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1. Ensemble-based estimates of eigenvector error for empirical covariance matrices

   https://doi.org/10.1093/imaiai/iay010  

   Taylor, Dane; Restrepo, Juan G; Meyer, François G (July 2018, Information and Inference: A Journal of the IMA)

   Abstract Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $$\{\lambda _i\}$$ and eigenvectors $$\{\boldsymbol{u}_i\}$$ of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $$n$$ to obtain empirical eigenvalues $$\{\tilde{\lambda }_i\}$$ and eigenvectors $$\{\tilde{\boldsymbol{u}}_i\}$$, and therefore understanding the error so introduced is of central importance. We analyse eigenvector error $$\|\boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2$$ while leveraging the assumption that the true covariance matrix having size $$p$$ is drawn from a matrix ensemble with known spectral properties—particularly, we assume the distribution of population eigenvalues weakly converges as $$p\to \infty $$ to a spectral density $$\rho (\lambda )$$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $$p$$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $$\lambda $$ and approximate the distribution of the expected square error $$r= \mathbb{E}\left [\| \boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2\right ]$$ across the matrix ensemble for all $$\boldsymbol{u}_i$$ associated with $$\lambda _i=\lambda $$. We find, for example, that for sufficiently large matrix size $$p$$ and sample size $n> p$, the probability density of $$r$$ scales as $1/nr^2$. This power-law scaling implies that the eigenvector error is extremely heterogeneous—even if $$r$$ is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments. 

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2. √2-estimation for smooth eigenvectors of matrix-valued functions

   https://doi.org/10.1093/biomet/asad018  

   Motta, Giovanni; Wu, Wei Biao; Pourahmadi, Mohsen (March 2023, Biometrika)

   Summary Modern statistical methods for multivariate time series rely on the eigendecomposition of matrix-valued functions such as time-varying covariance and spectral density matrices. The curse of indeterminacy or misidentification of smooth eigenvector functions has not received much attention. We resolve this important problem and recover smooth trajectories by examining the distance between the eigenvectors of the same matrix-valued function evaluated at two consecutive points. We change the sign of the next eigenvector if its distance with the current one is larger than the square root of 2. In the case of distinct eigenvalues, this simple method delivers smooth eigenvectors. For coalescing eigenvalues, we match the corresponding eigenvectors and apply an additional signing around the coalescing points. We establish consistency and rates of convergence for the proposed smooth eigenvector estimators. Simulation results and applications to real data confirm that our approach is needed to obtain smooth eigenvectors. 

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3. Effects of H2O–CO2 Fluids, Temperature, and Peridotite Fertility on Partial Melting in Mantle Wedges and Generation of Primary Arc Basalts

   https://doi.org/10.1093/petrology/egad047  

   Lara, Michael; Dasgupta, Rajdeep (July 2023, Journal of Petrology)

   Abstract Many lines of evidence from high P–T experiments, thermodynamic models, and natural observations suggest that slab-derived aqueous fluids, which flux mantle wedges contain variable amounts of dissolved carbon. However, constraints on the effects of H2O–CO2 fluids on mantle melting, particularly at mantle wedge P–T conditions, are limited. Here, we present new piston cylinder experiments on fertile and depleted peridotite compositions with 3.5 wt.% H2O and XCO2 [= molar CO2 / (CO2 + H2O)] of 0.04–0.17. Experiments were performed at 2–3 GPa and 1350°C to assess how temperature, peridotite fertility, and XCO2 of slab-derived fluid affects partial melting in mantle wedges. All experiments produce olivine + orthopyroxene +7 to 41 wt.% partial melt. Our new data, along with previous lower temperature data, show that as mantle wedge temperature increases, primary melts become richer in SiO2, FeO*, and MgO and poorer CaO, Al2O3, and alkalis when influenced by H2O–CO2 fluids. At constant P–T and bulk H2O content, the extent of melting in the mantle wedge is largely controlled by peridotite fertility and XCO2 of slab-fluid. High XCO2 depleted compositions generate ~7 wt.% melt, whereas, at identical P–T, low XCO2 fertile compositions generate ~30 to 40 wt.% melt. Additionally, peridotite fertility and XCO2 have significant effects on peridotite partial melt compositions. At a constant P–T–XCO2, fertile peridotites generate melts richer in CaO and Al2O3 and poorer in SiO2, MgO + FeO, and alkalis. Similar to previous experimental studies, at a constant P–T fertility condition, as XCO2 increases, SiO2 and CaO of melts systematically decrease and increase, respectively. Such distinctive effects of oxidized form of dissolved carbon on peridotite partial melt compositions are not observed if the carbon-bearing fluid is reduced, such as CH4-bearing. Considering the large effect of XCO2 on melt SiO2 and CaO concentrations and the relatively oxidized nature of arc magmas, we compare the SiO2/CaO of our experimental melts and melts from previous peridotite + H2O ± CO2 studies to the SiO2/CaO systematics of primitive arc basalts and ultra-calcic, silica-undersaturated arc melt inclusions. From this comparison, we demonstrate that across most P–T–fertility conditions predicted for mantle wedges, partial melts from bulk compositions with XCO2 ≥ 0.11 have lower SiO2/CaO than all primitive arc melts found globally, even when correcting for olivine fractionation, whereas partial melts from bulk compositions with XCO2 = 0.04 overlap the lower end of the SiO2/CaO field defined by natural data. These results suggest that the upper XCO2 limit of slab-fluids influencing primary arc magma formation is 0.04 < XCO2 < 0.11, and this upper limit is likely to apply globally. Lastly, we show that the anomalous SiO2/CaO and CaO/Al2O3 signatures observed in ultra-calcic arc melt inclusions can be reproduced by partial melting of either CO2-bearing hydrous fertile and depleted peridotites with 0 < XCO2 < 0.11 at 2–3 GPa, or from nominally CO2-free hydrous fertile peridotites at P > 3 GPa. 

