TODO

整理学习

PUT THE FOLLOWING NOTES INTO CORRECT POSTS:

  • “evidence framework” and “expectation maximization (EM) algorithm” sole the same problem: maximize marginal (over w\bm{w}) likelihood function p(tX,α,β)p(\bm{t}|\bm{X},\alpha,\beta). (Bishop, 2006, p166)

notes

复习完善

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OneWeb

~648 satellites

global Internet broadband service

as early as 2019

proposed by the company WorldVu Satellites Ltd.

The 648 communication satellites will operate in circular low Earth orbit, at approximately 750 miles (1,200 km) altitude,[2] transmitting and receiving in the Ku band of the radio frequency spectrum.

进展

  1. Most of the capacity of the initial 648 satellites has been sold, and OneWeb is considering nearly quadrupling the size of the satellite constellation by adding 1,972 additional satellites that it has priority rights to.

Different frames

EME2000 == J2000 (https://en.wikipedia.org/wiki/Earth-centered_inertial)

ICRF == J2000 (for JPL)

Geocentric Celestial Reference Frame (GCRF) is the Earth-centered counterpart of the International Celestial Reference Frame. (https://en.wikipedia.org/wiki/Earth-centered_inertial)

TEME:

http://itrf.ensg.ign.fr/

geocentric system, rotational, ECEF

ECEF: Earth-Centered, Earth-Fixed

ECR: Earth-Centered Rotational

  • == ECEF

ITRS: International Terrestrial Reference System

  • Realizations of the ITRS are produced by the IERS ITRS Product Center (ITRS-PC) under the name International Terrestrial Reference Frames (ITRF).

ITRF: International Terrestrial Reference Frame

WGS84: World Geodetic System

  • a standard for use in cartography, geodesy, and satellite navigation including GPS. (wikipedia)
  • It comprises (wikipedia)
    • a standard coordinate system for the Earth,
    • a standard spheroidal reference surface (the datum or reference ellipsoid) for raw altitude data,
    • a gravitational equipotential surface (the geoid) that defines the nominal sea level.

BCRS: barycentric celestial reference system (https://en.wikipedia.org/wiki/Barycentric_celestial_reference_system)

  • The focus of the BCRS is on astronomy: exploration of the Solar System and the universe.

geocentric system, inertial

IERS: International Earth Rotation and Reference Systems Service.

  • maintained ITRS and ITRF solutions

GCRS: Geocentric Celestial Reference System (https://en.wikipedia.org/wiki/Barycentric_celestial_reference_system)

  • BCRS centered at Earth
  • The focus of the GCRS is somewhat more on the navigation of Earth satellites and the geophysical applications they support. The proper functioning of the Global Positioning System (GPS) is directly dependent upon the accuracy of satellite measurements as supported by the GCRS.

GCRF: Geocentric Celestial Reference Frame

centered at the barycenter of the Solar System

ICRS: International Celestial Reference System

ICRF: International Celestial Reference Frame

others

Most excerpted from Orekit Frames: (https://www.orekit.org/site-orekit-9.2/architecture/frames.html)

MOD: Mean Of Date frame = J2000 + precession evolution

TOD: True Of Date frame = J2000 + precession evolution + nutation

GTOD: Greenwich True Of Date frame = J2000 + precession evolution + Greenwich Apparent Sidereal Time

GCRF --> EME2000

  • rotations along three axis, very tiny angles (from Orekit source code, EME2000Provider())

softwares

There are two software libraries of IAU-sanctioned algorithms for manipulating and transforming among the BCRS and other reference systems:

  • the Standards of Fundamental Astronomy (SOFA) system
  • the Naval Observatory Vector Astrometry Subroutines (NOVAS).

