Advanced methods for analysing and modelling multivariate palaeoclimatic time series

The separation of natural and anthropogenically caused climatic changes is an important task of contemporary climate research. For this purpose, a detailed knowledge of the natural variability of the climate during warm stages is a necessary prerequisite. Beside model simulations and historical documents, this knowledge is mostly derived from analyses of so-called climatic proxy data like tree rings or sediment as well as ice cores. In order to be able to appropriately interpret such sources of palaeoclimatic information, suitable approaches of statistical modelling as well as methods of time series analysis are necessary, which are applicable to short, noisy, and non-stationary uniand multivariate data sets. Correlations between different climatic proxy data within one or more climatological archives contain significant information about the climatic change on longer time scales. Based on an appropriate statistical decomposition of such multivariate time series, one may estimate dimensions in terms of the number of significant, linear independent components of the considered data set. In the presented work, a corresponding approach is introduced, critically discussed, and extended with respect to the analysis of palaeoclimatic time series. Temporal variations of the resulting measures allow to derive information about climatic changes. For an example of trace element abundances and grain-size distributions obtained near the Cape Roberts (Eastern Antarctica), it is shown that the variability of the dimensions of the investigated data sets clearly correlates with the Oligocene/Miocene transition about 24 million years before present as well as regional deglaciation events. Grain-size distributions in sediments give information about the predominance of different transportation as well as deposition mechanisms. Finite mixture models may be used to approximate the corresponding distribution functions appropriately. In order to give a complete description of the statistical uncertainty of the parameter estimates in such models, the concept of asymptotic uncertainty distributions is introduced. The relationship with the mutual component overlap as well as with the information missing due to grouping and truncation of the measured data is discussed for a particular geological example. An analysis of a sequence of grain-size distributions obtained in Lake Baikal reveals that there are certain problems accompanying the application of finite mixture models, which cause an extended climatological interpretation of the results to fail. As an appropriate alternative, a linear principal component analysis is used to decompose the data set into suitable fractions whose temporal variability correlates well with the variations of the average solar insolation on millenial to multi-millenial time scales. The abundance of coarse-grained material is obviously related to the annual snow cover, whereas a significant fraction of fine-grained sediments is likely transported from the Taklamakan desert via dust storms in the spring season.

[1]  C. McLaren.  Mixture models in haematology: a series of case studies , 1996, Statistical methods in medical research.

[2]  A. Mackay,et al.  A diatom record of centennial resolution for the Kazantsevo Interglacial stage in Lake Baikal (Siberia) , 2005 .

[3]  O. Torres,et al.  ENVIRONMENTAL CHARACTERIZATION OF GLOBAL SOURCES OF ATMOSPHERIC SOIL DUST IDENTIFIED WITH THE NIMBUS 7 TOTAL OZONE MAPPING SPECTROMETER (TOMS) ABSORBING AEROSOL PRODUCT , 2002 .

[4]  R Quian Quiroga,et al.  Event synchronization: a simple and fast method to measure synchronicity and time delay patterns. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  M. Yoneda,et al.  Temporal changes in the phytoplankton community of the southern basin of Lake Baikal over the last 24,000 years recorded by photosynthetic pigments in a sediment core , 2002 .

[6]  Michael Schulz,et al.  The Mid-Pleistocene climate transition: onset of 100 ka cycle lags ice volume build-up by 280 ka , 1997 .

[7]  H. Pfuhl,et al.  Integrated Age Models for the Early Oligocene–Early Miocene, Sites 1168 and 1170-1172 , 2004 .

[8]  A. Hine,et al.  Paleoclimate implications of high latitude precession-scale mineralogic fluctuations during early Oligocene Antarctic glaciation: the Great Australian Bight record , 2003 .

[9]  S. Colman,et al.  Evidence from Lake Baikal for Siberian Glaciation during Oxygen-Isotope Substage 5d , 1998, Quaternary Research.

[10]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[11]  G. McLachlan.  On Bootstrapping the Likelihood Ratio Test Statistic for the Number of Components in a Normal Mixture , 1987 .

[12]  J. Kalbfleisch,et al.  A modified likelihood ratio test for homogeneity in finite mixture models , 2001 .

[13]  M. B. Priestley,et al.  Non-linear and non-stationary time series analysis , 1990 .

[14]  I. D. Hill.  The Normal Integral , 1973 .

[15]  H. Cramér.  Mathematical methods of statistics , 1947 .

[16]  R. Folk,et al.  Brazos River bar [Texas]; a study in the significance of grain size parameters , 1957 .

[17]  Henry S. Greenside,et al.  KARHUNEN-LOEVE DECOMPOSITION OF EXTENSIVE CHAOS , 1996, chao-dyn/9610007.

[18]  J. M. Craddock,et al.  Eigenvectors for representing the 500 mb geopotential surface over the Northern Hemisphere , 1969 .

[19]  M. Paluš,et al.  Information theoretic test for nonlinearity in time series , 1993 .

[20]  U. Barcelona,et al.  Detection of superimposed periodic signals using wavelets , 2002, astro-ph/0202107.

[21]  T. Crowley,et al.  Tectonic boundary conditions for climate reconstructions , 1998 .

[22]  Richard J. Telford,et al.  The secret assumption of transfer functions: problems with spatial autocorrelation in evaluating model performance , 2005 .

[23]  Aapo Hyvärinen,et al.  Independent Component Analysis: Fast ICA by a fixed-point algorithm that maximizes non-Gaussianity , 2001 .

[24]  O. Barndorff-Nielsen,et al.  The pattern of natural size distributions , 1980 .

[25]  G. McLachlan,et al.  Fitting mixture distributions to phenylthiocarbamide (PTC) sensitivity. , 1991, American journal of human genetics.

[26]  V. Plerou,et al.  Universal and Nonuniversal Properties of Cross Correlations in Financial Time Series , 1999, cond-mat/9902283.

