The Theory Behind Overfitting, Cross Validation, Regularization, Bagging, and Boosting: Tutorial

In this tutorial paper, we first define mean squared error, variance, covariance, and bias of both random variables and classification/predictor models. Then, we formulate the true and generalization errors of the model for both training and validation/test instances where we make use of the Stein's Unbiased Risk Estimator (SURE). We define overfitting, underfitting, and generalization using the obtained true and generalization errors. We introduce cross validation and two well-known examples which are $K$-fold and leave-one-out cross validations. We briefly introduce generalized cross validation and then move on to regularization where we use the SURE again. We work on both $\ell_2$ and $\ell_1$ norm regularizations. Then, we show that bootstrap aggregating (bagging) reduces the variance of estimation. Boosting, specifically AdaBoost, is introduced and it is explained as both an additive model and a maximum margin model, i.e., Support Vector Machine (SVM). The upper bound on the generalization error of boosting is also provided to show why boosting prevents from overfitting. As examples of regularization, the theory of ridge and lasso regressions, weight decay, noise injection to input/weights, and early stopping are explained. Random forest, dropout, histogram of oriented gradients, and single shot multi-box detector are explained as examples of bagging in machine learning and computer vision. Finally, boosting tree and SVM models are mentioned as examples of boosting.Comment: 23 pages, 9 figure

Using minimum tillage to improve the efficiency of ecosystem service delivery on organic farms

Organic farming practices aim to maximise the delivery of ecosystem services in the agricultural landscape. However, in order to maintain optimal crop productivity the mouldboard plough is often used to control weeds and this can have negative effects on a range of soil parameters, thereby jeopardizing delivery of these services. Reduced tillage (RT) can be beneficial to soils and could improve both the efficiency of production and the delivery of ecosystem services on organic farms. However, abandoning the plough on organic farms is challenging due to impaired weed control. Here we report on a two year trial where an RT system with the Ecodyn, with duck feet shares operating at a depth of 7.6 cm in combination with seed drilling, was compared with mouldboard ploughing. Spring oat and spring barley establishment was improved under RT. Weed cover and biomass was greater under RT, but there was no difference in cereal grain yields in either year. The RT system used 71% less fuel and tillage operations took 72% less time that the plough system

Early Social-Emotional Functioning and Public Health: The Relationship Between Kindergarten Social Competence and Future Wellness

A new 20-year study shows a link between children's social skills in kindergarten and their well-being in early adulthood.Researchers from Pennsylvania State and Duke Universities analyzed what happened to nearly 800 kindergarteners from four locations after their teachers measured their social competency skills in 1991. The children were evaluated on a range of social behaviors, such as whether they resolve peer problems, listen to others, share materials, cooperate, and are helpful. Each student then received a composite score representing his or her overall level of positive social skills/behavior, on a scale from ("not at all") to 4 ("very well"). The research team monitored these students and the positive and negative milestones each obtained until they turned 25.Using a variety of data sources, including official records; reports from parents; and self-reporting by the participants, researchers recorded whether the students obtained high school diplomas, college degrees, and full-time jobs. They also kept track of whether students developed a criminal record or substance abuse problems, among other negative outcomes

The Poincaré-Hopf Theorem for line fields revisited

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The Topological Period-Index Conjecture for spinc 6-manifolds

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Apatite Reference Materials for SIMS Microanalysis of Isotopes and Trace Elements