IERS: International Earth Rotation and Reference Systems Service

A Terrestrial Reference frame provides a set of coordinates of some points located on the Earth’s surface

GCRS = GCRF = ICRF = ICRS: small difference with EME2000 所以,skyfield得到的结果,应该不需要转换到EME2000中,可以直接在GCRF下使用 可以测试orekit中两个坐标系相差多少 https://space.stackexchange.com/questions/26259/what-is-the-difference-between-gcrs-and-j2000-frames

EOP: Earth Orientation Parameters

(201104–201206) Landsat-5 Conjunctions with the A-Train (Afternoon Constellation)

(20120404) Fermi - Cosmos 2251 close approach

(20130903) EOS Aura and Shijian (SJ)-11-02 satellite conjunction ppt (saved) web

On September 3, 2013, predicted close approach

Second in a series of SJ-11 satellites launched by China into an orbit very similar to that of the Morning and Afternoon Constellations.

Aura flight controllers prepared a RMM to avoid the close approach. They did not know whether SJ-11-02 was capable of maneuvering.

A request was sent through the US State Department to its Chinese counterpart to let their space agency know of NASA’s planned maneuver. A request was sent through the US State Department to its counterpart in China to let their space agency know of NASA’s planned maneuver, but there was no direct two-way coordination.

Both satellites maneuvered within hours of each other. Fortunately, the 2 maneuvers mitigated the risk.

Example of the need to improve communication with non-constellation satellites.

(20140204) EO-1 and HELIOS-2B close approach

(20140405) Sentinel 1A - ACRIMSAT close approach immediately after launch

Useful Links

Estimation theory

数学概念

似然函数 / likelihood function

L(θx)=f(xθ)\mathcal{L}(\theta \mid x) = f(x\mid\theta)

f(xθ)f(x|\theta) 作为 xx 的函数时,是概率密度函数; 当 f(xθ)f(x|\theta) 作为 θ\theta 的函数时,是似然函数。

后验分布 / posterior distribution

f(θx)=f(xθ)g(θ)Θf(xϑ)g(ϑ)dϑf(\theta \mid x) = \frac{f(x\mid \theta )\,g(\theta )}{\int _{\Theta }f(x\mid \vartheta )\,g(\vartheta )\,d\vartheta}


最大似然估计 / maximum likelihood (ML) estimation

θ^ML(x)=argmaxθL(θx)=argmaxθf(xθ)\hat{\theta}_{\rm ML}(x) = \arg \max_{\theta} \mathcal{L}(\theta|x) = \arg \max_{\theta} f(x|\theta)

θ^MAP(x)=argmaxθf(xθ)g(θ)Θf(xθ)g(θ)dθ=argmaxθf(xθ)g(θ)\hat{\theta}_{\rm MAP}(x) = \arg \max_{\theta}{\frac{f(x\mid\theta )\,g(\theta )}{\int_{\Theta}f(x\mid\theta ')\,g(\theta ')\,d\theta '}}=\arg \max_{\theta}f(x|\theta )\,g(\theta)

最大后验概率估计 / Maximum a posteriori (MAP) estimation

g(θ)g(\theta) 是假设的先验概率;分母与 θ\theta 的取值无关;g(θ)g(\theta) 为均匀分布时与 θ^ML\hat\theta_{\rm ML} 等价。

贝叶斯估计


算法

最大期望算法

(未完成)

12βσ212B(Φ(ΦΣ1)1m+121m-\frac{1}{2} \bm{\beta} \sigma^2 - \frac{1}{2} \bm{B} ( \Phi \circ (\Phi\Sigma^{-1} ) \bm{1}_m + \frac{1}{2} \cdot \bm{1}_m

Method of moments (statistics)

矩估计与最大似然估计

矩估计:将概率分布函数分散为不同次幂的期望来刻画。计算简便。

最大似然估计:使用 Bayesian 方法得到给定观测时的参数的分布–似然函数,然后最大化似然函数。理论上精度较高。

高阶矩

概率论中「矩」(moment)的实际含义是什么,高阶矩表示数据的哪些状态?

概率论中「矩」(moment)的实际含义是什么,高阶矩表示数据的哪些状态? - 荆哲的回答 - 知乎

未读

Moment matching algorithm

学习资料

Jorbe, 变分法和变分贝叶斯推断 (感觉文章最后缺少一些内容,为什么要得到 P(X,Z)P(X,Z) ? 以及得到后怎么使用,下一步操作是什么?

