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Samuel J. Cooper,
David Dye,
and
A. Freddie Page
This course offers a brief introduction to the multivariate calculus required to build many common machine learning techniques. We start at the very beginning with a refresher on the “rise over run” formulation of a slope, before converting this to the formal definition of the gradient of a function. We then start to build up a set of tools for making calculus easier and faster. Next, we learn how to calculate vectors that point up hill on multidimensional surfaces and even put this into action using an interactive game. We take a look at how we can use calculus to build approximations to functions, as well as helping us to quantify how accurate we should expect those approximations to be. We also spend some time talking about where calculus comes up in the training of neural networks, before finally showing you how it is applied in linear regression models. This course is intended to offer an intuitive understanding of calculus, as well as the language necessary to look concepts up yourselves when you get stuck. Hopefully, without going into too much detail, you’ll still come away with the confidence to dive into some more focused machine learning courses in future.
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What's inside
Syllabus
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.
Read more
Syllabus
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.
Read more
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.
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.
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.
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.
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.
Good to know
Good to know
Know what's good ,
what to watch for , and
possible dealbreakers
Develops the understanding of multivariate calculus, which is a foundational skill set for practicing data scientists
Explores multivariate calculus, which is applicable for working with high-dimensional data often encountered in machine learning
Introduces tools and techniques to enable students to understand and work on complex machine learning algorithms
Taught by instructors recognized for their work in machine learning and calculus
Provides a solid theoretical foundation in multivariate calculus for students pursuing careers in machine learning
Suitable for students with a background in mathematics, particularly in calculus
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Reviews summary
Multivariate calculus for machine learning
This course provides a comprehensive overview of the calculus concepts that are essential for understanding and applying machine learning algorithms. Learners say the course is well-structured and engaging, with clear explanations and helpful assignments, making it suitable for beginners and those looking to refresh their knowledge. Led by enthusiastic instructors, it guides students through topics such as derivatives, gradients, optimization techniques, and neural networks, building intuition and demonstrating practical applications. The course materials include videos, quizzes, coding exercises, and interactive simulations to enhance understanding. While the last two weeks of the course are slightly more challenging, the overall sentiment is positive, with learners appreciating the balance between theory and practice, and the opportunity to develop a strong mathematical foundation for machine learning.
concepts that are essential for understanding and applying machine learning algorithms
engaging, with clear explanations
"The course taught Multivariate Calculus from scratch. Quizzes and programming assignments were highly helpful in clearing concepts and applying acquired knowledge. Thanks a lot for this wonderful course."
well-structured
"I found this to be a valuable overview to calculus. The course could be improved by including more explanations for answers, as sometimes I wasn't sure how a given answer was derived, but it was a worthwhile use of my time overall."
Suitable for beginners and those looking to refresh their knowledge
"This course assumes some knowledge of Python. Some topics are taught quite quickly, and overall this is not a comprehensive course."
Helpful assignments
"The first 3 weeks is really good, the fourth week is okay but the last 2 weeks is kinda confusing."
enthusiastic instructors
learners say
"Very clear lessons by two excellent coachs who present a smooth progression of the fundamentals of optimizations for machine learning."
videos, quizzes, coding exercises, and interactive simulations
Activities
Be better prepared
before
your course. Deepen your understanding
during
and
after
it. Supplement your coursework and achieve mastery of the topics covered
in Mathematics for Machine Learning: Multivariate Calculus with these
activities:
Review the basics of algebra and trigonometry
Show steps
A strong foundation in algebra and trigonometry is essential for success in calculus.
Browse courses on
Algebra
Show steps
-
Review your notes or textbooks from previous algebra and trigonometry courses.
-
Complete some practice problems to refresh your memory.
Review the textbook
Show steps
Thoroughly reviewing the textbook will provide you with a solid foundation in calculus.
View
Algebra and Trigonometry
on Amazon
Show steps
-
Read each chapter carefully and take notes.
-
Complete all the practice problems at the end of each chapter.
Join a calculus study group
Show steps
Study calculus with a group of peers to reinforce your understanding and learn from others.
