課程信息
4.0
420 個評分
90 個審閱
專項課程

第 3 門課程(共 3 門),位於

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立即開始,按照自己的計劃學習。
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中級

中級

完成時間(小時)

完成時間大約為18 小時

建議:4 weeks of study, 4-5 hours/week...
可選語言

英語(English)

字幕:英語(English)...

您將獲得的技能

Python ProgrammingPrincipal Component Analysis (PCA)Projection MatrixMathematical Optimization
專項課程

第 3 門課程(共 3 門),位於

100% 在線

100% 在線

立即開始,按照自己的計劃學習。
可靈活調整截止日期

可靈活調整截止日期

根據您的日程表重置截止日期。
中級

中級

完成時間(小時)

完成時間大約為18 小時

建議:4 weeks of study, 4-5 hours/week...
可選語言

英語(English)

字幕:英語(English)...

教學大綱 - 您將從這門課程中學到什麼

1
完成時間(小時)
完成時間為 5 小時

Statistics of Datasets

Principal Component Analysis (PCA) is one of the most important dimensionality reduction algorithms in machine learning. In this course, we lay the mathematical foundations to derive and understand PCA from a geometric point of view. In this module, we learn how to summarize datasets (e.g., images) using basic statistics, such as the mean and the variance. We also look at properties of the mean and the variance when we shift or scale the original data set. We will provide mathematical intuition as well as the skills to derive the results. We will also implement our results in code (jupyter notebooks), which will allow us to practice our mathematical understand to compute averages of image data sets....
Reading
8 個視頻(共 27 分鐘), 5 個閱讀材料, 4 個測驗
Video8 個視頻
Welcome to module 1分鐘
Mean of a dataset4分鐘
Variance of one-dimensional datasets4分鐘
Variance of higher-dimensional datasets5分鐘
Effect on the mean4分鐘
Effect on the (co)variance3分鐘
See you next module!分鐘
Reading5 個閱讀材料
About Imperial College & the team5分鐘
How to be successful in this course5分鐘
Grading policy5分鐘
Additional readings & helpful references5分鐘
Symmetric, positive definite matrices10分鐘
Quiz3 個練習
Mean of datasets15分鐘
Variance of 1D datasets15分鐘
Covariance matrix of a two-dimensional dataset15分鐘
2
完成時間(小時)
完成時間為 4 小時

Inner Products

Data can be interpreted as vectors. Vectors allow us to talk about geometric concepts, such as lengths, distances and angles to characterise similarity between vectors. This will become important later in the course when we discuss PCA. In this module, we will introduce and practice the concept of an inner product. Inner products allow us to talk about geometric concepts in vector spaces. More specifically, we will start with the dot product (which we may still know from school) as a special case of an inner product, and then move toward a more general concept of an inner product, which play an integral part in some areas of machine learning, such as kernel machines (this includes support vector machines and Gaussian processes). We have a lot of exercises in this module to practice and understand the concept of inner products....
Reading
8 個視頻(共 36 分鐘), 1 個閱讀材料, 5 個測驗
Video8 個視頻
Dot product4分鐘
Inner product: definition5分鐘
Inner product: length of vectors7分鐘
Inner product: distances between vectors3分鐘
Inner product: angles and orthogonality5分鐘
Inner products of functions and random variables (optional)7分鐘
Heading for the next module!分鐘
Reading1 個閱讀材料
Basis vectors20分鐘
Quiz4 個練習
Dot product10分鐘
Properties of inner products20分鐘
General inner products: lengths and distances20分鐘
Angles between vectors using a non-standard inner product20分鐘
3
完成時間(小時)
完成時間為 4 小時