当概率质量分布f(x)为正态分布时,信息熵最大。

可以通过最大这个下界来达到最小化KL(Q||P)的目的,这样做的一个好处在于,如果直接优化KL(Q||P),由于真实后验分布往往事先不知道,而且如果用贝叶斯公式来计算P(Z|X)的复杂度特别高,因此不好直接优化KL(Q||P)。

为了优化L(Q),需要引入一种叫做平均场的方法(mean field theory),即假设每个隐变量之间服从独立同分布:

Gal, Yarin, “Uncertainty in Deep Learning,” Doctor of Philosophy, University of Cambridge, 2016.

全文的主要贡献

(p15) We will thus concentrate on the development of practical techniques to obtain model confidence in deep learning, techniques which are also well rooted within the theoretical foundations of probability theory and Bayesian modelling. Specifically, we will make use of stochastic regularisation techniques (SRTs).

These techniques adapt the model output stochastically as a way of model regularisation (hence the name stochastic regularisation). This results in the loss becoming a random quantity, which is optimised using tools from the stochastic non-convex optimisation literature. Popular SRTs include dropout [Hinton et al., 2012], multiplicative Gaussian noise [Srivastava et al., 2014], dropConnect [Wan et al., 2013], and countless other recent techniques4,5.

作者对 NN 的一些讨论

CNN

经常用在图像识别。卷积层处理空间信息;pooling 层缩减维度。

Convolutional neural networks (CNNs). CNNs [LeCun et al., 1989; Rumelhart et al., 1985] are popular deep learning tools for image processing, which can solve tasks that until recently were considered to lie beyond our reach [Krizhevsky et al., 2012; Szegedy et al., 2014]. The model is made of a recursive application of convolution and pooling layers, followed by inner product layers at the end of the network (simple NNs as described above). A convolution layer is a linear transformation that preserves spatial information in the input image (depicted in figure 1.1). Pooling layers simply take the output of a convolution layer and reduce its dimensionality (by taking the maximum of each (2, 2) block of pixels for example). The convolution layer will be explained in more detail in section §3.4.1.

RNN

擅长处理序列数据,比如自然语言识别、语言生成、视频处理、其它(?)。

Recurrent neural networks (RNNs). RNNs [Rumelhart et al., 1985; Werbos, 1988] are sequence-based models of key importance for natural language understanding, language generation, video processing, and many other tasks [Kalchbrenner and Blunsom, 2013; Mikolov et al., 2010; Sundermeyer et al., 2012; Sutskever et al., 2014].

PILCO

PILCO [Deisenroth and Rasmussen, 2011], for example, is a data-efficient probabilistic model-based policy search algorithm. PILCO analytically propagates uncertain state distributions through a Gaussian process dynamics model. This is done by recursively feeding the output state distribution (output uncertainty) of one time step as the input state distribution (input uncertainty) of the next time step, until a fixed time horizon T.

与 GP 的关系

使用无穷个 neuron,每个 weight 都取为高斯分布,则成为 GP。

对有限个 weights,则是 BNN。

(p14) Even though modern deep learning models used in practice do not capture model confidence, they are closely related to a family of probabilistic models which induce probability distributions over functions: the Gaussian process. Given a neural network, by placing a probability distribution over each weight (a standard normal distribution for example), a Gaussian process can be recovered in the limit of infinitely many weights (see Neal [1995] or Williams [1997]). For a finite number of weights, model uncertainty can still be obtained by placing distributions over the weights—these models are called Bayesian neural networks.