Browse courses on
Multivariate Calculus
Show steps
-
Find a calculus study group to join.
-
Attend the study group meetings regularly.
Four other activities
Expand to see all activities and additional details
Show all seven activities
Practice calculus drills
Show steps
Review the basics of calculus by completing practice drills to strengthen your understanding.
Browse courses on
Multivariate Calculus
Show steps
-
Find an online calculus practice drill resource.
-
Complete a set of calculus practice drills.
Find a calculus mentor
Show steps
A calculus mentor can provide you with personalized guidance and support.
Browse courses on
Multivariate Calculus
Show steps
-
Identify the specific areas of calculus where you need help.
-
Contact a professor, tutor, or experienced professional in the field.
Apply calculus to real-world scenarios
Show steps
Apply the calculus concepts you've learned to solve problems in the real world.
Browse courses on
Multivariate Calculus
Show steps
-
Find a real-world problem that can be solved using calculus.
-
Apply the appropriate calculus concepts to solve the problem.
-
Check your solution and reflect on the process.
Create a visual representation of a calculus concept
Show steps
Creating a visual representation of a calculus concept will help you understand it more deeply.
Browse courses on
Multivariate Calculus
Show steps
-
Choose a calculus concept that you want to understand better.
-
Brainstorm different ways to represent the concept visually.
-
Create your visual representation.
-
Explain your visual representation to someone else.
Review the basics of algebra and trigonometry
Show steps
A strong foundation in algebra and trigonometry is essential for success in calculus.
Browse courses on
Algebra
Show steps
Show steps
Show steps
- Review your notes or textbooks from previous algebra and trigonometry courses.
- Complete some practice problems to refresh your memory.
Review the textbook
Show steps
Thoroughly reviewing the textbook will provide you with a solid foundation in calculus.
View
Algebra and Trigonometry
on Amazon
Show steps
Show steps
Show steps
- Read each chapter carefully and take notes.
- Complete all the practice problems at the end of each chapter.
Join a calculus study group
Show steps
Study calculus with a group of peers to reinforce your understanding and learn from others.
Browse courses on
Multivariate Calculus
Show steps
Show steps
Show steps
- Find a calculus study group to join.
- Attend the study group meetings regularly.
Four other activities
Expand to see all activities and additional details
Show all seven activities
Practice calculus drills
Show steps
Review the basics of calculus by completing practice drills to strengthen your understanding.
Browse courses on
Multivariate Calculus
Show steps
Show steps
Show steps
- Find an online calculus practice drill resource.
- Complete a set of calculus practice drills.
Find a calculus mentor
Show steps
A calculus mentor can provide you with personalized guidance and support.
Browse courses on
Multivariate Calculus
Show steps
Show steps
Show steps
- Identify the specific areas of calculus where you need help.
- Contact a professor, tutor, or experienced professional in the field.
Apply calculus to real-world scenarios
Show steps
Apply the calculus concepts you've learned to solve problems in the real world.
Browse courses on
Multivariate Calculus
Show steps
Show steps
Show steps
- Find a real-world problem that can be solved using calculus.
- Apply the appropriate calculus concepts to solve the problem.
- Check your solution and reflect on the process.
Create a visual representation of a calculus concept
Show steps
Creating a visual representation of a calculus concept will help you understand it more deeply.
Browse courses on
Multivariate Calculus
Show steps
Show steps
Show steps
- Choose a calculus concept that you want to understand better.
- Brainstorm different ways to represent the concept visually.
- Create your visual representation.
- Explain your visual representation to someone else.
Career center
Learners who complete Mathematics for Machine Learning: Multivariate Calculus will develop knowledge and skills
that may be useful to these careers:
Machine Learning Engineer
Machine Learning Engineer
A Machine Learning Engineer designs, develops, and deploys machine learning models. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Machine Learning Engineer because it would provide a solid foundation in the mathematical concepts and techniques used in machine learning, such as calculus, linear algebra, and probability. This course would also help Machine Learning Engineers develop the skills necessary to build and interpret machine learning models.
