Abstract
The nonlinear scaling of meteorological processes is an issue of much interest. The objectives of the present work were (a) to investigate cross-correlations between pairs of meteorological time series using different resolutions and (b) to explore the long-range cross-correlations through different scaling exponents. We used 13 years of daily records of rainfall, relative humidity, cloudiness and vapor pressure ranging from January 1st 1996 to December 31st 2009. Data sets were compiled from Veguita agro-meteorological station at Granma province, Cuba. Detrended cross-correlation analysis, multiscale sample entropy, Lévy-stable laws and Hurst–Kolmogorov dynamics were the main methodological and theoretical tools. The detrended cross-correlation coefficient showed significant cross-correlation between rainfall, relative humidity, cloudiness and actual vapor pressure at all investigated time scales. The individual Hurst exponents were in the range 0.62 ≤ H ≤ 0.72 which is consistent with long-range correlated patterns. Bivariate Hurst exponents (Hxy) were larger than the average exponents of the separate processes (Hx and Hy, respectively). The Hurst–Kolmogorov exponents estimated from the climacograms were in the range 0.6 ≤ H ≤ 0.7 (0.603 ≤ β ≤ 0.798) consistent with the values estimated from detrended fluctuation analysis. Each pair of meteorological variables fitted reasonably well bistable distributions with approximately the same Lévy index (α ≅ 0.736). Hurst–Kolmogorov and infinite variance processes are important drivers of the atmospheric dynamics which can explain the persistence of extreme events (droughts) usually observed in the studied region. The multivariate multiscale sample entropy method and multivariate stable distributions could be valuable candidates for describing daily atmospheric processes.
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Abbreviations
- A :
-
Density location parameter
- B :
-
Skewness parameter
- F :
-
Dispersion parameter
- d :
-
Maximum norm distance
- G :
-
Embedding delay vector
- e°:
-
Saturated vapor pressure (kPa)
- e a :
-
Actual vapor pressure (kPa)
- τ :
-
Time lag vector
- H ρ :
-
Reduced form of 2Hxy–Hx–Hy
- H x :
-
Hurst exponent of the residual time series of x(t)
- H xy :
-
Bivariate Hurst exponent
- H y :
-
Hurst exponent of the residual time series of y(t)
- LRCC:
-
Long-range cross-correlation coefficient
- m :
-
Scale factor
- p :
-
Number of variables in the multivariate multiscale entropy
- s :
-
Local scale length in the DCCA method
- S(G,τ,ε):
-
Entropy-based metric as a function of the G-length vector
- β :
-
Scaling exponent of the climacogram
- T :
-
Superscript indicating matrix transpose
- T a :
-
Air temperature
- \(\theta\) :
-
Fourier variable
- α :
-
Scaling exponent of the Lévy-stable distribution
- γ :
-
Exponent of the autocorrelation function
- ε :
-
Distance parameter (usually 0.15*SD)
- ρ[S(x,y,G,τ,ε)]:
-
Entropy correlation coefficient
- ρ DCCA :
-
Detrended cross-correlation coefficient
- \(\sigma ({y_{l,j}^m })\) :
-
Variance of the coarse-grained time series
- \(C^g \left( \varepsilon \right)\) :
-
Probability that two vectors are within ε in the multivariate multiscale entropy analysis
- \(\hat{f}\left( \theta \right)\) :
-
Fourier transform of the Lévy-stable probability density function f(x)
- \(y_{l,j}^m\) :
-
Coarse-grained time series corresponding to the scale factor m
References
Ahmed MU, Mandic DP (2012) Multivariate multiscale entropy analysis. IEEE Signal Process Lett 19:91–94
Allen RG, Pereira LS, Raes D, Smith M (1998) Crop evapotranspiration: Guidelines for computing crop water requirements. FAO irrigation and drainage paper no 56, FAO, Rome, p 300