Bayesian modeling 的一些基础知识

在未进行观测前,假设存在关于函数 y=fω(x)\bm{y}=f^\bm{\omega}(\bm{x})

先验分布 prior distribution: p(ω)p(\bm{\omega})

代表了我们对先验情况的主观猜测。


当获得一些观测后,可以定义

似然函数 likelihood distribution: p(yx,ω)p(\bm{y}|\bm{x},\bm{\omega})

反映了在当前假设的函数参数 ω\bm{\omega} 下,x\bm{x} 给出观测值 y\bm{y} 的概率。

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Monte Carlo integration

对于我这样的统计外行,最大的问题是在一开始没有搞明白为什么要采样,或者说采样的结果应该如何使用,而不是MCMC的算法细节。

其实我想知道的东西,名字叫 Monte Carlo integration,参考这里的提问或者这里的PPT

Assuming θi\theta_i is sampled from the distribution p(θD)p(\theta|D), the Monte Carlo integreation formula is:

Eθp(θD)[g(θ)]=g(θ)p(θD)dθ1nθip(θD)g(θi)+O(n)\mathbb{E}_{\theta\sim p(\theta|D)}[g(\theta)] = \int g(\theta) p(\theta|D) d\theta \approx \frac{1}{n} \sum_{\theta_i\sim p(\theta|D)} g(\theta_i) + O(\sqrt{n})

The following discussion provides a very clear interpretation about Bayes Inferencing, but I’m not sure it’s exact and 100% correct. Need to do more reading.
Can a posterior expectation be used as a approximate for the true (prior) expectation?

学习资料

按照我的阅读顺序排序

[ref-1] Dylon 大仙, MCMC基本原理与应用(一), 2015-06-03

[ref-2] daniel-D, 从随机过程到马尔科夫链蒙特卡洛方法 (不太好,讲得比较混乱)

[ref-3] 靳志辉, LDA-math-MCMC 和 Gibbs Sampling (我从这里开始仔细看算法,细致平稳条件)

[ref-4] shenxiaolu1984, 蒙特卡洛-马尔科夫链(MCMC)初步 (简要介绍了4种采用方法,具体算法的公式挂了)

[ref-5] shenxiaolu1984, 蒙特卡洛-马尔科夫链(MCMC)的混合速度

[ref-6] qy20115549, HMC(Hamiltonian Monte Carlo抽样算法详细介绍) (未看)

[ref-7] 随机模拟-Monte Carlo积分及采样(详述直接采样、接受-拒绝采样、重要性采样) (讲了 Monte Carlo 积分与几种常见的采样方式的解释比较直观和深刻。MCMC 的主要作用之一是用来 支持 Monte Carlo 积分,其中涉及到了对某概率 f(x)f(x) 的采样。)

[ref-8] Bin的专栏, 随机采样方法整理与讲解(MCMC、Gibbs Sampling等) (推荐。基本是最正确的理解顺序。)

[ref-9] 再谈MCMC方法
讲到用MCMC破解凯撒密码

[ref-wiki-MCMC] Markov chain Monte Carlo(未看)

[ref-wiki-Gibbs]

HMC

如何简单地理解「哈密尔顿蒙特卡洛 (HMC)」?/ 写得比较简略,对入门不是很有帮助。

我的整理

理解这一算法的关键在于对以下几个概念的理解:

  • 随机过程
  • Markov 性,无后效性
  • Markov Chain 的极限和平稳分布
  • 概率分布的采样,数值方法
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steinwart_support_2008

Statistical Learning Theory 的本质

与经典的参数化方法不同,参数化方法假设 x 和 y 的关系服从某个确定性的函数。 (p3) this is a fundamental difference from parametric models, in which the relationship between the inputs x and the outputs y is assumed to follow some unknown function f ∈ F from a known, finite-dimensional set of functions F.

  • assuming that the output value y to a given x is stochastically generated by P( · |x) accommodates the fact that in general the information contained in x may not be sufficient to determine a single response in a deterministic manner.
  • assuming that the conditional probability P( · |x) is unknown contributes to the fact that we assume that we do not have a reasonable description of the relationship between the input and output values.

SVM 和 GP 的关系

For a brief description of kernel ridge regression and Gaussian processes, see Cristianini and Shawe-Taylor (2000, Section 6.2).

We refer to Wahba (1999) for the relationship between SVMs and Gaussian processes.