使用 Hatchfill2
配合 legendflex 对 lagend 进行控制

SPACEofPHD 上的推送v


科研绘图 | 提取两个曲面构成的交曲面(两种方法)

科研绘图 | Matlab绘制精美的势能函数剖面图

link properties of graphic objects:
linkaxes only links XLim or YLim or both。 内部是用 linkprop 实现的。


问题:fill会重写之前的所有axis属性(as of 2019-05-01 18:17:59)

科研绘图 | Matlab绘制精美的势能函数剖面图

Christopher M. Bishop, Pattern recognition and machine learning, New York: Springer, 2006.

This reading note is served as a quick reference and short summary of the PRML book. Read the book first. It is incorrect to trying to understand the book through reading this post.

Basically, I was just trying to simplify the book by extracting only useful definitions, equations, formulas, and explanations. So, to understand the context here, one should study the PRML book first.

(p23) This book places a strong emphasis on the Bayesian viewpoint, reflecting the huge growth in the practical importance of Bayesian methods in the past few years, while also discussing useful frequentist concepts as required.

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Special for Hexo-theme-next

Use this line to include image:

<img src="{% asset_path FPKE_history.png%}" width="400px" title="caption">

General HTML

Add color to text

<span style="color:red">TEXT_COLORFUL</span>

Add background color to text

<span style="background-color:green">BACKGROUND_COLORFUL</span>

AngularRaDec(final GroundStation station, final Frame referenceFrame, ... 给出的是 topocentric Ra/Dec。

theoreticalEvaluation()的代码中可以看出, Station-satellite vector expressed in inertial frame 是一个 vector (not a state),转换到 Field transform from inertial to reference frame at station’s reception date 后,对得到的 state 求解 Ra/Dec。
实质上,等价于,把 reference frame 平移到 station’s location,然后在这个新的frame中求解 Ra/Dec,即 topocentric Ra/Dec。

另一方面,参考 Vallado1997 书中 6.4 Angle-only observations 的说明,angle-only observation 没有办法直接得到 range information,所以得到的只能是 topocentric Ra/Dec。



ARD的kernel的引入使得高斯过程模型自然自带feature selection的能力,这就是GP相对于其他机器学习模型的有优势之一\

基于Python的GPy(以前记得没有的哈,不过最近在其中发现了gp_multiout_regression.py, 因而想必也是可以多维输出的哦!由于写着是回归,想必应该也是只能做多维输出的回归吧!),

第一步:所有training output构成矩阵后(因为每一个点是一个向量,所以堆叠后所有training可以构成矩阵)向量化,当然也可以说直接就把每个点直接堆叠构成一个更长的向量。只不过这个一个更长的向量的可以分为d段,每一段对应这一个output。
第二部:核函数重构,即采取kronecker product的方式将核函数与output相关性矩阵结合,构造一个新的大的covariance matrix。

[2] Bonilla, Edwin V., Kian M. Chai, and Christopher Williams. "Multi-task Gaussian process prediction."Advances in neural information processing systems. 2008.
[3] Boyle, Phillip, and Marcus Frean. "Dependent gaussian processes."Advances in neural information processing systems. 2005.
[4] Alvarez, Mauricio A., Lorenzo Rosasco, and Neil D. Lawrence. "Kernels for vector-valued functions: A review."Foundations and Trends® in Machine Learning4.3 (2012): 195-266.
[5] Wang, Bo, and Tao Chen. "Gaussian process regression with multiple response variables."Chemometrics and Intelligent Laboratory Systems142 (2015): 159-165.




It has two good publications:

  • Chen, Zexun, and Bo Wang. “How priors of initial hyperparameters affect Gaussian process regression models.” Neurocomputing 275 (2018): 1702-1710.
  • Chen, Zexun, Bo Wang, and Alexander N. Gorban. “Multivariate Gaussian and Student $-t $ Process Regression for Multi-output Prediction.” arXiv preprint arXiv:1703.04455 (2017).

Both STP and GP, both univariate and multivariate.

This guy (Zexun Chen (陈泽汛)) maintains the collection in Zhihu:
蓦风星吟,高斯世界下的Machine Learning, https://zhuanlan.zhihu.com/gpml2016

Authors’ own comments at Zhihu:




The Sheffield group summarizes all their independent repos since 2015 in this big one.

  • The GPmat toolbox is the ‘one stop shop’ on github for a number of dependent toolboxes, each of which used to be released independently. Since May 2015 each toolbox is a sub-directory within GPmat. They are included as subtrees from the relevant repositories.



Seems not integrated into the above GPmat yet.


MTGP - A multi-task Gaussian Process Toolbox

蓦风星吟:另外就是最近有小伙伴推荐的一个MTGP - A multi-task Gaussian Process Toolbox,同样也木有用过,不过看上去还不错呢,至少功能明确。

Covariance/noise matrix 在 body frame 和 inertial frame 之间的转换

这部分代码出现在 UnivariateProcessNoise 中,see
#537: `FIXME` in codes
#403: Add a provider generating process noise increasing in time, for better Kalman filtering
for more discussions.

RSW === LVLH in Orekit.

lofCartesianProcessNoiseMatrix Local orbital frame (RSW, or LVLH) 中的 covariance QRSWQ^{RSW}

  • GP-prediction covariance, a diagonal 6 x 6 matrix. (in my codes)

jacLofToInertial JR2I=JRSWIner.=XIner.XRSWJ_{R2I} = J_{RSW\rightarrow Iner.} = \frac{\partial X^{Iner.}}{\partial X^{RSW}}

  • Question? Is there a loss of information at this step?

jacParametersWrtCartesian JP2C=JPar.Cart.=Par.Cart.=Par.XIner.J_{P2C} = J_{Par.\rightarrow Cart.} = \frac{\partial Par.}{\partial Cart.} = \frac{\partial Par.}{\partial X^{Iner.}}

  • If the parameters used to represent the orbit is inertial Cartesian, then this should be I6I_6.
  • Otherwise, it’s not.

Finial conversion:

QIner.=QPar.=Par.XIner.XIner.XRSWQRSW(XIner.XRSW)T(Par.XIner.)T=JP2CJR2IQRSWJR2ITJP2CT\begin{aligned} Q^{Iner.} &= Q^{Par.} \\ &= \frac{\partial Par.}{\partial X^{Iner.}} \frac{\partial X^{Iner.}}{\partial X^{RSW}} Q^{RSW} \left(\frac{\partial X^{Iner.}}{\partial X^{RSW}}\right)^T \left(\frac{\partial Par.}{\partial X^{Iner.}}\right)^T \\ &= J_{P2C} \cdot J_{R2I} \cdot Q^{RSW} J_{R2I}^T \cdot J_{P2C}^T \end{aligned}

inverse: from Inertial to RSW

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  • “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)



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~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.


  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)



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


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())


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