Uploaded February 2025 | Updated September 2026, 2 weeks ago
Oxford Mathematician Dr Tom Crawford explains the process of Principal Component Analysis and how it is used to find patterns in data. Use code TRM20 for 20% off the Simplilearn Data Analyst Masters Course: bit.ly/DataAnalyst-TomRockMaths
This video is part of the 'Oxford Statistics Lecture Series' which is based on the first year undergraduate mathematics course that Tom teaches at the University of Oxford.
The video begins with a discussion of the purpose of PCA and why it is helpful for finding patterns in data. We then look at a simple example to determine what a principal component is, and the properties that we want it to have.
Once we have determined the concept of a principal component as the linear combination of the variables which maximises the variance, the variance is expressed using vector notation via the covariance matrix. We then set up a Lagrange multiplier problem which is solved to give the eigenvector equation of the covariance matrix.
This is solved using the characteristic polynomial to find the eigenvalues, where the eigenvector corresponding to the largest eigenvalue gives the first principal component. Finally, the scores matrix is introduced as the data matrix multiplied by the matrix of the ordered eigenvectors.
Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.uk/profiles/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
A HUGE thank you to all of my patrons for their support:
Dr Peet Morris
Jeryagor
John Hanson
Rodhern
Denise
Hiro
Delicious Rose
Bin Liu
Check out the Tom Rocks Maths Patreon page here: patreon.com/tomrocksmaths
Oxford Mathematician Dr Tom Crawford explains the process of Principal Component Analysis and how it is used to find patterns in data. Use code TRM20 for 20% off the Simplilearn Data Analyst Masters Course: bit.ly/DataAnalyst-TomRockMaths
This video is part of the 'Oxford Statistics Lecture Series' which is based on the first year undergraduate mathematics course that Tom teaches at the University of Oxford.
The video begins with a discussion of the purpose of PCA and why it is helpful for finding patterns in data. We then look at a simple example to determine what a principal component is, and the properties that we want it to have.
Once we have determined the concept of a principal component as the linear combination of the variables which maximises the variance, the variance is expressed using vector notation via the covariance matrix. We then set up a Lagrange multiplier problem which is solved to give the eigenvector equation of the covariance matrix.
This is solved using the characteristic polynomial to find the eigenvalues, where the eigenvector corresponding to the largest eigenvalue gives the first principal component. Finally, the scores matrix is introduced as the data matrix multiplied by the matrix of the ordered eigenvectors.
Produced by Dr Tom Crawford at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: conted.ox.ac.uk/profiles/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
A HUGE thank you to all of my patrons for their support:
Dr Peet Morris
Jeryagor
John Hanson
Rodhern
Denise
Hiro
Delicious Rose
Bin Liu
Check out the Tom Rocks Maths Patreon page here: patreon.com/tomrocksmaths
