This intermediate-level course introduces the mathematical foundations to derive Principal Component Analysis (PCA), a fundamental dimensionality reduction technique. We'll cover some basic statistics of data sets, such as mean values and variances, we'll compute distances and angles between vectors using inner products and derive orthogonal projections of data onto lower-dimensional subspaces. Using all these tools, we'll then derive PCA as a method that minimizes the average squared reconstruction error between data points and their reconstruction.
Ce cours fait partie de la Spécialisation Mathématiques pour l'apprentissage automatique
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Spécialisation Mathématiques pour l'apprentissage automatiqueImperial College LondonÀ propos de ce cours
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Niveau intermédiaire
Approx. 19 heures pour terminer
Anglais
Sous-titres : Anglais
Ce que vous allez apprendre
Implement mathematical concepts using real-world data
Derive PCA from a projection perspective
Understand how orthogonal projections work
Master PCA
Compétences que vous acquerrez
Dimensionality ReductionPython ProgrammingLinear Algebra
Résultats de carrière des étudiants
50%
ont commencé une nouvelle carrière après avoir terminé ce cours
48%
ont bénéficié d'un avantage concret dans leur carrières grâce à ce cours
Certificat partageable
Obtenez un Certificat lorsque vous terminez
100 % en ligne
Commencez dès maintenant et apprenez aux horaires qui vous conviennent.
Cours 3 sur 3 dans le
Dates limites flexibles
Réinitialisez les dates limites selon votre disponibilité.
Niveau intermédiaire
Approx. 19 heures pour terminer
Anglais
Sous-titres : Anglais
Offert par
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.
Programme du cours : ce que vous apprendrez dans ce cours
5 heures pour terminer
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.
5 heures pour terminer
8 vidéos (Total 27 min), 6 lectures, 4 quiz
8 vidéos
Welcome to module 141s
Mean of a dataset4 min
Variance of one-dimensional datasets4 min
Variance of higher-dimensional datasets5 min
Effect on the mean4 min
Effect on the (co)variance3 min
See you next module!27s
6 lectures
About Imperial College & the team5 min
How to be successful in this course5 min
Grading policy5 min
Additional readings & helpful references5 min
Set up Jupyter notebook environment offline10 min
Symmetric, positive definite matrices10 min
3 exercices pour s'entraîner
Mean of datasets15 min
Variance of 1D datasets15 min
Covariance matrix of a two-dimensional dataset15 min
4 heures pour terminer
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.
4 heures pour terminer
8 vidéos (Total 36 min), 1 lecture, 5 quiz
8 vidéos
Welcome to module 21 min
Dot product4 min
Inner product: definition5 min
Inner product: length of vectors7 min
Inner product: distances between vectors3 min
Inner product: angles and orthogonality5 min
Inner products of functions and random variables (optional)7 min
Heading for the next module!35s
1 lecture
Basis vectors20 min
4 exercices pour s'entraîner
Dot product10 min
Properties of inner products20 min
General inner products: lengths and distances20 min
Angles between vectors using a non-standard inner product20 min
4 heures pour terminer
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.
4 heures pour terminer
6 vidéos (Total 25 min), 1 lecture, 3 quiz
6 vidéos
Projection onto 1D subspaces7 min
Example: projection onto 1D subspaces3 min
Projections onto higher-dimensional subspaces8 min
Example: projection onto a 2D subspace3 min
This was module 3!32s
1 lecture
Full derivation of the projection20 min
2 exercices pour s'entraîner
Projection onto a 1-dimensional subspace25 min
Project 3D data onto a 2D subspace40 min
5 heures pour terminer
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.
5 heures pour terminer
10 vidéos (Total 52 min), 5 lectures, 2 quiz
10 vidéos
Welcome to module 41 min
Problem setting and PCA objective7 min
Finding the coordinates of the projected data5 min
Reformulation of the objective10 min
Finding the basis vectors that span the principal subspace7 min
Steps of PCA4 min
PCA in high dimensions5 min
Other interpretations of PCA (optional)7 min
Summary of this module42s
This was the course on PCA56s
5 lectures
Vector spaces20 min
Orthogonal complements10 min
Multivariate chain rule10 min
Lagrange multipliers10 min
Did you like the course? Let us know!10 min
1 exercice pour s'entraîner
Chain rule practice20 min
Avis
Meilleurs avis pour MATHEMATICS FOR MACHINE LEARNING: PCA
par JS16 juil. 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.
par NS18 juin 2020
Relatively tougher than previous two courses in the specialization. I'd suggest giving more time and being patient in pursuit of completing this course and understanding the concepts involved.
par CJ28 févr. 2019
Great capstone for the three-class Mathematics for Machine Learning series. Assignments were way harder and programming debugging skills had to be appropiate in order to finish the class.
par JV30 avr. 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!
À propos du Spécialisation Mathématiques pour l'apprentissage automatique
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 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.
Foire Aux Questions
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The course may offer 'Full Course, No Certificate' instead. This option lets you see all course materials, submit required assessments, and get a final grade. This also means that you will not be able to purchase a Certificate experience.
What will I get if I subscribe to this Specialization?
When you enroll in the course, you get access to all of the courses in the Specialization, and you earn a certificate when you complete the work. Your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.
What is the refund policy?
If you subscribed, you get a 7-day free trial during which you can cancel at no penalty. After that, we don’t give refunds, but you can cancel your subscription at any time. See our full refund policy.
Is financial aid available?
Yes, Coursera provides financial aid to learners who cannot afford the fee. Apply for it by clicking on the Financial Aid link beneath the "Enroll" button on the left. You'll be prompted to complete an application and will be notified if you are approved. You'll need to complete this step for each course in the Specialization, including the Capstone Project. Learn more.
What level of programming is required to do this course?
You will need good python knowledge to get through the course.
How difficult is this course in comparison to the other two of this specialization?
This course is significantly harder and different in style: it uses more abstract concepts and requires much more programming experience than the other two courses. Therefore, when you complete the full specialization, you will be equipped with a much more diverse set of skills.
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