Data Scientist
Data Scientist
A Data Scientist uses scientific methods, processes, algorithms, and systems to extract knowledge and insights from data in various forms, both structured and unstructured. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Data Scientist because it would provide a solid foundation in the mathematical concepts and techniques used in data science, such as calculus, linear algebra, and probability. The course would also help Data Scientists develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Quantitative Analyst
Quantitative Analyst
A Quantitative Analyst uses mathematical and statistical methods to analyze data and make investment decisions. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Quantitative Analyst because it would provide a solid foundation in the mathematical concepts and techniques used in quantitative analysis, such as calculus, linear algebra, and probability. This course would also help Quantitative Analysts develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Statistician
Statistician
A Statistician collects, analyzes, interprets, and presents data. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Statistician because it would provide a solid foundation in the mathematical concepts and techniques used in statistics, such as calculus, linear algebra, and probability. This course would also help Statisticians develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Data Analyst
Data Analyst
A Data Analyst collects, transforms, and analyzes data to extract meaningful insights. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Data Analyst because it would provide a solid foundation in the mathematical concepts and techniques used in data analysis, such as calculus, linear algebra, and probability. This course would also help Data Analysts develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Financial Analyst
Financial Analyst
A Financial Analyst provides financial advice and guidance to individuals and organizations. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Financial Analyst because it would provide a solid foundation in the mathematical concepts and techniques used in finance, such as calculus, linear algebra, and probability. This course would also help Financial Analysts develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Actuary
Actuary
An Actuary evaluates and manages financial risk. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to an Actuary because it would provide a solid foundation in the mathematical concepts and techniques used in actuarial science, such as calculus, linear algebra, and probability. This course would also help Actuaries develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Business Analyst
Business Analyst
A Business Analyst analyzes an organization's business processes and makes recommendations for improvements. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Business Analyst because it would provide a solid foundation in the mathematical concepts and techniques used in business analysis, such as calculus, linear algebra, and probability. This course would also help Business Analysts develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Operations Research Analyst
Operations Research Analyst
An Operations Research Analyst uses mathematical and analytical methods to solve business problems. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to an Operations Research Analyst because it would provide a solid foundation in the mathematical concepts and techniques used in operations research, such as calculus, linear algebra, and probability. This course would also help Operations Research Analysts develop the skills necessary to build and interpret machine learning models, which are increasingly used to solve business problems.
Insurance Analyst
Insurance Analyst
An Insurance Analyst evaluates and manages insurance risks. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to an Insurance Analyst because it would provide a solid foundation in the mathematical concepts and techniques used in insurance analysis, such as calculus, linear algebra, and probability. This course would also help Insurance Analysts develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Financial Planner
Financial Planner
A Financial Planner helps individuals and families plan and manage their finances. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Financial Planner because it would provide a solid foundation in the mathematical concepts and techniques used in financial planning, such as calculus, linear algebra, and probability. This course would also help Financial Planners develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Economist
Economist
An Economist studies the production, distribution, and consumption of goods and services. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to an Economist because it would provide a solid foundation in the mathematical concepts and techniques used in economics, such as calculus, linear algebra, and probability. This course would also help Economists develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Software Developer
Software Developer
A Software Developer designs, develops, and maintains software applications. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Software Developer because it would provide a solid foundation in the mathematical concepts and techniques used in software development, such as calculus, linear algebra, and probability. This course would also help Software Developers develop the skills necessary to build and interpret machine learning models, which are increasingly used to develop software applications.
Market Researcher
Market Researcher
A Market Researcher conducts research to understand consumer behavior and market trends. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Market Researcher because it would provide a solid foundation in the mathematical concepts and techniques used in market research, such as calculus, linear algebra, and probability. This course would also help Market Researchers develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze data and make predictions.
Systems Analyst
Systems Analyst
A Systems Analyst analyzes and designs computer systems. A course in Mathematics for Machine Learning: Multivariate Calculus could be helpful to a Systems Analyst because it would provide a solid foundation in the mathematical concepts and techniques used in systems analysis, such as calculus, linear algebra, and probability. This course would also help Systems Analysts develop the skills necessary to build and interpret machine learning models, which are increasingly used to analyze and design computer systems.