Orthogonal Projections

In this module, we will look at orthogonal projections of vectors, which live in a high-dimensional vector space, onto lower-dimensional subspaces. This will play an important role in the next module when we derive PCA. We will start off with a geometric motivation of what an orthogonal projection is and work our way through the corresponding derivation. We will end up with a single equation that allows us to project any vector onto a lower-dimensional subspace. However, we will also understand how this equation came about. As in the other modules, we will have both pen-and-paper practice and a small programming example with a jupyter notebook....
Reading
6 個視頻(共 25 分鐘), 1 個閱讀材料, 3 個測驗
Video6 個視頻
Projection onto 1D subspaces7分鐘
Example: projection onto 1D subspaces3分鐘
Projections onto higher-dimensional subspaces8分鐘
Example: projection onto a 2D subspace3分鐘
This was module 3!分鐘
Reading1 個閱讀材料
Full derivation of the projection20分鐘
Quiz2 個練習
Projection onto a 1-dimensional subspace25分鐘
Project 3D data onto a 2D subspace40分鐘
4
完成時間(小時)
完成時間為 5 小時

Principal Component Analysis

We can think of dimensionality reduction as a way of compressing data with some loss, similar to jpg or mp3. Principal Component Analysis (PCA) is one of the most fundamental dimensionality reduction techniques that are used in machine learning. In this module, we use the results from the first three modules of this course and derive PCA from a geometric point of view. Within this course, this module is the most challenging one, and we will go through an explicit derivation of PCA plus some coding exercises that will make us a proficient user of PCA. ...
Reading
10 個視頻(共 52 分鐘), 5 個閱讀材料, 2 個測驗
Video10 個視頻
Problem setting and PCA objective7分鐘
Finding the coordinates of the projected data5分鐘
Reformulation of the objective10分鐘
Finding the basis vectors that span the principal subspace7分鐘
Steps of PCA4分鐘
PCA in high dimensions5分鐘
Other interpretations of PCA (optional)7分鐘
Summary of this module分鐘
This was the course on PCA分鐘
Reading5 個閱讀材料
Vector spaces20分鐘
Orthogonal complements10分鐘
Multivariate chain rule10分鐘
Lagrange multipliers10分鐘
Did you like the course? Let us know!10分鐘
Quiz1 個練習
Chain rule practice20分鐘
4.0

熱門審閱

創建者 JSJul 17th 2018

This is one hell of an inspiring course that demystified the difficult concepts and math behind PCA. Excellent instructors in imparting the these knowledge with easy-to-understand illustrations.

創建者 JVMay 1st 2018

This course was definitely a bit more complex, not so much in assignments but in the core concepts handled, than the others in the specialisation. Overall, it was fun to do this course!

講師

Avatar

Marc P. Deisenroth

Lecturer in Statistical Machine Learning
Department of Computing

關於 Imperial College London

Imperial College London is a world top ten university with an international reputation for excellence in science, engineering, medicine and business. located in the heart of London. Imperial is a multidisciplinary space for education, research, translation and commercialisation, harnessing science and innovation to tackle global challenges. Imperial students benefit from a world-leading, inclusive educational experience, rooted in the College’s world-leading research. Our online courses are designed to promote interactivity, learning and the development of core skills, through the use of cutting-edge digital technology....

關於 Mathematics for Machine Learning 專項課程

For a lot of higher level courses in Machine Learning and Data Science, you find you need to freshen up on the basics in mathematics - stuff you may have studied before in school or university, but which was taught in another context, or not very intuitively, such that you struggle to relate it to how it’s used in Computer Science. This specialization aims to bridge that gap, getting you up to speed in the underlying mathematics, building an intuitive understanding, and relating it to Machine Learning and Data Science. In the first course on Linear Algebra we look at what linear algebra is and how it relates to data. Then we look through what vectors and matrices are and how to work with them. The second course, Multivariate Calculus, builds on this to look at how to optimize fitting functions to get good fits to data. It starts from introductory calculus and then uses the matrices and vectors from the first course to look at data fitting. The third course, Dimensionality Reduction with Principal Component Analysis, uses the mathematics from the first two courses to compress high-dimensional data. This course is of intermediate difficulty and will require basic Python and numpy knowledge. At the end of this specialization you will have gained the prerequisite mathematical knowledge to continue your journey and take more advanced courses in machine learning....
Mathematics for Machine Learning

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