Selection bias and the cross-validation of regression models for prediction

Strategies are compared for the development of a linear regression model with stochastic (multivariate normal) regressor variables and the subsequent assessment of its predictive ability. Bias and mean squared error of four estimators of predictive performance are evaluated in simulated samples of 32 population correlation matrices. Models including all of the available predictors are compared with those obtained using selected subsets. The subset selection procedures investigated include two stopping rules, C$\sb{\rm p}$ and S$\sb{\rm p}$, each combined with an 'all possible subsets' or 'forward selection' of variables. The estimators of performance utilized include parametric (MSEP$\sb{\rm m}$) and non-parametric (PRESS) assessments in the entire sample, and two data splitting estimates restricted to a random or balanced (Snee's DUPLEX) 'validation' half sample. The simulations were performed as a designed experiment, with population correlation matrices representing a broad range of data structures.^ The techniques examined for subset selection do not generally result in improved predictions relative to the full model. Approaches using 'forward selection' result in slightly smaller prediction errors and less biased estimators of predictive accuracy than 'all possible subsets' approaches but no differences are detected between the performances of C$\sb{\rm p}$ and S$\sb{\rm p}$. In every case, prediction errors of models obtained by subset selection in either of the half splits exceed those obtained using all predictors and the entire sample.^ Only the random split estimator is conditionally (on $\\beta$) unbiased, however MSEP$\sb{\rm m}$ is unbiased on average and PRESS is nearly so in unselected (fixed form) models. When subset selection techniques are used, MSEP$\sb{\rm m}$ and PRESS always underestimate prediction errors, by as much as 27 percent (on average) in small samples. Despite their bias, the mean squared errors (MSE) of these estimators are at least 30 percent less than that of the unbiased random split estimator. The DUPLEX split estimator suffers from large MSE as well as bias, and seems of little value within the context of stochastic regressor variables.^ To maximize predictive accuracy while retaining a reliable estimate of that accuracy, it is recommended that the entire sample be used for model development, and a leave-one-out statistic (e.g. PRESS) be used for assessment. ^

Slope estimation in a parametric measurement error model: A simulation study

In regression analysis, covariate measurement error occurs in many applications. The error-prone covariates are often referred to as latent variables. In this proposed study, we extended the study of Chan et al. (2008) on recovering latent slope in a simple regression model to that in a multiple regression model. We presented an approach that applied the Monte Carlo method in the Bayesian framework to the parametric regression model with the measurement error in an explanatory variable. The proposed estimator applied the conditional expectation of latent slope given the observed outcome and surrogate variables in the multiple regression models. A simulation study was presented showing that the method produces estimator that is efficient in the multiple regression model, especially when the measurement error variance of surrogate variable is large.^

Analysis of life expectancy using linear and nonlinear models

Life expectancy has consistently increased over the last 150 years due to improvements in nutrition, medicine, and public health. Several studies found that in many developed countries, life expectancy continued to rise following a nearly linear trend, which was contrary to a common belief that the rate of improvement in life expectancy would decelerate and was fit with an S-shaped curve. Using samples of countries that exhibited a wide range of economic development levels, we explored the change in life expectancy over time by employing both nonlinear and linear models. We then observed if there were any significant differences in estimates between linear models, assuming an auto-correlated error structure. When data did not have a sigmoidal shape, nonlinear growth models sometimes failed to provide meaningful parameter estimates. The existence of an inflection point and asymptotes in the growth models made them inflexible with life expectancy data. In linear models, there was no significant difference in the life expectancy growth rate and future estimates between ordinary least squares (OLS) and generalized least squares (GLS). However, the generalized least squares model was more robust because the data involved time-series variables and residuals were positively correlated. ^

Mean dynamic topography and oceanographic parameters estimated from an inverse model and satellite geodesy, with link to model result in one single NetCDF file (392 MB), including the inverse data error covariance

