27] Ogutu, JO; Piepho, HP; Dublin, HT; Bhola, N.; Reid, RS, El Niño Southern Oscillation, rainfall, temperature and Normalized Difference Vegetation Index fluctuations in the Mara-Serengeti ecosystem, African Journal of Ecology, 46, 132-143, (2008)
[28] Ogutu, JO; Piepho, HP; Dublin, HT; Bhola, N.; Reid, RS, Rainfall influences on ungulate population abundance in the Mara-Serengeti ecosystem, Journal of Animal Ecology, 77, 814-829, (2008)
[29] Ogutu, JO; Piepho, HP; Dublin, HT; Bhola, N.; Reid, RS, Rainfall extremes explain interannual shifts in timing and synchrony of calving in topi and warthog, Population Ecology, 52, 89-102, (2010)
[30] Okuto, E. O. A. (2013), Bayesian spatial and spatio-temporal modelling (applied to precipitation dataset),, Technical Report I56/68975/2011, University of Nairobi, College of Biological and Physical Sciences, School of Mathematics.
[31] Pennycuick, L.; Norton-Griffiths, M., Fluctuations in the rainfall of the Serengeti ecosystem, Tanzania, Journal of Biogeography, 3, 125-140, (1976)
[32] Prins, HHT, Plant phenology patterns in Lake Manyara National Park, Tanzania, Journal of Biogeography, 15, 455-480, (1988)
[33] Prins, HHT; Loth, PE, Rainfall pattern as background to plant phenology in northern Tanzania, Journal of Biogeography, 15, 451-463, (1988)
[34] Ritchie, M. E. (2008), Global environmental changes and their impact on the Serengeti.Serengeti III: human impacts on ecosystem dynamics, Chicago and London: The University of Chicago Press.
[35] Sahu, SK; Bakar, KS, A comparison of bayesian models for daily ozone concentration levels, Statistical Methodology, 9, 144-157, (2012) · Zbl 1248.86004
[36] Sahu, SK; Challenor, P., A space-time model for joint modeling of ocean temperature and salinity levels as measured by Argo floats, Environmetrics, 19, 509-528, (2008)
[37] Sahu, SK; Gelfand, AE; Holland, DM, Spatio-temporal modeling of fine particulate matter, Journal of Agricultural, Biological, and Environmental Statistics, 11, 61-86, (2006)
[38] Sahu, SK; Gelfand, AE; Holland, DM, High-resolution space-time ozone modeling for assessing trends, Journal of the American Statistical Association, 102, 1221-1234, (2007) · Zbl 1332.86014
[39] Sahu, SK; Lasinio, GJ; Orasi, A.; Mardia, KV, A comparison of spatio-temporal bayesian models for reconstruction of rainfall fields in a cloud seeding experiment, Journal of Mathematics and Statistics, 1, 282-290, (2001) · Zbl 1143.62358
[40] Sahu, S. K., and Mukhopadhyay, S. (2015), On generating a flexible class of anisotropic spatial models using Gaussian predictive processes,, Technical report, University of Southampton.
[41] Sansó, B.; Guenni, L., Venezuelan rainfall data analysed by using a Bayesian space-time model, Journal of Royal Statistical Society, Series C, 48, 345-362, (1999) · Zbl 0939.62124
[42] Simpson, D.; Rue, H.; Riebler, A.; Martins, TG; Sørbye, SH, Penalising model component complexity: a principled, practical approach to constructing priors, Statistical Science, 32, 1-28, (2017) · Zbl 1442.62060
[43] Sinclair, ARE, The resource limitation of trophic levels in tropical grassland ecosystems, The Journal of Animal Ecology, 44, 497-520, (1975)
[44] Sinclair, ARE; Mduma, SA; Arcese, P., What determines phenology and synchrony of ungulate breeding in Serengeti?, Ecology, 81, 2100-2111, (2000)
[45] Tobin, J., Estimation of relationships for limited dependent variables, Econometrica, 6, 24-36, (1958) · Zbl 0088.36607
[46] Willmott, CJ; Matsuura, K., Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance, Climate Research, 30, 79-82, (2005)
[47] Zhang, H., Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics, Journal of the American Statistical Association, 99, 250-261, (2004) · Zbl 1089.62538
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.