[27]  A. Bowker,et al.  Statistical Theory with Engineering Applications. , 1953 .

[28]  M. Thiel,et al.  Cross recurrence plot based synchronization of time series , 2002, physics/0201062.

[29]  Martienssen,et al.  Characterization of spatiotemporal chaos from time series. , 1993, Physical review letters.

[30]  J. Jouzel,et al.  Evidence for general instability of past climate from a 250-kyr ice-core record , 1993, Nature.

[31]  Franz Dieter Fischer,et al.  Fracture statistics of brittle materials: Weibull or normal distribution. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  J. Laskar,et al.  Astronomical calibration of Oligocene--Miocene time , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[33]  J. Aitchison,et al.  Biplots of Compositional Data , 2002 .

[34]  D. Inman,et al.  Measures for describing the size distribution of sediments , 1952 .

[35]  V. Hasselblad.  Finite mixtures of distributions from the exponential family , 1969 .

[36]  A. Roberts,et al.  Integrated chronostratigraphic calibration of the Oligocene-Miocene boundary at 24.0 ± 0.1 Ma from the CRP-2A drill core, Ross Sea, Antarctica , 2002 .

[37]  G. P. King,et al.  Extracting qualitative dynamics from experimental data , 1986 .

[38]  G. McLachlan,et al.  Fitting mixture models to grouped and truncated data via the EM algorithm. , 1988, Biometrics.

[39]  S. Cande,et al.  Revised calibration of the geomagnetic polarity timescale for the Late Cretaceous and Cenozoic , 1995 .

[40]  C L Webber,et al.  Dynamical assessment of physiological systems and states using recurrence plot strategies. , 1994, Journal of applied physiology.

[41]  C. Biernacki,et al.  Degeneracy in the maximum likelihood estimation of univariate Gaussian mixtures with EM , 2003 .

[42]  S.Drozdz,et al.  Dynamics of competition between collectivity and noise in the stock market , 1999, cond-mat/9911168.

[43]  D. Bates,et al.  Newton-Raphson and EM Algorithms for Linear Mixed-Effects Models for Repeated-Measures Data , 1988 .

[44]  William H. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[45]  D. Hartmann,et al.  The hyperbolic shape triangle as a tool for discriminating populations of sediment samples of closely connected origin , 1992 .

[46]  T. Williams,et al.  A high‐resolution record of early Miocene Antarctic glacial history from ODP Site 1165, Prydz Bay , 2005 .

[47]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[48]  John C. Davis.  Information contained in sediment-size analyses , 1970 .

[49]  G. Parisi,et al.  A Theory of Stochastic Resonance in Climatic Change , 1983 .

[50]  Wilfried Seidel,et al.  On the Power of Different Versions of the Likelihood Ratio Test for Homogeneity in an Exponential Mixture Model , 2000 .

[51]  Stefan Rahmstorf,et al.  Abrupt glacial climate changes due to stochastic resonance. , 2002, Physical review letters.

[52]  W. C. Krumbein,et al.  Size distribution of source rocks of sediments , 1940 .

[53]  J. Zbilut,et al.  Embeddings and delays as derived from quantification of recurrence plots , 1992 .

[54]  Katharina Krischer,et al.  Transitions to electrochemical turbulence. , 2005, Physical review letters.

[55]  A. Zhisheng,et al.  A Quaternary climate record based on grain size analysis from the Luochuan loess section on the Central Loess Plateau, China , 2004 .

[56]  R. Yuretich,et al.  Clay minerals in the sediments of Lake Baikal; a useful climate proxy , 1999 .

[57]  C. McLaren,et al.  Detection of two-component mixtures of lognormal distributions in grouped, doubly truncated data: analysis of red blood cell volume distributions. , 1991, Biometrics.

[58]  M. Clark.  Some methods for statistical analysis of multimodal distributions and their application to grain-size data , 1976 .

[59]  R. Sutherland,et al.  Application of the Log-Hyperbolic Distribution to Hawai'ian Beach Sands , 1994 .

[60]  M. Mudelsee,et al.  Exploring the structure of the mid-Pleistocene revolution with advanced methods of time-series analysis , 1997 .

[61]  R. Renner.  An examination of the use of the logratio transformation for the testing of endmember hypotheses , 1991 .

[62]  M. Batist,et al.  Sedimentary dynamics on isolated highs in Lake Baikal: evidence from detailed high-resolution geophysical data and sediment cores , 2005 .

[63]  R. A. Groeneveld,et al.  Practical Nonparametric Statistics (2nd ed). , 1981 .

[64]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[65]  B. Chao,et al.  Wavelet Spectral Analysis of the Earth’s Orbital Variations and Paleoclimatic Cycles , 1998 .

[66]  Astrid Ruck.  Calculating the (Asymptotic) Distribution of the Log-LRT Statistic in a Contamination Mixture Model , 2001 .

[67]  Sandy P. Harrison,et al.  DIRTMAP: the geological record of dust , 2001 .

[68]  P. Rosin.  The Laws Governing the Fineness of Powdered Coal , 1933 .

[69]  John Aitchison,et al.  The Statistical Analysis of Compositional Data , 1986 .

[70]  Astrid Ruck.  Calculating the Asymptotic Distribution of the Log-LRT Statistic for Testing One Against Two Populations in Normal Mixtures , 2002 .

[71]  Peter Frick,et al.  Wavelet Analysis of Stellar Chromospheric Activity Variations , 1997 .

[72]  D. D. Gilbertson,et al.  A new method for environmental analysis of particle size distribution data from shoreline sediments , 1984, Nature.

[73]  J. Friedman,et al.  Estimating Optimal Transformations for Multiple Regression and Correlation. , 1985 .

[74]  Geoffrey J. McLachlan,et al.  Modelling mass−size particle data by finite mixtures , 1989 .