For more career information including salaries, visit:
OpenCourser.com/course/7cq9gs/mathematics
Reading list
We've selected 14 books
that we think will supplement your
learning. Use these to
develop background knowledge, enrich your coursework, and gain a
deeper understanding of the topics covered in
Mathematics for Machine Learning: Multivariate Calculus.
Serves as a comprehensive textbook for machine learning, covering mathematical foundations, statistical techniques, and algorithmic approaches in detail, making it a highly relevant reference for the course.
Provides a probabilistic perspective on machine learning, covering a wide range of topics including supervised learning, unsupervised learning, and Bayesian inference, making it a valuable resource for understanding the probabilistic foundations of machine learning.
Provides a comprehensive introduction to statistical learning, covering a wide range of topics including supervised learning, unsupervised learning, and statistical modeling, making it a valuable resource for understanding the statistical techniques used in machine learning.
Save
Provides a comprehensive introduction to deep learning, covering a wide range of topics including neural networks, convolutional neural networks, and recurrent neural networks, making it a valuable resource for understanding the deep learning techniques used in machine learning.
Provides a practical introduction to deep learning using the Python programming language, covering a wide range of topics including neural networks, convolutional neural networks, and recurrent neural networks, making it a valuable resource for understanding the deep learning techniques used in machine learning.
Provides a practical introduction to machine learning using the Python programming language, covering a wide range of topics including supervised learning, unsupervised learning, and deep learning, making it a valuable resource for applying machine learning techniques in practice.
Provides a practical introduction to machine learning using the Python programming language, covering a wide range of topics including supervised learning, unsupervised learning, and deep learning, making it a valuable resource for applying machine learning techniques in practice.
Provides a practical introduction to machine learning using the R programming language, covering a wide range of topics including supervised learning, unsupervised learning, and statistical modeling, making it a valuable resource for applying statistical techniques in machine learning.
Provides a practical introduction to machine learning, covering a wide range of topics including supervised learning, unsupervised learning, and deep learning, making it a valuable resource for understanding the practical aspects of machine learning.
Provides a practical introduction to statistical learning using the R programming language, covering a wide range of topics including supervised learning, unsupervised learning, and statistical modeling, making it a valuable resource for applying statistical techniques in machine learning.
Provides a comprehensive introduction to numerical optimization, covering a wide range of algorithms and techniques, making it a valuable resource for understanding the numerical methods used in machine learning.
Provides a comprehensive treatment of convex optimization, covering both theoretical foundations and practical algorithms, making it a valuable resource for understanding the optimization techniques used in machine learning.
Provides a thorough introduction to calculus with a focus on applications in the life sciences, suitable for students and those with little prior experience in the subject.
Provides a clear and concise introduction to linear algebra, focusing on practical applications, making it a valuable resource for understanding the linear algebra concepts used in multivariate calculus.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/7cq9gs/mathematics
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Effort
6 weeks of study, 2-5 hours/week
Level
Introductory
Via
Coursera
Institution
Imperial College London
Instructors
Samuel J. Cooper
David Dye
A. Freddie Page
Language
English
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This course belongs to a
collection
called
Mathematics for Machine Learning. It's comprised of three courses:
- Mathematics for Machine Learning: PCA
- Mathematics for Machine Learning: Linear Algebra
- Mathematics for Machine Learning: Multivariate Calculus (viewing)
Good to know
Develops the understanding of multivariate calculus, which is a foundational skill set for practicing data scientists
Explores multivariate calculus, which is applicable for working with high-dimensional data often encountered in machine learning
Introduces tools and techniques to enable students to understand and work on complex machine learning algorithms
Taught by instructors recognized for their work in machine learning and calculus
Provides a solid theoretical foundation in multivariate calculus for students pursuing careers in machine learning
Suitable for students with a background in mathematics, particularly in calculus
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