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.

提供方

## 课程信息

### 学生职业成果

## 50%

## 48%

### 您将获得的技能

### 学生职业成果

## 50%

## 48%

#### 可分享的证书

#### 100% 在线

#### 第 3 门课程（共 3 门）

#### 可灵活调整截止日期

#### 中级

#### 完成时间大约为18 小时

#### 英语（English）

### 提供方

#### 伦敦帝国学院

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.

## 教学大纲 - 您将从这门课程中学到什么

**完成时间为 5 小时**

## 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.

**完成时间为 5 小时**

**8 个视频**

**6 个阅读材料**

**3 个练习**

**完成时间为 4 小时**

## 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.

**完成时间为 4 小时**

**8 个视频**

**1 个阅读材料**

**4 个练习**

**完成时间为 4 小时**

## 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.

**完成时间为 4 小时**

**6 个视频**

**1 个阅读材料**

**2 个练习**

**完成时间为 5 小时**

## 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.

**完成时间为 5 小时**

**10 个视频**

**5 个阅读材料**

**1 个练习**

### 审阅

#### 4.0

##### 来自MATHEMATICS FOR MACHINE LEARNING: PCA的热门评论

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.

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.

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!

Challenging, but doable. Has some bugs in coding assignments, but clearing them out makes you understand things better. Get ready to spend extra time understanding the concepts.

This course is well worth the time. I have a better understanding of one of the most foundational and biologically plausible machine learning algorithms used today! Love it.

Programming assignment for week 1 wastes to much time due to lack of instructions.\n\nThe notebook also does not work...(maybe locally , but I have other things to do).

Course content tackles a difficult topic well. Only flaw is that programming assignments are poorly designed in some places and are quite difficult to pick up at times.

This course demystifies the Principal Components Analysis through practical implementation. It gives me solid foundations for learning further data science techniques.

Teaching pacing is good, and clear in explanation. It will be good if there are some examples about how we should apply all these theories to some real problems.

I found this course really excellent. Very clear explanations with very hepful illustrations.\n\nI was looking for course on PCA, thank you for this one

I learned a lot in this course, though the last week was somehow hurried and the lecturer didn't spend enough time to piece the whole stuff together.

Solid conceptual explanations of PCA make this course stand out. The thorough review of this content is a must for any serious data researcher.

Course addresses important subject, but I worth like to have more in-depth explanation of the mathematics by the instructors and more examples.

Felt like explanations in this course were a bit confusing, but otherwise, it was a very interesting course. Thank you so much for doing this.

Excellent course. The fourth week material is the hardest for folks not comfortable with linear algebra and vectorization in numpy and scipy.

What a great opportunity this course offers to learn from the best in this simplified manner. Thank you Coursera and Imperial College London!

This was a very hard course for me. But I think the instructor has done the best possible he can with presenting and explaining the course

The final Notebook contains some errors (Xbar instead of X, passed as an argument). Otherwise a very well organized course. Thanks a lot!

Highly informative course! Loved the depth of the material. Found this course content highly useful in my current project based on PCA.

Coding assignment is hard for people who are not familiar with numpy. Would appreciate some material at least going over the basis.

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