[75]  W. Mitchell,et al.  6. DETERMINATION OF BIOGENIC OPAL IN PELAGIC MARINE SEDIMENTS: A SIMPLE METHOD REVISITED 1 , 2002 .

[76]  C. Wunsch,et al.  Rectification and precession signals in the climate system , 2003 .

[77]  Eckehard Olbrich,et al.  Analysing local observations of weakly coupled maps , 1998 .

[78]  M. Woodbury.  A missing information principle: theory and applications , 1972 .

[79]  G. Plaut,et al.  Spells of Low-Frequency Oscillations and Weather Regimes in the Northern Hemisphere. , 1994 .

[80]  N. E. Day.  Estimating the components of a mixture of normal distributions , 1969 .

[81]  J. Zachos,et al.  Climate Response to Orbital Forcing Across the Oligocene-Miocene Boundary , 2001, Science.

[82]  Kunihiko Kaneko,et al.  Spatiotemporal chaos in one-and two-dimensional coupled map lattices , 1989 .

[83]  L. J. Poppe,et al.  A Visual Basic Program to Generate Sediment Grain-Size Statistics and to Extrapolate Particle Distributions , 2004, Comput. Geosci..

[84]  T. Louis.  Finding the Observed Information Matrix When Using the EM Algorithm , 1982 .

[85]  A Comparison between Log-hyperbolic and Model-independent Grain Size Distributions in Sediment Trend Analysis! (STA®) , 2001 .

[86]  Xiao-Li Meng,et al.  Using EM to Obtain Asymptotic Variance-Covariance Matrices: The SEM Algorithm , 1991 .

[87]  C. Wunsch,et al.  A Depth-Derived Pleistocene Age-Model: Uncertainty Estimates, Sedimentation Variability, and Nonlinear Climate Change , 2002 .

[88]  Theiler,et al.  Generating surrogate data for time series with several simultaneously measured variables. , 1994, Physical review letters.

[89]  B. Everitt,et al.  Finite Mixture Distributions , 1981 .

[90]  J. Kurths,et al.  Recurrence-plot-based measures of complexity and their application to heart-rate-variability data. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[91]  D. Heslop,et al.  Quantification of magnetic coercivity components by the analysis of acquisition curves of isothermal remanent magnetisation , 2001 .

[92]  C Raab,et al.  Estimation of large-scale dimension densities. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[93]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[94]  Helen V. Atkinson,et al.  A model-based analysis of particle size distributions in composite materials , 2003 .

[95]  M. Kramer.  Nonlinear principal component analysis using autoassociative neural networks , 1991 .

[96]  L. R. Kittleman.  Application of Rosin's distribution in size-frequency analysis of clastic rocks , 1964 .

[97]  M. Paluš.  Detecting Nonlinearity in Multivariate Time Series , 1996 .

[98]  Bruce M. Hill,et al.  Information for Estimating the Proportions in Mixtures of Exponential and Normal Distributions , 1963 .

[99]  Brian Everitt,et al.  Maximum Likelihood Estimation of the Parameters in a Mixture of Two Univariate Normal Distributions; a Comparison of Different Algorithms , 1984 .

[100]  W. C. Krumbein,et al.  Size frequency distributions of sediments and the normal phi curve , 1938 .

[101]  B. Lindsay.  The Geometry of Mixture Likelihoods: A General Theory , 1983 .

[102]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[103]  R. R. Hocking,et al.  The analysis of incomplete data. , 1971 .

[104]  E. Vrba.  Paleoclimate and evolution, with emphasis on human origins , 1995 .

[105]  Patience A. Cowie.  Fractals and Dynamic Systems in Geoscience , 1996 .

[106]  A. Ioannides,et al.  Temporal correlations versus noise in the correlation matrix formalism: an example of the brain auditory response. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[107]  Jose A. Rial,et al.  Understanding nonlinear responses of the climate system to orbital forcing , 2000 .

[108]  J. Zachos,et al.  Latest Oligocene through early Miocene isotopic stratigraphy and deep-water paleoceanography of the western equatorial Atlantic : Sites 926 and 929 , 1997 .

[109]  J. Jouzel,et al.  Comparison of oxygen isotope records from the GISP2 and GRIP Greenland ice cores , 1993, Nature.

[110]  M. Mudelsee,et al.  Plio-/Pleistocene climate modeling based on oxygen isotope time series from deep-sea sediment cores: The Grassberger-Procaccia algorithm and chaotic climate systems , 1994 .

[111]  J. Mélice,et al.  Amplitude and frequency modulations of the Earth's obliquity for the last million years , 2001 .

[112]  Maureen E. Raymo,et al.  A methane-based time scale for Vostok ice , 2003 .

[113]  D. C. Erbach,et al.  Effect of wetting and drying on soil physical properties 1 Joint contribution USDAAgricultural Rese , 1999 .

[114]  Dimitris Karlis,et al.  Improving the EM algorithm for mixtures , 1999, Stat. Comput..

[115]  M. Schulz.  The tempo of climate change during Dansgaard‐Oeschger interstadials and its potential to affect the manifestation of the 1470‐year climate cycle , 2002 .

[116]  James Theiler,et al.  Testing for nonlinearity in time series: the method of surrogate data , 1992 .

[117]  Alternation of different fluctuation regimes in the stock market dynamics , 2003, cond-mat/0306608.

[118]  Michael Schulz,et al.  Spectrum: spectral analysis of unevenly spaced paleoclimatic time series , 1997 .

[119]  M. Schulz,et al.  Amplitude variations of 1470‐year climate oscillations during the last 100,000 years linked to fluctuations of continental ice mass , 1999 .

[120]  D. Demske,et al.  Late Pliocene vegetation and climate of the Lake Baikal region, southern East Siberia, reconstructed from palynological data , 2002 .

