scholarly journals The Generalised Plug-in Algorithm for the Diffeomorphism Kernel Estimate

Author(s):  
Molka Troudi ◽  
Faouzi Ghorbel

The optimal value of the smoothing parameter of the Kernel estimator can be obtained by the well known Plug-in algorithm. The optimality is realised in the sense of Mean Integrated Square Error (MISE). In this paper, we propose to generalise this algorithm to the case of the difficult problem of the estimation of a distribution which has a bounded support. The proposed algorithm consists in searching the optimal smoothing parameter by iterations from the expression of MISE of the kernel-diffeomorphism estimator. By some simulations applied to some distribution having a support bounded and semi bounded, we show that the support of the pdf estimator respects the one of the theoretical distribution. We also prove that the proposed method minimizes the Gibbs phenomenon.

2016 ◽  
Vol 5 (2) ◽  
pp. 29
Author(s):  
Mounir ARFI

We give the rate of the uniform convergence for the kernel estimate of the regression function over a sequence of compact sets which increases to $\mathbb{R}^{d}$ when $n$ approaches the infinity and when the observed process is $\varphi$-mixing. The used estimator for the regression function is the kernel estimator proposed by Nadaraya, Watson (1964).


1949 ◽  
Vol 45 (2) ◽  
pp. 263-274 ◽  
Author(s):  
H. S. Green

The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.


Author(s):  
Bryan G. Norton

Albert Hochbaum, whom we met in Chapter 3, was Leopold’s student and friend; Director of the Delta Duck Station in Manitoba, Canada; and a part-time collaborator on A Sand County Almanac. He also had an admirable talent for succinctly hitting the nail on the head. He summed up Leopold’s message in four words. “The lesson you wish to put across is the lesson that must be taught,” he said, “preservation of the natural.” So much for succinctness; the difficult problem, of course, is to explain what is meant by “preservation” and by “natural.” Thomas McNamee, writing forty years later, uses the same basic approach: “I believe that the true object of conservation is nature,” he says. “What is nature?” The answer cannot help but be complicated, he notes, because “our conception of nature springs from the darkest depths of our culture’s unconscious sense of life itself, and ancient irrational urges and fears give the concept its power.’” But that is only half of the story: “At the same time,” he says, “nature must also have an objective, rational, manageable, thinkable value.” And thus we have the paradox of modern land use theory: Americans love nature; our values were formed in nature’s womb, a huge, wonderful, and horrible wild place. Our values are freedom and independence, “split rail values,” as Leopold called them. But our activities, as builders and consumers, transform our environment into something not-wild; we manipulate and control and artificialize nature; we make it not-nature. As the song says, you always hurt the one you love. But the paradox has also an optimistic face: As we have built and consumed, we have become wealthy by exploiting nature. Wildness has become valuable, objectively, according even to economists, because our wealthy society is now willing to pay to preserve nature. But here is the bitter pill to swallow: We all must admit that, at least in some sense, “nature” preservation is a sham—we’ve gone too far to “free” nature, as we might free a wild animal, release it from captivity.


1955 ◽  
Vol 8 (1) ◽  
pp. 35-40
Author(s):  
J. G. Porter

Most people know something about space ships nowadays, and probably think that navigation in space is quite a simple matter; at any rate, it is a subject that is glossed over very briefly in most books on the subject. In my view, space navigation is not a simple matter, and it has certainly not received the attention it deserves. Navigation on the Earth is easy, because of the one important fact that you are on the surface of the Earth. A couple of sights, measuring the angles from two stars down to the horizon, together with the azimuths of the stars and the distance from the centre of the Earth, will give an exact statement of position. But out in space there is no Earth, no horizon—in fact nothing whatever to use as a basis of measurement. Clearly then, two angles are not enough; a third one is needed, to give a sort of tripod of sights—two of the legs being anchored to two planets (or the Sun and a planet) because their positions in space at any time are known, and the distance between them can be used as a base-line. The solution of all the triangles involved is indeed a difficult problem, but there is also the impossibility of making three simultaneous observations. It might be thought that one could do as at sea and take one sight followed later by others, making allowance for the motion of the ship in the intervals. However, this involves the idea of dead reckoning, which, although a useful concept at sea, is quite impossible to apply in space, as the following example shows.


Stats ◽  
2019 ◽  
Vol 2 (4) ◽  
pp. 416-425 ◽  
Author(s):  
George Livadiotis

The paper shows how the linear regression depends on the selection of the reference frame. The slope of the fitted line and the corresponding Pearson’s correlation coefficient are expressed in terms of the rotation angle. The correlation coefficient is found to be maximized for a certain optimal angle, for which the slope attains a special optimal value. The optimal angle, the value of the optimal slope, and the corresponding maximum correlation coefficient were expressed in terms of the covariance matrix, but also in terms of the values of the slope, derived from the fitting at the nonrotated and right-angle-rotated axes. The potential of the new method is to improve the derived values of the fitting parameters by detecting the optimal rotation angle, that is, the one that maximizes the correlation coefficient. The presented analysis was applied to the linear regression of density and temperature measurements characterizing the proton plasma in the inner heliosheath, the outer region of our heliosphere.


1988 ◽  
Vol 25 (3) ◽  
pp. 544-552 ◽  
Author(s):  
Masami Yasuda

This paper treats stopping problems on Markov chains in which the OLA (one-step look ahead) policy is optimal. Its associated optimal value can be explicitly expressed by a potential for a charge function of the difference between the immediate reward and the one-step-after reward. As an application to the best choice problem, we shall obtain the value of three problems: the classical secretary problem, a problem with a refusal probability and a problem with a random number of objects.


2012 ◽  
Vol 616-618 ◽  
pp. 619-623 ◽  
Author(s):  
Chun Mei Wang ◽  
Gu Zhang Zhuang ◽  
Hai Yin

In this paper, The Copper-bearing minerals Powder (≤2mm) in Dahongshan is used in conical ball grinding, the experimental study shows that the effect of ball material ratio on the grinding efficiency is obvious. The critical value and the optimal value range of the ball material ratio are different on different grinding stages, the feasible ball material ratio of coarse grinding is bigger than the one of fine grinding. In the test conditions, in terms of the ball material ratio, the critical value of ores’ over crushing in coarse grinding is 13, and the optimal range is 6-8, which is beneficial to the next beneficiation; while the critical value in fine grinding is 7 and the optimal range is 5-7. Although the ball material ratios are different, the change rule of grinding fineness corresponding to different particle size fractions is consistent, which can be used as a reference in the site operation optimization.


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