Geostrophic surface velocities can be derived from the gradients of the mean dynamic topography-the difference between the mean sea surface and the geoid. Therefore, independently observed mean dynamic topography data are valuable input parameters and constraints for ocean circulation models. For a successful fit to observational dynamic topography data, not only the mean dynamic topography on the particular ocean model grid is required, but also information about its inverse covariance matrix. The calculation of the mean dynamic topography from satellite-based gravity field models and altimetric sea surface height measurements, however, is not straightforward. For this purpose, we previously developed an integrated approach to combining these two different observation groups in a consistent way without using the common filter approaches (Becker et al. in J Geodyn 59(60):99-110, 2012, doi:10.1016/j.jog.2011.07.0069; Becker in Konsistente Kombination von Schwerefeld, Altimetrie und hydrographischen Daten zur Modellierung der dynamischen Ozeantopographie, 2012, http://nbn-resolving.de/nbn:de:hbz:5n-29199). Within this combination method, the full spectral range of the observations is considered. Further, it allows the direct determination of the normal equations (i.e., the inverse of the error covariance matrix) of the mean dynamic topography on arbitrary grids, which is one of the requirements for ocean data assimilation. In this paper, we report progress through selection and improved processing of altimetric data sets. We focus on the preprocessing steps of along-track altimetry data from Jason-1 and Envisat to obtain a mean sea surface profile. During this procedure, a rigorous variance propagation is accomplished, so that, for the first time, the full covariance matrix of the mean sea surface is available. The combination of the mean profile and a combined GRACE/GOCE gravity field model yields a mean dynamic topography model for the North Atlantic Ocean that is characterized by a defined set of assumptions. We show that including the geodetically derived mean dynamic topography with the full error structure in a 3D stationary inverse ocean model improves modeled oceanographic features over previous estimates.

Planktonic foraminiferal oxygen isotopes of ODP Site 143-865 samples

Cool tropical sea surface temperatures (SSTs) are reported for warm Paleogene greenhouse climates based on the d18O of planktonic foraminiferal tests. These results are difficult to reconcile with models of greenhouse gas-forced climate. It has been suggested that this "cool tropics paradox" arises from postdepositional alteration of foraminiferal calcite, yielding erroneously high d18O values. Recrystallization of foraminiferal tests is cryptic and difficult to quantify, and the compilation of robust d18O records from moderately altered material remains challenging. Scanning electron microscopy of planktonic foraminiferal chamber-wall cross sections reveals that the basal area of muricae, pustular outgrowths on the chamber walls of species belonging to the genus Morozovella, contain no mural pores and may be less susceptible to postdepositional alteration. We analyzed the d18O in muricae bases of morozovellids from the central Pacific (Ocean Drilling Program Site 865) by ion microprobe using 10 ?m pits with an analytical reproducibility of ±0.34 per mil (2 standard deviations). In situ measurements of d18O in these domains yield consistently lower values than those published for conventional multispecimen analyses. Assuming that the original d18O is largely preserved in the basal areas of muricae, this new d18O record indicates Early Paleogene (~49-56 Ma) tropical SSTs in the central Pacific were 4°-8°C higher than inferred from the previously published d18O record and that SSTs reached at least ~33°C during the Paleocene-Eocene thermal maximum. This study demonstrates the utility of ion microprobe analysis for generating more reliable paleoclimate records from moderately altered foraminiferal tests preserved in deep-sea sediments.