[121]  I. Meilijson.  A fast improvement to the EM algorithm on its own terms , 1989 .

[122]  P. Grassberger,et al.  Characterization of Strange Attractors , 1983 .

[123]  J. Laskar.  Long-term solution for the insolation quantities of the Earth , 2006, Proceedings of the International Astronomical Union.

[124]  K. Kashiwaya,et al.  Long‐term climato‐limnological oscillation during the past 2.5 million years printed in Lake Baikal sediments , 1998 .

[125]  Mikhael Gorokhovski,et al.  Analyses of Kolmogorov’s model of breakup and its application into Lagrangian computation of liquid sprays under air-blast atomization , 2003 .

[126]  S. Loukas,et al.  Ml estimation in the poisson binomial distribution with grouped data via the em algorithm , 1993 .

[127]  Jürgen Kurths,et al.  Multivariate recurrence plots , 2004 .

[128]  D. J. Doeglas.  Interpretation of the Results of Mechanical Analyses , 1946 .

[129]  B. Efron,et al.  Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information , 1978 .

[130]  H. Voss,et al.  Reconstruction of non-linear time delay models from data by the use of optimal transformations , 1997 .

[131]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[132]  D. Demske,et al.  Late glacial and Holocene vegetation and regional climate variability evidenced in high-resolution pollen records from Lake Baikal , 2005 .

[133]  A. Roberts,et al.  Glaciation across the Oligocene-Miocene boundary in southern McMurdo Sound, Antarctica: New chronology from the CIROS-1 drill hole , 2003 .

[134]  Schreiber,et al.  Improved Surrogate Data for Nonlinearity Tests. , 1996, Physical review letters.

[135]  D. Paillard.  Glacial cycles: Toward a new paradigm , 2001 .

[136]  C. Wunsch.  On sharp spectral lines in the climate record and the millennial peak , 2000 .

[137]  Heidi Cullen,et al.  A Pervasive Millennial-Scale Cycle in North Atlantic Holocene and Glacial Climates , 1997 .

[138]  Lorraine E. Lisiecki,et al.  Application of dynamic programming to the correlation of paleoclimate records , 2002 .

[139]  William E. Griffiths,et al.  Small Sample Properties of Probit Model Estimators , 1987 .

[140]  A. Mackay,et al.  DIATOM SUCCESSION TRENDS IN RECENT SEDIMENTS FROM LAKE BAIKAL AND THEIR RELATION TO ATMOSPHERIC POLLUTION AND TO CLIMATE CHANGE , 1998 .

[141]  Jaroslaw Kwapien,et al.  Quantifying the dynamics of financial correlations , 2001 .

[142]  M. Batist,et al.  Seismic evidence of small-scale lacustrine drifts in Lake Baikal (Russia) , 2001 .

[143]  Geoffrey J. McLachlan.  On the choice of starting values for the EM algorithm in fitting mixture models , 1988 .

[144]  Jörn H. Kruhl,et al.  Fractals and dynamic systems in geoscience , 1994 .

[145]  V. Hasselblad,et al.  Statistical and graphical evaluation of erythrocyte volume distributions. , 1987, The American journal of physiology.

[146]  E. Bard.  Greenhouse effect and ice ages: historical perspective , 2004 .

[147]  B. Pompe.  Measuring statistical dependences in a time series , 1993 .

[148]  P. Jones.  On collagen fibril diameter distributions. , 1991, Connective tissue research.

[149]  M. G. Smith.  Survival of E. coli and Salmonella after Chilling and Freezing in Liquid Media , 1995 .

[150]  Gordon K. Smyth,et al.  On Using The Log-hyperbolic Distribution To Describe The Textural Characteristics Of Eolian Sediments , 1988 .

[151]  W. C. Krumbein,et al.  Size frequency distributions of sediments , 1934 .

[152]  Rory A. Fisher,et al.  Theory of Statistical Estimation , 1925, Mathematical Proceedings of the Cambridge Philosophical Society.

[153]  W. K. Brown,et al.  Particle size distributions and the sequential fragmentation/transport theory applied to volcanic ash , 1989 .

[154]  N. N. Greenman.  The Mechanical Analysis of Sediments from Thin-Section Data , 1951, The Journal of Geology.

[155]  Philippe Gaillot,et al.  Wavelet analysis of relative geomagnetic paleointensity at ODP Site 983 , 2000 .

[156]  K. Holmgren,et al.  Highly variable Northern Hemisphere temperatures reconstructed from low- and high-resolution proxy data , 2005, Nature.

[157]  G. McLachlan,et al.  GASTROENTEROLOGY 1998;114:543–549 Distribution of Transferrin Saturation in an Australian Population: Relevance to the Early Diagnosis , 2022 .

[158]  P. Kyle,et al.  Geochemical indicators of weathering and Cenozoic palaeoclimates in sediments from CRP-1 and CIROS-1, McMurdo Sound, Antarctica , 1998 .

[159]  Michael Schulz,et al.  REDFIT: estimating red-noise spectra directly from unevenly spaced paleoclimatic time series , 2002 .

[160]  I. N. McCave,et al.  Holocene periodicity in North Atlantic climate and deep-ocean flow south of Iceland , 1999, Nature.

[161]  Manfred Mudelsee,et al.  TAUEST: a computer program for estimating persistence in unevenly spaced weather/climate time series , 2002 .

[162]  Ximing Wu,et al.  Calculation of Maximum Entropy Densities with Application to Income Distribtuions , 2002 .

[163]  Kunihiko Kaneko,et al.  Spatiotemporal chaos and noise , 1989 .

[164]  Stefan Rahmstorf,et al.  Timing of abrupt climate change: A precise clock , 2003 .

[165]  W. M. Smith,et al.  Spatial organization, predictability, and determinism in ventricular fibrillation. , 1998, Chaos.