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4. Oxygen diffusion in garnet: Experimental calibration and implications for timescales of metamorphic processes and retention of primary O isotopic signatures

   https://doi.org/10.2138/am-2022-7970  

   Scicchitano, Maria Rosa; Jollands, Michael C.; Williams, Ian S.; Hermann, Jörg; Rubatto, Daniela; Kita, Noriko T.; Nachlas, William O.; Valley, John W.; Escrig, Stéphane; Meibom, Anders (July 2022, American Mineralogist)

   Abstract Knowledge of oxygen diffusion in garnet is crucial for a correct interpretation of oxygen isotope signatures in natural samples. A series of experiments was undertaken to determine the diffusivity of oxygen in garnet, which remains poorly constrained. The first suite included high-pressure (HP), nominally dry experiments performed in piston-cylinder apparatus at: (1) T = 1050–1600 °C and P = 1.5 GPa and (2) T = 1500 °C and P = 2.5 GPa using yttrium aluminum garnet (YAG; Y3Al5O12) cubes. Second, HP H2O-saturated experiments were conducted at T = 900 °C and P = 1.0–1.5 GPa, wherein YAG crystals were packed into a YAG + Corundum powder, along with 18O-enriched H2O. Third, 1 atm experiments with YAG cubes were performed in a gas-mixing furnace at T = 1500–1600 °C under Ar flux. Finally, an experiment at T = 900 °C and P = 1.0 GPa was done using a pyrope cube embedded into pyrope powder and 18O-enriched H2O. Experiments using grossular were not successful. Profiles of 18O/(18O+16O) in the experimental charges were analyzed with three different secondary ion mass spectrometers (SIMS): sensitive high-resolution ion microprobe (SHRIMP II and SI), CAMECA IMS-1280, and NanoSIMS. Considering only the measured length of 18O diffusion profiles, similar results were obtained for YAG and pyrope annealed at 900 °C, suggesting limited effects of chemical composition on oxygen diffusivity. However, in both garnet types, several profiles deviate from the error function geometry, suggesting that the behavior of O in garnet cannot be fully described as simple concentration-independent diffusion, certainly in YAG and likely in natural pyrope as well. The experimental results are better described by invoking O diffusion via two distinct pathways with an inter-site reaction allowing O to move between these pathways. Modeling this process yields two diffusion coefficients (D values) for O, one of which is approximately two orders of magnitude higher than the other. Taken together, Arrhenius relationships are:log⁡Dm2s-1=-7.2(±1.3)+(-321(±32)kJmol-12.303RT) for the slow pathway, andlog⁡Dm2s-1=-5.4(±0.7)+(-321(±20)kJmol-12.303RT) for the fast pathway. We interpret the two pathways as representing diffusion following vacancy and inter-stitial mechanisms, respectively. Regardless, our new data suggest that the slow mechanism is prevalent in garnet with natural compositions, and thus is likely to control the retentivity of oxygen isotopic signatures in natural samples. The diffusivity of oxygen is similar to Fe-Mn diffusivity in garnet at 1000–1100 °C and Ca diffusivity at 850 °C. However, the activation energy for O diffusion is larger, leading to lower diffusivities at P-T conditions characterizing crustal metamorphism. Therefore, original O isotopic signatures can be retained in garnets showing major element zoning partially re-equilibrated by diffusion, with the uncertainty caveat of extrapolating the experimental data to lower temperature conditions. 

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5. Formation of Amphibole‐Bearing Peridotite and Amphibole‐Bearing Pyroxenite Through Hydrous Melt‐Peridotite Reaction and In Situ Crystallization: An Experimental Study

   https://doi.org/10.1029/2020JB019382  

   Wang, Chunguang; Liang, Yan; Xu, Wenliang (March 2021, Journal of Geophysical Research: Solid Earth)

   Abstract Amphibole is a common hydrous mineral in mantle rocks. To better understand processes leading to the formation of amphibole‐bearing peridotites and pyroxenites in the lithospheric mantle, we conducted experiments by juxtaposing a lherzolite against hydrous basaltic melts in Au‐Pd capsules. Two melts were examined, a basaltic andesite and a basalt, each containing 4 wt% of water. The experiments were run at 1200°C and 1 GPa for 3 or 12 h, and then cooled to 880°C and 0.8 GPa over 49 h. The reaction at 1200°C produced a melt‐bearing orthopyroxenite‐dunite sequence. Crystallization of the partially reacted melts during cooling lead to the formation of an amphibole‐bearing gabbronorite‐orthopyroxenite‐peridotite sequence. Orthopyroxene in the peridotite and orthopyroxenite has a poikilitic texture enclosing olivines and spinels. Amphibole in the peridotite occurs interstitial to olivine, orthopyroxene, clinopyroxene, and spinel. Comparisons of texture and mineral compositions in the experimental products with those from field observations allow a better understanding of hydrous melt‐rock reaction in the lithospheric mantle. Amphibole‐bearing pyroxenite veins (or dikes) can be formed in the lithospheric mantle or at the crust‐mantle boundary by interaction between hydrous melt and peridotite and subsequent crystallization. Hornblendite or amphibole gabbronorite can be formed in the veins when the flux of hydrous melt is high. Differences in reacting melt and peridotite compositions are responsible for the variation in amphibole composition in mantle xenoliths from different tectonic settings. The extent of melt‐rock reaction is a factor that control amphibole composition across the amphibole‐bearing vein and the host peridotite. 

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