Strontium chemistry of anhydrite from DSDP/ODP Hole 504B

A unique record of the chemical evolution of seawater during hydrothermal recharge into oceanic crust is preserved by anhydrite from the volcanic sequences and sheeted dike complex in ODP Hole 504B. Chemical and isotopic analyses 87Sr/86Sr, delta18O, delta34S of anhydrite constrain the changing composition of fluids due to reaction with basalt. There is a general trend of decreasing 87Sr/86Sr of anhydrite, corresponding to the minor incorporation of basaltic strontium with depth in the volcanic rocks. 87Sr/86Sr ratios decrease rapidly with depth in the dikes to values identical to host basalt (0.7029). Sr/Ca ratios (<0.1 mmol/mol) suggest that recharge fluids have very low Sr concentrations and fluids evolve by first precipitating Sr-bearing phases before extensive exchange of Sr with the host basalt. There is a background trend of decreasing sulfate delta18O with depth from +12-13? in the lower volcanics to +7? in the lower sheeted dikes recording an increase in recharge fluid temperature from c. 150° to c. 250°C, and confirming the presence of sulfate in hydrothermal fluids at elevated temperatures. From the amount of anhydrite recovered from Hole 504B and the amount of seawater sulfur that has been reduced to sulfide, a minimum seawater recharge flux can be calculated. This value is 4-25 times lower than estimates of high-temperature fluid fluxes based on either thermal constraints or global chemical budgets and suggests that there is significant deficit of seawater-derived sulfur in the oceanic crust. Only a minor proportion of the seawater that percolates into the crust near the axis is heated to high temperatures and exits as black smoker-type fluids. A significant proportion of the axial heat loss must be advected at 200-250°C by sulfate-bearing hydrothermal solutions that egress diffusely from the crust. These fluids penetrate into the dikes and exchange both heat and chemical tracers without the extensive clogging of porosity by anhydrite precipitation, which would halt hydrothermal circulation for any reasonable fluid flux. The heating of the major proportion of hydrothermal fluids to only moderate temperatures (c. 250°C) reconciles estimates of hydrothermal fluxes derived from thermal models and global geochemical budgets. The flux of hydrothermal sulfate would be of a magnitude similar to the riverine input, and oxygen-isotopic exchange at 200-250°C between dissolved sulfate and recharge fluids during hydrothermal circulation provides a mechanism to continuously buffer seawater sulfate oxygen to the light isotopic composition observed.

Elevation Models of Antarctica and Greenland derived from CryoSat-2 in the period 2011 to 2013

This study focuses on the present-day surface elevation of the Greenland and Antarctic ice sheets. Based on 3 years of CryoSat-2 data acquisition we derived new elevation models (DEMs) as well as elevation change maps and volume change estimates for both ice sheets. Here we present the new DEMs and their corresponding error maps. The accuracy of the derived DEMs for Greenland and Antarctica is similar to those of previous DEMs obtained by satellite-based laser and radar altimeters. Comparisons with ICESat data show that 80% of the CryoSat-2 DEMs have an uncertainty of less than 3 m ± 15 m. The surface elevation change rates between January 2011 and January 2014 are presented for both ice sheets. We compared our results to elevation change rates obtained from ICESat data covering the time period from 2003 to 2009. The comparison reveals that in West Antarctica the volume loss has increased by a factor of 3. It also shows an anomalous thickening in Dronning Maud Land, East Antarctica which represents a known large-scale accumulation event. This anomaly partly compensates for the observed increased volume loss of the Antarctic Peninsula and West Antarctica. For Greenland we find a volume loss increased by a factor of 2.5 compared to the ICESat period with large negative elevation changes concentrated at the west and southeast coasts. The combined volume change of Greenland and Antarctica for the observation period is estimated to be -503 ± 107 km**3/yr. Greenland contributes nearly 75% to the total volume change with -375 ± 24 km**3/yr.

Helium isotope ratios and sedimentation rate models of ODP holes

A continuous age model for the brief climate excursion at the Paleocene-Eocene boundary has been constructed by assuming a constant flux of extraterrestrial 3He (3He[ET]) to the seafloor. 3He[ET] measurements from ODP Site 690 provide quantitative evidence for the rapid onset (models indicating extremely rapid release of isotopically light carbon, possibly from seafloor methane hydrate, as the proximal cause of the event. However, the 3He[ET] technique indicates a previously unrecognized and extreme increase in sedimentation rate coincident with the return of climate proxies to pre-event values. The 3He[ET]-based age model thus suggests a far more rapid recovery from the climatic perturbation than previously proposed or predicted on the basis of the modern carbon cycle, and so may indicate additional or accelerated mechanisms of carbon removal from the ocean-atmosphere system during this period. 3He[ET] was also measured at ODP Site 1051 to test the validity of the Site 690 chronology. Comparison of these data sets seems to require removal of several tens of kyr of sediment within the climatic excursion at Site 1051, an observation consistent with sediment structures and previous age modeling efforts. The Site 1051 age model shows a ~30 kyr period in which climate proxies return toward pre-event values, after which they remain invariant for ~80 kyr. If this rise represents the recovery interval identified at Site 690, then the 3HeET-based age models of the two sites are in good agreement. However, alternative interpretations are possible, and work on less disrupted sites is required to evaluate the reliability of the proposed new chronology of the climate excursion. Regardless of these details, this work shows that the 3HeET technique can provide useful independent evidence for the development and testing of astronomically calibrated age models.