[166]  J. Channell.  Geomagnetic paleointensity and directional secular variation at Ocean Drilling Program (ODP) Site 984 (Bjorn Drift) since 500 ka: Comparisons with ODP Site 983 (Gardar Drift) , 1999 .

[167]  Vadim A. Kravchinsky,et al.  A rock-magnetic record from Lake Baikal, Siberia: Evidence for Late Quaternary climate change , 1994 .

[168]  H. Kaufmann,et al.  Variation in Lake Baikal's phytoplankton distribution and fluvial input assessed by SeaWiFS satellite data , 2005 .

[169]  John F. Hart,et al.  Computer Approximations , 1978 .

[170]  Allou Samé,et al.  A Mixture Model Approach for Binned Data Clustering , 2003, IDA.

[171]  B. Lindsay,et al.  The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family , 1994 .

[172]  V. Plerou,et al.  Random matrix approach to cross correlations in financial data. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[173]  John W. Harbaugh,et al.  Numerical Experiments in Stratigraphy: Recent Advances in Stratigraphic and Sedimentologic Computer Simulations , 1999 .

[174]  D. Maraun,et al.  Epochs of phase coherence between El Niño/Southern Oscillation and Indian monsoon , 2005 .

[175]  N. Shackleton,et al.  The 100,000-year ice-Age cycle identified and found to lag temperature, carbon dioxide, and orbital eccentricity , 2000, Science.

[176]  Bruce G. Lindsay,et al.  Computer-assisted analysis of mixtures (C.A.MAN) statistical algorithms , 1992 .

[177]  Jonathan M. Lilly,et al.  A wavelet analysis of Plio-Pleistocene climate indicators: A new view of periodicity evolution , 1995 .

[178]  Marco Thiel,et al.  Long-term predictability of mean daily temperature data , 2005 .

[179]  G. Weltje,et al.  End-Member Modeling of Siliciclastic Grain-Size Distributions: The Late Quaternary Record of Eolian and Fluvial Sediment Supply to the Arabian Sea and Its Paleoclimatic Significance , 1999 .

[180]  L. Burlaga,et al.  Fractal structure of the interplanetary magnetic field , 1986 .

[181]  Bronwyn H Hall,et al.  Estimation and Inference in Nonlinear Structural Models , 1974 .

[182]  C. Wunsch.  The spectral description of climate change including the 100 ky energy , 2003 .

[183]  Ramesh P. Singh,et al.  Further evidences for the weakening relationship of Indian rainfall and ENSO over India , 2004 .

[184]  P. Rosin,et al.  Die Kornzusammensetzung des Mahlgutes im Lichte der Wahrscheinlichkeitslehre , 1934 .

[185]  Bose,et al.  Karhunen-Loeve local characterization of spatiotemporal chaos in a reaction-diffusion system , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[186]  Matthias Holschneider,et al.  Wavelets - an analysis tool , 1995, Oxford mathematical monographs.

[187]  S. Havlin,et al.  Phase synchronization in temperature and precipitation records , 2002, physics/0211092.

[188]  U. Stephani,et al.  Detection and characterization of changes of the correlation structure in multivariate time series. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[189]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[190]  V. Hasselblad.  Estimation of parameters for a mixture of normal distributions , 1966 .

[191]  E. Gassiat,et al.  Asymptotic distribution and local power of the log-likelihood ratio test for mixtures: bounded and unbounded cases , 2006 .

[192]  J. Rial,et al.  An outsider's review of the astronomical theory of the climate: is the eccentricity-driven insolation the main driver of the ice ages? , 2001 .

[193]  M. Loutre,et al.  Astronomical theory of climate change , 2004 .

[194]  J. Orford,et al.  Assessment of temporal changes in coastal sand dune environments using the log‐hyperbolic grain‐size method , 2002 .

[195]  S. Brewer,et al.  Quantitative reconstruction of the last interglacial vegetation and climate based on the pollen record from Lake Baikal, Russia , 2005 .

[196]  R. Preisendorfer,et al.  Principal Component Analysis in Meteorology and Oceanography , 1988 .

[197]  G. Kondolf,et al.  Weibull vs. Lognormal Distributions for Fluvial Gravels , 2000 .

[198]  Mark J. van der Laan,et al.  Fitting of mixtures with unspecified number of components using cross validation distance estimate , 2003, Comput. Stat. Data Anal..

[199]  Annette Witt,et al.  Holocene climate variability on millennial scales recorded in Greenland ice cores , 2005 .

[200]  Jeffrey D. Scargle,et al.  Wavelet methods in astronomical time series analysis , 1997 .

[201]  Peter C. M. Molenaar,et al.  A comparison of four methods of calculating standard errors of maximum likelihood estimates in the analysis of covariance structure. , 1991 .

[202]  D. Rubin,et al.  The ECME algorithm: A simple extension of EM and ECM with faster monotone convergence , 1994 .

[203]  J. Timmer,et al.  Tempting long-memory - on the interpretation of DFA results , 2004 .

[204]  Geoffrey J. McLachlan,et al.  Maximum Likelihood Estimation of Mixture Densities for Binned and Truncated Multivariate Data , 2002, Machine Learning.

[205]  J. Aitchison.  Principal component analysis of compositional data , 1983 .

[207]  S. Cande,et al.  A new geomagnetic polarity time scale for the Late Cretaceous and Cenozoic , 1992 .

[208]  W. J. Conover,et al.  Practical Nonparametric Statistics , 1972 .

[209]  Kurths,et al.  Phase synchronization of chaotic oscillators. , 1996, Physical review letters.

[210]  D. Heslop,et al.  Spectral analysis of unevenly spaced climatic time series using CLEAN: signal recovery and derivation of significance levels using a Monte Carlo simulation , 2002 .

[211]  J. Kiefer,et al.  CONSISTENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN THE PRESENCE OF INFINITELY MANY INCIDENTAL PARAMETERS , 1956 .