Anderson PL, Meerschaert MM (1998) Modeling river flows with heavy tails. Water Resour Res 34(9):2271–2280
Balzter H, Tate NJ, Kaduk J, Harper D, Page S, Morrison R, Muskulus M, Jones P (2015) Multi-scale entropy analysis as a method for time-series analysis of climate data. Climate 3:227–240
Ben Alaya MA, Zwiers FW, Zhang X (2020) A bivariate approach to estimating the probability of very extreme precipitation events. Weather Clim Extrem 30:100290
Benson DA, Schumer R, Meerschaert MM, Wheatcraft SW (2001) Fractional dispersion, Lévy motion, and the MADE tracer tests. Transp Porous Med 42:211–240
Betts AK, Desjardins R, Worth D, Beckage B (2014) Climate coupling between temperature, humidity, precipitation and cloud cover over the Canadian Prairies. J Geophys Res Atmos 119:13305–13326
Bony S, Stevens B, Frierson D et al (2015) Clouds, circulation and climate sensitivity. Nat Geosci 8:261–268
Cahill ND (2010) Normalized measures of mutual information with general definitions of entropy for multimodal image registration in biomedical image registration. In: Lecture notes in computer science, vol 6204. Springer, Berlin, pp 258–268
Chen C, Sun S, Cao Z, Shi Y, Sun B, Zhang XD (2019) A comprehensive comparison and overview of R packages for calculating sample entropy. Biol Methods Protoc 4(1):1–8
Chou C-M (2012) Applying multiscale entropy to the complexity analysis of rainfall–runoff relationships. Entropy 14:945–957
Costa M, Goldberger AL, Peng C-K (2005) Multiscale entropy analysis of biological signals. Phys Rev E 71:021906
da Silva-Filho AM, Zebende GF, de Castro APN, Guedes EF (2021) Statistical test for multiple detrended cross-correlation coefficient. Physica A 562:125285
Dawley S, Zhang Y, Liu X, Jiang P, Tick GR, Sun HG, Zheng C, Chen L (2019) Statistical analysis of extreme events in precipitation, stream discharge, and groundwater head fluctuation: distribution, memory, and correlation. Water 11:707
de Almeida BA, Alves de Araújo H, Zebende GF (2019) Detrended multiple cross-correlation coefficient applied to solar radiation, air temperature and relative humidity. Sci Rep 9:19764. https://doi.org/10.1038/s41598-019-56114-6
Fu Z, Piao L (2016) Quantifying distinct associations on different temporal scales: comparison of DCCA and Pearson methods. Sci Rep 6:36759. https://doi.org/10.1038/srep36759
Graves T, Gramacy R, Watkins N, Franzke Ch (2017) A brief history of long-memory: Hurst, Mandelbrot and the road to ARFIMA, 1951–1980. Entropy 19:437
He L-Y, Chen S-P (2011) Nonlinear bivariate dependency of pricevolume relationships in agricultural commodity future markets: a perspective from multifractal detrended cross-correlation analysis. Physica A 390:297–308
Hottovy S, Stechmann S (2015) A spatiotemporal stochastic model for tropical precipitation and water vapor dynamics. J Atmos Sci 72(12):4721–4738
Huang F, Chunyu X, Wang Y, Wu Y, Qian B, Guo L, Zhao D, Xia Z (2017) Investigation into multi-temporal scale complexity of streamflows and water levels in the Poyang Lake Basin. China Entropy 19:67. https://doi.org/10.3390/e19020067
Jamin A, Humeau-Heurtier A (2020) (Multiscale) cross-entropy methods: a review. Entropy 22:45. https://doi.org/10.3390/e22010045
Koutsoyiannis D (2009) A random walk on water. Hydrol Earth Syst Sci Discuss 6:6611–6658
Koutsoyiannis D (2011) Hurst–Kolmogorov dynamics and uncertainty. J Am Water Resourc Assoc 47(3):481–495. https://doi.org/10.1111/j.1752-1688.2011.00543.x
Kristoufek L (2015) Can the bivariate Hurst exponent be higher than an average of the separate Hurst exponents? Physica A 431:124–127
Kristoufek L (2016) Power-law cross-correlations estimation under heavy tails. Comm Nonlinear Sci Numer Sim 40:163–172
