download - Alexandre Devert

60 Pages·2012·1.44 MB·English

by  Devert Alexandre

UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC
Parametric regression
Finding parameters of a model
Devert Alexandre
SchoolofSoftwareEngineeringofUSTC
December13,2012—Slide1/51
UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC
Introduction
DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—Parametricregression—Slide2/51
UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC
Orbits parameters of Ceres
Parametric regression technics have been pioneered by
Carl Friedrich Gauss in 1801, to find the dwarf planet
Ceres
Figure : This is not Carl
Friedrich Gauss
Figure : This is not Ceres
DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—Parametricregression—Slide3/51
UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC
Orbits parameters of Ceres
In 1801, astronomer Giuseppe Piazzi discovered Ceres.
But then he lost its track, as Ceres moved behind the
Sun.
So far, the available data where
• A few positions of Ceres in the sky
• Newton’s law of gravitation
• Trajectory of the Earth
How to find Ceres again ?
DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—Parametricregression—Slide4/51
UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC
Orbits parameters of Ceres
Made with 1801’s technology, the Ceres measurements
were of limited precision
Figure : Giuseppe Piazzi’s data
DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—Parametricregression—Slide5/51
UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC
Orbits parameters of Ceres
Carl Friedrich Gauss (24 years old) solved the problem
this way
• Make a model which can describe the position in the
sky of Ceres, with 6 unknown parameters
• Use the available data to find the 6 parameters such
as the model is the most likely explanation for the
data
• Use the model to find where was Ceres
210 years ago ⇒ no computers, all hand-made
calculations
DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—Parametricregression—Slide6/51
UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC
Parametric regression
What Gauss did is called parametric regression
• A model F(X,V) = Y
• A dataset of pairs (X,Y), from which we assume
they are generated by the model, with some noise.
• Finding the parameters V of the model from the
dataset
DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—Parametricregression—Slide7/51
UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC
Parametric regression
Since data are likely noisy, the best possible parameters
for a model will not fit exactly the data
N
(cid:88)
e = ||F(X ,V)−Y ||2
i i
i=1
e is the error of the model, which a regression will try to
minimize
DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—Parametricregression—Slide8/51
UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC
Linear regression
DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—Parametricregression—Slide9/51
UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC
Linear regression
Linear regression ⇒ the model of the data is a linear
combination of the variables
Y = AX +B
for a dataset with points in the form of {y ,x ,...,x }
i 1 n
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