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.
Dieser Kurs ist Teil der Spezialisierung Spezialisierung Mathematik für maschinelles Lernen
von
Spezialisierung Mathematik für maschinelles LernenImperial College LondonÜber diesen Kurs
242,851 kürzliche Aufrufe
Karriereergebnisse der Lernenden
50%
nahm einen neuen Beruf nach Abschluss dieser Kurse auf
48%
ziehen Sie für Ihren Beruf greifbaren Nutzen aus diesem Kurs
Zertifikat zur Vorlage
Erhalten Sie nach Abschluss ein Zertifikat
100 % online
Beginnen Sie sofort und lernen Sie in Ihrem eigenen Tempo.
Kurs 3 von 3 in
Flexible Fristen
Setzen Sie Fristen gemäß Ihrem Zeitplan zurück.
Stufe „Mittel“
Ca. 19 Stunden zum Abschließen
Englisch
Untertitel: Englisch
Was Sie lernen werden
Implement mathematical concepts using real-world data
Derive PCA from a projection perspective
Understand how orthogonal projections work
Master PCA
Kompetenzen, die Sie erwerben
Dimensionality ReductionPython ProgrammingLinear Algebra
Karriereergebnisse der Lernenden
50%
nahm einen neuen Beruf nach Abschluss dieser Kurse auf
48%
ziehen Sie für Ihren Beruf greifbaren Nutzen aus diesem Kurs
Zertifikat zur Vorlage
Erhalten Sie nach Abschluss ein Zertifikat
100 % online
Beginnen Sie sofort und lernen Sie in Ihrem eigenen Tempo.
Kurs 3 von 3 in
Flexible Fristen
Setzen Sie Fristen gemäß Ihrem Zeitplan zurück.
Stufe „Mittel“
Ca. 19 Stunden zum Abschließen
Englisch
Untertitel: Englisch
von
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.
Lehrplan - Was Sie in diesem Kurs lernen werden
5 Stunden zum Abschließen
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 Stunden zum Abschließen
8 Videos (Gesamt 27 min), 6 Lektüren, 4 Quiz
8 Videos
Welcome to module 141
Mean of a dataset4m
Variance of one-dimensional datasets4m
Variance of higher-dimensional datasets5m
Effect on the mean4m
Effect on the (co)variance3m
See you next module!27
6 Lektüren
About Imperial College & the team5m
How to be successful in this course5m
Grading policy5m
Additional readings & helpful references5m
Set up Jupyter notebook environment offline10m
Symmetric, positive definite matrices10m
3 praktische Übungen
Mean of datasets15m
Variance of 1D datasets15m
Covariance matrix of a two-dimensional dataset15m
4 Stunden zum Abschließen
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 Stunden zum Abschließen
8 Videos (Gesamt 36 min), 1 Lektüre, 5 Quiz
8 Videos
Dot product4m
Inner product: definition5m
Inner product: length of vectors7m
Inner product: distances between vectors3m
Inner product: angles and orthogonality5m
Inner products of functions and random variables (optional)7m
Heading for the next module!35
1 Lektüre
Basis vectors20m
4 praktische Übungen
Dot product10m
Properties of inner products20m
General inner products: lengths and distances20m
Angles between vectors using a non-standard inner product20m
4 Stunden zum Abschließen
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 Stunden zum Abschließen
6 Videos (Gesamt 25 min), 1 Lektüre, 3 Quiz
6 Videos
Projection onto 1D subspaces7m
Example: projection onto 1D subspaces3m
Projections onto higher-dimensional subspaces8m
Example: projection onto a 2D subspace3m
This was module 3!32
1 Lektüre
Full derivation of the projection20m
2 praktische Übungen
Projection onto a 1-dimensional subspace25m
Project 3D data onto a 2D subspace40m
5 Stunden zum Abschließen
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 Stunden zum Abschließen
10 Videos (Gesamt 52 min), 5 Lektüren, 2 Quiz
10 Videos
Problem setting and PCA objective7m
Finding the coordinates of the projected data5m
Reformulation of the objective10m
Finding the basis vectors that span the principal subspace7m
Steps of PCA4m
PCA in high dimensions5m
Other interpretations of PCA (optional)7m
Summary of this module42
This was the course on PCA56
5 Lektüren
Vector spaces20m
Orthogonal complements10m
Multivariate chain rule10m
Lagrange multipliers10m
Did you like the course? Let us know!10m
1 praktische Übung
Chain rule practice20m
Bewertungen
Top-Bewertungen von MATHEMATICS FOR MACHINE LEARNING: PCA
von JS16. Juli 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.
von NS18. Juni 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.
von CJ28. Feb. 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.
von JV30. Apr. 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!
Über den Spezialisierung Mathematik für maschinelles Lernen
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.
Häufig gestellte Fragen
Wann erhalte ich Zugang zu den Vorträgen und Aufgaben?
Access to lectures and assignments depends on your type of enrollment. If you take a course in audit mode, you will be able to see most course materials for free. To access graded assignments and to earn a Certificate, you will need to purchase the Certificate experience, during or after your audit. If you don't see the audit option:
- The course may not offer an audit option. You can try a Free Trial instead, or apply for Financial Aid.
- 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.
Was bekomme ich, wenn ich diese Spezialisierung abonniere?
Wenn Sie sich für den Kurs anmelden, erhalten Sie Zugriff auf alle Kurse der Spezialisierung und Sie erhalten nach Abschluss aller Arbeiten ein Zertifikat. Ihr elektronisches Zertifikat wird zu Ihrer Seite „Errungenschaften“ hinzugefügt – von dort können Sie Ihr Zertifikat ausdrucken oder es zu Ihrem LinkedIn Profil hinzufügen. Wenn Sie nur lesen und den Inhalt des Kurses anzeigen möchten, können Sie kostenlos als Gast an dem Kurs teilnehmen.
Wie erfolgen Rückerstattungen?
Wenn Sie ein Abonnement abgeschlossen haben, erhalten Sie eine 7-tägige, kostenlose Testversion, die Sie gebührenfrei wieder kündigen können. Danach gewähren wir keine Rückerstattungen mehr, aber Sie können Ihr Abonnement jederzeit kündigen. Lesen Sie unsere vollständige Rückerstattungsrichtlinie.
Ist finanzielle Unterstützung möglich?
Ja, Coursera bietet für Kursteilnehmer, die sich die Kursgebühr nicht leisten können, finanzielle Unterstützung an. Bewerben Sie sich dafür, indem Sie auf den Link für finanzielle Unterstützung links unter der Schaltfläche „Anmelden“ klicken. Sie werden zum Ausfüllen eines Antrags aufgefordert und werden bei Genehmigung benachrichtigt. Diesen Schritt müssen Sie für jeden Kurs der Spezialisierung ausführen, auch für das Abschlussprojekt. Mehr erfahren
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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