Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGroping Toward Linear Regression Analysis: Newtons Analysis of Hipparchus Equinox Observations
126
0
0.0
(
0
)
Authors
Ari Belenkiy
Ask ChatGPT about the research
No Arabic abstract
In February 1700, Isaac Newton needed a precise tropical year to design a new universal calendar that would supersede the Gregorian one. However, 17th-Century astronomers were uncertain of the long-term variation in the inclination of the Earths axis and were suspicious of Ptolemys equinox observations. As a result, they produced a wide range of tropical years. Facing this problem, Newton attempted to compute the length of the year on his own, using the ancient equinox observations reported by a famous Greek astronomer Hipparchus of Rhodes, ten in number. Though Newton had a very thin sample of data, he obtained a tropical year only a few seconds longer than the correct length. The reason lies in Newtons application of a technique similar to modern regression analysis. Newton wrote down the first of the two so-called normal equations known from the ordinary least-squares (OLS) method. In that procedure, Newton seems to have been the first to employ the mean (average) value of the data set, while the other leading astronomers of the era (Tycho Brahe, Galileo, and Kepler) used the median. Fifty years after Newton, in 1750, Newtons method was rediscovered and enhanced by Tobias Mayer. Remarkably, the same regression method served with distinction in the late 1920s when the founding fathers of modern cosmology, Georges Lemaitre (1927), Edwin Hubble (1929), and Willem de Sitter (1930), employed it to derive the Hubble constant.
rate research
Read More
In his Chronology, Newton uses astronomical evidence to support its extreme rejuvenation of ancient times. These elements, having a scientific varnish, provide some credibility to the work. They have been fiercely debated for a century, with a gradual undermining of Newtons assumptions. However, this has not dented the prestige of the English scientist. ----- Dans sa Chronologie, Newton utilise des preuves astronomiques pour appuyer son rajeunissement extreme des epoques anciennes. Ces elements, au vernis scientifique, donnent une credibilite certaine a lensemble. Ils ont donc ete aprement discutes, les debats sapant petit a petit les hypotheses du savant anglais pour finalement porter un coup mortel a lensemble. Cela na toutefois pas entame le prestige du savant anglais.
Adversarially robust learning aims to design algorithms that are robust to small adversarial perturbations on input variables. Beyond the existing studies on the predictive performance to adversarial samples, our goal is to understand statistical properties of adversarially robust estimates and analyze adversarial risk in the setup of linear regression models. By discovering the statistical minimax rate of convergence of adversarially robust estimators, we emphasize the importance of incorporating model information, e.g., sparsity, in adversarially robust learning. Further, we reveal an explicit connection of adversarial and standard estimates, and propose a straightforward two-stage adversarial learning framework, which facilitates to utilize model structure information to improve adversarial robustness. In theory, the consistency of the adversarially robust estimator is proven and its Bahadur representation is also developed for the statistical inference purpose. The proposed estimator converges in a sharp rate under either low-dimensional or sparse scenario. Moreover, our theory confirms two phenomena in adversarially robust learning: adversarial robustness hurts generalization, and unlabeled data help improve the generalization. In the end, we conduct numerical simulations to verify our theory.
The sensitivity to angular rotation of the top class Sagnac gyroscope GINGERINO is carefully investigated with standard statistical means, using 103 days of continuous operation and the available geodesic measurements of the Earth angular rotation rate. All features of the Earth rotation rate are correctly reproduced. The sensitivity of fractions of frad/s is attained for long term runs. This excellent sensitivity and stability put Sagnac gyroscopes at the forefront for fundamental physics, in particular for tests of general relativity and Lorentz violation, where the sensitivity plays the key role to provide reliable data for deeper theoretical investigations. The achieved sensitivity overcomes the conventionally expected one for Sagnac ring laser gyroscopes.
Correlated data are ubiquitous in todays data-driven society. A fundamental task in analyzing these data is to understand, characterize and utilize the correlations in them in order to conduct valid inference. Yet explicit regression analysis of correlations has been so far limited to longitudinal data, a special form of correlated data, while implicit analysis via mixed-effects models lacks generality as a full inferential tool. This paper proposes a novel regression approach for modelling the correlation structure, leveraging a new generalized z-transformation. This transformation maps correlation matrices that are constrained to be positive definite to vectors with un-restricted support, and is order-invariant. Building on these two properties, we develop a regression model to relate the transformed parameters to any covariates. We show that coupled with a mean and a variance regression model, the use of maximum likelihood leads to asymptotically normal parameter estimates, and crucially enables statistical inference for all the parameters. The performance of our framework is demonstrated in extensive simulation. More importantly, we illustrate the use of our model with the analysis of the classroom data, a highly unbalanced multilevel clustered data with within-class and within-school correlations, and the analysis of the malaria immune response data in Benin, a longitudinal data with time-dependent covariates in addition to time. Our analyses reveal new insights not previously known.
We revisited an analysis made by Euler in a memoir of 1753 Sur une nouvelle maniere delever de leau propose par M. de Mour (On a new method to raise water proposed by Mr. de Mour), addressing a type of water sprinkler, consisting of an inclined tube with its lower end immersed into water, and discharging water at its top by turning a vertical axis to which the inclined tube is attached.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