Age models for sediment cores in the eastern tropical Pacific

We present new high-resolution N isotope records from the Gulf of Tehuantepec and the Nicaragua Basin spanning the last 50-70 ka. The Tehuantepec site is situated within the core of the north subtropical denitrification zone while the Nicaragua site is at the southern boundary. The d15N record from Nicaragua shows an 'Antarctic' timing similar to denitrification changes observed off Peru-Chile but is radically different from the northern records. We attribute this to the leakage of isotopically heavy nitrate from the South Pacific oxygen minimum zone (OMZ) into the Nicaragua Basin. The Nicaragua record leads the other eastern tropical North Pacific (ETNP) records by about 1000 years because denitrification peaks in the eastern tropical South Pacific (ETSP) before denitrification starts to increase in the Northern Hemisphere OMZ, i.e., during warming episodes in Antarctica. We find that the influence of the heavy nitrate leakage from the ETSP is still noticeable, although attenuated, in the Gulf of Tehuantepec record, particularly at the end of the Heinrich events, and tends to alter the recording of millennial timescale denitrification changes in the ETNP. This implies (1) that sedimentary d15N records from the southern parts of the ETNP cannot be used straightforwardly as a proxy for local denitrification and (2) that denitrification history in the ETNP, like in the Arabian Sea, is synchronous with Greenland temperature changes. These observations reinforce the conclusion that on millennial timescales during the last ice age, denitrification in the ETNP is strongly influenced by climatic variations that originated in the high-latitude North Atlantic region, while commensurate changes in Southern Ocean hydrography more directly, and slightly earlier, affected oxygen concentrations in the ETSP. Furthermore, the d15N records imply ongoing physical communication across the equator in the shallow subsurface continuously over the last 50-70 ka.

Amino acid racemization of Neogloboquadrina pachyderma from the Arctic Ocean

The long-term rate of racemization for amino acids preserved in planktonic foraminifera was determined by using independently dated sediment cores from the Arctic Ocean. The racemization rates for aspartic acid (Asp) and glutamic acid (Glu) in the common taxon, Neogloboquadrina pachyderma, were calibrated for the last 150 ka using 14C ages and the emerging Quaternary chronostratigraphy of Arctic Ocean sediments. An analysis of errors indicates realistic age uncertainties of about ±12% for Asp and ±17% for Glu. Fifty individual tests are sufficient to analyze multiple subsamples, identify outliers, and derive robust sample mean values. The new age equation can be applied to verify and refine age models for sediment cores elsewhere in the Arctic Ocean, a critical region for understanding the dynamics of global climate change.

Age models, organic analyses and major-element analyses of sediment cores on a transect of the tropical African rainbelt

The distribution of rainfall in tropical Africa is controlled by the African rainbelt**1, which oscillates on a seasonal basis. The rainbelt has varied on centennial to millennial timescales along with changes in Northern Hemisphere high-latitude climate**2, 3, 4, 5, the Atlantic meridional overturning circulation**6 and low-latitude insolation**7 over the past glacial-interglacial cycle. However, the overall dynamics of the African rainbelt remain poorly constrained and are not always consistent with a latitudinal migration**2, 4, 5, 6, as has been proposed for other regions**8, 9. Here we use terrestrially derived organic and sedimentary markers from marine sediment cores to reconstruct the distribution of vegetation, and hence rainfall, in tropical Africa during extreme climate states over the past 23,000 years. Our data indicate that rather than migrating latitudinally, the rainbelt contracted and expanded symmetrically in both hemispheres in response to changes in climate. During the Last Glacial Maximum and Heinrich Stadial 1, the rainbelt contracted relative to the late Holocene, which we attribute to a latitudinal compression of atmospheric circulation associated with lower global mean temperatures**10. Conversely, during the mid-Holocene climatic optimum, the rainbelt expanded across tropical Africa. In light of our findings, it is not clear whether the tropical rainbelt has migrated latitudinally on a global scale, as has been suggested**8,9.