Twelve apatite samples have been tested as secondary ion mass spectrometry (SIMS) reference materials. Laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS) analysis shows that the SLAP, NUAN and GR40 apatite gems are internally homogeneous, with most trace element mass fractions having 2 standard deviations (2s) ≤ 2.0%. BR2, BR5, OL2, AFG2 and AFB1, which have U \u3e 63 μg g-1, 206Pb/204Pb \u3e 283, and homogeneous SIMS U-Pb data, have respective isotope dilution thermal ionisation mass spectrometry (ID-TIMS) ages of 2053.83 ± 0.21 Ma, 2040.34 ± 0.09 Ma, 868.87 ± 0.25 Ma, 478.71 ± 0.22 Ma and 473.25 ± 0.09 Ma. Minor U-Pb heterogeneity exists and accurate SIMS results require correction with the 3D Concordia-constrained common Pb composition. Among the studied samples, AFG2 and BR5 are the most homogeneous U-Pb reference materials. The SIMS sulfur isotopic compositions of eight of the apatites shows they are homogeneous, with 2s for both 103δ34S and 103δ33S \u3c 0.55‰. One apatite, BR96, has Δ33S = -0.36 ± 0.2‰. The apatite samples have ID-TIMS 87Sr/86Sr between 0.704214 ± 0.000030 and 0.723134 ± 0.000035

The Temporal Evolution of Subduction Initiation in the Samail Ophiolite: High-Precision U–Pb Zircon Petrochronology of the Metamorphic Sole

High-precision dating of the metamorphic sole of ophiolites can provide insight into the tectonic evolution of ophiolites and subduction zone processes. To understand subduction initiation beneath a young, well-preserved and well-characterized ophiolite, we performed coupled zircon laser-ablation inductively coupled mass spectrometry trace element analyses and high-precision isotope dilution-thermal ionization mass spectrometry U–Pb dating on 25 samples from the metamorphic sole of the Samail ophiolite (Oman-United Arab Emirates). Zircon grains from amphibolite- to granulite-facies (0.8–1.3 GPa, ~700–900°C), garnet- and clinopyroxene-bearing amphibolite samples (n = 18) show systematic trends of decreasing heavy rare earth element slope (HREE; Yb/Dy) with decreasing Yb concentration, reflecting progressive depletion of the HREE during prograde garnet growth. For half of the garnet-clinopyroxene amphibolite samples, Ti-in-zircon temperatures increase, and U–Pb dates young with decreasing HREE slope, consistent with coupled zircon and garnet growth during prograde metamorphism. In the remaining samples, there is no apparent variation in Ti-in-zircon temperature with decreasing HREE slope, and the combined U–Pb and geochemical data suggest zircon crystallization along either the prograde to peak or prograde to initial retrograde portions of the metamorphic P–T–t path. The new data bracket the timing of prograde garnet and zircon growth in the highest grade rocks of the metamorphic sole between 96.698 ± 0.094 and 95.161 ± 0.064 Ma, in contrast with previously published geochronology suggesting prograde metamorphism at ~104 Ma. Garnet-free amphibolites and leucocratic pods from lower grade (but still upper amphibolite facies) portions of the sole are uniformly HREE enriched (Yb/Dy \u3e 5) and are ~0.5–1.3 Myr younger than the higher grade rocks from the same localities, constraining the temporal offset between the metamorphism and juxtaposition of the higher and lower grade units. Positive zircon εHf (+6.5 to +14.6) for all but one of the dated amphibolites are consistent with an oceanic basalt protolith for the sole. Our new data indicate that prograde sole metamorphism (96.7–95.2 Ma) immediately predated and overlapped growth of the overlying ophiolite crust (96.1–95.2 Ma). The ~600 ky offset between the onset of sole metamorphism in the northern portion of the ophiolite versus the start of ophiolite magmatism is an order of magnitude shorter than previously proposed (~8 Ma) and is consistent with either spontaneous subduction initiation or an abbreviated period of initial thrusting during induced subduction initiation. Taken together, the sole and ophiolite crust preserve a record of the first ~1.5 Myr of subduction. A gradient in the initiation of high-grade metamorphism from the northwest (96.7 Ma) to southeast (96.0–95.7 Ma) may record propagation of the nascent subduction zone and/or variations in subduction rate along the length of the ophiolite

Eigenvalue and Generalized Eigenvalue Problems: Tutorial

This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also provide examples from machine learning, including principal component analysis, kernel supervised principal component analysis, and Fisher discriminant analysis, which result in eigenvalue and generalized eigenvalue problems. Finally, we introduce the solutions to both eigenvalue and generalized eigenvalue problems.Comment: 8 pages, Tutorial pape