[212]  New York Dover,et al.  ON THE CONVERGENCE PROPERTIES OF THE EM ALGORITHM , 1983 .

[213]  Carin Andersson,et al.  Biases in the estimation of transfer function prediction errors , 2004 .

[214]  R. Muller,et al.  Constructing a stacked benthic δ18O record , 2002 .

[215]  Geoffrey J. McLachlan,et al.  Improving the convergence rate of the em algorithm for a mixture model fitted to grouped truncated data , 1992 .

[216]  P. Barker,et al.  The opening of Drake Passage , 1977 .

[217]  Anupam Sahay,et al.  The search for a low-dimensional characterization of a local climate system , 1996, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[218]  Gerald M. Friedman,et al.  Determination of Sieve-Size Distribution from Thin-Section Data for Sedimentary Petrological Studies , 1958, The Journal of Geology.

[219]  C. Kissel,et al.  Geomagnetic paleointensity and environmental record from Labrador Sea core MD95-2024: global marine sediment and ice core chronostratigraphy for the last 110 kyr , 2000 .

[220]  C. J. Butler,et al.  Air temperatures at Armagh Observatory, Northern Ireland, from 1796 to 2002 , 2005 .

[221]  A. Witt,et al.  High-resolution magnetostratigraphy of late quaternary sediments from Lake Baikal, Siberia: timing of intracontinental paleoclimatic responses , 2005 .

[222]  W. Seidel,et al.  Tools for analyzing and maximizing likelihood functions in mixture models , 2002 .

[223]  J. Bouchaud,et al.  Noise Dressing of Financial Correlation Matrices , 1998, cond-mat/9810255.

[224]  James C. Zachos,et al.  Orbitally paced climate oscillations across the Oligocene/Miocene boundary , 1997, Nature.

[225]  J. Jouzel,et al.  Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica , 1999, Nature.

[226]  Multiresolution approach for period determination on unevenly sampled data , 2004, astro-ph/0403066.

[227]  Sergei Maslov.  Measures of globalization based on cross-correlations of world financial indices , 2001 .

[228]  R. Haszeldine,et al.  Quantification of Illite Content in Sedimentary Rocks Using Magnetic Susceptibility—A Rapid Complement or Alternative to X-Ray Diffraction , 2004 .

[229]  O. Peles,et al.  Stability of terrestrial planets in the habitable zone of Gl 777 A, HD 72659, Gl 614, 47 Uma and HD 4208 , 2004 .

[230]  Michael Schulz,et al.  Coherent Resonant Millennial-Scale Climate Oscillations Triggered by Massive Meltwater Pulses , 2003 .

[231]  Analysis of red blood cell volume distributions using the ICSH reference method: detection of sequential changes in distributions determined by hydrodynamic focusing , 2008 .

[232]  P. Yiou,et al.  Interhemispheric space-time attributes of the Dansgaard-Oeschger oscillations between 100 and 0 ka , 2002 .

[233]  D. N. Geary.  Mixture Models: Inference and Applications to Clustering , 1989 .

[234]  D. Demske,et al.  Extraction and AMS Radiocarbon Dating of Pollen from Lake Baikal Sediments , 2004, Radiocarbon.

[235]  K. Wohletz,et al.  Discrimination of grain-size subpopulations in pyroclastic deposits , 1987 .

[236]  M. Stuiver,et al.  Oxygen 18/16 variability in Greenland snow and ice with 10 -3- to 105-year time resolution , 1997 .

[237]  Detecting components in censored and truncated meteorological data , 1998 .

[238]  Karl Mosler,et al.  A Cautionary Note on Likelihood Ratio Tests in Mixture Models , 2000 .

[239]  Department of Physics,et al.  Extensive scaling and nonuniformity of the Karhunen-Loève decomposition for the spiral-defect chaos state , 1998, chao-dyn/9808006.

[240]  T. Schreiber,et al.  Surrogate time series , 1999, chao-dyn/9909037.

[241]  E. Martin,et al.  Circulation in the Southern Ocean during the Paleogene inferred from neodymium isotopes , 2004 .

[242]  Geoffrey J. McLachlan,et al.  Hierarchical Models for Screening of Iron Deficiency Anemia , 1999, ICML.

[243]  Milan Paluš,et al.  Sunspot Cycle: A Driven Nonlinear Oscillator? , 1999 .

[244]  J. D. Hays,et al.  Age Dating and the Orbital Theory of the Ice Ages: Development of a High-Resolution 0 to 300,000-Year Chronostratigraphy , 1987, Quaternary Research.

[245]  J. Bloemendal,et al.  Grain-size distribution function of polymodal sediments in hydraulic and aeolian environments, and numerical partitioning of the sedimentary components , 2002 .

[246]  Aradhna K. Tripati,et al.  Orbitally Induced Climate and Geochemical Variability Across the Oligocene/Miocene Boundary , 2000 .

[247]  M. Paluš.  Detecting phase synchronization in noisy systems , 1997 .

[248]  Anson W. Mackay,et al.  Introduction to “Progress towards reconstructing past climate in Central Eurasia, with special emphasis on Lake Baikal” , 2005 .

[249]  D. J. Robertson.  Discrimination of remanence-carrying minerals in mixtures, using isothermal remanent magnetisation acquisition curves , 1994 .

[250]  N. Loboda,et al.  Orbital forcings of the Earth's climate in wavelet domain , 2005 .

[251]  Juergen Kurths,et al.  Detection of synchronization for non-phase-coherent and non-stationary data , 2005 .

[252]  Peter J. Webster,et al.  Monsoon and Enso: Selectively Interactive Systems , 1992 .

[253]  Xin Liu,et al.  Asymptotics for the likelihood ratio test in a two-component normal mixture model , 2004 .

[254]  Jin Zhang.  Powerful goodness‐of‐fit tests based on the likelihood ratio , 2002 .