Late Pliocene age models of Agulhas Plateau sediment cores

An ensemble of new, high-resolution records of surface ocean hydrography from the Indian-Atlantic oceanic gateway, south of Africa, demonstrates recurrent and high-amplitude salinity oscillations in the Agulhas Leakage area during the penultimate glacial-interglacial cycle. A series of millennial-scale salinification events, indicating strengthened salt leakage into the South Atlantic, appear to correlate with abrupt changes in the North Atlantic climate and Atlantic Meridional Overturning Circulation (AMOC). This interhemispheric coupling, which plausibly involved changes in the Hadley Cell and midlatitude westerlies that impacted the interocean transport at the tip of Africa, suggests that the Agulhas Leakage acted as a source of negative buoyancy for the perturbed AMOC, possibly aiding its return to full strength. Our finding points to the Indian-to-Atlantic salt transport as a potentially important modulator of the AMOC during the abrupt climate changes of the Late Pleistocene.

Age models, ostracods and trace elements of five gravity cores from the NW Black Sea

New Mg/Ca, Sr/Ca, and published stable oxygen isotope and 87Sr/86Sr data obtained on ostracods from gravity cores located on the northwestern Black Sea slope were used to infer changes in the Black Sea hydrology and water chemistry for the period between 30 to 8 ka B.P. (calibrated radiocarbon years). The period prior to 16.5 ka B.P. was characterized by stable conditions in all records until a distinct drop in d18O values combined with a sharp increase in 87Sr/86Sr occurred between 16.5 and 14.8 ka B.P. This event is attributed to an increased runoff from the northern drainage area of the Black Sea between Heinrich Event 1 and the onset of the Bølling warm period. While the Mg/Ca and Sr/Ca records remained rather unaffected by this inflow; they show an abrupt rise with the onset of the Bølling/Allerød warm period. This rise was caused by calcite precipitation in the surface water, which led to a sudden increase of the Sr/Ca and Mg/Ca ratios of the Black Sea water. The stable oxygen isotopes also start to increase around 15 ka B.P., although in a more gradual manner, due to isotopically enriched meteoric precipitation. While Sr/Ca remains constant during the following interval of the Younger Dryas cold period, a decrease in the Mg/Ca ratio implies that the intermediate water masses of the Black Sea temporarily cooled by 1-2°C during the Younger Dryas. The 87Sr/86Sr values drop after the cessation of the water inflow at 15 ka B.P. to a lower level until the Younger Dryas, where they reach values similar to those observed during the Last Glacial Maximum. This might point to a potential outflow to the Mediterranean Sea via the Sea of Marmara during this period. The inflow of Mediterranean water started around 9.3 ka B.P., which is clearly detectable in the abruptly increasing Mg/Ca, Sr/Ca, and 87Sr/86Sr values. The accompanying increase in the d18O record is less pronounced and would fit to an inflow lasting ~100 a.

Putting prey and predator into the CO2 equation-qualitative and quantitative effects of ocean acidification on predator-prey interactions

Little is known about the impact of ocean acidification on predator-prey dynamics. Herein, we examined the effect of carbon dioxide (CO(2)) on both prey and predator by letting one predatory reef fish interact for 24 h with eight small or large juvenile damselfishes from four congeneric species. Both prey and predator were exposed to control or elevated levels of CO(2). Mortality rate and predator selectivity were compared across CO(2) treatments, prey size and species. Small juveniles of all species sustained greater mortality at high CO(2) levels, while large recruits were not affected. For large prey, the pattern of prey selectivity by predators was reversed under elevated CO(2). Our results demonstrate both quantitative and qualitative consumptive effects of CO(2) on small and larger damselfish recruits respectively, resulting from CO(2)-induced behavioural changes likely mediated by impaired neurological function. This study highlights the complexity of predicting the effects of climate change on coral reef ecosystems.