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推奨:4 weeks of study, 4-5 hours/week...

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Python ProgrammingPrincipal Component Analysis (PCA)Projection MatrixMathematical Optimization

100%オンライン

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中級レベル

約18時間で修了

推奨:4 weeks of study, 4-5 hours/week...

英語

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シラバス - 本コースの学習内容

1
5時間で修了

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....
8件のビデオ (合計27分), 6 readings, 4 quizzes
8件のビデオ
Welcome to module 141
Mean of a dataset4 分
Variance of one-dimensional datasets4 分
Variance of higher-dimensional datasets5 分
Effect on the mean4 分
Effect on the (co)variance3 分
See you next module!27
6件の学習用教材
About Imperial College & the team5 分
How to be successful in this course5 分
Grading policy5 分
Additional readings & helpful references5 分
Set up Jupyter notebook environment offline10 分
Symmetric, positive definite matrices10 分
3の練習問題
Mean of datasets15 分
Variance of 1D datasets15 分
Covariance matrix of a two-dimensional dataset15 分
2
4時間で修了

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....
8件のビデオ (合計36分), 1 reading, 5 quizzes
8件のビデオ
Dot product4 分
Inner product: definition5 分
Inner product: length of vectors7 分
Inner product: distances between vectors3 分
Inner product: angles and orthogonality5 分
Inner products of functions and random variables (optional)7 分
Heading for the next module!35
1件の学習用教材
Basis vectors20 分
4の練習問題
Dot product10 分
Properties of inner products20 分
General inner products: lengths and distances20 分
Angles between vectors using a non-standard inner product20 分
3
4時間で修了

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....
6件のビデオ (合計25分), 1 reading, 3 quizzes
6件のビデオ
Projection onto 1D subspaces7 分
Example: projection onto 1D subspaces3 分
Projections onto higher-dimensional subspaces8 分
Example: projection onto a 2D subspace3 分
This was module 3!32
1件の学習用教材
Full derivation of the projection20 分
2の練習問題
Projection onto a 1-dimensional subspace25 分
Project 3D data onto a 2D subspace40 分
4
5時間で修了

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. ...
10件のビデオ (合計52分), 5 readings, 2 quizzes
10件のビデオ
Problem setting and PCA objective7 分
Finding the coordinates of the projected data5 分
Reformulation of the objective10 分
Finding the basis vectors that span the principal subspace7 分
Steps of PCA4 分
PCA in high dimensions5 分
Other interpretations of PCA (optional)7 分
Summary of this module42
This was the course on PCA56
5件の学習用教材
Vector spaces20 分
Orthogonal complements10 分
Multivariate chain rule10 分
Lagrange multipliers10 分
Did you like the course? Let us know!10 分
1の練習問題
Chain rule practice20 分
4.0
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人気のレビュー

by 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.

by 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!

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Marc P. Deisenroth

Lecturer in Statistical Machine Learning
Department of Computing

インペリアル・カレッジ・ロンドン(Imperial College London)について

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....

Mathematics for Machine Learningの専門講座について

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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