[255]  H C Lukaski,et al.  Patient-specific analysis of sequential haematological data by multiple linear regression and mixture distribution modelling. , 2000, Statistics in medicine.

[256]  Tore P°Sse.  Grain size distribution expressed as tanh‐functions , 1997 .

[257]  André Berger,et al.  Long-term variations of daily insolation and Quaternary climatic changes , 1978 .

[258]  R. Vannucci,et al.  Sand provenance from major and trace element analyses of bulk rock and sand grains , 1998 .

[259]  A. Giuliani,et al.  Detecting deterministic signals in exceptionally noisy environments using cross-recurrence quantification , 1998 .

[260]  Warren Galke,et al.  Analysis of Coarsely Grouped Data from the Lognormal Distribution , 1980 .

[261]  J. Zachos,et al.  28. MILANKOVITCH-SCALE CLIMATE VARIABILITY RECORDED NEAR THE OLIGOCENE/MIOCENE BOUNDARY 1 , 1997 .

[262]  C. Elger,et al.  Measuring nonstationarity by analyzing the loss of recurrence in dynamical systems. , 2002, Physical review letters.

[263]  Sheila A. Farmer.  An Investigation into the Results of Principal Component Analysis of Data Derived from Random Numbers , 1971 .

[264]  Geoffrey J. McLachlan,et al.  On a Resampling Approach to Choosing the Number of Components in Normal Mixture Models , 2007 .

[265]  Pierre Francus,et al.  An image-analysis technique to measure grain-size variation in thin sections of soft clastic sediments , 1998 .

[266]  A. Prokopenko,et al.  Insolation regime in Siberia as a major factor controlling diatom production in Lake Baikal during the past 800,000 years , 2001 .

[267]  K. Lam,et al.  REML and ML estimation for clustered grouped survival data , 2003, Statistics in medicine.

[268]  R. Jennrich,et al.  Conjugate Gradient Acceleration of the EM Algorithm , 1993 .

[269]  R. Hegger,et al.  Estimation of Lyapunov spectra from space–time data , 1999, chao-dyn/9908004.

[270]  P. Seba,et al.  Random matrix analysis of human EEG data. , 2003, Physical review letters.

[271]  R. Carver.  Procedures in sedimentary petrology , 1971 .

[272]  N. Fieller,et al.  Statistics of Particle Size Data , 1992 .

[273]  L. Lirer,et al.  Deconvolution of pyroclastic grain‐size spectra for interpretation of transport mechanisms: an application to the AD 79 Vesuvio deposits , 1996 .

[274]  G. Weltje.  End-member modeling of compositional data: Numerical-statistical algorithms for solving the explicit mixing problem , 1997 .

[275]  Peter Arcidiacono,et al.  Finite Mixture Distributions, Sequential Likelihood and the EM Algorithm , 2003 .

[276]  A. Roberts,et al.  Orbitally induced oscillations in the East Antarctic ice sheet at the Oligocene/Miocene boundary , 2001, Nature.

[277]  J. Kurths,et al.  Estimation of dynamical invariants without embedding by recurrence plots. , 2004, Chaos.

[278]  V. Hasselblad,et al.  Relationship between transferrin saturation and iron stores in the African American and US Caucasian populations: analysis of data from the third National Health and Nutrition Examination Survey. , 2001, Blood.

[279]  Milan Paluš,et al.  Testing for Nonlinearity in Weather Records , 1994 .

[280]  Sandy P. Harrison,et al.  The role of dust in climate changes today, at the last glacial maximum and in the future , 2001 .

[281]  P. Barker,et al.  Scotia Sea regional tectonic evolution: implications for mantle flow and palaeocirculation , 2001 .

[282]  Konstantinos Adamidis.  Theory & Methods: An EM algorithm for estimating negative binomial parameters , 1999 .

[283]  Wilfried Seidel,et al.  Types of likelihood maxima in mixture models and their implication on the performance of tests , 2004 .

[284]  G. Parisi,et al.  Stochastic resonance in climatic change , 1982 .

[285]  J. Kennett.  Cenozoic evolution of Antarctic glaciation the Circum-Antarctic Ocean and their impact on global paleoceanography , 1977 .

[286]  R. Schleyer.  The Goodness-of-fit to Ideal Gauss and Rosin Distributions: A New Grain-size Parameter , 1987 .

[287]  N. Shackleton,et al.  Constraints on astronomical parameters from the geological record for the last 25 Myr , 2000 .

[288]  V. Hasselblad,et al.  Analysis of the volume of red blood cells: application of the expectation-maximization algorithm to grouped data from the doubly-truncated lognormal distribution. , 1986, Biometrics.

[289]  P. F. Barkera,et al.  Origin , signature and palaeoclimatic influence of the Antarctic Circumpolar Current , 2004 .

[290]  P. Francus,et al.  A Computer-assisted Thin-section Study of Lake Baikal Sediments: a Tool for Understanding Sedimentary Processes and Deciphering Their Climatic Signal , 1998 .

[291]  D. Heslop,et al.  Analysis of isothermal remanent magnetization acquisition curves using the expectation-maximization algorithm , 2002 .

[292]  Peter Hall,et al.  Theoretical analysis of power in a two-component normal mixture model , 2005 .

[293]  R. Wirth,et al.  Detrital input and early diagenesis in sediments from Lake Baikal revealed by rock magnetism , 2005 .

[294]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[295]  S. Ciliberto,et al.  Estimating the Number of Degrees of Freedom in Spatially Extended Systems , 1991 .

[296]  Fraser,et al.  Independent coordinates for strange attractors from mutual information. , 1986, Physical review. A, General physics.

[297]  Paul J. Roebber,et al.  The architecture of the climate network , 2004 .

[298]  J. Laskar,et al.  Orbital, precessional, and insolation quantities for the earth from -20 Myr to +10 Myr. , 1993 .