Krutto A (2016) Parameter estimation in stable law. Risks 4:43. https://doi.org/10.3390/risks4040043
Li H, Choy S, Zaminpardaz S, Carter B, Sun C, Purwar S, Liang H, Li L, Wang X (2023) Investigating the inter-relationships among multiple atmospheric variables and their responses to precipitation. Atmosphere 14:571. https://doi.org/10.3390/atmos14030571
Looney D, Tricia A, Mandic DP (2018) A novel multivariate sample entropy algorithm for modeling time series synchronization. Entropy 20:82. https://doi.org/10.3390/e20020082
Lovejoy S (2015) A voyage through scales, a missing quadrillion and why the climate is not what you expect. Clim Dyn 44:3187–3210. https://doi.org/10.1007/s00382-014-2324-0
Lovejoy S, Schertzer D, Varon D (2013) Do GCMs predict the climate….or macroweather? Earth Sys Dynam 4:439–454
Menabde M, Sivapalan M (2020) Modeling of rainfall time series and extremes using bounded cascades and Levy-stable distributions. Water Resour Res 36(11):3293–3300
Mendoza V, Pazos M, Garduño R, Mendoza B (2021) Thermodynamics of climate change between cloud cover, atmospheric temperature and humidity. Sci Rep 11:21244. https://doi.org/10.1038/s41598-021-00555-5
Millán H, Kalauzi A, Llerena G, Sucoshañay J, Piedra D (2009) Meteorological complexity in the Amazonian area of Ecuador: an approach based on dynamical system theory. Ecol Compl 6:278–285
Millán H, Rodríguez J, Ghanbarian-Alavijeh B, Biondi R, Llerena G (2011) Temporal complexity of daily precipitation records from different atmospheric environments: chaotic and Lévy stable parameters. Atmos Res 101:879–892
Mohammadi M, Mohammadpour A, Ogata H (2015) On estimating the tail index and the spectral measure of multivariate α-stable distributions. Metrika 78(5):549–561
Mollaei S, Darooneh AH, Karimi S (2019) Multi-scale entropy analysis and Hurst exponent. Physica A 528:121292. https://doi.org/10.1016/j.physa.2019.121292
Mudelsee M (2010) Climate time series analysis: classical statistics and bootstrap methods. Springer, Dordrecht
Neelin JD, Martinez-Villalobos C, Stechmann SN, Ahmed F, Chen G, Norris JM, Kuo Y-H, Lenderink G (2022) Precipitation extremes and water vapor: relationships in current climate and implications for climate change. Curr Clim Change Rep 8:17–33
Nolan JP (1997) Numerical computation of stable densities and distribution functions. Stoch Model 13:759–774
Nolan JP (1998) Parameterizations and modes of stable distributions. Stat Probl Lett 38:187–195
Nolan JP (2013) Financial modeling with heavy-tailed stable distributions. Wires Comput Stat 6(1):45–55
O’Connell E, O’Donnell G, Koutsoyiannis D (2023) On the spatial scale dependence of long-term persistence in global annual precipitation data and the Hurst Phenomenon. Water Resourc Res 59:e2022WR033133. https://doi.org/10.1029/2022WR033133
Peel MC, Finlayson BL, McMahon TA (2007) Updated world map of the Köppen–Geiger climate classification. Hydrol Earth Sys Sci 11:1633–1644
Peng CK, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL (1994) Mosaic organization of DNA nucleotides. Phys Rev E 49(2):1685–1689
Podobnik B, Stanley HE (2008) Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series. Phys Rev Lett 100:084102–084104
Podobnik B, Horvatic D, Petersen AM, Stanley E (2009) Cross-correlations between volume change and price change. PNAS 106(52):22079–22084
Prass TS, Pumi G (2020) DCCA: detrended fluctuation and detrended cross-correlation analysis. R Package, version 0.1.1. https://CRAN.R-project.org/package=DCCA
R Core Team (2023) R: a language and environment for statistical computing version 4.3.1. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org
Richardson AD, Denny EG, Siccama TG, Lee X (2003) Evidence for a rising cloud ceiling in eastern North America. J Clim 16:2093–2098
