4.7
1,431 ratings
211 reviews

#### 100% online

Start instantly and learn at your own schedule.

#### Approx. 21 hours to complete

Suggested: 6 weeks of study, 2-5 hours/week...

#### English

Subtitles: English, Greek, Spanish

### Skills you will gain

#### 100% online

Start instantly and learn at your own schedule.

#### Approx. 21 hours to complete

Suggested: 6 weeks of study, 2-5 hours/week...

#### English

Subtitles: English, Greek, Spanish

### Syllabus - What you will learn from this course

Week
1
4 hours to complete

## What is calculus?

Understanding calculus is central to understanding machine learning! You can think of calculus as simply a set of tools for analysing the relationship between functions and their inputs. Often, in machine learning, we are trying to find the inputs which enable a function to best match the data. We start this module from the basics, by recalling what a function is and where we might encounter one. Following this, we talk about the how, when sketching a function on a graph, the slope describes the rate of change of the output with respect to an input. Using this visual intuition we next derive a robust mathematical definition of a derivative, which we then use to differentiate some interesting functions. Finally, by studying a few examples, we develop four handy time saving rules that enable us to speed up differentiation for many common scenarios. ...
10 videos (Total 46 min), 4 readings, 6 quizzes
10 videos
Welcome to Module 1!1m
Functions4m
Rise Over Run4m
Definition of a derivative10m
Differentiation examples & special cases7m
Product rule4m
Chain rule5m
Taming a beast5m
See you next module!39s
About Imperial College & the team5m
How to be successful in this course5m
6 practice exercises
Matching functions visually20m
Matching the graph of a function to the graph of its derivative20m
Let's differentiate some functions20m
Practicing the product rule20m
Practicing the chain rule20m
Unleashing the toolbox20m
Week
2
3 hours to complete

## Multivariate calculus

Building on the foundations of the previous module, we now generalise our calculus tools to handle multivariable systems. This means we can take a function with multiple inputs and determine the influence of each of them separately. It would not be unusual for a machine learning method to require the analysis of a function with thousands of inputs, so we will also introduce the linear algebra structures necessary for storing the results of our multivariate calculus analysis in an orderly fashion. ...
9 videos (Total 41 min), 5 quizzes
9 videos
Variables, constants & context7m
Differentiate with respect to anything4m
The Jacobian5m
Jacobian applied6m
The Sandpit4m
The Hessian5m
Reality is hard4m
See you next module!23s
5 practice exercises
Practicing partial differentiation20m
Calculating the Jacobian20m
Bigger Jacobians!20m
Calculating Hessians20m
Assessment: Jacobians and Hessians20m
Week
3
3 hours to complete

## Multivariate chain rule and its applications

Having seen that multivariate calculus is really no more complicated than the univariate case, we now focus on applications of the chain rule. Neural networks are one of the most popular and successful conceptual structures in machine learning. They are build up from a connected web of neurons and inspired by the structure of biological brains. The behaviour of each neuron is influenced by a set of control parameters, each of which needs to be optimised to best fit the data. The multivariate chain rule can be used to calculate the influence of each parameter of the networks, allow them to be updated during training. ...
6 videos (Total 19 min), 4 quizzes
6 videos
Multivariate chain rule2m
More multivariate chain rule5m
Simple neural networks5m
More simple neural networks4m
See you next module!34s
3 practice exercises
Multivariate chain rule exercise20m
Simple Artificial Neural Networks20m
Training Neural Networks25m
Week
4
2 hours to complete

## Taylor series and linearisation

The Taylor series is a method for re-expressing functions as polynomial series. This approach is the rational behind the use of simple linear approximations to complicated functions. In this module, we will derive the formal expression for the univariate Taylor series and discuss some important consequences of this result relevant to machine learning. Finally, we will discuss the multivariate case and see how the Jacobian and the Hessian come in to play. ...
9 videos (Total 41 min), 5 quizzes
9 videos
Building approximate functions3m
Power series3m
Power series derivation9m
Power series details6m
Examples5m
Linearisation5m
Multivariate Taylor6m
See you next module!28s
5 practice exercises
Matching functions and approximations20m
Applying the Taylor series15m
Taylor series - Special cases10m
2D Taylor series15m
Taylor Series Assessment20m
Week
5
2 hours to complete

## Intro to optimisation

If we want to find the minimum and maximum points of a function then we can use multivariate calculus to do this, say to optimise the parameters (the space) of a function to fit some data. First we’ll do this in one dimension and use the gradient to give us estimates of where the zero points of that function are, and then iterate in the Newton-Raphson method. Then we’ll extend the idea to multiple dimensions by finding the gradient vector, Grad, which is the vector of the Jacobian. This will then let us find our way to the minima and maxima in what is called the gradient descent method. We’ll then take a moment to use Grad to find the minima and maxima along a constraint in the space, which is the Lagrange multipliers method....
4 videos (Total 28 min), 4 quizzes
4 videos
Constrained optimisation8m
See you next module!2m
4 practice exercises
Newton-Raphson in one dimension20m
Checking Newton-Raphson10m
Lagrange multipliers20m
Optimisation scenarios20m
Week
6
2 hours to complete

## Regression

In order to optimise the fitting parameters of a fitting function to the best fit for some data, we need a way to define how good our fit is. This goodness of fit is called chi-squared, which we’ll first apply to fitting a straight line - linear regression. Then we’ll look at how to optimise our fitting function using chi-squared in the general case using the gradient descent method. Finally, we’ll look at how to do this easily in Python in just a few lines of code, which will wrap up the course....
4 videos (Total 25 min), 1 reading, 3 quizzes
4 videos
General non linear least squares7m
Doing least squares regression analysis in practice6m
Wrap up of this course48s
Did you like the course? Let us know!10m
2 practice exercises
Linear regression25m
Fitting a non-linear function15m
4.7
211 Reviews

## 25%

started a new career after completing these courses

## 21%

got a tangible career benefit from this course

### Top Reviews

By DPNov 26th 2018

Great course to develop some understanding and intuition about the basic concepts used in optimization. Last 2 weeks were a bit on a lower level of quality then the rest in my opinion but still great.

By JTNov 13th 2018

Excellent course. I completed this course with no prior knowledge of multivariate calculus and was successful nonetheless. It was challenging and extremely interesting, informative, and well designed.

## Instructors

### Samuel J. Cooper

Lecturer
Dyson School of Design Engineering

### David Dye

Professor of Metallurgy
Department of Materials

### A. Freddie Page

Strategic Teaching Fellow
Dyson School of Design Engineering

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....

## About the Mathematics for Machine Learning Specialization

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....