[299]  M. Raymo,et al.  A Pliocene‐Pleistocene stack of 57 globally distributed benthic δ18O records , 2005 .

[300]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[301]  Murray Aitkin,et al.  A hybrid EM/Gauss-Newton algorithm for maximum likelihood in mixture distributions , 1996, Stat. Comput..

[302]  Stuart G. Baker,et al.  A Simple Method for Computing the Observed Information Matrix When Using the EM Algorithm with Categorical Data , 1992 .

[303]  D. Oakes.  Direct calculation of the information matrix via the EM , 1999 .

[304]  G. McLachlan,et al.  An algorithm for the likelihood ratio test of one versus two components in a normal mixture model fitted to grouped and truncated data , 1995 .

[305]  M. Schulz.  On the 1470‐year pacing of Dansgaard‐Oeschger warm events , 2002 .

[306]  E. Mosley‐Thompson,et al.  Holocene—Late Pleistocene Climatic Ice Core Records from Qinghai-Tibetan Plateau , 1989, Science.

[307]  K. Pye,et al.  GRADISTAT: a grain size distribution and statistics package for the analysis of unconsolidated sediments , 2001 .

[308]  R. Chambers,et al.  Multivariate analysis of sedimentary environments using grain-size frequency distributions , 1979 .

[309]  P. Kyle,et al.  Geochemical indicators of weathering, Cenozoic palaeoclimates, and provenance from fine-grained sediments in CRP-2/2A, Victoria Land Basin, Antarctica , 2000 .

[310]  B. Lindsay,et al.  Alternative EM methods for nonparametric finite mixture models , 2001 .

[311]  J. Vandenberghe,et al.  Lithogenesis and geomorphological processes of the Pleisto¬cene deposits at Maastricht-Belvédère. , 1993 .

[312]  Kirk A. Maasch,et al.  Calculating climate attractor dimension from δ18O records by the Grassberger-Procaccia algorithm , 1989 .

[313]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.

[314]  M. S. Santhanam,et al.  Statistics of atmospheric correlations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[315]  Dankmar Böhning,et al.  The EM algorithm with gradient function update for discrete mixtures with known (fixed) number of components , 2003, Stat. Comput..

[316]  Rajagopalan,et al.  On the weakening relationship between the indian monsoon and ENSO , 1999, Science.

[317]  K. Kashiwaya.  Long continental records from Lake Baikal , 2003 .

[318]  Dimitris Karlis,et al.  Choosing Initial Values for the EM Algorithm for Finite Mixtures , 2003, Comput. Stat. Data Anal..

[319]  Isabella Raffi,et al.  Astronomical calibration age for the Oligocene-Miocene boundary , 2000 .

[320]  R. Alley,et al.  Stochastic resonance in the North Atlantic , 2001 .

[322]  R. Jennrich,et al.  Acceleration of the EM Algorithm by using Quasi‐Newton Methods , 1997 .

[323]  Julia C. Hargreaves,et al.  Timing of ice‐age terminations determined by wavelet methods , 2003 .

[324]  Klaus Lehnertz,et al.  Improved statistical test for nonstationarity using recurrence time statistics. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[325]  Christophe Biernacki,et al.  Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models , 2003, Comput. Stat. Data Anal..

[326]  D. Hartmann,et al.  Efficiency of the Log-Hyperbolic Distribution-A Case Study: Pattern of Sediment Sorting in a Small Tidal-Inlet-Het Zwin, The Netherlands , 1993 .

[327]  P. Tchamitchian,et al.  Wavelet analysis of signals with gaps , 1998 .

[328]  K. Kashiwaya,et al.  Orbit-related long-term climate cycles revealed in a 12-Myr continental record from Lake Baikal , 2001, Nature.

[329]  Grant Foster,et al.  Wavelets for period analysis of unevenly sampled time series , 1996 .

[330]  Bernd Kromer,et al.  Persistent Solar Influence on North Atlantic Climate During the Holocene , 2001, Science.

[331]  Christophe Biernacki,et al.  Degeneracy in the Maximum Likelihood Estimation of Univariate Gaussian Mixtures for Grouped Data and Behaviour of the EM Algorithm , 2007 .

[332]  J. Kurths,et al.  Three types of transitions to phase synchronization in coupled chaotic oscillators. , 2003, Physical review letters.

[333]  M. Agha,et al.  Maximum Likelihood Estimation of Mixtures of Distributions , 1984 .

[334]  H. Oberhaensli,et al.  Late Quaternary clay mineral record in Central Lake Baikal (Academician Ridge, Siberia) , 2003 .

[335]  Geoffrey J. McLachlan,et al.  Standard errors of fitted component means of normal mixtures , 1997 .

[336]  H. Kantz,et al.  Nonlinear time series analysis , 1997 .

[337]  L. Gahagan,et al.  Evolution of Cenozoic seaways in the circum-Antarctic region , 2003 .

[338]  G. Weltje,et al.  Genetically meaningful decomposition of grain-size distributions. , 2007 .

[339]  Paul R. Cohen,et al.  Very Predictive Ngrams for Space-Limited Probabilistic Models , 2003, IDA.

[340]  J. Kurths,et al.  Large-scale dimension densities for heart rate variability analysis , 2005, Computers in Cardiology, 2005.

[341]  Jürgen Kurths,et al.  Detection of n:m Phase Locking from Noisy Data: Application to Magnetoencephalography , 1998 .

[342]  Heiko Pälike,et al.  Astronomic calibration of the late Oligocene through early Miocene geomagnetic polarity time scale , 2004 .

[343]  T. Higuchi.  Approach to an irregular time series on the basis of the fractal theory , 1988 .

[344]  Dankmar Böhning,et al.  Asymptotic properties of the EM algorithm estimate for normal mixture models with component specific variances , 2003, Comput. Stat. Data Anal..