Ryu C (2023) dlookr: tools for data diagnosis, exploration, transformation. R Package version 0.6.2. https://CRAN.R-project.org/package=dlookr
Samorodnitsky G, Taqqu M (1994) Stable non-Gaussian random processes: stochastic models with infinite variance. Chapman and Hall, New York
Sankaran A, Krzyszczak J, Baranowski P, Devarajan Sindhu A, Kumar N, Lija Jayaprakash N, Thankamani V, Ali M (2020) Multifractal cross correlation analysis of agro-meteorological datasets (including reference evapotranspiration) of california, United States. Atmosphere 11: 1116
Schneider T, O’Gorman PA, Levine X (2009) Water vapor and the dynamics of climate changes. Rev Geophys 48(3):1–23
Sela R, Hurvich C (2012) The average periodogram estimator for a power law in coherency. J Time Series Anal 33:340–363
Teimouri M, Rezakhah S, Mohammadpour A (2018) Parameter estimation using the EM algorithm for symmetric stable random variables and sub-Gaussian random vectors. J Stat Theor Appl 17(3):439–461
Teimouri M, Mohammadpour A, Nadarajah S (2019) Alphastable: inference for stable distribution. R Package version 0.2.1. https://CRAN.R-project.org/package=alphastable
The MathWorks Inc (2007) MATLAB: the language of technical computing, version 7.5.0.342 (R2007b). MathWorks, Natick
Valjarević A, Popovici C, Štilić A, Radojković M (2022) Cloudiness and water from cloud seeding in connection with plants distribution in the Republic of Moldova. Appl Water Sci 12:262. https://doi.org/10.1007/s13201-022-01784-3
Vassoler RT, Zebende GF (2012) DCCA cross-correlation coefficient apply in time series of air temperature and air relative humidity. Physica A 391:2438–2443
Vu TM, Mishra AK, Konapala G (2018) Information entropy suggested stronger nonlinear associations between hydro-meteorological variables and ENSO. Entropy 20:38
Wang G-J, Xie C, He L-Y, Chen S (2014) Detrended minimum-variance hedge ratio: a new method for hedge ratio at different time scales. Physica A 405:70–79
Wei Q, Xu J, Liao L, Yu Y, Liu W, Zhou J, Ding Y (2021) Indicators for evaluating trends of air humidification in arid regions under circumstance of climate change: relative humidity (RH) vs. Actual water vapour pressure (ea). Ecol Indic 121:107043. https://doi.org/10.1016/j.ecolind.2020.107043
Zebende GF (2011) DCCA cross-correlation coefficient: quantifying level of cross-correlation. Physica A 390:614–618
Zhang XD, Zhang Z, Wang D (2018) CGManalyzer: an R package for analyzing continuous glucose monitoring studies. Bioinform 34:1609–1611
Zhou X, Lin J-S, Liang X, Xu W (2022) Rainfall patterns from multiscale sample entropy analysis. Front Water 4:885456. https://doi.org/10.3389/frwa.2022.885456
Acknowledgements
H. M. wants to acknowledge Dr. John Nolan (American University) for his valuable comments on the statistics of Lévy-stable distributions. We also thank Dr. Danilo P. Mandic (Imperial College London) for providing the MATLAB code MMSEV3 used in the present work.
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Conceptualization: [HM], [EF-G], [RC], [RB]; methodology: [HM], [EF-G]; formal analysis and investigation: [HM], [RC], [RB], [EF-G]; writing—original draft preparation: [HM]; writing—review and editing: [RB], [RC]; resources: [HM], [RC]; supervision: [EF-G], [RB], [RC], [HM].
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Millán, H., Biondi, R., Cumbrera, R. et al. Associating daily meteorological variables of a local climate using DCCA, sample entropy, Lévy-index and Hurst–Kolmogorov exponents: a case study. Meteorol Atmos Phys 136, 7 (2024). https://doi.org/10.1007/s00703-024-01006-2
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DOI: https://doi.org/10.1007/s00703-024-01006-2